16392 lines
308 KiB
Plaintext
16392 lines
308 KiB
Plaintext
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LRFTubes documentation - v1.1
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Dr.ir.
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J.A.
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Ir.
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C.
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ASCEE
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Nikola Teslastraat 1-11
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+31 6 18971622
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2
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2018-02-21: rev.
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1
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|
||
description "Characteristic length scale of a fluid space \\nomunit{\\si{\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$N$"
|
||
description "Number of\\nomunit{-}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$\\mathbf{n}$"
|
||
description "Normal vector pointing from the solid into the fluid\\nomunit{-}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$p$"
|
||
description "Pressure, acoustic pressure \\nomunit{\\si{\\pascal}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$r_h$"
|
||
description "Hydraulic radius \\nomunit{\\si{\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$\\mathbf{r}$"
|
||
description "Transverse position vector\\nomunit{-}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$S$"
|
||
description "Cross-sectional area, surface area\\nomunit{\\si{\\square\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$t$"
|
||
description "Time \\nomunit{\\si{\\second}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$T$"
|
||
description "Temperature\\nomunit{\\si{\\kelvin}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$\\mathbf{u}$"
|
||
description "Velocity vector\\nomunit{\\si{\\metre\\per\\second}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$u$"
|
||
description "Velocity in wave propagation direction\\nomunit{\\si{\\metre\\per\\second}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$U$"
|
||
description "Volume flow\\nomunit{\\si{\\cubic\\metre\\per\\second}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$V$"
|
||
description "Volume \\nomunit{\\si{\\cubic\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$\\mathbf{x}$"
|
||
description "Position vector \\nomunit{\\si{\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$z$"
|
||
description "Specific acoustic impedance\\nomunit{\\si{\\pascal\\second\\per\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$Z$"
|
||
description "Volume flow impedance\\nomunit{\\si{\\pascal\\second\\per\\cubic\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
% Greek (G)
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Unused:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "M"
|
||
symbol "$\\Delta$"
|
||
description "Difference\\nonomunit"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "G"
|
||
symbol "$\\gamma$"
|
||
description "Ratio of specific heats\\nomunit{-}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "G"
|
||
symbol "$\\Gamma$"
|
||
description "Viscothermal wave number for a prismatic duct \\nomunit{\\si{\\radian\\per\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "G"
|
||
symbol "$\\delta_{\\kappa}$"
|
||
description "Thermal penetration depth\\nomunit{\\si{\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "G"
|
||
symbol "$\\delta_{\\nu}$"
|
||
description "Viscous penetration depth\\nomunit{\\si{\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "G"
|
||
symbol "$\\epsilon_s$"
|
||
description "Ideal stack correction factor \\nomunit{-}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "G"
|
||
symbol "$\\lambda$"
|
||
description "Wavelength \\nomunit{\\si{\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "G"
|
||
symbol "$\\pi$"
|
||
description "Ratio of the circumference to the diameter of a circle \\nomunit{-}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "G"
|
||
symbol "$\\Pi$"
|
||
description "Wetted perimeter (contact length between solid and fluid) \\nomunit{\\si{\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
% Miscellaneous symbols and operators (M)
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Unused:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "M"
|
||
symbol "$\\otimes$"
|
||
description "Dyadic product\\nonomunit"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "M"
|
||
symbol "$\\Re$"
|
||
description "Real part\\nonomunit"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "M"
|
||
symbol "$\\Im$"
|
||
description "Imaginary part\\nonomunit"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "M"
|
||
symbol "$\\nabla$"
|
||
description "Gradient \\nomunit{\\si{\\per\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "M"
|
||
symbol "$\\nabla^2$"
|
||
description "Laplacian\\nomunit{\\si{\\per\\square\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "M"
|
||
symbol "$\\sim$"
|
||
description "Same order of magnitude\\nonomunit"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "M"
|
||
symbol "$\\left\\Vert \\bullet \\right\\Vert $"
|
||
description "Eucledian norm\\nonomunit"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "M"
|
||
symbol "d"
|
||
description "Infinitesimal\\nonomunit"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "M"
|
||
symbol "$\\partial$"
|
||
description "Infinitesimal\\nonomunit"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "M"
|
||
symbol "$\\bullet$"
|
||
description "Placeholder for an operand\\nonomunit"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
% Subscripts (S)
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "S"
|
||
symbol "wall"
|
||
description "At the wall"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "S"
|
||
symbol "0"
|
||
description "Evaluated at the reference condition"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "S"
|
||
symbol "$f$"
|
||
description "Fluid"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "S"
|
||
symbol "$s$"
|
||
description "Solid"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "S"
|
||
symbol "$w$"
|
||
description "Wall"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "S"
|
||
symbol "$R$"
|
||
description "Right side"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "S"
|
||
symbol "$L$"
|
||
description "Left side"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "S"
|
||
symbol "$s$"
|
||
description "Solid"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "S"
|
||
symbol "$s$"
|
||
description "Squeeze"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "S"
|
||
symbol "$i$"
|
||
description "Inner"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "S"
|
||
symbol "$o$"
|
||
description "Outer"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "S"
|
||
symbol "$t$"
|
||
description "Tube"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
% Often used abbreviations (O)
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "O"
|
||
symbol "Sec(s)."
|
||
description "Section(s)"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "O"
|
||
symbol "Eq(s)."
|
||
description "Equation(s)"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "O"
|
||
symbol "LRF"
|
||
description "Low Reduced Frequency"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
printnomenclature[1.8cm]
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Overview of
|
||
\begin_inset ERT
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubes
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Introduction
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Welcome to the documentation of
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubes
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\begin_inset ERT
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubes
|
||
\backslash
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
is a numerical code to solve one-dimensional acoustic duct systems using
|
||
the transfer matrix method.
|
||
Segments can be connected to generate simple one-dimensional acoustic systems
|
||
to model acoustic propagation problems in ducts in the frequency domain.
|
||
Viscothermal dissipation mechanisms are taken into account such that the
|
||
damping effects can be modeled accurately, below the cut-on frequency of
|
||
the duct.
|
||
For more information regarding the models and the theory behind the models,
|
||
the reader is referred to the work of
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "van_der_eerden_noise_2000"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "kampinga_viscothermal_2010"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "ward_deltaec_2017"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
This documentation serves as a reference for the implemented models.
|
||
For examples on how to use the code, please take a look at the example
|
||
models as worked out in the IPython Notebooks.
|
||
For installation instructions, please refer the the
|
||
\begin_inset CommandInset href
|
||
LatexCommand href
|
||
name "README"
|
||
target "https://code.ascee.nl/ASCEE/lrftubes/raw/branch/master/LICENSE"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
in the main repository.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
This document is very brief on the theory and it is assumed that the reader
|
||
has some knowledge on the basics of acoustics in general and viscothermal
|
||
acoustics as well.
|
||
If you are not falling in this category, I would please refer you first
|
||
to the book of Swift
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "swift_thermoacoustics:_2003"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
A more detailed introduction to the notation used in this documentation
|
||
can be found in the PhD thesis of de Jong
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "de_jong_numerical_2015"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Besides that, if you find the work interesting, but you are not sure how
|
||
to apply it, please contact ASCEE for more information.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
License and disclaimer
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Redistribution and use in source and binary forms are permitted provided
|
||
that the above copyright notice and this paragraph are duplicated in all
|
||
such forms and that any documentation, advertising materials, and other
|
||
materials related to such distribution and use acknowledge that the software
|
||
was developed by the ASCEE.
|
||
The name of the ASCEE may not be used to endorse or promote products derived
|
||
from this software without specific prior written permission.
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR IMPLIED WARRANTIE
|
||
S, INCLUDING, WITHOUT LIMITATION, THE IMPLIED WARRANTIES OF MERCHANTABILITY
|
||
AND FITNESS FOR A PARTICULAR PURPOSE.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Features
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Currently the
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubes
|
||
\backslash
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
code provides acoustic models for the following physical entities:
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Prismatic ducts with circular cross section,
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Prismatic ducts with triangular cross section,
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Prismatic ducts with parallel plate cross section,
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Prismatic ducts with square cross section,
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Acoustic compliance volumes
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Discontinuity correction
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
End correction for a baffled piston
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Lumped series impedance
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
These segments can be connected to form one-dimensional acoustic systems
|
||
to model wave propagation below the cut-on frequency of higher order modes.
|
||
For a circular cross section, the cut-on frequency is
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "van_der_eerden_noise_2000"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
f_{c}\approx\frac{c_{0}}{3.4r},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $r$
|
||
\end_inset
|
||
|
||
is the tube radius and
|
||
\begin_inset Formula $c_{o}$
|
||
\end_inset
|
||
|
||
is the speed of sound.
|
||
Above the cut-on frequency, besides evanescent waves, there are also propagatin
|
||
g waves with a non-constant pressure distribution along the cross section
|
||
of the duct.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Limitations and future features
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The current version of has some limitations that will be resolved in a future
|
||
release.
|
||
These are:
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Ducts with (turbulent) flow
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For thermoacoustic and HVAC (Heating, ventilation and Air Conditioning)
|
||
duct modeling it is imperative that mean flows can be taken into account.
|
||
An acoustic wave superimposed on a mean flow results in asymmetric wave
|
||
propagation.
|
||
More specifically, the phase velocity is higher in the direction of the
|
||
mean flow, and slower in the opposite direction.
|
||
In a future release, we will provide models for ducts including a mean
|
||
flow.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Porous acoustic absorbers
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
To model absorption of sound, a one-dimensional porous material model should
|
||
be implemented.
|
||
This work has been postponed to a later stage.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Prismatic and spherical ducts filled with porous material are defined in
|
||
dbmduct.py.
|
||
These use the Delaney-Bazley-Miki model.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Overview of this documentation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The next chapter of this documentation will describe the basic framework
|
||
of the
|
||
\begin_inset ERT
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubess
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
code: the transfer matrix method.
|
||
After that, in Chapter
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "chap:Provided-acoustic-models"
|
||
|
||
\end_inset
|
||
|
||
, an overview of the provided acoustic models is given, with which acoustic
|
||
networks can be built.
|
||
For each of the segments, the resulting transfer matrix model is derived.
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Material properties
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Air
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Nonlinearity parameter:
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Exhaust gas
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Composition
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Definitions:
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $\omega_{i}$
|
||
\end_inset
|
||
|
||
mass fraction of species
|
||
\begin_inset Formula $i$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $x_{i}$
|
||
\end_inset
|
||
|
||
molar / volume fraction of species
|
||
\begin_inset Formula $i$
|
||
\end_inset
|
||
|
||
(assuming ideal gas behavior)
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $\overline{M}$
|
||
\end_inset
|
||
|
||
average molar mass of (exhaust gas) mixture
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $M_{i}$
|
||
\end_inset
|
||
|
||
molar mass of species
|
||
\begin_inset Formula $i$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The following equations hold in a mixture:
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\sum_{i}\omega_{i} & =1\\
|
||
\sum_{i}x_{i} & =1\\
|
||
\overline{M} & =\sum\nolimits _{i}x_{i}M_{i}\label{eq:molar_mass_comp}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
We can convert mass fractions to mole fractions with the following rule:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
The total mass is (
|
||
\begin_inset Formula $N$
|
||
\end_inset
|
||
|
||
) is the total number of moles
|
||
\begin_inset Formula
|
||
\[
|
||
m=x_{i}M_{i}N
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
The total number of moles is:
|
||
\begin_inset Formula
|
||
\[
|
||
N=\frac{m}{\overline{M}}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
The average molar mass is:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\overline{M}=\frac{m}{N}=\sum_{i}x_{i}M_{i}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
The mass fraction to mole fraction is:
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\omega_{i}=x_{i}\frac{M_{i}}{\overline{M}}\qquad\Longleftrightarrow\qquad x_{i}=\omega_{i}\frac{\overline{M}}{M_{i}}\label{eq:massfr_to_molarfr_viceversa}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Henceforth, what is often used, is to compute the average molar mass given
|
||
only the mass fractions:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\omega_{i}m=N_{i}M_{i}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\omega_{i}}{M_{i}}=\frac{N_{i}}{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\sum_{i}\frac{\omega_{i}}{M_{i}}=\frac{N}{m}=\frac{1}{\overline{M}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\overline{M}=\frac{1}{\sum\nolimits _{i}\frac{\omega_{i}}{M_{i}}}\label{eq:molar_mass_vs_massfr}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Mixing of mixtures
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Suppose we mix two mixtures of substances, mixture 1, and mixture 2.
|
||
We want to know the final concentrations / mass fraction in the mixed mixture.
|
||
Mix 1 comprises mass fractions
|
||
\begin_inset Formula $\omega_{1,i}$
|
||
\end_inset
|
||
|
||
, and mix 2 comprises mass fractions
|
||
\begin_inset Formula $\omega_{2,j}$
|
||
\end_inset
|
||
|
||
.
|
||
We assume that
|
||
\begin_inset Formula $i$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $j$
|
||
\end_inset
|
||
|
||
can interfere.
|
||
For example, mixing air with Dutch natural gas, both contain nitrogen.
|
||
The first step is to determine the mass flow of the two, called
|
||
\begin_inset Formula $m_{1}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $m_{2}$
|
||
\end_inset
|
||
|
||
.
|
||
Then, assuming mass conservation under chemically inert conditions:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
m_{1}\omega_{1,i}+m_{2}\omega_{2,i}=m\omega_{i}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection*
|
||
Mixing air with natural gas
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The air factor
|
||
\begin_inset Formula $\lambda$
|
||
\end_inset
|
||
|
||
(not to be confused with wavelength in an acoustic context), is defined
|
||
as the ratio of air to the stoichiometric ratio.
|
||
The stoichiometric ratio can be determined by calculating the required
|
||
moles of oxygen such that all carbon atoms can become CO
|
||
\begin_inset Formula $_{2}$
|
||
\end_inset
|
||
|
||
, and
|
||
\emph on
|
||
half of
|
||
\emph default
|
||
all hydrogen atoms can become H
|
||
\begin_inset Formula $_{2}$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Ideal gas mixtures
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For an ideal gas, the components of a gas mixture can be represented by
|
||
their
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
partial pressure
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
, which is the total pressure times the volume fraction of the component
|
||
in the mixture.
|
||
For an ideal gas, the volume fraction equals to mole fraction.
|
||
Hence:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{V_{i}}{V}\overset{\mathrm{ideal\,gas}}{=}x_{i}=\frac{p_{i}}{R_{u}T}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The mass fraction can be computed from the mole fraction.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Transport properties
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float table
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Tabular
|
||
<lyxtabular version="3" rows="5" columns="5">
|
||
<features booktabs="true" tabularvalignment="middle">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top" width="0pt">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Substance
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $M$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $T_{c}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $G$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $C_{r}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Carbon dioxide
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
SI{44.01e-3}{kg
|
||
\backslash
|
||
per
|
||
\backslash
|
||
mole}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
SI{304}{
|
||
\backslash
|
||
K}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
44.6
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
0.766
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Oxygen
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
SI{32.00e-3}{kg
|
||
\backslash
|
||
per
|
||
\backslash
|
||
mole}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
SI{154}{
|
||
\backslash
|
||
K}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
32.8
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
0.712
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Nitrogen
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
SI{28.02e-3}{kg
|
||
\backslash
|
||
per
|
||
\backslash
|
||
mole}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
SI{126}{
|
||
\backslash
|
||
K}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
24.6
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
0.881
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Water vapor
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
SI{18.02e-3}{kg
|
||
\backslash
|
||
per
|
||
\backslash
|
||
mole}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
SI{647}{
|
||
\backslash
|
||
K}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
52.2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
1.018
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Critical values and constants of common diatomic gases
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "tab:crit_values_diatom_gas"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Dynamic viscosity of pure gases
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Here we assume the dynamic viscosity of a pure substance can be modeled
|
||
using Sutherland's equation:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mu=\mu_{c}\left(\frac{T_{0}+C}{T+C}\right)\left(\frac{T}{T_{0}}\right)^{3/2},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where the subscript
|
||
\begin_inset Formula $c$
|
||
\end_inset
|
||
|
||
denotes the value at its
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
critical point
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
.
|
||
In convenient form we solve:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mu=\mu_{c}\mu_{r},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\mu_{c}$
|
||
\end_inset
|
||
|
||
is the critical viscosity and
|
||
\begin_inset Formula $\mu_{r}$
|
||
\end_inset
|
||
|
||
is the
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
reduced viscosity defined as
|
||
\begin_inset Formula $\mu/\mu_{c}$
|
||
\end_inset
|
||
|
||
.
|
||
For
|
||
\begin_inset Formula $\mu_{c}$
|
||
\end_inset
|
||
|
||
we have the reduced form of Sutherland's equation:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mu_{c}=\frac{1+C_{r}}{T_{r}+C_{r}}T_{r}^{3/2}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
The value for
|
||
\begin_inset Formula $\mu_{c}$
|
||
\end_inset
|
||
|
||
can be calculated as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mu_{c}=\num{3.5e-6}G
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Values for
|
||
\begin_inset Formula $T_{r}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $C_{r}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $G$
|
||
\end_inset
|
||
|
||
are listed in Table
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "tab:crit_values_diatom_gas"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "licht_variation_1944"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Dynamic viscosity of a gas mixture
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The dynamic viscosity of a gas mixture can be derived from the dynamic viscositi
|
||
es of pure gases as
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "p. 27"
|
||
key "bird_transport_2007"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mu_{\mathrm{mix}}=\sum_{α=0}^{N-1}\frac{x_{α}\mu_{α}}{\sum_{β=0}^{N-1}\Phi_{αβ}x_{β}},\label{eq:mumix}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\mu_{α}$
|
||
\end_inset
|
||
|
||
is the dynamic viscosity of pure chemical species
|
||
\begin_inset Formula $α$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $x_{α}$
|
||
\end_inset
|
||
|
||
denotes its mole fraction in the mixture.
|
||
|
||
\begin_inset Formula $\Phi_{αβ}$
|
||
\end_inset
|
||
|
||
is defined as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Phi_{αβ}=\frac{1}{\sqrt{8}}\left(1+\frac{M_{α}}{M_{β}}\right)^{-1/2}\left[1+\left(\frac{\mu_{α}}{\mu_{β}}\right)^{1/2}\left(\frac{M_{β}}{M_{α}}\right)^{1/4}\right]^{2},\label{eq:Phi_mn}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $M_{α}$
|
||
\end_inset
|
||
|
||
is the molar mass of species
|
||
\begin_inset Formula $α$
|
||
\end_inset
|
||
|
||
.
|
||
The denominator of Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:mumix"
|
||
|
||
\end_inset
|
||
|
||
can efficiently be solved by noting that
|
||
\begin_inset Formula $d_{α}=\sum_{β=0}^{N-1}\Phi_{αβ}x_{β}$
|
||
\end_inset
|
||
|
||
is a matrix-vector product, which can be written as
|
||
\begin_inset Formula $\boldsymbol{d}=\boldsymbol{\Phi}\cdot\boldsymbol{x}$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Thermal conductivity of a gas mixture
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The thermal conductivity of a gas mixture can be derived from the thermal
|
||
conductivities of pure gases as
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "p. 276"
|
||
key "bird_transport_2007"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
k_{\mathrm{mix}}=\sum_{α=0}^{N-1}\frac{x_{α}k_{α}}{\sum_{β=0}^{N-1}\Phi_{αβ}x_{β}}\label{eq:kappamix}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
where
|
||
\begin_inset Formula $k_{α}$
|
||
\end_inset
|
||
|
||
is the thermal conductivity of pure chemical species
|
||
\begin_inset Formula $α$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $x_{α}$
|
||
\end_inset
|
||
|
||
denotes its mole fraction in the mixture and
|
||
\begin_inset Formula $\Phi_{αβ}$
|
||
\end_inset
|
||
|
||
is identical to that appearing in the viscosity equation, see
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:Phi_mn"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Combustion
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
To compute the gas constant, first the mixture components of the exhaust
|
||
gas need to be computed.
|
||
We assume that the oxidizer is air with 79% vol of nitrogen (molecules)
|
||
and 21% oxygen molecules.
|
||
The tiny part of argon and other components is ignored.
|
||
Then, the gross formula for combustion is:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\underbrace{x_{f,C}C+x_{f,O}O+x_{H,f}H+x_{f,N}N}_{\mathrm{fuel}}+\underbrace{y_{\mathrm{ox}}\left(0.79N_{2}+0.21O_{2}\right)}_{\mathrm{oxidizer}}\rightarrow\underbrace{y_{g,\mathrm{water}}H_{2}O+y_{g,CO_{2}}CO_{2}+y_{g,N_{2}}N_{2}}_{\mathrm{exhaust\,gas}}.\label{eq:combustion}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Above reaction formula can be read as:
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
take
|
||
\begin_inset Formula $x_{f,C}$
|
||
\end_inset
|
||
|
||
moles of carbon in the fuel, add
|
||
\begin_inset Formula $y_{\mathrm{ox}}$
|
||
\end_inset
|
||
|
||
moles of air, and it should result in
|
||
\begin_inset Formula $y_{g,CO_{2}}$
|
||
\end_inset
|
||
|
||
moles of
|
||
\begin_inset Formula $CO_{2}$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
And so on for the other elements.
|
||
The mole fractions in the fuel composition can be derived from its mass
|
||
fractions, upon utilizing Eqs.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:massfr_to_molarfr_viceversa"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:molar_mass_vs_massfr"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
From Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:combustion"
|
||
|
||
\end_inset
|
||
|
||
, the following system of equations can be created:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left\{ \begin{array}{c}
|
||
x_{f,C}\\
|
||
x_{f,O}\\
|
||
x_{f,H}\\
|
||
x_{f,N}
|
||
\end{array}\right\} +\left[\begin{array}{cccc}
|
||
0 & 0 & -1 & 0\\
|
||
2\times0.21 & -1 & -2 & 0\\
|
||
0 & -2 & 0 & 0\\
|
||
2\times0.79 & 0 & 0 & -2
|
||
\end{array}\right]\left\{ \begin{array}{c}
|
||
y_{\mathrm{ox}}\\
|
||
y_{g,\mathrm{water}}\\
|
||
y_{g,CO_{2}}\\
|
||
y_{g,N_{2}}
|
||
\end{array}\right\} =\left\{ \begin{array}{c}
|
||
0\\
|
||
0\\
|
||
0\\
|
||
0
|
||
\end{array}\right\}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Solving this results in:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $x_{f,O}+2\times0.21y_{\mathrm{ox}}-y_{g,\mathrm{water}}-2y_{g,CO_{2}}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $2\times0.21y_{\mathrm{ox}}=\frac{1}{2}x_{f,H}+2x_{f,C}+x_{f,O}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
–
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $x_{f,N}+2\times0.79y_{\mathrm{ox}}-2y_{g,N_{2}}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $y_{g,N_{2}}=0.79y_{\mathrm{ox}}+\frac{1}{2}x_{f,N}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
y_{g,CO_{2}} & =x_{f,C}\\
|
||
y_{g,\mathrm{water}} & =\frac{1}{2}x_{f,H}\\
|
||
y_{\mathrm{ox}}= & \frac{\frac{1}{2}x_{f,H}+2x_{f,C}-x_{f,O}}{2\times0.21}\\
|
||
y_{g,N_{2}}= & 0.79y_{\mathrm{ox}}+\frac{1}{2}x_{f,N}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Note that the mole fractions are
|
||
\emph on
|
||
unnormalized
|
||
\emph default
|
||
(that is why we use symbol
|
||
\begin_inset Formula $y$
|
||
\end_inset
|
||
|
||
, not
|
||
\begin_inset Formula $x$
|
||
\end_inset
|
||
|
||
): they denote the number of moles required to burn 1 mole of fuel.
|
||
To compute the mole fractions in the exhaust gas,
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
x_{g,\mathrm{water}}=\frac{y_{1}}{y_{1}+y_{2}+y_{3}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Table
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "tab:fuel_components"
|
||
|
||
\end_inset
|
||
|
||
gives an overview of the composition of typical combustion fuels.
|
||
Once the molar fractions of the exhaust gas are known, the average molar
|
||
mass can be computed using Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:molar_mass_comp"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Then, the specific gas constant can be computed according to:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
R_{s}=\frac{R_{u}}{\overline{M}},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $R_{u}$
|
||
\end_inset
|
||
|
||
is the universal gas constant.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float table
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Tabular
|
||
<lyxtabular version="3" rows="5" columns="3">
|
||
<features booktabs="true" tabularvalignment="middle">
|
||
<column alignment="center" valignment="top" width="0pt">
|
||
<column alignment="right" valignment="top" width="0pt">
|
||
<column alignment="right" valignment="top" width="0pt">
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Mass fraction
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="right" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Wood
|
||
\begin_inset Foot
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
https://www.engineeringtoolbox.com/co2-emission-fuels-d_1085.html
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="right" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Dutch Natural gas
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Carbon
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="right" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
50 %
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="right" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Oxygen
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="right" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
42 %
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="right" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
0 %
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Hydrogen
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="right" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
6 %
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="right" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Nitrogen
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
0 %
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Mixture mass composition of fuels
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "tab:fuel_components"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Specific heat ratio
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The specific heat is build-up according to mass percentages of the flue
|
||
gas.
|
||
Carbon dioxide has a
|
||
\begin_inset Formula $c_{p}$
|
||
\end_inset
|
||
|
||
of 840 J/kg/K, water vapor of 1930:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\overline{c}_{p}=\sum\nolimits _{i}\omega_{i}c_{p,i}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Sound absorbing solid materials
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
High porosity soft materials can be modeled adequately with the Delaney-Bazley-M
|
||
iki model.
|
||
The model has a single input, namely the static flow resistivity.
|
||
Table
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float table
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Tabular
|
||
<lyxtabular version="3" rows="5" columns="2">
|
||
<features booktabs="true" tabularvalignment="middle">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Name
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Basotect TG
|
||
\begin_inset Foot
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
A.k.a.Flamex Basic (akoestiekwinkel.nl)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Description
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Melamine resin foam (fire retardant)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Density [
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
si{
|
||
\backslash
|
||
kg
|
||
\backslash
|
||
per
|
||
\backslash
|
||
cubic
|
||
\backslash
|
||
m}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Foot
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
https://www.forman.co.nz/media/emizen_banner/b/a/basf_basotect_datasheet.pdf
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Flow resistivity [
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
si{
|
||
\backslash
|
||
pascal
|
||
\backslash
|
||
s
|
||
\backslash
|
||
per
|
||
\backslash
|
||
meter}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
]
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
num{8.5e3}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
, source:
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "kino_investigation_2009"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
, Table 2 average value.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Resistivity values are given for room temperature
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
Conversion
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
The transfer matrix method
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Introduction
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Each part of an acoustic system in
|
||
\begin_inset ERT
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubess
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
is modeled using a so-called transfer matrix.
|
||
A transfer matrix maps the state quantities on one side of the segment
|
||
(node) to the other side of the segment (node).
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For one-dimensional wave propagation, analytical solutions for the velocity,
|
||
temperature and density field in the transverse direction can be found.
|
||
The state variables in frequency domain satisfy a system of first order
|
||
ordinary differential equations.
|
||
Once the solution is known on one end of a segment, the solution on the
|
||
other end can be deduced.
|
||
The transfer matrix couples the state variables
|
||
\begin_inset Formula $\boldsymbol{\phi}$
|
||
\end_inset
|
||
|
||
on one end of a segment to the other end, in frequency domain:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\boldsymbol{\phi}_{R}(\omega)=\boldsymbol{T}(\omega)\boldsymbol{\phi}_{L}(\omega)+\mathbf{s}(\omega),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $L$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $R$
|
||
\end_inset
|
||
|
||
denote the left and right side, respectively,
|
||
\begin_inset Formula $\boldsymbol{T}$
|
||
\end_inset
|
||
|
||
denotes the transfer matrix and
|
||
\begin_inset Formula $\boldsymbol{s}$
|
||
\end_inset
|
||
|
||
is a source term.
|
||
In the code and in this documentation
|
||
\begin_inset Formula $e^{{\color{red}+}i\omega t}$
|
||
\end_inset
|
||
|
||
convention is used.
|
||
A common choice of state variables is such that their product has the unit
|
||
of power.
|
||
For all systems in this code, the state variables satisfy this property.
|
||
For example in an acoustic segment, the power is the product of acoustic
|
||
pressure
|
||
\begin_inset Formula $p\left(\omega\right)$
|
||
\end_inset
|
||
|
||
and volume flow
|
||
\begin_inset Formula $U\left(\omega\right)$
|
||
\end_inset
|
||
|
||
.
|
||
For complex phasors and, the acoustic power flow can then be computed as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
E=\frac{1}{2}\Re\left[pU^{*}\right],
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\Re[\bullet]$
|
||
\end_inset
|
||
|
||
denotes the real part of
|
||
\begin_inset Formula $\bullet$
|
||
\end_inset
|
||
|
||
, and * denotes the complex conjugation.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Example transfer matrix of an acoustic duct
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
This section will provide the derivation of the transfer matrix of a simple
|
||
acoustic duct.
|
||
Starting with the isentropic acoustic continuity and momentum equation
|
||
:
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\frac{1}{c_{0}^{2}}\frac{\partial\hat{p}}{\partial\hat{t}}+\rho_{0}\nabla\cdot\hat{\boldsymbol{u}} & =0,\\
|
||
\rho_{0}\frac{\partial\hat{\boldsymbol{u}}}{\partial t}+\nabla\hat{p} & =0.
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
The next step is to transform these equations to frequency domain and assuming
|
||
only wave propagation in the
|
||
\begin_inset Formula $x-$
|
||
\end_inset
|
||
|
||
direction, integrating over the cross section we find:
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\frac{i\omega}{c_{0}^{2}}p+\frac{\rho_{0}}{S_{f}}\frac{\mathrm{d}U}{\mathrm{d}x} & =0,\label{eq:contU}\\
|
||
\rho_{0}i\omega U+S_{f}\frac{\mathrm{d}p}{\mathrm{d}x} & =0,\label{eq:momU}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $U$
|
||
\end_inset
|
||
|
||
denotes the acoustic volume flow in
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
si{
|
||
\backslash
|
||
cubic
|
||
\backslash
|
||
metre
|
||
\backslash
|
||
per
|
||
\backslash
|
||
second}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Eqs.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
(
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:contU"
|
||
|
||
\end_inset
|
||
|
||
-
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:momU"
|
||
|
||
\end_inset
|
||
|
||
) is a coupled set of ordinary differential equations, which can be solved
|
||
for the acoustic pressure to find
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p(x)=A\exp\left(-ikx\right)+B\exp\left(ikx\right),\label{eq:HH_sol_prismaticinviscid}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $A$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $B$
|
||
\end_inset
|
||
|
||
are constants, to be determined from the boundary conditions.
|
||
Setting
|
||
\begin_inset Formula $p=p_{L}$
|
||
\end_inset
|
||
|
||
, and
|
||
\begin_inset Formula $U=U_{L}$
|
||
\end_inset
|
||
|
||
at
|
||
\begin_inset Formula $x=0$
|
||
\end_inset
|
||
|
||
, we can solve for the acoustic pressure, upon using Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:momU"
|
||
|
||
\end_inset
|
||
|
||
as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p(x)=p_{L}\cos\left(kx\right)-iZ_{0}\sin\left(kx\right)U_{L},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and for the acoustic volume flow we find:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
U(x)=U_{L}\cos\left(kx\right)-\frac{i}{Z_{0}}\sin\left(kx\right)p_{L}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Now, we have all ingredients to derive the transfer matrix of an acoustic
|
||
duct.
|
||
Setting
|
||
\begin_inset Formula $p(x=L)=p_{R}$
|
||
\end_inset
|
||
|
||
, and
|
||
\begin_inset Formula $U(x=L)=U_{R}$
|
||
\end_inset
|
||
|
||
, we find the following two-port coupling between the pressure and the velocity
|
||
from the left side of the duct to the right side of the duct:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left\{ \begin{array}{c}
|
||
p_{R}\\
|
||
U_{R}
|
||
\end{array}\right\} =\left[\begin{array}{cc}
|
||
\cos\left(kL\right) & -iZ_{0}\sin\left(kL\right)\\
|
||
-iZ_{0}^{-1}\sin\left(kL\right) & \cos\left(kL\right)
|
||
\end{array}\right]\left\{ \begin{array}{c}
|
||
p_{L}\\
|
||
U_{L}
|
||
\end{array}\right\} .\label{eq:transfer_inviscid}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Setting up the system of equations
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubes
|
||
\backslash
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
has been set up to solve systems of acoustic segments such as this prismatic
|
||
duct.
|
||
The advantage of the transfer matrix method is the ease with which mixed
|
||
(impedance/pressure/velocity) boundary conditions can be implemented.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In this section, the assembly of the global system of equations is explained.
|
||
The state variables of each segment are stacked in a column vector
|
||
\series bold
|
||
|
||
\begin_inset Formula $\boldsymbol{\phi}_{\mbox{sys}}$
|
||
\end_inset
|
||
|
||
|
||
\series default
|
||
, which has the size of
|
||
\begin_inset Formula $4N_{\mbox{segs}}$
|
||
\end_inset
|
||
|
||
, where
|
||
\begin_inset Formula $N_{\mbox{segs}}$
|
||
\end_inset
|
||
|
||
denotes the number of segments in the system.
|
||
The coupling equations between the nodes of each segment, are the transfer
|
||
matrices.
|
||
Since the transfer matrices are
|
||
\begin_inset Formula $2\times2$
|
||
\end_inset
|
||
|
||
, this fills only half of the required amount of equations.
|
||
The other half is filled with boundary conditions.
|
||
Each segments transfer matrix can be regarded as the element matrix, which
|
||
all have a form like:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\boldsymbol{\phi}_{R}=\boldsymbol{T}\cdot\boldsymbol{\phi}_{L}+\boldsymbol{s},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\boldsymbol{\phi}_{L},\boldsymbol{\phi}_{R}$
|
||
\end_inset
|
||
|
||
are the state vectors on the left and right sides of the segment, respectively,
|
||
|
||
\begin_inset Formula $\boldsymbol{T}$
|
||
\end_inset
|
||
|
||
is the transfer matrix, and
|
||
\begin_inset Formula $\boldsymbol{s}$
|
||
\end_inset
|
||
|
||
is a source term.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
There are two kind of boundary conditions, called external and internal
|
||
boundary conditions.
|
||
External boundary conditions apply where a prescribed condition is given,
|
||
such as a prescribed pressure, voltage, volume flow, current or acoustic/electr
|
||
ic impedance.
|
||
Internal boundary conditions are used to couple different segments at a
|
||
connection point, which is recognized by a shared node number.
|
||
At a connection point, the effort variable is shared, which means that
|
||
the pressure at the node is equal for each connected segment sharing the
|
||
node.
|
||
The flow variable is conserved, so the sum of the volume flow out of all
|
||
segments connected at the node is 0.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection*
|
||
Example: two ducts
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/tfm_expl.pdf
|
||
width 80text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Example of two simple duct segments connected together.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:coupling_example"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
This procedure of creating a system matrix is explained by an example where
|
||
only two ducts are coupled.
|
||
A schematic of the situation is depicted in Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:coupling_example"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
For the example situation, at the left node of segment (1), an impedance
|
||
boundary
|
||
\begin_inset Formula $Z_{L}$
|
||
\end_inset
|
||
|
||
is prescribed.
|
||
The right node of segment (1) is connected to the left node of segment
|
||
(2), and at the right side of segment (2), a volume flow boundary condition
|
||
is prescribed of
|
||
\begin_inset Formula $U_{R}$
|
||
\end_inset
|
||
|
||
.
|
||
The corresponding system of equations for this case is
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left[\begin{array}{cccc}
|
||
\mathbf{T}_{1} & -\mathbf{I} & \mathbf{0} & \mathbf{0}\\
|
||
\mathbf{0} & \mathbf{0} & \mathbf{T}_{2} & -\mathbf{I}\\
|
||
\mathbf{0} & \left[\begin{array}{cc}
|
||
1 & 0\\
|
||
0 & 1
|
||
\end{array}\right] & \left[\begin{array}{cc}
|
||
-1 & 0\\
|
||
0 & -1
|
||
\end{array}\right] & \mathbf{0}\\
|
||
\left[\begin{array}{cc}
|
||
1 & Z_{L}\\
|
||
0 & 0
|
||
\end{array}\right] & \mathbf{0} & \mathbf{0} & \left[\begin{array}{cc}
|
||
0 & 0\\
|
||
0 & 1
|
||
\end{array}\right]
|
||
\end{array}\right]\left\{ \begin{array}{c}
|
||
p_{1L}\\
|
||
U_{1L}\\
|
||
p_{1R}\\
|
||
U_{1R}\\
|
||
p_{2L}\\
|
||
U_{2L}\\
|
||
p_{2R}\\
|
||
U_{2R}
|
||
\end{array}\right\} =\left\{ \begin{array}{c}
|
||
0\\
|
||
0\\
|
||
0\\
|
||
0\\
|
||
0\\
|
||
0\\
|
||
0\\
|
||
U_{R}
|
||
\end{array}\right\} ,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In this system matrix,
|
||
\begin_inset Formula $\mathbf{0}$
|
||
\end_inset
|
||
|
||
denotes a
|
||
\begin_inset Formula $2\times2$
|
||
\end_inset
|
||
|
||
sub matrix of zeros and
|
||
\begin_inset Formula $\mathbf{I}$
|
||
\end_inset
|
||
|
||
denotes a
|
||
\begin_inset Formula $2\times2$
|
||
\end_inset
|
||
|
||
identity sub matrix.
|
||
|
||
\begin_inset Formula $\mathbf{T}_{i}$
|
||
\end_inset
|
||
|
||
is the transfer matrix of the
|
||
\begin_inset Formula $i$
|
||
\end_inset
|
||
|
||
-th segment.
|
||
The solution can be obtained by Gaussian elimination, for which in
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubess
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
the
|
||
\family typewriter
|
||
numpy.linalg.solve()
|
||
\family default
|
||
solver is used.
|
||
Once the solution on the nodes is known, the solution in each segment can
|
||
be computed as a post processing step.
|
||
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubess
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
provides some post processing routines to aid in visualization of the acoustic
|
||
field inside a non-lumped segment, such as an acoustic duct.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Input impedance, output impedance
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The acoustic input impedance
|
||
\begin_inset Formula $Z_{\mathrm{in}}\equiv p_{L}/U_{L}$
|
||
\end_inset
|
||
|
||
on the left side of a segment is defined as the impedance a connecting
|
||
segment
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
feels
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
for a certain boundary condition on the right side.
|
||
|
||
\begin_inset Foot
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Note that the definitions of open and closed below are relating to electrical
|
||
circuits, not open or closed in the acoustical sense.
|
||
I.e.
|
||
an open impedance corresponds to a hard acoustic wall (which is acoustically
|
||
closed).
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
There are two special load cases for the segment, either on the right side,
|
||
the circuit is open, resulting in
|
||
\begin_inset Formula $U_{R}=0$
|
||
\end_inset
|
||
|
||
, or the circuit is shorted, which results in
|
||
\begin_inset Formula $p_{R}=0$
|
||
\end_inset
|
||
|
||
.
|
||
For the open circuit, the input impedance can be computed from the transfer
|
||
matrix as:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Open case: (
|
||
\begin_inset Formula $U_{R}=0$
|
||
\end_inset
|
||
|
||
_
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p_{R}=T_{11}p_{L}+T_{12}U_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U_{R}=0=T_{21}p_{L}+T_{22}U_{L}\Rightarrow\frac{p_{L}}{U_{L}}=-\frac{T_{22}}{T_{21}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Shorted case (
|
||
\begin_inset Formula $p_{R}=0$
|
||
\end_inset
|
||
|
||
):
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $0=T_{11}p_{L}+T_{12}U_{L}\Rightarrow\frac{p_{L}}{U_{L}}=-\frac{T_{12}}{T_{11}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
Z_{\mathrm{in},\mathrm{open}} & =-\frac{T_{22}}{T_{21}}\\
|
||
Z_{\mathrm{in},\mathrm{short}} & =-\frac{T_{12}}{T_{11}}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For a passive component (and passive load on the right side), the real part
|
||
of the input impedance should be positive:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Re\left[Z_{\mathrm{in}}\right]\geq0.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The acoustic output impedance
|
||
\begin_inset Formula $Z_{\mathrm{out}}\equiv p_{R}/U_{R}$
|
||
\end_inset
|
||
|
||
on the right side of a segment is defined as the impedance a connecting
|
||
segment
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
feels
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
for a certain boundary condition on the left side.
|
||
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
Z_{\mathrm{out},\mathrm{open}} & =\frac{T_{11}}{T_{21}}\\
|
||
Z_{\mathrm{out},\mathrm{short}} & =\frac{T_{\mathrm{12}}}{T_{22}}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Open case left side, means
|
||
\begin_inset Formula $U_{L}=0$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p_{R}=T_{11}p_{L}+T_{12}U_{L}$
|
||
\end_inset
|
||
|
||
–>
|
||
\begin_inset Formula $p_{R}=T_{11}p_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U_{R}=T_{21}p_{L}+T_{22}U_{L}$
|
||
\end_inset
|
||
|
||
–>
|
||
\begin_inset Formula $U_{R}=T_{21}p_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
==================================
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Shorted case, means
|
||
\begin_inset Formula $p_{L}=0$
|
||
\end_inset
|
||
|
||
,
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p_{R}=T_{11}p_{L}+T_{12}U_{L}$
|
||
\end_inset
|
||
|
||
–>
|
||
\begin_inset Formula $p_{R}=T_{12}U_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U_{R}=T_{21}p_{L}+T_{22}U_{L}$
|
||
\end_inset
|
||
|
||
–>
|
||
\begin_inset Formula $U_{R}=T_{22}U_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For passive segments, the real part of the output impedance should be
|
||
\emph on
|
||
negative:
|
||
\emph default
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Re\left[Z_{\mathrm{out}}\right]\leq0.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Segment properties and arguments
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "chap:Segment-properties"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Dit is meer een gebruikershandleiding van de technische achtergrond.
|
||
Misschien kunnen we hier een apart deel voor maken in de documentatie.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Introduction
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Each segment has various properties and user-set parameters.
|
||
This chapter provides a basic reference to these items.
|
||
Which properties and parameters are available differs for each segment.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Parameters
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Radius, diameter, area
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Most segments require a measure for the cross-sectional area.
|
||
For ease of use, this can be entered as either the radius
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $r$
|
||
\end_inset
|
||
|
||
, diameter
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $D$
|
||
\end_inset
|
||
|
||
or cross-sectional area
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
.
|
||
The letter is followed by either a capital
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $L$
|
||
\end_inset
|
||
|
||
for the left node or
|
||
\begin_inset Formula $R$
|
||
\end_inset
|
||
|
||
for the right, e.g.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $rL=0.2$
|
||
\end_inset
|
||
|
||
.
|
||
To avoid interpretation errors, they should be entered as keyword arguments,
|
||
e.g.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $\mathrm{ConeDuct}(L=1,rL=0.2,rR=0.3)$
|
||
\end_inset
|
||
|
||
instead of
|
||
\begin_inset Formula $\mathrm{ConeDuct}(1,0.2,0.3)$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Properties
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Once a segment has been constructed, its arguments become properties and
|
||
can be adjusted later on.
|
||
For example, say 'duct' is an instance of the class PrsDuct and we want
|
||
to change its radius.
|
||
Then
|
||
\begin_inset Formula $\mathrm{duct}.r=2$
|
||
\end_inset
|
||
|
||
will set it to
|
||
\begin_inset Formula $2$
|
||
\end_inset
|
||
|
||
m.
|
||
Retrieving any other measure of cross-sectional area is also possible:
|
||
|
||
\begin_inset Formula $x=\mathrm{duct}.D$
|
||
\end_inset
|
||
|
||
will retrieve the diameter and save it to
|
||
\begin_inset Formula $x$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
On top of that, segments contain calculated properties.
|
||
These are listed in Table
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "tab:calc_seg_properties"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float table
|
||
wide false
|
||
sideways false
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Tabular
|
||
<lyxtabular version="3" rows="4" columns="2">
|
||
<features tabularvalignment="middle">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Property
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Explanation
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $T(\omega)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Transfer matrix
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $Vf$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Volume of segment filled with fluid [
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
si{m}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
]
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $Z(\omega)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
(series) impedance [
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
si{Pa*s/m^3}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
]
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Calculated properties of segments
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "tab:calc_seg_properties"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Provided acoustic models
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "chap:Provided-acoustic-models"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Introduction
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
This chapter provides a concise overview of the provided acoustic models
|
||
implemented in
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubes
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Prismatic duct
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "subsec:Prismatic-duct"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/prsduct.pdf
|
||
width 80text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Geometry of the prismatic duct
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:prsduct"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A prismatic duct is used to model one-dimensional acoustic wave propagation.
|
||
The prismatic duct is implemented in
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubess
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
in the
|
||
\family typewriter
|
||
PrsDuct
|
||
\family default
|
||
class.
|
||
Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:prsduct"
|
||
|
||
\end_inset
|
||
|
||
shows this segment schematically.
|
||
In the thermal boundary layer, heat and momentum diffuse to the wall.
|
||
The thermal boundary layer can be a small layer w.r.t.
|
||
to the transverse characteristic length scale of the tube, or can fully
|
||
occupy the tube.
|
||
In the latter case, the solution converges to the classic laminar Poisseuille
|
||
flow solution.
|
||
The basic assumptions behind this model are
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Prismatic cross sectional area.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $L\gg r_{h}$
|
||
\end_inset
|
||
|
||
, (tube is long compared to its transverse length scale).
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Radius is much smaller than the wave length.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Wave length is much larger than viscous penetration depth.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
End effects and entrance effects are negligible.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For a formal derivation of the model for prismatic cylindrical tubes, the
|
||
reader is referred to the work of Tijdeman
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "tijdeman_propagation_1975"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
and Nijhof
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "nijhof_viscothermal_2010"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
For a somewhat more pragmatic derivation, we would like to refer to the
|
||
work of Swift
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "swift_thermoacoustics:_2003,swift_thermoacoustic_1988"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
and Rott
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "rott_damped_1969"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\frac{\mathrm{d}p}{\mathrm{d}x} & =\frac{\omega\rho_{0}}{i\left(1-f_{\nu}\right)S_{f}}U,\label{eq:momentum_LRF}\\
|
||
\frac{\mathrm{d}U}{\mathrm{d}x} & =\frac{k}{iZ_{0}}\left(1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\varepsilon_{s}}\right)p,\label{eq:continuity_LRF}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $S_{f}$
|
||
\end_inset
|
||
|
||
is the cross-sectional area filled with fluid,
|
||
\begin_inset Formula $k$
|
||
\end_inset
|
||
|
||
is the inviscid wave number, and
|
||
\begin_inset Formula $Z_{0}$
|
||
\end_inset
|
||
|
||
the inviscid characteristic impedance of a tube (
|
||
\begin_inset Formula $Z_{0}=z_{0}/S_{f}$
|
||
\end_inset
|
||
|
||
).
|
||
|
||
\begin_inset Formula $f_{\nu}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $f_{\kappa}$
|
||
\end_inset
|
||
|
||
are the viscous and thermal Rott functions, respectively
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "rott_damped_1969"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
They model the viscous and thermal effects with the wall.
|
||
For circular tubes, the
|
||
\begin_inset Formula $f$
|
||
\end_inset
|
||
|
||
's are defined as
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "p. 88"
|
||
key "swift_thermoacoustics:_2003"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
f_{j,\mathrm{circ}}=\frac{J_{1}\left[\left(i-1\right)\frac{2r_{h}}{\delta_{j}}\right]}{\left(i-1\right)\frac{r_{h}}{\delta}J_{0}\left[\left(i-1\right)\frac{2r_{h}}{\delta_{j}}\right]},\label{eq:f_cylindrical}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$j$"
|
||
description "Index, subscript placeholder\\nomunit{-}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\delta_{j}=\delta_{\nu}$
|
||
\end_inset
|
||
|
||
for
|
||
\begin_inset Formula $f_{\nu,\mathrm{circ}}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\delta_{j}=\delta_{\kappa}$
|
||
\end_inset
|
||
|
||
for
|
||
\begin_inset Formula $f_{\kappa,\mathrm{circ}}$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\begin_inset Formula $J_{\alpha}$
|
||
\end_inset
|
||
|
||
denotes the cylindrical Bessel function of the first kind and order
|
||
\begin_inset Formula $\alpha$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\begin_inset Formula $r_{h}$
|
||
\end_inset
|
||
|
||
is the hydraulic radius, defined as the ratio of the cross sectional area
|
||
to the
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
wetted perimeter
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
r_{h}=S_{f}/\Pi.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Note that for a circular tube with diameter
|
||
\begin_inset Formula $D$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $r_{h}=\nicefrac{D}{4}$
|
||
\end_inset
|
||
|
||
.
|
||
The parameter
|
||
\begin_inset Formula $\epsilon_{s}$
|
||
\end_inset
|
||
|
||
in Eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:continuity_LRF"
|
||
|
||
\end_inset
|
||
|
||
is the ideal solid correction factor, which corrects for solids that have
|
||
a finite heat capacity.
|
||
This parameter is dependent on the thermal properties and the geometry
|
||
of the solid.
|
||
An example of
|
||
\begin_inset Formula $\epsilon_{s}$
|
||
\end_inset
|
||
|
||
is derived in Section
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "subsec:Thermal-relaxation-effect"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
For the case of an thermally ideal solid,
|
||
\begin_inset Formula $\epsilon_{s}$
|
||
\end_inset
|
||
|
||
can be set to 0.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Other cross-sectional geometries
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Rectangular duct
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Analytical functions exist for prismatic geometries, such as parallel plates,
|
||
rectangular holes, and even triangular holes.
|
||
For parallel plates with sides
|
||
\begin_inset Formula $2y_{0}\times2z_{0}$
|
||
\end_inset
|
||
|
||
, the Rott function reads:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
f=1-\frac{64}{\pi^{4}}\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{1}{\left(2m-1\right)^{2}}\frac{1}{\left(2n-1\right)^{2}C_{mn}},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
C_{mn}=1-\frac{i\pi^{2}\delta^{2}}{8y_{0}^{2}z_{0}^{2}}\left(\left(2m-1\right)^{2}z_{0}^{2}+\left(2n-1\right)^{2}y_{0}^{2}\right).
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
The hydraulic radius is related to
|
||
\begin_inset Formula $y_{0}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $z_{0}$
|
||
\end_inset
|
||
|
||
as:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $r_{h}=\frac{S}{\Pi}=\frac{4y_{0}z_{0}}{4y_{0}+4z_{0}}=$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
r_{h}=\frac{y_{0}z_{0}}{y_{0}+z_{0}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Defining the aspect ratio as
|
||
\begin_inset Formula $\AR=z_{0}/y_{0}$
|
||
\end_inset
|
||
|
||
, a useful equation is to derive
|
||
\begin_inset Formula $y_{0}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $z_{0}$
|
||
\end_inset
|
||
|
||
from
|
||
\begin_inset Formula $r_{h}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\AR$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $r_{h}=\frac{y_{0}A}{\left(1+A\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $y_{0}=r_{h}\frac{\left(1+A\right)}{A}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $z_{0}=r_{h}\left(1+A\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
y_{0} & =r_{h}\frac{\left(1+\AR\right)}{\AR}\\
|
||
z_{0} & =r_{h}\left(1+\AR\right)
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Annular ring
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $i\omega\rho_{0}u=-\frac{\mathrm{d}p}{\mathrm{d}x}+\mu_{0}\nabla_{\perp}^{2}u$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Fill in:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
u=\frac{i}{\omega\rho_{0}}\left(1-h_{\nu}\right)\frac{\mathrm{d}p}{\mathrm{d}x}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Note that
|
||
\begin_inset Formula $h_{\nu}|_{\mathrm{wall}}\equiv1$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $0=h_{\nu}+\frac{i\delta_{\nu}^{2}}{2}\nabla_{\perp}^{2}h_{\nu}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $h_{\nu}+\frac{i\mu_{0}}{\omega\rho_{0}}\nabla_{\perp}^{2}h_{\nu}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
-
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{i\mu_{0}}{\omega\rho_{0}}\nabla_{\perp}^{2}h_{\nu}+h_{\nu}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The differential equation that is required to be solved
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{i\mu_{0}}{\omega\rho_{0}}\nabla_{\perp}^{2}h_{\nu}+h_{\nu}=0,\qquad h_{\nu|\mathrm{wall}}=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For an annular duct the Rott function reads:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
h_{\nu}=\frac{\left(J_{0}\left(\frac{r_{0}\left(1-i\right)}{\delta_{\nu}}\right)-J_{0}\left(\frac{r_{1}\left(1-i\right)}{\delta_{\nu}}\right)\right)Y_{0}\left(\frac{r\left(1-i\right)}{\delta_{\nu}}\right)+\left(Y_{0}\left(\frac{r_{1}\left(1-i\right)}{\delta_{\nu}}\right)-Y_{0}\left(\frac{r_{0}\left(1-i\right)}{\delta_{\nu}}\right)\right)J_{0}\left(\frac{r\left(1-i\right)}{\delta_{\nu}}\right)}{J_{0}\left(\frac{r_{0}\left(1-i\right)}{\delta_{\nu}}\right)Y_{0}\left(\frac{r_{1}\left(1-i\right)}{\delta_{\nu}}\right)-J_{0}\left(\frac{r_{1}\left(1-i\right)}{\delta_{\nu}}\right)Y_{0}\left(\frac{r_{0}\left(1-i\right)}{\delta_{\nu}}\right)}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Where
|
||
\begin_inset Formula
|
||
\begin{align*}
|
||
\alpha_{0} & =\frac{r_{0}\left(1-i\right)}{\delta_{i}}\\
|
||
\alpha_{1} & =\frac{r_{1}\left(1-i\right)}{\delta_{i}}
|
||
\end{align*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
And:
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
C_{1} & =\frac{Y_{0}\left(\alpha_{1}\right)-Y_{0}\left(\alpha_{0}\right)}{J_{0}\left(\alpha_{0}\right)Y_{0}\left(\alpha_{1}\right)-J_{0}\left(\alpha_{1}\right)Y_{0}\left(\alpha_{0}\right)}\\
|
||
C_{2} & =\frac{J_{0}\left(\alpha_{0}\right)-J_{0}\left(\alpha_{1}\right)}{J_{0}\left(\alpha_{0}\right)Y_{0}\left(\alpha_{1}\right)-J_{0}\left(\alpha_{1}\right)Y_{0}\left(\alpha_{0}\right)}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
f_{i} & =\delta_{i}\left(1+i\right)\left[\frac{\left\{ H_{0}^{(1)}\left(\alpha_{0}\right)-H_{0}^{(1)}\left(\alpha_{1}\right)\right\} \left[r_{0}H_{-1}^{(2)}\left(\alpha_{0}\right)-r_{1}H_{-1}^{(2)}\left(\alpha_{1}\right)\right]}{\left(r_{1}^{2}-r_{0}^{2}\right)\left[H_{0}^{(1)}\left(\alpha_{0}\right)H_{0}^{(2)}\left(\alpha_{1}\right)-H_{0}^{(1)}\left(\alpha_{1}\right)H_{0}^{(2)}\left(\alpha_{0}\right)\right]}+\right.\\
|
||
& \qquad\qquad\qquad\left.\frac{\left\{ H_{0}^{(2)}\left(\alpha_{0}\right)-H_{0}^{(2)}\left(\alpha_{1}\right)\right\} \left[r_{1}H_{-1}^{(1)}\left(\alpha_{1}\right)-r_{0}H_{-1}^{(1)}\left(\alpha_{0}\right)\right]}{\left(r_{1}^{2}-r_{0}^{2}\right)\left[H_{0}^{(1)}\left(\alpha_{0}\right)H_{0}^{(2)}\left(\alpha_{1}\right)-H_{0}^{(1)}\left(\alpha_{1}\right)H_{0}^{(2)}\left(\alpha_{0}\right)\right]}\right]
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Transfer matrix
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Upon solving for Eqs.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:momentum_LRF"
|
||
|
||
\end_inset
|
||
|
||
-
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:continuity_LRF"
|
||
|
||
\end_inset
|
||
|
||
, a transfer matrix can be derived which couples the pressure and volume
|
||
flow on the left side to the right side as:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{align*}
|
||
\frac{\mathrm{d}p}{\mathrm{d}x} & =\frac{\omega\rho_{0}}{i\left(1-f_{\nu}\right)S_{f}}U,\\
|
||
\frac{\mathrm{d}U}{\mathrm{d}x} & =\frac{k}{iZ_{0}}\left(1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\varepsilon_{s}}\right)p,
|
||
\end{align*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
We know the solution for
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
is
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p=A\exp\left(-i\Gamma x\right)+B\exp\left(i\Gamma x\right)$
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\Gamma^{2}=k^{2}\frac{\left(1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\varepsilon_{s}}\right)}{1-f_{\nu}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Then
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U=\frac{i\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{0}}\frac{\mathrm{d}p}{\mathrm{d}x}=\frac{i\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{0}}\Gamma i\left(-A\exp\left(-i\Gamma x\right)+B\exp\left(i\Gamma x\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U=-\frac{\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{0}}\Gamma\left(B\exp\left(i\Gamma x\right)-A\exp\left(-i\Gamma x\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Now:
|
||
\begin_inset Formula $p(x=0)=p_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
And:
|
||
\begin_inset Formula $U(x=0)=U_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Then:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U_{L}=\frac{\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{0}}\Gamma\left(A-B\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p_{L}=A+B\Rightarrow B=p_{L}-A$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Hence:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U_{L}=\frac{\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{0}}\Gamma\left(2A-p_{L}\right)$
|
||
\end_inset
|
||
|
||
or
|
||
\begin_inset Formula $A=\frac{1}{2}p_{L}+\frac{1}{2}\frac{\omega\rho_{0}}{\left(1-f_{\nu}\right)S_{f}\Gamma}U_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
And:
|
||
\begin_inset Formula $B=p_{L}-A=\frac{1}{2}p_{L}-\frac{1}{2}\frac{\omega\rho_{0}}{\left(1-f_{\nu}\right)S_{f}\Gamma}U_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
So, finally for
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
we find:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p=\left(\frac{1}{2}p_{L}+\frac{1}{2}\frac{\omega\rho_{0}}{\left(1-f_{\nu}\right)S_{f}\Gamma}U_{L}\right)\exp\left(-i\Gamma x\right)+\left(\frac{1}{2}p_{L}-\frac{1}{2}\frac{\omega\rho_{0}}{\left(1-f_{\nu}\right)S_{f}\Gamma}U_{L}\right)\exp\left(i\Gamma x\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p=\left(\frac{1}{2}p_{L}+\frac{1}{2}Z_{c}U_{L}\right)\exp\left(-i\Gamma x\right)+\left(\frac{1}{2}p_{L}-\frac{1}{2}Z_{c}U_{L}\right)\exp\left(i\Gamma x\right)$
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $Z_{c}=\frac{kZ_{0}}{\left(1-f_{\nu}\right)\Gamma}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Or, working to transfer matrices
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p=\frac{1}{2}p_{L}\exp\left(-i\Gamma x\right)+\frac{1}{2}Z_{c}U_{L}\exp\left(-i\Gamma x\right)+\frac{1}{2}p_{L}\exp\left(i\Gamma x\right)-Z_{c}U_{L}\exp\left(i\Gamma x\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p=p_{L}\cos\left(\Gamma x\right)+\frac{1}{2}Z_{c}U_{L}\exp\left(-i\Gamma x\right)-Z_{c}U_{L}\exp\left(i\Gamma x\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Using the rule:
|
||
\begin_inset Formula $\sin\left(x\right)=\frac{1}{2i}\left(e^{ix}-e^{-ix}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p=p_{L}\cos\left(\Gamma x\right)-iZ_{c}U_{L}\sin\left(\Gamma x\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Using
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U=\frac{i\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{0}}\frac{\mathrm{d}p}{\mathrm{d}x}=\frac{i}{Z_{c}}\left[-p_{L}\sin\left(\Gamma x\right)-iZ_{c}U_{L}\cos\left(\Gamma x\right)\right]=\left[-\frac{i}{Z_{c}}p_{L}\sin\left(\Gamma x\right)+U_{L}\cos\left(\Gamma x\right)\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left\{ \begin{array}{c}
|
||
p_{R}\\
|
||
U_{R}
|
||
\end{array}\right\} =\left[\begin{array}{cc}
|
||
\cos\left(\Gamma L\right) & -iZ_{c}\sin\left(\Gamma L\right)\\
|
||
-iZ_{c}^{-1}\sin\left(\Gamma L\right) & \cos\left(\Gamma L\right)
|
||
\end{array}\right]\left\{ \begin{array}{c}
|
||
p_{L}\\
|
||
U_{L}
|
||
\end{array}\right\} ,\label{eq:transfer_matrix_prismatic_duct}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $Z_{c}$
|
||
\end_inset
|
||
|
||
is the characteristic impedance of the duct, i.e.
|
||
the impedance
|
||
\begin_inset Formula $p/U$
|
||
\end_inset
|
||
|
||
of a plane (although damped) propagating wave:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{c}=\frac{kZ_{0}}{\left(1-f_{\nu}\right)\Gamma}.\label{eq:Z_c_prismduct}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
The parameter
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
in Eqs.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:transfer_matrix_prismatic_duct"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:Z_c_prismduct"
|
||
|
||
\end_inset
|
||
|
||
is the viscothermal wave number, i.e.
|
||
the wave number corrected for viscothermal losses:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Gamma=\frac{\omega}{c_{0}}\sqrt{\frac{1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\epsilon_{s}}}{1-f_{\nu}}}.\label{eq:Gamma}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Due to the numerical implementation of the Bessel functions in many libraries,
|
||
the
|
||
\begin_inset Formula $f_{j}$
|
||
\end_inset
|
||
|
||
function for cylindrical ducts (Eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:f_cylindrical"
|
||
|
||
\end_inset
|
||
|
||
) cannot be computed for high
|
||
\begin_inset Formula $r_{h}/\delta$
|
||
\end_inset
|
||
|
||
by computing this ratio
|
||
\begin_inset Formula $J_{1}/J_{0}$
|
||
\end_inset
|
||
|
||
.
|
||
The numerical result starts to break down at
|
||
\begin_inset Formula $r_{h}/\delta\sim100$
|
||
\end_inset
|
||
|
||
.
|
||
To resolve this problem, the
|
||
\begin_inset ERT
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubess
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
code applies a smooth transition from the Bessel function ratio to the
|
||
boundary layer limit solution for
|
||
\begin_inset Formula $f$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
f_{j,\mathrm{bl}}=\frac{\left(1-i\right)\delta_{j}}{2r_{h}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
in the range of
|
||
\begin_inset Formula $100<r_{h}/\delta\leq200$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Note that in the limit of
|
||
\begin_inset Formula $r_{h}\to\infty$
|
||
\end_inset
|
||
|
||
, or
|
||
\begin_inset Formula $\kappa$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\mu$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $\to0$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\Re\left[\Gamma\right]\to k$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\Re\left[Z_{c}\right]\to Z_{0}$
|
||
\end_inset
|
||
|
||
whereas
|
||
\begin_inset Formula $\Im\left[\Gamma\right]$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\Im\left[Z_{c}\right]$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $\to0$
|
||
\end_inset
|
||
|
||
.
|
||
Hence in these limits the lossless wave equation is resolved from the result.
|
||
This is not true in the limit of
|
||
\begin_inset Formula $\omega\to\infty$
|
||
\end_inset
|
||
|
||
, as in that limit it can be computed that
|
||
\begin_inset Formula $\Re\left[\Gamma\right]\to k$
|
||
\end_inset
|
||
|
||
, while the imaginary part
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Gamma=k\sqrt{\frac{1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\epsilon_{s}}}{1-f_{\nu}}}$
|
||
\end_inset
|
||
|
||
filling in
|
||
\begin_inset Formula $f_{\mathrm{bl}}$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $f_{\nu}=\frac{\left(1-i\right)\delta_{\nu}}{2r_{h}}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $f_{\kappa}=\frac{\left(1-i\right)\delta_{\nu}}{2\sqrt{\Pr}r_{h}}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\epsilon_{s}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Gamma^{2}=k^{2}\frac{1+\left(\gamma-1\right)\frac{\left(1-i\right)\delta_{\nu}}{2\sqrt{\Pr}r_{h}}}{1-\frac{\left(1-i\right)\delta_{\nu}}{2r_{h}}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Using
|
||
\begin_inset Formula $\alpha=\frac{1}{\sqrt{\Pr}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Gamma^{2}=k^{2}\frac{r_{h}+\frac{1}{2}\alpha\left(\gamma-1\right)\left(1-i\right)\delta_{\nu}}{r_{h}-\frac{1}{2}\left(1-i\right)\delta_{\nu}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Multiply numerator and denominator with
|
||
\begin_inset Formula $r_{h}+\frac{1}{2}\left(-i-1\right)\delta_{\nu}$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Gamma^{2}=k^{2}\frac{\left(r_{h}+\frac{1}{2}\left(-i-1\right)\delta_{\nu}\right)\left(r_{h}+\frac{1}{2}\left(-i-1\right)\delta_{\nu}\right)}{\left[r_{h}-\frac{1}{2}\left(1-i\right)\delta_{\nu}\right]\left(r_{h}+\frac{1}{2}\left(-i-1\right)\delta_{\nu}\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Gamma^{2}=k^{2}\frac{r_{h}^{2}+\frac{1}{2}r_{h}\delta_{\nu}\left[\alpha\left(\gamma-1\right)-1-i\left(1+\alpha\left(\gamma-1\right)\right)\right]+-\frac{1}{2}\alpha\delta_{\nu}^{2}\left(\gamma-1\right)}{r_{h}^{2}-r_{h}\delta_{\nu}+\frac{1}{2}\delta_{\nu}^{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Leaving terms of
|
||
\begin_inset Formula $\mathcal{O}\left(\delta_{\nu}^{0}\right)$
|
||
\end_inset
|
||
|
||
in the denominator and
|
||
\begin_inset Formula $\mathcal{O}\left(\delta_{\nu}^{1}\right)$
|
||
\end_inset
|
||
|
||
in the numerator:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Gamma^{2}=k^{2}\frac{r_{h}^{2}+\frac{1}{2}r_{h}\delta_{\nu}\left[\alpha\left(\gamma-1\right)-1-i\left(1+\alpha\left(\gamma-1\right)\right)\right]}{r_{h}^{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Removing from the real part the small stuff:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Gamma^{2}=k^{2}\left(1-i\frac{1}{2}\frac{\delta_{\nu}}{r_{h}}\left(1+\alpha\left(\gamma-1\right)\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Gamma^{2}=k^{2}\left(1-ix\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
where
|
||
\begin_inset Formula $x=\frac{\delta_{\nu}}{2r_{h}}\left[\left(1+\left(\gamma-1\right)\sqrt{\Pr^{-1}}\right)\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Taking the square root:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Gamma=\sqrt{k^{2}\left(1-ix\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Take the imaginary part:
|
||
\begin_inset Formula $\Im\left[\sqrt{a}\right]=\sqrt{|a|\frac{\Im\left[a\right]}{|a|}}=\sqrt{|a|}\frac{\Im\left[a\right]}{2|a|}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Now we assume:
|
||
\begin_inset Formula $\Im\left[a\right]/|a|\ll1$
|
||
\end_inset
|
||
|
||
, such that:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Im\left[\sqrt{a}\right]\approx\frac{1}{2}\sqrt{|a|}\frac{\Im\left[a\right]}{|a|}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Im\left[\Gamma\right]\approx k\frac{1}{2}\frac{-k^{2}x}{k^{2}}=-\frac{1}{2}kx=-k\frac{\delta_{\nu}}{4r_{h}}\left[1+\frac{\left(\gamma-1\right)}{\sqrt{\Pr}}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
-\Im\left[\Gamma\right]\to\sqrt{\omega}\frac{\sqrt{\frac{1}{8}\frac{\mu}{\rho_{0}}}}{c_{0}r_{h}}\left[1+\frac{\left(\gamma-1\right)}{\sqrt{\Pr}}\right].\label{eq:hf_limit_im_gamma}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
In other words the imaginary part of the wave number keeps growing, although
|
||
with a smaller rate than real part of the wave number.
|
||
So the higher the frequency, the smaller the viscothermal damping per wavelengt
|
||
h, but the higher the viscothermal damping per meter of duct.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:im_gamma"
|
||
|
||
\end_inset
|
||
|
||
shows the imaginary part of the wave number as a function of the frequency.
|
||
As visible, the magnitude of the viscothermal damping grows monotonically
|
||
with frequency.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/im_Gamma.pdf
|
||
width 80text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Logarithmic plot of the negative of imaginary part of the viscothermal wave
|
||
number
|
||
\begin_inset Formula $\left(-\Im\left[\Gamma\right]\right)$
|
||
\end_inset
|
||
|
||
, for a tube with a diameter of 1 mm.
|
||
In blue, the full
|
||
\begin_inset Formula $f_{\nu}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $f_{\kappa}$
|
||
\end_inset
|
||
|
||
of Eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:Gamma"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:f_cylindrical"
|
||
|
||
\end_inset
|
||
|
||
is used.
|
||
The orange curve corresponds to Eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:hf_limit_im_gamma"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:im_gamma"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
|
||
\series bold
|
||
Duct with varying cross-sectional area
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For ducts with variation in the cross-sectional area, an approximately valid
|
||
ordinary differential equation can be derived, which is a viscothermal
|
||
correction to Webster's horn equation
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "p. 181"
|
||
key "rienstra_introduction_2015"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}+\frac{1}{S_{f}}\frac{\mathrm{d}S_{f}}{\mathrm{d}x}\frac{\mathrm{d}p}{\mathrm{d}x}+\Gamma^{2}p=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Exponential duct (horn)
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
S_{f}=\exp\left(\alpha x\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}+\alpha\frac{\mathrm{d}p}{\mathrm{d}x}+\Gamma^{2}p=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Filling in:
|
||
\begin_inset Formula $p=a\exp\left(\beta x\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\beta^{2}+\alpha\beta+\Gamma^{2}=0
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Solving for
|
||
\begin_inset Formula $\beta$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\[
|
||
\beta=\frac{1}{2}\left(-\alpha\pm\sqrt{\alpha^{2}-4\Gamma^{2}}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Conical ducts
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For conical ducts, i.e.
|
||
ducts with quadratic variation in the cross-sectional area (linear variation
|
||
in the diameter, or cross-sectional length scale),
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\frac{dp_{1}}{dx} & = & \frac{\omega\rho_{m}}{i\left(1-f_{\nu}\right)S_{f}}U_{1},\\
|
||
\frac{dU_{1}}{dx} & = & \frac{\omega S_{f}}{i\gamma p_{m}}\left(1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\varepsilon_{s}}\right)p_{1},\\
|
||
& & +\tfrac{f_{\kappa}-f_{\nu}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\frac{1}{T_{m}}\frac{dT_{m}}{dx}U_{1},
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Neglect dTmdx part, assume Sf not consant:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{dp_{1}}{dx}=\frac{\omega\rho_{m}}{i\left(1-f_{\nu}\right)S_{f}}U_{1}$
|
||
\end_inset
|
||
|
||
so
|
||
\begin_inset Formula $U_{1}=\frac{i\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{m}}\frac{dp_{1}}{dx}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\lang english
|
||
\begin_inset Formula $\frac{d^{2}p_{1}}{dx^{2}}=\frac{\omega\rho_{m}}{i\left(1-f_{\nu}\right)}\left(\frac{1}{S_{f}}\frac{dU_{1}}{dx}-\frac{U_{1}}{S_{f}^{2}}\frac{dS_{f}}{dx}\right)$
|
||
\end_inset
|
||
|
||
———< fill in one below
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\lang english
|
||
\begin_inset Formula $\frac{dU_{1}}{dx}=\frac{\omega S_{f}}{i\gamma p_{m}}\left(1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\varepsilon_{s}}\right)p_{1}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
————-
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\lang english
|
||
\begin_inset Formula $\frac{d^{2}p_{1}}{dx^{2}}=\frac{\omega\rho_{m}}{i\left(1-f_{\nu}\right)}\left(\frac{1}{S_{f}}\left(\frac{\omega S_{f}}{i\gamma p_{m}}\left(1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\varepsilon_{s}}\right)p_{1}\right)-\frac{1}{S_{f}^{2}}\frac{dS_{f}}{dx}\left(\frac{i\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{m}}\frac{dp_{1}}{dx}\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\lang english
|
||
\begin_inset Formula $\frac{d^{2}p_{1}}{dx^{2}}=\frac{\omega\rho_{m}}{i\left(1-f_{\nu}\right)}\frac{\omega}{i\gamma p_{m}}\left(1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\varepsilon_{s}}\right)p_{1}-\frac{\omega\rho_{m}}{i\left(1-f_{\nu}\right)}\frac{1}{S_{f}}\frac{i\left(1-f_{\nu}\right)}{\omega\rho_{m}}\frac{dS_{f}}{dx}\frac{dp_{1}}{dx}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\lang english
|
||
\begin_inset Formula $\frac{d^{2}p_{1}}{dx^{2}}+\frac{1}{S_{f}}\frac{dS_{f}}{dx}\frac{dp_{1}}{dx}+\frac{\omega^{2}}{c_{m}^{2}}\frac{\left(1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\varepsilon_{s}}\right)}{\left(1-f_{\nu}\right)}p_{1}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Makes:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
\frac{d^{2}p_{1}}{dx^{2}}+\frac{1}{S_{f}}\frac{dS_{f}}{dx}\frac{dp_{1}}{dx}+\Gamma^{2}p_{1}=0
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\Gamma^{2}=\frac{\omega^{2}}{c_{m}^{2}}\frac{\left(1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\varepsilon_{s}}\right)}{\left(1-f_{\nu}\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $r=r_{0}+\alpha x$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $S=\pi\left(r_{0}+\alpha x\right)^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{dS_{f}}{dx}=2\alpha\pi\left(r_{0}+\alpha x\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{1}{S_{f}}\frac{dS_{f}}{dx}=\frac{2\alpha}{\left(r_{0}+\alpha x\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
For this horn,
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
\frac{d^{2}p_{1}}{dx^{2}}+\frac{2\alpha}{\left(r_{0}+\alpha x\right)}\frac{dp_{1}}{dx}+\Gamma^{2}p_{1}=0
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
And we find volume flow from
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{dp_{1}}{dx}=\frac{\omega\rho_{m}}{i\left(1-f_{\nu}\right)S_{f}}U_{1}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{i\left(1-f_{\nu}\right)\pi\left(r_{0}+\alpha x\right)^{2}}{\omega\rho_{m}}\frac{dp_{1}}{dx}=U_{1}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
such that for a conical tube the radius
|
||
\begin_inset Formula $r(x)$
|
||
\end_inset
|
||
|
||
varies as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
r(x)=r_{0}+\eta x,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\eta=\frac{x}{L}\left(r_{1}-r_{0}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Filling in for
|
||
\begin_inset Formula $S_{f}=\pi\left(r_{0}+\eta x\right)^{2}$
|
||
\end_inset
|
||
|
||
yields
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}+\frac{2\eta}{r_{0}+\eta x}\frac{\mathrm{d}p}{\mathrm{d}x}+\Gamma^{2}p=0,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
for which the solution is:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p=\frac{C_{1}\exp\left(-i\Gamma x\right)+C_{1}\exp\left(-i\Gamma x\right)}{r_{0}+\eta x}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
If we assume
|
||
\begin_inset Formula $S_{f}=\pi\left(r_{0}+\eta x\right)^{2}$
|
||
\end_inset
|
||
|
||
, where
|
||
\begin_inset Formula $\eta$
|
||
\end_inset
|
||
|
||
is the radius variation factor, this can be written as
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}+\frac{2\eta}{\left(r_{0}+\eta x\right)}\frac{\mathrm{d}p}{\mathrm{d}x}+\Gamma^{2}p=0.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Now assume that
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Gamma(x)\approx\Gamma(x=0)\equiv\Gamma_{0},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
or, the variation in the viscothermal wave number is negligible.
|
||
We can find the solution to this differential equation to be
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Solution:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Try:
|
||
\begin_inset Formula $p_{1}=Ae^{kx}\frac{1}{r_{0}+\alpha x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{d}{dx}p_{1}=Ae^{kx}\left(\frac{k}{r_{0}+\alpha x}-\frac{\alpha}{\left(r_{0}+\alpha x\right)^{2}}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{d^{2}}{dx^{2}}p_{1}=Ae^{kx}\left(\frac{k^{2}}{r_{0}+\alpha x}-\frac{\alpha k}{\left(r_{0}+\alpha x\right)^{2}}\right)+Ae^{kx}\left(-\frac{\alpha k}{\left(r_{0}+\alpha x\right)^{2}}+\frac{2\alpha^{2}}{\left(r_{0}+\alpha x\right)^{3}}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
——————-Substitution in
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{d^{2}p_{1}}{dx^{2}}+\frac{2\alpha}{\left(r_{0}+\alpha x\right)}\frac{dp_{1}}{dx}+\Gamma^{2}p_{1}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{k^{2}}{r_{0}+\alpha x}-\frac{2\alpha k}{\left(r_{0}+\alpha x\right)^{2}}+\frac{2\alpha^{2}}{\left(r_{0}+\alpha x\right)^{3}}+\frac{2\alpha}{\left(r_{0}+\alpha x\right)}\left(\frac{k}{r_{0}+\alpha x}-\frac{\alpha}{\left(r_{0}+\alpha x\right)^{2}}\right)+\Gamma^{2}\frac{1}{r_{0}+\alpha x}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{k^{2}}{r_{0}+\alpha x}-\frac{2\alpha k}{\left(r_{0}+\alpha x\right)^{2}}+\frac{2\alpha^{2}}{\left(r_{0}+\alpha x\right)^{3}}+\frac{2\alpha k}{\left(r_{0}+\alpha x\right)^{2}}-\frac{2\alpha^{2}}{\left(r_{0}+\alpha x\right)^{3}}+\Gamma^{2}\frac{1}{r_{0}+\alpha x}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{k^{2}}{r_{0}+\alpha x}+\Gamma^{2}\frac{1}{r_{0}+\alpha x}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $k^{2}=-\Gamma^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Resulting in:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{1}=p^{+}\frac{e^{-i\Gamma x}}{r_{0}+\alpha x}+p^{-}\frac{e^{i\Gamma x}}{r_{0}+\alpha x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{1}=C_{1}\frac{e^{-i\Gamma_{0}x}}{r_{0}+\eta x}+C_{2}\frac{e^{i\Gamma_{0}x}}{r_{0}+\eta x},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $C_{1}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $C_{2}$
|
||
\end_inset
|
||
|
||
are constants to be determined from the boundary conditions.
|
||
Upon filling in the boundary conditions, we can derive a transfer matrix
|
||
for a conical tube:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Derivation transfer matrix:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $U_{1}=\frac{i\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{m}}\frac{dp_{1}}{dx}=\frac{i\left(1-f_{\nu}\right)}{kZ_{0}}\frac{dp_{1}}{dx}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
And:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{1}=C_{1}\frac{e^{-i\Gamma x}}{r_{0}+\alpha x}+C_{2}\frac{e^{i\Gamma x}}{r_{0}+\alpha x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{dp_{1}}{dx}=C_{1}\left(-i\Gamma\frac{e^{-i\Gamma x}}{r_{0}+\alpha x}-\alpha\frac{e^{-i\Gamma x}}{\left(r_{0}+\alpha x\right)^{2}}\right)+C_{2}\left(i\Gamma\frac{e^{i\Gamma x}}{r_{0}+\alpha x}-\frac{\alpha e^{i\Gamma x}}{\left(r_{0}+\alpha x\right)^{2}}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
So:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $U_{1}=\frac{i\left(1-f_{\nu}\right)\pi}{\omega\rho_{m}}\left(-C_{1}\left(i\Gamma\left(r_{0}+\alpha x\right)e^{-i\Gamma x}+\alpha e^{-i\Gamma x}\right)+C_{2}\left(\left(r_{0}+\alpha x\right)i\Gamma e^{i\Gamma x}-\alpha e^{i\Gamma x}\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
————-
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $U_{1}=\frac{i\left(1-f_{\nu}\right)\pi}{\omega\rho_{m}}\left(-C_{1}\left(i\Gamma\left(r_{0}+\alpha x\right)e^{-i\Gamma x}+\alpha e^{-i\Gamma x}\right)+C_{2}\left(\left(r_{0}+\alpha x\right)i\Gamma e^{i\Gamma x}-\alpha e^{i\Gamma x}\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
———————-
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{1}=C_{1}\frac{e^{-i\Gamma x}}{r_{0}+\alpha x}+C_{2}\frac{e^{i\Gamma x}}{r_{0}+\alpha x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{L}=\frac{1}{r_{0}}\left(C_{1}+C_{2}\right)\Rightarrow C_{2}=r_{0}p_{L}-C_{1}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $U_{L}=U_{1}(0)=\frac{i\left(1-f_{\nu}\right)\pi}{\omega\rho_{m}}\left(r_{0}p_{L}\left(r_{0}i\Gamma-\alpha\right)-2C_{1}r_{0}i\Gamma\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $r_{0}p_{L}\left(r_{0}i\Gamma-\alpha\right)-\frac{\omega\rho_{m}}{i\left(1-f_{\nu}\right)\pi}U_{L}=2C_{1}r_{0}i\Gamma$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
So:
|
||
\begin_inset Formula $C_{1}=\frac{r_{0}p_{L}\left(r_{0}i\Gamma-\alpha\right)-\frac{\omega\rho_{m}}{i\left(1-f_{\nu}\right)\pi}U_{L}}{2i\Gamma r_{0}}=\frac{p_{L}\left(r_{0}i\Gamma-\alpha\right)}{2i\Gamma}+\frac{\omega\rho_{m}}{2\Gamma\pi r_{0}\left(1-f_{\nu}\right)}U_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
And:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $C_{2}=r_{0}p_{L}-C_{1}=r_{0}p_{L}-\frac{p_{L}\left(r_{0}i\Gamma-\alpha\right)}{2i\Gamma}-\frac{\omega\rho_{m}}{2\Gamma\pi r_{0}\left(1-f_{\nu}\right)}U_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Makes finally:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{1}=C_{1}\frac{e^{-i\Gamma x}}{r_{0}+\alpha x}+C_{2}\frac{e^{i\Gamma x}}{r_{0}+\alpha x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{1}=\left(\frac{p_{L}\left(r_{0}i\Gamma-\alpha\right)}{2i\Gamma}+\frac{\omega\rho_{m}}{2\Gamma\pi r_{0}\left(1-f_{\nu}\right)}U_{L}\right)\frac{e^{-i\Gamma x}}{r_{0}+\alpha x}+\left(r_{0}p_{L}-\frac{p_{L}\left(r_{0}i\Gamma-\alpha\right)}{2i\Gamma}-\frac{\omega\rho_{m}}{2\Gamma\pi r_{0}\left(1-f_{\nu}\right)}U_{L}\right)\frac{e^{i\Gamma x}}{r_{0}+\alpha x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{1}=\frac{p_{L}\left(r_{0}i\Gamma-\alpha\right)}{2i\Gamma}\frac{e^{-i\Gamma x}}{r_{0}+\alpha x}+\frac{\omega\rho_{m}}{2\Gamma\pi r_{0}\left(1-f_{\nu}\right)}U_{L}\frac{e^{-i\Gamma x}}{r_{0}+\alpha x}+r_{0}p_{L}\frac{e^{i\Gamma x}}{r_{0}+\alpha x}-\frac{p_{L}\left(r_{0}i\Gamma-\alpha\right)}{2i\Gamma}\frac{e^{i\Gamma x}}{r_{0}+\alpha x}-\frac{\omega\rho_{m}}{2\Gamma\pi r_{0}\left(1-f_{\nu}\right)}U_{L}\frac{e^{i\Gamma x}}{r_{0}+\alpha x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{1}=p_{L}\left[\frac{r_{0}\cos\left(\Gamma x\right)}{r_{0}+\alpha x}+\frac{\alpha}{\Gamma}\frac{\sin\left(\Gamma x\right)}{r_{0}+\alpha x}\right]-\frac{i\omega\rho_{m}}{\Gamma\pi r_{0}\left(1-f_{\nu}\right)}U_{L}\frac{\sin\left(\Gamma x\right)}{r_{0}+\alpha x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Check with prismatic:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\lang english
|
||
\begin_inset Formula $p_{1}=p_{L}\cos\left(\Gamma x\right)-\frac{\omega\rho_{m}i}{\left(1-f_{\nu}\right)S_{f}\Gamma}U_{L}\sin\left(\Gamma x\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Check!
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Check one: p(0):
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{1}\left(0\right)=p_{L}$
|
||
\end_inset
|
||
|
||
check
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Now: U1:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{dp_{1}}{dx}=-\Gamma p_{L}\frac{r_{0}\sin\left(\Gamma x\right)}{r_{0}+\alpha x}-p_{L}\alpha\frac{r_{0}\cos\left(\Gamma x\right)}{\left(r_{0}+\alpha x\right)^{2}}+p_{L}\alpha\frac{\cos\left(\Gamma x\right)}{r_{0}+\alpha x}-p_{L}\frac{\alpha^{2}}{\Gamma}\frac{\sin\left(\Gamma x\right)}{\left(r_{0}+\alpha x\right)^{2}}-\frac{i\omega\rho_{m}}{\pi r_{0}\left(1-f_{\nu}\right)}U_{L}\frac{\cos\left(\Gamma x\right)}{r_{0}+\alpha x}+\frac{i\omega\rho_{m}}{\Gamma\pi r_{0}\left(1-f_{\nu}\right)}\alpha U_{L}\frac{\sin\left(\Gamma x\right)}{\left(r_{0}+\alpha x\right)^{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\frac{dp_{1}}{dx}=-\Gamma p_{L}\frac{r_{0}\sin\left(\Gamma x\right)}{r_{0}+\alpha x}-p_{L}\frac{\alpha^{2}}{\Gamma}\frac{\sin\left(\Gamma x\right)}{\left(r_{0}+\alpha x\right)^{2}}-p_{L}\alpha\frac{r_{0}\cos\left(\Gamma x\right)}{\left(r_{0}+\alpha x\right)^{2}}+p_{L}\alpha\frac{\cos\left(\Gamma x\right)}{r_{0}+\alpha x}-\frac{i\omega\rho_{m}}{\pi r_{0}\left(1-f_{\nu}\right)}U_{L}\frac{\cos\left(\Gamma x\right)}{r_{0}+\alpha x}+\frac{i\omega\rho_{m}}{\Gamma\pi r_{0}\left(1-f_{\nu}\right)}\alpha U_{L}\frac{\sin\left(\Gamma x\right)}{\left(r_{0}+\alpha x\right)^{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $U_{1}=\frac{i\left(1-f_{\nu}\right)\pi\left(r_{0}+\alpha x\right)^{2}}{\omega\rho_{m}}\frac{dp_{1}}{dx}=\frac{i\left(1-f_{\nu}\right)\pi}{\omega\rho_{m}}\left(-p_{L}\left(\Gamma\left(r_{0}^{2}+\alpha r_{0}x\right)+\frac{\alpha^{2}}{\Gamma}\right)\sin\left(\Gamma x\right)+p_{L}\alpha^{2}x\cos\left(\Gamma x\right)+U_{L}\left(-\left(r_{0}+\alpha x\right)\frac{i\omega\rho_{m}}{\pi r_{0}\left(1-f_{\nu}\right)}\cos\left(\Gamma x\right)+\frac{i\omega\rho_{m}}{\Gamma\pi r_{0}\left(1-f_{\nu}\right)}\alpha\sin\left(\Gamma x\right)\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $U_{1}=-p_{L}\left(\Gamma\left(r_{0}^{2}+\alpha r_{0}x\right)+\frac{\alpha^{2}}{\Gamma}\right)\frac{i\left(1-f_{\nu}\right)\pi}{\omega\rho_{m}}\sin\left(\Gamma x\right)+p_{L}\frac{i\left(1-f_{\nu}\right)\pi}{\omega\rho_{m}}\alpha^{2}x\cos\left(\Gamma x\right)+U_{L}\left(\frac{\left(r_{0}+\alpha x\right)}{r_{0}}\cos\left(\Gamma x\right)-\frac{\alpha}{\Gamma r_{0}}\sin\left(\Gamma x\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\lang english
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\lang english
|
||
Introducing:
|
||
\begin_inset Formula $\delta=\frac{i\omega\rho_{m}}{\left(1-f_{\nu}\right)S_{f}\Gamma}\Rightarrow\delta_{0}=\frac{i\omega\rho_{m}}{\left(1-f_{\nu}\right)\pi r_{0}^{2}\Gamma}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Check for
|
||
\begin_inset Formula $U_{1}(0)$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $U_{1}(0)=U_{L}$
|
||
\end_inset
|
||
|
||
check!!
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
————————– Simpler form of
|
||
\begin_inset Formula $U_{1}$
|
||
\end_inset
|
||
|
||
?
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $U_{1}=p_{L}\left(\Gamma\left(r_{0}^{2}+\alpha r_{0}x\right)+\frac{\alpha^{2}}{\Gamma}\right)\frac{\left(1-f_{\nu}\right)\pi}{i\omega\rho_{m}}\sin\left(\Gamma x\right)+p_{L}\frac{i\left(1-f_{\nu}\right)\pi}{\omega\rho_{m}}\alpha^{2}x\cos\left(\Gamma x\right)+U_{L}\left(\frac{\left(r_{0}+\alpha x\right)}{r_{0}}\cos\left(\Gamma x\right)-\frac{\alpha}{\Gamma r_{0}}\sin\left(\Gamma x\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Prismatic tube check:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $U_{1pris}=p_{L}\frac{\Gamma\left(1-f_{\nu}\right)\pi r_{0}^{2}}{i\omega\rho_{m}}\sin\left(\Gamma x\right)+U_{L}\cos\left(\Gamma x\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
with:
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
|
||
\begin_inset Formula $\frac{\left(1-f_{\nu}\right)\Gamma S_{f}}{i\omega\rho_{m}}p_{L}\sin\left(\Gamma x\right)+U_{L}\cos\left(\Gamma x\right)$
|
||
\end_inset
|
||
|
||
<< from previous derivation!
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $U_{1}=p_{L}\left[\left(1+\frac{\alpha x}{r_{0}}+\frac{\alpha^{2}}{\Gamma^{2}r_{0}^{2}}\right)\frac{1}{\delta_{0}}\sin\left(\Gamma x\right)-\frac{\alpha^{2}x\cos\left(\Gamma x\right)}{r_{0}^{2}\Gamma\delta_{0}}\right]+U_{L}\left[\left(1+\frac{\alpha x}{r_{0}}\right)\cos\left(\Gamma x\right)-\frac{\alpha}{\Gamma r_{0}}\sin\left(\Gamma x\right)\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{1}=p_{L}\left[\frac{r_{0}\cos\left(\Gamma x\right)}{r_{0}+\alpha x}+\frac{\alpha}{\Gamma}\frac{\sin\left(\Gamma x\right)}{r_{0}+\alpha x}\right]-\delta_{0}U_{L}\frac{r_{0}\sin\left(\Gamma x\right)}{r_{0}+\alpha x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $U_{1}=p_{L}\left[\left(1+\frac{\alpha x}{r_{0}}+\frac{\alpha^{2}}{\Gamma^{2}r_{0}^{2}}\right)\frac{1}{\delta_{0}}\sin\left(\Gamma x\right)-\frac{\alpha^{2}x\cos\left(\Gamma x\right)}{r_{0}^{2}\Gamma\delta_{0}}\right]+U_{L}\left[\left(1+\frac{\alpha x}{r_{0}}\right)\cos\left(\Gamma x\right)-\frac{\alpha}{\Gamma r_{0}}\sin\left(\Gamma x\right)\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{1}=p_{L}\left[\frac{r_{0}\cos\left(\Gamma x\right)}{r_{0}+\alpha x}+\frac{\alpha}{\Gamma}\frac{\sin\left(\Gamma x\right)}{r_{0}+\alpha x}\right]-\delta_{0}U_{L}\frac{r_{0}\sin\left(\Gamma x\right)}{r_{0}+\alpha x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\left\{ \begin{array}{c}
|
||
p_{1}\\
|
||
U_{1}
|
||
\end{array}\right\} _{R}=\left[\begin{array}{cc}
|
||
\left[\frac{r_{0}\cos\left(\Gamma L\right)}{r_{0}+\alpha L}+\frac{\alpha}{\Gamma}\frac{\sin\left(\Gamma L\right)}{r_{0}+\alpha L}\right] & \left[-\delta_{0}\frac{r_{0}\sin\left(\Gamma L\right)}{r_{0}+\alpha L}\right]\\
|
||
\left[\left(1+\frac{\alpha L}{r_{0}}+\frac{\alpha^{2}}{\Gamma^{2}r_{0}^{2}}\right)\frac{1}{\delta_{0}}\sin\left(\Gamma L\right)-\frac{\alpha^{2}L\cos\left(\Gamma L\right)}{r_{0}^{2}\Gamma\delta_{0}}\right] & \left[\left(1+\frac{\alpha L}{r_{0}}\right)\cos\left(\Gamma L\right)-\frac{\alpha}{\Gamma r_{0}}\sin\left(\Gamma L\right)\right]
|
||
\end{array}\right]\left\{ \begin{array}{c}
|
||
p_{1}\\
|
||
U_{1}
|
||
\end{array}\right\} _{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\delta_{0}=i\frac{\omega\rho_{m}}{\left(1-f_{\nu}\right)\pi r_{0}^{2}\Gamma}=iZ_{c,0}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
According to sympy, for pR:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\Gamma p_{L}r_{0}\cos\left(\Gamma L\right)+\eta p_{L}\sin\left(\Gamma L\right)}{\Gamma r_{1}}-iZ_{c,0}\frac{r_{0}\sin\left(\Gamma L\right)}{r_{1}}U_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
, klopt!
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
According to sympy, for UR:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{align*}
|
||
U_{R} & =-i\frac{\Gamma\left(L\eta+r_{0}\right)\left(\Gamma p_{L}r_{0}\sin{\left(\Gamma L\right)}+\left(i\Gamma U_{L}Z_{c0}r_{0}-\eta p_{L}\right)\cos{\left(\Gamma L\right)}\right)+\eta\left(\Gamma p_{L}r_{0}\cos{\left(\Gamma L\right)}-\left(i\Gamma U_{L}Z_{c0}r_{0}-\eta p_{L}\right)\sin{\left(\Gamma L\right)}\right)}{\Gamma^{2}Z_{c0}r_{0}^{2}}\\
|
||
\end{align*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathbf{T}_{\mbox{cone}}=\left[\begin{array}{cc}
|
||
\frac{\Gamma r_{0}\cos\left(\Gamma L\right)+\eta\sin\left(\Gamma L\right)}{\Gamma r_{1}} & -iZ_{c,0}\frac{kr_{0}\sin\left(\Gamma L\right)}{\Gamma r_{1}}\\
|
||
\frac{iL\eta^{2}\cos\left(\Gamma L\right)}{\Gamma Z_{c0}r_{0}^{2}}-\frac{i}{Z_{c0}}\left(\frac{r_{1}}{r_{0}}+\frac{\eta^{2}}{\Gamma^{2}r_{0}^{2}}\right)\sin\left(\Gamma L\right)\,\,\,\, & \frac{r_{1}}{r_{0}}\cos\left(\Gamma L\right)-\frac{\eta\sin\left(\Gamma L\right)}{\Gamma r_{0}}
|
||
\end{array}\right],
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{c,0}=\frac{\omega\rho_{0}}{\left(1-f_{\nu}\right)S_{f,0}\Gamma_{0}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
and
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
is defined in eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:Gamma"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
and can be approximated as
|
||
\begin_inset Formula $\Gamma\approx k$
|
||
\end_inset
|
||
|
||
under some undefined and guessed circumstances AANVULLEN, in which
|
||
\begin_inset Formula $k=\frac{\omega}{c_{0}}$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Prismatic lined circular duct
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The Fourier transformed wave equation in axisymmetric cylindrical coordinates
|
||
can be written as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial^{2}p}{\partial r^{2}}+\frac{1}{r}\frac{\partial p}{\partial r}+\frac{\partial^{2}p}{\partial x^{2}}+k^{2}p=0,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Using separation of variables:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p=\rho(r)\xi(x),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
this can be written as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\rho^{''}}{\rho}+\frac{1}{r}\frac{\rho'}{\rho}+\frac{\xi^{''}}{\xi}+k^{2}=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Solutions:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{1}{r}\frac{\rho'}{\rho}+\frac{\rho^{''}}{\rho}=-k^{2}-\frac{\xi^{''}}{\xi}=-\epsilon^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Try:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\xi=A\exp\left(\alpha x\right)$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $-k^{2}-\alpha^{2}=-\epsilon^{2}$
|
||
\end_inset
|
||
|
||
Or:
|
||
\begin_inset Formula $\alpha^{2}=\epsilon^{2}-k^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
And
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{1}{r}\frac{\rho'}{\rho}+\frac{\rho^{''}}{\rho}=-\epsilon^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Means:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $r\frac{\rho'}{\rho}+r^{2}\frac{\rho^{''}}{\rho}+r^{2}\epsilon^{2}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Which has solution:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\rho=J_{0}\left(\epsilon r\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\xi & =\exp\left(-i\alpha x\right),\\
|
||
\rho & =J_{0}\left(\epsilon r\right),
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
such that the solution for the pressure is:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p=J_{0}\left(\epsilon r\right)\exp\left(\alpha x\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
under the condition:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\alpha^{2}=k^{2}-\epsilon^{2}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
At
|
||
\begin_inset Formula $r=R$
|
||
\end_inset
|
||
|
||
we have the boundary condition that
|
||
\begin_inset Formula $Z_{0}\zeta_{R}u=p$
|
||
\end_inset
|
||
|
||
.
|
||
After filling in and using the rule
|
||
\begin_inset Formula $J_{0}'(x)=J_{-1}(x)$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{i\zeta_{R}}{k}\frac{\partial p}{\partial x}|_{r=R}=p|r=R$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{i\zeta_{R}}{k}\epsilon J'_{0}\left(\epsilon r\right)=J_{0}\left(\epsilon r\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Or:
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\epsilon R\frac{J_{-1}\left(\epsilon R\right)}{J_{0}\left(\epsilon R\right)}=-i\upsilon,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\upsilon=\frac{kR}{\zeta_{R}}$
|
||
\end_inset
|
||
|
||
.
|
||
This is the characteristic equation for
|
||
\begin_inset Formula $\epsilon R$
|
||
\end_inset
|
||
|
||
.
|
||
Solutions for
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Im\left[\epsilon R\right]<3$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
And
|
||
\begin_inset Formula $\Re\left[2\right]<2$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Using
|
||
\begin_inset Formula $\upsilon=\frac{kR}{\zeta_{R}}$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Solution:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(\epsilon R\right)^{2}\approx\frac{96+36i\upsilon\pm\sqrt{9216+2304i\upsilon-912\upsilon^{2}}}{12+i\upsilon}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Filling in for
|
||
\begin_inset Formula $U$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\epsilon\approx+\frac{1}{R}\sqrt{\frac{96+36i\upsilon\pm\sqrt{9216+2304i\upsilon-912\upsilon^{2}}}{12+i\upsilon}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $0\leq\Re[\epsilon R]\leq2$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $0\leq\Im\left[\epsilon R\right]\leq3$
|
||
\end_inset
|
||
|
||
should be satisfied in order to guarantee precision, see Mechel, p.
|
||
630.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Prismatic duct with flow
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Assuming fully developed plug flow in a duct the linearized governing equations
|
||
in frequency domain read:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
i\omega\rho+\rho_{0}\frac{\mathrm{d}u}{\mathrm{d}x}+u_{0}\frac{\mathrm{d}\rho}{\mathrm{d}x} & =0\\
|
||
i\rho_{0}\omega u+\rho_{0}u_{0}\frac{\mathrm{d}u}{\mathrm{d}x}+\frac{\mathrm{d}p}{\mathrm{d}x} & =0\\
|
||
p & =c_{0}^{2}\rho
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
With subscript 0 are the mean flow variables.
|
||
Eliminating
|
||
\begin_inset Formula $\rho$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\frac{1}{c_{0}^{2}}\left(i\omega p+u_{0}\frac{\mathrm{d}p}{\mathrm{d}x}\right)+\rho_{0}\frac{\mathrm{d}u}{\mathrm{d}x} & =0\\
|
||
\rho_{0}\left(i\omega u+u_{0}\frac{\mathrm{d}u}{\mathrm{d}x}\right)+\frac{\mathrm{d}p}{\mathrm{d}x} & =0
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Taking spatial derivative of momentum and subtracting the convective derivative
|
||
of the continuity equation from it yields the convective wave equation:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $i\omega\rho_{0}\frac{\mathrm{d}u}{\mathrm{d}x}+u_{0}\rho_{0}\frac{\mathrm{d}^{2}u}{\mathrm{d}^{2}x}+\frac{\mathrm{d}^{2}p}{\mathrm{d}^{2}x}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Take the convected time derivative of the continuity:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(i\omega+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\right)^{2}\frac{1}{c_{0}^{2}}p+\rho_{0}\left(i\omega+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\right)\frac{\mathrm{d}u}{\mathrm{d}x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
–
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Subtract:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(i\omega+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\right)^{2}\frac{1}{c_{0}^{2}}p-\frac{\mathrm{d}^{2}p}{\mathrm{d}^{2}x}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Try
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left(i\omega+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\right)^{2}\frac{1}{c_{0}^{2}}p-\frac{\mathrm{d}^{2}p}{\mathrm{d}^{2}x}=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
For constant
|
||
\begin_inset Formula $u_{0}$
|
||
\end_inset
|
||
|
||
, we try solutions of the form:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p=A\exp\left(\alpha x\right),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
which yields the characteristic equation for
|
||
\begin_inset Formula $\alpha$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\underbrace{\left(M^{2}-1\right)}_{a}\alpha^{2}+\underbrace{2Mki}_{b}\alpha\underbrace{-k^{2}}_{c}=0,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $M$
|
||
\end_inset
|
||
|
||
denotes the Mach number
|
||
\begin_inset Formula $u_{0}/c_{0}$
|
||
\end_inset
|
||
|
||
.
|
||
The solutions for
|
||
\begin_inset Formula $\alpha$
|
||
\end_inset
|
||
|
||
are:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\alpha=i\frac{Mk\pm k}{1-M^{2}}=\pm ik\frac{1}{1\mp M}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Written out:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(i\omega+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\right)^{2}\frac{1}{c_{0}^{2}}A\exp\left(\alpha x\right)-\frac{\mathrm{d}^{2}A\exp\left(\alpha x\right)}{\mathrm{d}^{2}x}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(i\omega+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\right)\left(i\omega\frac{1}{c_{0}^{2}}A\exp\left(\alpha x\right)+\frac{u_{0}}{c_{0}^{2}}\alpha A\exp\left(\alpha x\right)\right)-\frac{\mathrm{d}^{2}A\exp\left(\alpha x\right)}{\mathrm{d}^{2}x}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(i\omega i\omega\frac{1}{c_{0}^{2}}\exp\left(\alpha x\right)+u_{0}\alpha i\omega\frac{1}{c_{0}^{2}}\exp\left(\alpha x\right)\right)+\left(i\omega\frac{u_{0}}{c_{0}^{2}}\alpha\exp\left(\alpha x\right)+u_{0}\frac{u_{0}}{c_{0}^{2}}\alpha^{2}\exp\left(\alpha x\right)\right)-\frac{\mathrm{d}^{2}\exp\left(\alpha x\right)}{\mathrm{d}^{2}x}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(M^{2}-1\right)\alpha^{2}+2Mk\alpha i-k^{2}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Regular form:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\underbrace{\left(M^{2}-1\right)}_{a}\alpha^{2}+\underbrace{2Mki}_{b}\alpha\underbrace{-k^{2}}_{c}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Solutions are:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\alpha=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{-2Mki\pm\sqrt{\left(2Mki\right)^{2}+4\left(M^{2}-1\right)k^{2}}}{2\left(M^{2}-1\right)}=\frac{\pm ik-Mki}{\left(M^{2}-1\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
p=A\exp\left(-\frac{ik}{1+M}x\right)+B\exp\left(\frac{ik}{1-M}x\right),
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
and the volume flow:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\mathrm{d}u}{\mathrm{d}x}=-\frac{1}{\rho_{0}c_{0}^{2}}\left(i\omega\left(A\exp\left(-\frac{ik}{1+M}x\right)+B\exp\left(\frac{ik}{1-M}x\right)\right)+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\left(A\exp\left(-\frac{ik}{1+M}x\right)+B\exp\left(\frac{ik}{1-M}x\right)\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\mathrm{d}u}{\mathrm{d}x}=-\frac{1}{\rho_{0}c_{0}^{2}}i\omega\left(A\exp\left(-\frac{ik}{1+M}x\right)+B\exp\left(\frac{ik}{1-M}x\right)\right)+-\frac{1}{\rho_{0}c_{0}^{2}}u_{0}\left(-\frac{ik}{1+M}A\exp\left(-\frac{ik}{1+M}x\right)+B\frac{ik}{1-M}\exp\left(\frac{ik}{1-M}x\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $u=\frac{A}{z_{0}}\exp\left(-\frac{ik}{1+M}x\right)-\frac{B}{z_{0}}\exp\left(\frac{ik}{1-M}x\right)+C$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Cremers impedance
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{kR}{\zeta}=2.9803824+1.2796025i
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Or:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\zeta=y_{cr}\pi\frac{kR}{y_{cr}\pi}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\zeta=kR\left(0.28-0.12i\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Attenuation reached when the liner impedance equals Cremer's impedance is
|
||
around 15 dB per unit of radius maximum.
|
||
It decreases with increasing frequency, when
|
||
\begin_inset Formula $fR\approx100$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Locally reacting lining with back-volume
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Impedance of concentric liner, outer radius is
|
||
\begin_inset Formula $R_{o}$
|
||
\end_inset
|
||
|
||
, inner radius is
|
||
\begin_inset Formula $R_{i}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\zeta_{\mathrm{back}}=i\frac{H_{0}^{(1)}\left(kR_{i}\right)-\frac{H_{1}^{(1)}\left(kR_{o}\right)}{H_{1}^{(2)}\left(kR_{o}\right)}H_{0}^{(2)}\left(kR_{i}\right)}{H_{1}^{(1)}\left(kR_{i}\right)-\frac{H_{1}^{(1)}\left(kR_{o}\right)}{H_{1}^{(2)}\left(kR_{o}\right)}H_{1}^{(2)}\left(kR_{i}\right)}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Such that the total impedance is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\zeta=\zeta_{\mathrm{back}}+\zeta_{\mathrm{MPP}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Cavity silencer
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
-
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Compliance volume
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "subsec:Compliance-volume"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/volume.pdf
|
||
width 30text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Schematic of the compliance volume segment.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:compliance"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:compliance"
|
||
|
||
\end_inset
|
||
|
||
gives a schematic of the compliance volume.
|
||
A compliance volume is implemented in the
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubess
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
code in the
|
||
\family typewriter
|
||
Volume
|
||
\family default
|
||
class.
|
||
A compliance volume is a volume (tank) which is small compared to the wavelengt
|
||
h.
|
||
Hence, we can assume that the acoustic pressure is constant throughout
|
||
the volume
|
||
\begin_inset Formula $V$
|
||
\end_inset
|
||
|
||
.
|
||
As thermal relaxation still occurs, the model for this segment takes into
|
||
account thermal relaxation due to temperature oscillations.
|
||
The basic assumptions behind the model are:
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
The characteristic length scale of volume is small compared to the wavelength.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
The characteristic length scale of volume is large compared to thermal penetrati
|
||
on depth.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The lower the frequency, the more the second assumption is violated, while
|
||
the higher the frequency, the more the first assumption is violated.
|
||
In practice, violating the first assumption has a larger impact.
|
||
For a compliance, the following governing equations can be derived
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "p. 156"
|
||
key "ward_deltaec_2017"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Derivation of the capacitance:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U_{R}=U_{L}-i\frac{k}{z_{0}}\left(V-\frac{i}{2}\frac{\left(\gamma-1\right)}{1+\epsilon_{s,0}}S\delta_{\kappa}\right)p,$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{U_{R}}{i\omega}=\frac{U_{L}}{i\omega}-\frac{1}{z_{0}c_{0}}\left(V-\frac{i}{2}\frac{\left(\gamma-1\right)}{1+\epsilon_{s,0}}S\delta_{\kappa}\right)p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{U_{R}}{i\omega}=\frac{U_{L}}{i\omega}-\frac{1}{z_{0}c_{0}}\left(V-\frac{i}{2}\frac{\left(\gamma-1\right)}{1+\epsilon_{s,0}}S\delta_{\kappa}\right)p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\xi_{L}-\xi_{R}=\frac{1}{z_{0}c_{0}}\left(V-\frac{i}{2}\frac{\left(\gamma-1\right)}{1+\epsilon_{s,0}}S\delta_{\kappa}\right)p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Using
|
||
\begin_inset Formula $C=\frac{\xi_{L}-\xi_{R}}{p}$
|
||
\end_inset
|
||
|
||
in that case:
|
||
\begin_inset Formula $C=\frac{1}{z_{0}c_{0}}\left(V-\frac{i}{2}\frac{\left(\gamma-1\right)}{1+\epsilon_{s,0}}S\delta_{\kappa}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Such that:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U_{R}=U_{L}-i\omega C_{c}p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
p_{L} & =p=p_{R},\\
|
||
U_{R} & =U_{L}-i\omega C_{c}p,
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
in which
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$C_c$"
|
||
description "Acoustic capacitance of a compliance volume\\nomunit{\\si{\\cubic\\metre\\per\\pascal}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $C_{c}$
|
||
\end_inset
|
||
|
||
is the acoustic
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
capacitance
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
C_{c}=\frac{1}{z_{0}c_{0}}\left(V+\frac{1}{2}\frac{\left(1-i\right)\left(\gamma-1\right)}{1+\epsilon_{s,0}}S\delta_{\kappa}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $V$
|
||
\end_inset
|
||
|
||
is the volume,
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
the surface area of the volume in contact with a wall, and
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\epsilon_{s,0}=\sqrt{\frac{\kappa\rho_{0}c_{p}}{\kappa_{s}\rho_{s}c_{s}}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
It should be noticed that in practice, a compliance volume often functions
|
||
as the end of an acoustic system.
|
||
In that case, either
|
||
\begin_inset Formula $U_{L}$
|
||
\end_inset
|
||
|
||
or
|
||
\begin_inset Formula $U_{R}$
|
||
\end_inset
|
||
|
||
is 0.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Membrane
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A membrane is a mechanical
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Circular plate membrane
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
series_impedance/class CircPlateMembrane(SeriesImpedance)
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A thin circular plate can be modeled using CircPlateMembrane.
|
||
It behaves like an acoustic compliance.
|
||
A typical use is the attenuation of acoustic pressure by combining it with
|
||
an enclosed volume.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Two boundary condition cases can be applied: fixed/clamped edges and simply
|
||
supported edges.
|
||
The general equation for the static displacement of the plate is given
|
||
by
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "p. 487"
|
||
key "young_roarks_2002"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
y\left(r\right)=y_{c}+\frac{M_{c}r^{2}}{2D\left(1+\nu\right)}+LT_{y}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $y_{c}$
|
||
\end_inset
|
||
|
||
and
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
|
||
\begin_inset Formula $M_{c}$
|
||
\end_inset
|
||
|
||
are
|
||
\family default
|
||
\series default
|
||
\shape default
|
||
\size default
|
||
\emph default
|
||
\bar default
|
||
\strikeout default
|
||
\xout default
|
||
\uuline default
|
||
\uwave default
|
||
\noun default
|
||
\color inherit
|
||
the displacement and moment at the center of the plate,
|
||
\begin_inset Formula $LT_{y}$
|
||
\end_inset
|
||
|
||
is the load term in the y-direction,
|
||
\begin_inset Formula $\nu$
|
||
\end_inset
|
||
|
||
is the Poisson's ratio of the plate material and
|
||
\begin_inset Formula $D$
|
||
\end_inset
|
||
|
||
is the flexural stiffness of the plate, which is given by the equation:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
D=\frac{Et^{3}}{12\left(1-\nu^{2}\right)}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $E$
|
||
\end_inset
|
||
|
||
is the Young's modulus of the plate material and
|
||
\begin_inset Formula $t$
|
||
\end_inset
|
||
|
||
is the plate thickness.
|
||
Substituting
|
||
\begin_inset Formula $D$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $y_{c}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $M_{c}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $LT_{y}$
|
||
\end_inset
|
||
|
||
for this specific load case (uniform load/pressure) and boundary conditions
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "p. 458 & p. 488"
|
||
key "young_roarks_2002"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
and simplifying yields the following equations for the static plate deflection:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
In these equations the distributed load
|
||
\begin_inset Formula $q$
|
||
\end_inset
|
||
|
||
is replaced by
|
||
\begin_inset Formula $-p$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
y_{ss}\left(r\right)=\frac{3p\left(1-\nu^{2}\right)}{16Et^{3}\left(1+\nu\right)}\left(a^{2}\left[a^{2}\left\{ 5+\nu\right\} -2r^{2}\left\{ 3+\nu\right\} \right]+r^{4}\left[1+\nu\right]\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
y_{fix}\left(r\right)=\frac{3p\left(1-\nu^{2}\right)}{16Et^{3}}\left(a^{4}-2a^{2}r^{2}+r^{4}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In which
|
||
\begin_inset Formula $a$
|
||
\end_inset
|
||
|
||
is the radius of the plate and
|
||
\begin_inset Formula $r$
|
||
\end_inset
|
||
|
||
is the radial coordinate.
|
||
The static acoustic compliance of the plate is given by the equation:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
c_{stat}\left(r\right)=\frac{y\left(r\right)}{p}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The static acoustic volume compliance for both cases can be calculated by
|
||
integrating over the surface of the plate:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
C_{stat}=2\pi\int_{0}^{a}c_{stat}\left(r\right)rdr
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Performing this integration for both boundary condition cases yields:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
C_{stat,ss}=\frac{\pi a^{6}}{16Et^{3}}\left(7-6\nu-\nu^{2}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
C_{stat,fix}=\frac{\pi a^{6}}{16Et^{3}}\left(1-\nu^{2}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The dynamic acoustic volume compliance of the plate is given by the equation:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
C_{dyn}\left(f\right)=\frac{C_{stat}}{1-\left(\frac{f}{f_{r}}\right)^{2}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $f$
|
||
\end_inset
|
||
|
||
is the frequency in Hz and
|
||
\begin_inset Formula $f_{r}$
|
||
\end_inset
|
||
|
||
is the resonance frequency of the plate in Hz.
|
||
The resonance frequency for the simply supported plate is given by the
|
||
equation
|
||
\begin_inset CommandInset citation
|
||
LatexCommand citeyear
|
||
key "calcdevice"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
This is an approximation from an online calculator.
|
||
A more exact equation like the one for the fxed case should be found.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
f_{r,ss}=\frac{0.8}{a^{2}}\sqrt{\frac{D}{\rho t}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $\rho$
|
||
\end_inset
|
||
|
||
is the density of the plate material.
|
||
The resonance frequency for the fixed plate is given by the equation
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "p. 430"
|
||
key "leniowska_plate_resonance_1999"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
f_{r,fix}=\frac{\gamma_{1}^{2}}{a^{2}}\sqrt{\frac{D}{\rho t}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $\gamma_{1}$
|
||
\end_inset
|
||
|
||
is the first solution to the following equation:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
J_{0}\left(\gamma_{m}\right)I_{1}\left(\gamma_{m}\right)+J_{1}\left(\gamma_{m}\right)I_{0}\left(\gamma_{m}\right)=0\label{eq:gamma}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $J_{n}\left(\gamma_{m}\right)$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $I_{n}\left(\gamma_{m}\right)$
|
||
\end_inset
|
||
|
||
are the Bessel function of the first kind and modified Bessel functions
|
||
of order
|
||
\begin_inset Formula $n$
|
||
\end_inset
|
||
|
||
.
|
||
Solving equation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:gamma"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
yields
|
||
\begin_inset Formula $\gamma_{1}=3.196$
|
||
\end_inset
|
||
|
||
.
|
||
The impedance is given by the equation:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{cpm}\left(f\right)=\frac{1}{i2\pi fC_{dyn}\left(f\right)}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Exponential horn
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
hon.py/class ExpHorn(Duct)
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
An exponential horn can be modelled using ExpHorn.
|
||
The transfer matrix is derived from Webster's horn equation.
|
||
The solution for an exponential horn is
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p\left(x\right)=C_{1}e^{E_{1}x}+C_{2}e^{E_{2}x},\qquad U\left(x\right)=i\frac{S\left(x\right)}{\rho\omega}\left(C_{1}E_{1}e^{E_{1}x}+C_{2}E_{2}e^{E_{2}x}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $S(x)$
|
||
\end_inset
|
||
|
||
is the local cross-sectional area,
|
||
\begin_inset Formula $\rho$
|
||
\end_inset
|
||
|
||
is the air density,
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
is the angular frequency.
|
||
The exponent terms
|
||
\begin_inset Formula $E_{1}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $E_{2}$
|
||
\end_inset
|
||
|
||
are given by
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
E_{1}=\frac{-m+\sqrt{-4k^{2}+m^{2}}}{2},\qquad E_{2}=-\frac{m+\sqrt{-4k^{2}+m^{2}}}{2}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $k$
|
||
\end_inset
|
||
|
||
is the wave number and
|
||
\begin_inset Formula $m$
|
||
\end_inset
|
||
|
||
is the flare rate, which is defined as
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
m=\frac{1}{L}\ln\left(\frac{S_{R}}{S_{L}}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
with
|
||
\begin_inset Formula $L$
|
||
\end_inset
|
||
|
||
being the horn length and
|
||
\begin_inset Formula $S_{R}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $S_{L}$
|
||
\end_inset
|
||
|
||
being the cross-sectional areas on the right-hand and left-hand side of
|
||
the horn respectively.
|
||
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
By setting the solutions on
|
||
\begin_inset Formula $x=0$
|
||
\end_inset
|
||
|
||
to
|
||
\begin_inset Formula $p_{L}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $U_{L}$
|
||
\end_inset
|
||
|
||
and those on
|
||
\begin_inset Formula $x=L$
|
||
\end_inset
|
||
|
||
to
|
||
\begin_inset Formula $p_{R}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $U_{R}$
|
||
\end_inset
|
||
|
||
, the following transfer matrix can be derived:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
T_{ExpHorn}=\frac{1}{E_{2}-E_{1}}\left[\begin{array}{cc}
|
||
E_{2}e^{E_{1}L}-E_{1}e^{E_{2}L} & i\frac{\rho\omega}{S_{L}}\left(e^{E_{1}L}-e^{E_{2}L}\right)\\
|
||
i\frac{S_{R}}{\rho\omega}E_{1}E_{2}\left(e^{E_{1}L}-e^{E_{2}L}\right) & -\frac{S_{R}}{S_{L}}\left(E_{1}e^{E_{1}L}-E_{2}e^{E_{2}L}\right)
|
||
\end{array}\right]
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p_{L}=p\left(0\right)=C_{1}+C_{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U_{L}=U\left(0\right)=i\frac{S_{L}}{\rho\omega}\left(C_{1}E_{1}+C_{2}E_{2}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p_{R}=p\left(L\right)=C_{1}e^{E_{1}L}+C_{2}e^{E_{2}L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U_{R}=U\left(L\right)=i\frac{S_{L}}{\rho\omega}\left(C_{1}E_{1}e^{E_{1}L}+C_{2}E_{2}e^{E_{2}L}\right)$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
Solve for constants
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $C_{1}=P_{L}-C_{2}\rightarrow U_{L}=i\frac{S_{L}}{\rho\omega}\left[E_{1}p_{L}+\left(E_{2}-E_{1}\right)C_{2}\right]$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $C_{1}=\frac{1}{E_{2}-E_{1}}\left(i\frac{\rho\omega}{S_{L}}U_{L}+E_{2}P_{L}\right),\qquad C_{2}=-\frac{1}{E_{2}-E_{1}}\left(i\frac{\rho\omega}{S_{L}}U_{L}+E_{1}P_{L}\right)$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
Fill in in solution:
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $p_{R}=\frac{1}{E_{2}E_{1}}\left[\left(E_{2}e^{E_{1}L}-E_{1}e^{E_{2}L}\right)P_{L}+i\frac{\rho\omega}{S_{L}}\left(e^{E_{1}L}-e^{E_{2}L}\right)U_{L}\right]$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $U_{R}=\frac{1}{E_{2}E_{1}}\left[i\frac{S_{R}}{\rho\omega}E_{1}E_{2}\left(e^{E_{1}L}-e^{E_{2}L}\right)P_{L}-\frac{S_{R}}{S_{L}}\left(E_{1}e^{E_{1}L}-E_{2}e^{E_{2}L}\right)U_{L}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Holes in plate
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
series_impedance.py/class CircHoleNeck(SeriesImpedance)
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A plate with several holes can be modelled using CircHoleNeck.
|
||
It behaves like an acoustic mass with losses and can represent the neck
|
||
of a Helmholtz resonator.
|
||
Typical uses are to connect volumes to eachother or volumes to ducts, to
|
||
form Helmholtz resonators.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Limitations are that hole-hole interaction is neglected and that the resistance
|
||
term is an approximation for holes with diameter >> length.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Impedance is given by the equation:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\noindent
|
||
\align center
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{holes}=\frac{1}{N_{h}}\left(R_{v}+i\omega M_{A}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $N_{h}$
|
||
\end_inset
|
||
|
||
is the number of holes,
|
||
\begin_inset Formula $R_{v}$
|
||
\end_inset
|
||
|
||
the acoustic resistance as described in equation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:Rv_hole"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
the angular frequency and
|
||
\begin_inset Formula $m_{a}$
|
||
\end_inset
|
||
|
||
the acoustic mass as described in equation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:acoustic_mass"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, except without Karal's discontinuity factor.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
End corrections and discontinuities
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "subsec:End-corrections-and"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/discontinuity.pdf
|
||
width 60text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Schematic of a waveguide discontinuity.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:karal"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
For discontinuities in the cross section of a waveguide, and the case of
|
||
inviscid adiabatic wave propagation, an exact expression is available for
|
||
the added acoustic mass
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "karal_analogous_1953"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:karal"
|
||
|
||
\end_inset
|
||
|
||
gives a schematic of the situation.
|
||
The model is implemented in the
|
||
\family typewriter
|
||
Discontinuity
|
||
\family default
|
||
class in the
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubess
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
code.
|
||
The assumptions behind the model are:
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Both tubes on either side of the discontinuity are cylindrical.
|
||
The tubes are co-axially connected.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
The wavelength is larger than transverse characteristic length scale (no
|
||
propagating modes expect for the plane waves).
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Other discontinuities are far away from the current one.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Inviscid and adiabatic wave propagation (Helmholtz equation).
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The ratio of tube radii
|
||
\begin_inset Formula $a_{L}/a_{R}$
|
||
\end_inset
|
||
|
||
is denoted by
|
||
\begin_inset Formula $\alpha$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "G"
|
||
symbol "$\\alpha$"
|
||
description "Ratio of tube radii\\nomunit{-}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
It turns out that a surface area discontinuity only generates an acoustic
|
||
pressure discontinuity.
|
||
The volume flow is preserved.
|
||
Hence:
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
U_{R} & =U_{L}\\
|
||
p_{R} & =p_{L}-i\omega M_{A}U_{L}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $M_{A}$
|
||
\end_inset
|
||
|
||
is the so-called added acoustic mass in
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
si{
|
||
\backslash
|
||
kg
|
||
\backslash
|
||
per
|
||
\backslash
|
||
metre
|
||
\backslash
|
||
tothe{4}}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
, which equals
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$M_A$"
|
||
description "Acoustic mass\\nomunit{\\si{\\kg\\per\\metre\\tothe{4}}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$a$"
|
||
description "Tube radius\\nomunit{\\si{\\metre}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
M_{A}=\chi(\alpha,k)\frac{8\rho_{0}}{3\pi^{2}a_{L}},\label{eq:acoustic_mass}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\chi$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "G"
|
||
symbol "$\\chi$"
|
||
description "Karal's discontinuity factor\\nomunit{-}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
is Karal's discontinuity factor, which is in general a function of the tube
|
||
radii and the wave number.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For
|
||
\begin_inset Formula $\lambda\gg a_{R}$
|
||
\end_inset
|
||
|
||
, the dependency of
|
||
\begin_inset Formula $\chi$
|
||
\end_inset
|
||
|
||
on the wave number
|
||
\begin_inset Formula $k$
|
||
\end_inset
|
||
|
||
can be neglected, which lowers the computational burden significantly,
|
||
as
|
||
\begin_inset Formula $\chi$
|
||
\end_inset
|
||
|
||
has to be computed only once.
|
||
For the case
|
||
\begin_inset Formula $\alpha\to0$
|
||
\end_inset
|
||
|
||
(by letting
|
||
\begin_inset Formula $a_{R}\to\infty$
|
||
\end_inset
|
||
|
||
),
|
||
\begin_inset Formula $\chi\to1$
|
||
\end_inset
|
||
|
||
.
|
||
In case of
|
||
\begin_inset Formula $\alpha\to1$
|
||
\end_inset
|
||
|
||
, the acoustic mass gradually reduces to zero as
|
||
\begin_inset Formula $\chi\to0$
|
||
\end_inset
|
||
|
||
.
|
||
When
|
||
\begin_inset Formula $\alpha=1$
|
||
\end_inset
|
||
|
||
, there is no continuity left, such that
|
||
\begin_inset Formula $M_{A}=0$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The derivation of the coefficient
|
||
\begin_inset Formula $\chi$
|
||
\end_inset
|
||
|
||
is documented in Appendix
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "chap:Derivation-of-Karal's"
|
||
|
||
\end_inset
|
||
|
||
, except of the following information.
|
||
To solve the curve of
|
||
\begin_inset Formula $\chi$
|
||
\end_inset
|
||
|
||
, a system of infinite equations has to be solved for an infinite number
|
||
of unknowns.
|
||
In the
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubes
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
code, as a standard this system is truncated up to
|
||
\begin_inset Formula $N=$
|
||
\end_inset
|
||
|
||
100 equations and 100 unknowns.
|
||
Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:chi_vs_alpha"
|
||
|
||
\end_inset
|
||
|
||
shows the effect of truncating this infinite system of equations.
|
||
As visible for the case of 100 equations, the curves start to deviate from
|
||
each other for lower values of
|
||
\begin_inset Formula $\alpha$
|
||
\end_inset
|
||
|
||
.
|
||
Assuming that convergence is obtained as
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$N$"
|
||
description "Number\\nomunit{-}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $N\to\infty$
|
||
\end_inset
|
||
|
||
, the curve of
|
||
\begin_inset Formula $N=100$
|
||
\end_inset
|
||
|
||
has acceptable accuracy for
|
||
\begin_inset Formula $\alpha>0.07$
|
||
\end_inset
|
||
|
||
.
|
||
To limit possible faulty results, the
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubess
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
code gives a warning when the tube ratio is chosen such that an invalid
|
||
|
||
\begin_inset Formula $\chi$
|
||
\end_inset
|
||
|
||
is computed.
|
||
When an
|
||
\begin_inset Formula $\alpha<0.07$
|
||
\end_inset
|
||
|
||
is desired, the user should choose a higher value of
|
||
\begin_inset Formula $N$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/chi_vs_alpha.pdf
|
||
width 90text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\chi$
|
||
\end_inset
|
||
|
||
vs
|
||
\begin_inset Formula $\alpha$
|
||
\end_inset
|
||
|
||
for different truncations
|
||
\begin_inset Formula $\left(N\right)$
|
||
\end_inset
|
||
|
||
of the infinite system of equations.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:chi_vs_alpha"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Hard wall
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A hard wall is the wall perpendicular to the wave propagation direction.
|
||
Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:hardwall"
|
||
|
||
\end_inset
|
||
|
||
shows the schematic configuration for this segment.
|
||
Due to thermal relaxation a hard wall consumes acoustic energy is consumed.
|
||
The hard wall segment models this thermal relaxation loss.
|
||
The assumptions behind the model are:
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Normal incident waves.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Uniform normal velocity.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
The wavelength is much larger than the thermal penetration depth (
|
||
\begin_inset Formula $\lambda\gg\delta_{\kappa}$
|
||
\end_inset
|
||
|
||
).
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
We can derive the following impedance boundary condition
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "p. 157"
|
||
key "ward_deltaec_2017"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Delta EC User guide:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
U_{R}=U_{L}-\frac{\omega p}{\rho_{0}c_{0}^{2}}\frac{\gamma-1}{1+\epsilon_{s}}S\frac{\delta_{\kappa}}{2}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Or:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
U_{L}=\frac{k}{z_{0}}\frac{\gamma-1}{1+\epsilon_{s}}S\frac{\delta_{\kappa}}{2}p
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
U=k\delta_{\kappa}\frac{S}{z_{0}}\frac{\left(\gamma-1\right)\left(1+i\right)}{2\left(1+\epsilon_{s}\right)}p.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Hence the impedance of a hard wall scales with
|
||
\begin_inset Formula $Z\sim Z_{0}\frac{\lambda}{\delta_{\kappa}}$
|
||
\end_inset
|
||
|
||
.
|
||
For 1 kHz, this results in
|
||
\begin_inset Formula $\sim4100Z_{0}$
|
||
\end_inset
|
||
|
||
, which is practically already close to
|
||
\begin_inset Formula $\infty$
|
||
\end_inset
|
||
|
||
.
|
||
Except for really high frequencies this segment can often be replaced with
|
||
a boundary condition of
|
||
\begin_inset Formula $U=0$
|
||
\end_inset
|
||
|
||
.
|
||
An important point to make here is that this boundary condition is inconsistent
|
||
with the LRF solution for 1D wave propagation in ducts, as the velocity
|
||
profile in a duct is not uniform.
|
||
This is especially true for the case of small ducts where
|
||
\begin_inset Formula $r_{h}\sim\delta$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/hardwall.pdf
|
||
width 50text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Schematic of a hard acoustic wall where the thermal boundary layer dissipates
|
||
a bit of the acoustic energy (
|
||
\begin_inset Formula $Z\neq\infty$
|
||
\end_inset
|
||
|
||
).
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:hardwall"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Spherical wave propagation models
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For spherical waves, the Helmholtz equation reads
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left(\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}+\frac{2}{r}\frac{\mathrm{d}}{\mathrm{d}r}+\Gamma^{2}\right)p=0.\label{eq:hh_spher}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
The solution of Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:hh_spher"
|
||
|
||
\end_inset
|
||
|
||
reads:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p=\frac{C_{1}\exp\left(-i\Gamma r\right)+C_{2}\exp\left(-i\Gamma r\right)}{r}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
The acoustic volume flow can be computed as
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $u_{r}=\frac{i}{\omega\rho_{0}}\frac{\mathrm{d}p}{\mathrm{d}r}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $u_{r}=\frac{i}{kz_{0}}\frac{\mathrm{d}p}{\mathrm{d}r}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U_{r}=4\pi r^{2}\alpha\frac{i}{kz_{0}}\frac{\mathrm{d}p}{\mathrm{d}r}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
U=i\frac{\alpha4\pi r^{2}}{\Gamma z_{c}}\frac{\mathrm{d}p}{\mathrm{d}r},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\alpha=1$
|
||
\end_inset
|
||
|
||
for a full sphere and
|
||
\begin_inset Formula $\alpha=\frac{1}{2}$
|
||
\end_inset
|
||
|
||
for a hemisphere.
|
||
We can derive the following transfer matrix for
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $U$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{C_{1}\exp\left(-i\Gamma r_{L}\right)+C_{2}\exp\left(-i\Gamma r_{L}\right)}{\Gamma r_{L}}=p_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{C_{1}\exp\left(-i\Gamma r_{R}\right)+C_{2}\exp\left(-i\Gamma r_{R}\right)}{\Gamma r_{R}}=p_{R}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(U_{L}\frac{e^{-i\Gamma\left(r_{L}+r_{R}\right)}}{8\pi\Gamma\alpha r_{L}r_{R}}\left(i\Gamma z_{c}e^{2i\Gamma r_{L}}-i\Gamma z_{c}e^{2i\Gamma r_{R}}\right)+p_{L}\left(\frac{r_{L}e^{i\Gamma\left(r_{L}+r_{R}\right)}}{2r_{R}}+\frac{i}{2\Gamma r_{R}}\left(e^{i\Gamma\left(r_{L}-r_{R}\right)}-e^{i\Gamma\left(r_{R}+r_{L}\right)}\right)\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p_{R}=\frac{iU_{L}z_{c}}{4\pi\alpha r_{L}r_{R}}\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right)+p_{L}\left[\frac{r_{L}}{r_{R}}\cos\left(\Gamma\left(r_{L}-r_{R}\right)\right)-\frac{1}{\Gamma r_{R}}\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right)\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
and:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U_{R}=U_{L}\left(\frac{r_{R}}{r_{L}}\cos\left(\Gamma\left(r_{L}-r_{R}\right)\right)+\frac{1}{\Gamma r_{L}}\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right)\right)+\frac{4i\pi\alpha}{z_{c}}p_{L}\left[\left(r_{L}r_{R}+\frac{1}{\Gamma^{2}}\right)\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right)+\frac{\left(r_{R}-r_{L}\right)}{\Gamma}\cos\left(\Gamma\left(r_{L}-r_{R}\right)\right)\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left\{ \begin{array}{c}
|
||
p\\
|
||
U
|
||
\end{array}\right\} _{R}=\left[\begin{array}{cc}
|
||
M_{11} & M_{12}\\
|
||
M_{21} & M_{22}
|
||
\end{array}\right]\left\{ \begin{array}{c}
|
||
p\\
|
||
U
|
||
\end{array}\right\} _{L},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
M_{11} & =\frac{r_{L}}{r_{R}}\cos\left(\Gamma\left(r_{L}-r_{R}\right)\right)-\frac{1}{\Gamma r_{R}}\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right),\\
|
||
M_{12} & =\frac{iz_{c}\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right)}{4\pi\alpha r_{L}r_{R}},\\
|
||
M_{21} & =\frac{4\pi i\alpha}{z_{c}}\left[\left(r_{L}r_{R}+\frac{1}{\Gamma^{2}}\right)\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right)+\frac{r_{R}-r_{L}}{\Gamma}\cos\left(\Gamma\left(r_{L}-r_{R}\right)\right)\right]\\
|
||
M_{22} & =\frac{r_{R}}{r_{L}}\cos\left(\Gamma\left(r_{L}-r_{R}\right)\right)+\frac{1}{\Gamma r_{L}}\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right),
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Boundary conditions
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Radiation impedance of a baffled piston
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $a$
|
||
\end_inset
|
||
|
||
: radius of the exit [m]
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula $\pi a^{2}$
|
||
\end_inset
|
||
|
||
cross sectional area [m
|
||
\begin_inset Formula $^{2}$
|
||
\end_inset
|
||
|
||
]
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p=Z_{\mathrm{rad}}U,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{\mathrm{rad}}=\frac{z_{0}}{S}\left[1-\frac{2J_{1}\left(2ka\right)}{2ka}+i\frac{2H_{1}(2ka)}{2ka}\right]
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In the low frequency range, a power series expansion of
|
||
\begin_inset Formula $H_{1}$
|
||
\end_inset
|
||
|
||
yields [Aarts]:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
H_{1}(x)=\frac{2}{\pi}\left[\frac{x^{2}}{3}-\frac{x^{4}}{45}+\frac{x^{6}}{1575}-\dots\right]
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Filling this in, we obtain the following low-frequency approximation to
|
||
|
||
\begin_inset Formula $Z_{\mathrm{rad}}$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{\mathrm{rad}}=\frac{z_{0}}{S}\left[i\frac{8ka}{3\pi}+\frac{1}{2}\left(ka\right)^{2}+\mathcal{O}\left(\left(ka\right)^{3}\right)\right]\label{eq:Zrad-baffled-piston}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Incident plane wave on small port in infinite baffle
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Situation: an acoustic system, which is connected to the outside world though
|
||
a port, ending in an infinite wall
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:bc_planewave_port"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
There is an incident plane wave with specified amplitude and frequency.
|
||
It would be beneficial for computing time to replace the outside world
|
||
by a boundary condition on the port.
|
||
Here it is approached as a scattering problem.
|
||
More information is described in
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "p. 132-134"
|
||
key "zwikker_sound_1949"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
The pressure field can be written as:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{t}=p_{i}+p_{s}\label{eq:scattering-problem}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $p_{t}$
|
||
\end_inset
|
||
|
||
is the total pressure field,
|
||
\begin_inset Formula $p_{i}$
|
||
\end_inset
|
||
|
||
the incident pressure field (the field as if there were only an infinite
|
||
wall) and
|
||
\begin_inset Formula $p_{s}$
|
||
\end_inset
|
||
|
||
the scattered pressure field.
|
||
The combination of the incident and scattered field combined result in
|
||
the total pressure field.
|
||
All depend on both position and time (or frequency).
|
||
If only the infinite wall is taken into account and the port and system
|
||
behind it are ignored, the amplitude of the incident plane wave is:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{i}(x,\omega)=\begin{cases}
|
||
P_{i}\cdot\cos(kx) & x<=0\\
|
||
0 & x>0
|
||
\end{cases}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $P_{i}$
|
||
\end_inset
|
||
|
||
is the amplitude of the incident plane wave at the wall (resulting in sound
|
||
pressure
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
|
||
\begin_inset Formula $P_{i}$
|
||
\end_inset
|
||
|
||
|
||
\family default
|
||
\series default
|
||
\shape default
|
||
\size default
|
||
\emph default
|
||
\bar default
|
||
\strikeout default
|
||
\xout default
|
||
\uuline default
|
||
\uwave default
|
||
\noun default
|
||
\color inherit
|
||
on the surface of a reflecting wall),
|
||
\begin_inset Formula $k$
|
||
\end_inset
|
||
|
||
is the wave number and
|
||
\begin_inset Formula $x$
|
||
\end_inset
|
||
|
||
the position into the wall.
|
||
There is no scattered pressure field, so this is the total pressure field
|
||
right away.
|
||
When the port and system behind it are added, the total pressure field
|
||
is no longer equal to the incident pressure field: a correction must be
|
||
added, which is captured in
|
||
\begin_inset Formula $p_{s}$
|
||
\end_inset
|
||
|
||
.
|
||
The correction is due to the air slug within the port moving.
|
||
At
|
||
\begin_inset Formula $x<0$
|
||
\end_inset
|
||
|
||
, this has the same effect as a baffled piston.
|
||
On the condition that the wavelength is much larger than the port size,
|
||
the scattered field near the boundary (but still outside of the port) is
|
||
given by:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{s}(x=0^{-})=-Z_{\mathrm{rad}}U
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $Z_{\mathrm{rad}}$
|
||
\end_inset
|
||
|
||
is the radiation impedance of a baffled piston and
|
||
\begin_inset Formula $U$
|
||
\end_inset
|
||
|
||
is the acoustic volume flow rate.
|
||
Note the minus sign, which stems from the direction in which
|
||
\begin_inset Formula $U$
|
||
\end_inset
|
||
|
||
is defined.
|
||
The same convention is taken as in COMSOL: velocity
|
||
\begin_inset Formula $v$
|
||
\end_inset
|
||
|
||
is positive when inwards, so inwards
|
||
\begin_inset Formula $U$
|
||
\end_inset
|
||
|
||
is positive.
|
||
Filling in equation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:scattering-problem"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, just outside of the port at
|
||
\begin_inset Formula $x=0^{-}$
|
||
\end_inset
|
||
|
||
, yields:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{t}(x=0^{-})=P_{i}-Z_{rad}U
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
It is questionable whether the port acoustically ends at the boundary, so
|
||
this might be an approximation.
|
||
In COMSOL, the pressure is continuous, to it is fine to apply it at
|
||
\begin_inset Formula $x=0$
|
||
\end_inset
|
||
|
||
instead of
|
||
\begin_inset Formula $x=0^{-}$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\begin_inset Formula $U$
|
||
\end_inset
|
||
|
||
can be found by integrating the inner product of velocity and the normal
|
||
vector over the boundary, while adding a minus sign because the normal
|
||
vector points outwards.
|
||
In COMSOL it is more convenient to use
|
||
\emph on
|
||
specific
|
||
\emph default
|
||
impedances and
|
||
\emph on
|
||
velocities
|
||
\emph default
|
||
.
|
||
Then the equation is slightly modified to:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{t}(x=0)=P_{i}-z_{\mathrm{rad}}v\label{eq:bc-planewave-port-pressure}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $z_{\mathrm{rad}}$
|
||
\end_inset
|
||
|
||
is the specific radiation impedance of a baffled piston and
|
||
\begin_inset Formula $v$
|
||
\end_inset
|
||
|
||
the acoustic velocity (inwards).
|
||
This equation can be applied as a
|
||
\emph on
|
||
pressure
|
||
\emph default
|
||
boundary condition in COMSOL.
|
||
The required
|
||
\begin_inset Formula $v$
|
||
\end_inset
|
||
|
||
can be
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
measured
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
by averaging the normal component of the velocity and adding a minus sign
|
||
to make it inwards.
|
||
Alternatively, the equation can be solved for
|
||
\begin_inset Formula $v$
|
||
\end_inset
|
||
|
||
to obtain a
|
||
\emph on
|
||
velocity
|
||
\emph default
|
||
boundary condition:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
v=\frac{P_{i}-p_{t}(x=0)}{z_{\mathrm{rad}}}\label{eq:bc-planewave-port-velocity}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $p_{t}(x=0)$
|
||
\end_inset
|
||
|
||
can be
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
measured
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
by averaging it over the port's boundary.
|
||
The LRFTubes implementation of this
|
||
\emph on
|
||
mixed
|
||
\emph default
|
||
boundary condition is for a left wall:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{L}+Z_{\mathrm{rad}}U_{L}=P_{i},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and the same on a right wall:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{R}-Z_{\mathrm{rad}}U_{R}=P_{i}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
TO DO: redraw image and list what approximations are used
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/bc_planewave_port.jpg
|
||
lyxscale 10
|
||
width 50text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Schematic view of incident wave (green) on an infinite wall (blue) containing
|
||
a port with a system connected to it.
|
||
The location of the boundary condition is shown in red.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:bc_planewave_port"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Thermoacoustic segments
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For relatively small temperature gradients, Swift's thermoacoustic equations
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "p. 91"
|
||
key "swift_thermoacoustics:_2003"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\frac{\mathrm{d}p}{\mathrm{d}x} & =-\frac{-i\omega p_{m}}{R_{s}T_{m}S_{\mathrm{gas}}\left(1-f_{\nu}\right)}U,\\
|
||
\frac{\mathrm{d}U}{\mathrm{d}x} & =\frac{-i\omega S_{\mathrm{gas}}}{\gamma p_{m}}\left[1+\left(\gamma-1\right)f_{\kappa}\right]p+\frac{f_{\kappa}-f_{\nu}}{\left(1-f_{\nu}\right)\left(1-\Pr\right)}\frac{1}{T_{m}}\frac{\mathrm{d}T_{m}}{\mathrm{d}x}U,
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
can be integrated.
|
||
Assuming
|
||
\begin_inset Formula $\frac{\mathrm{d}T_{m}}{\mathrm{d}x}L\ll T_{m}$
|
||
\end_inset
|
||
|
||
.
|
||
Then we find for the solution::
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p(x)=C_{1}\exp\left(\Gamma_{1}x\right)+C_{2}\exp\left(\Gamma_{2}x\right),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Gamma_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a},\label{eq:Gammasol}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
wheren
|
||
\begin_inset Formula $1$
|
||
\end_inset
|
||
|
||
denotes to the
|
||
\begin_inset Formula $+$
|
||
\end_inset
|
||
|
||
and 2 to the
|
||
\begin_inset Formula $-$
|
||
\end_inset
|
||
|
||
sign.
|
||
In Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:Gammasol"
|
||
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $a,$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $b,$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $c$
|
||
\end_inset
|
||
|
||
are defined as:
|
||
\begin_inset Formula
|
||
\[
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\left(\Gamma_{1}f_{\nu}p_{L}-\Gamma_{1}p_{L}+iU_{L}\omega\rho_{m}\right)e^{\Gamma_{2}L}+\left(-\Gamma_{2}f_{\nu}p_{L}+\Gamma_{2}p_{L}-iU_{L}\omega\rho_{m}\right)e^{\Gamma_{1}L}}{\Gamma_{1}f_{\nu}-\Gamma_{1}-\Gamma_{2}f_{\nu}+\Gamma_{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\begin{array}{c}
|
||
p_{R}\\
|
||
U_{R}
|
||
\end{array}=\left[\begin{array}{cc}
|
||
\frac{\left(\Gamma_{1}f_{\nu}p_{L}-\Gamma_{1}p_{L}\right)e^{\Gamma_{2}L}+\left(-\Gamma_{2}f_{\nu}p_{L}+\Gamma_{2}p_{L}\right)e^{\Gamma_{1}L}}{\left(\Gamma_{2}-\Gamma_{1}\right)\left(1-f_{\nu}\right)} & i\frac{\left(iU_{L}\omega\rho_{m}\right)e^{\Gamma_{2}L}+\left(-iU_{L}\omega\rho_{m}\right)e^{\Gamma_{1}L}}{\left(\Gamma_{2}-\Gamma_{1}\right)\left(1-f_{\nu}\right)}\\
|
||
\\
|
||
\end{array}\right]\left\{ \begin{array}{c}
|
||
p_{L}\\
|
||
U_{L}
|
||
\end{array}\right\}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Speaker segment
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
As an active element, with voltage control
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/spk.pdf
|
||
width 100text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Electrical and mechanical model of the speaker
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The speaker generates electromotive force
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
F_{\mathrm{emf}} & =B\ell I,
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $B\ell$
|
||
\end_inset
|
||
|
||
is the
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
motor constant
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
, or force factor, in units
|
||
\begin_inset ERT
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
si{
|
||
\backslash
|
||
newton
|
||
\backslash
|
||
per
|
||
\backslash
|
||
ampere}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
, or
|
||
\begin_inset ERT
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
si{
|
||
\backslash
|
||
volt
|
||
\backslash
|
||
second
|
||
\backslash
|
||
per
|
||
\backslash
|
||
meter}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
.
|
||
The back-emf
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
force
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
V_{\mathrm{bemf}}=B\ell u
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
circuit equation
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
V_{\mathrm{in}}-V_{\mathrm{bemf}}=Z_{\mathrm{el}}I,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $Z_{\mathrm{el}}$
|
||
\end_inset
|
||
|
||
is the equivalent impedance of the electrical circuit in
|
||
\begin_inset Formula $\Omega$
|
||
\end_inset
|
||
|
||
.
|
||
The mechanical impedance comprises a stiffness part, a damping part and
|
||
a mass part.
|
||
The equation of motion is:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
z_{m}u=F_{\mathrm{emf}}+p_{l}S-p_{r}S,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
where
|
||
\begin_inset Formula $u$
|
||
\end_inset
|
||
|
||
denotes the velocity phasor of the membrane.
|
||
The mechanical impedance
|
||
\begin_inset Formula $z_{m}$
|
||
\end_inset
|
||
|
||
is defined as:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
z_{m}=i\omega m_{m}+r_{m}+\frac{k_{m}}{i\omega},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $m_{m}$
|
||
\end_inset
|
||
|
||
is the moving mass,
|
||
\begin_inset Formula $r_{m}$
|
||
\end_inset
|
||
|
||
the damping force and
|
||
\begin_inset Formula $k_{m}$
|
||
\end_inset
|
||
|
||
the spring constant.
|
||
|
||
\begin_inset Formula $z_{m}$
|
||
\end_inset
|
||
|
||
can equivalently be written as:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\begin_inset Formula $z_{m}=i\omega m+R_{m}+\frac{K_{m}}{i\omega}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\begin_inset Formula $z_{m}=m\left(i\omega+\frac{R_{m}}{m}+\frac{\omega_{r}^{2}}{i\omega}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\begin_inset Formula $z_{m}=m\left(i\omega+\frac{R_{m}}{m}+\frac{\omega_{r}^{2}}{i\omega}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
using:
|
||
\begin_inset Formula $\omega_{r}^{2}=\frac{K_{m}}{m}\Rightarrow m=\frac{K_{m}}{\omega_{r}^{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\begin_inset Formula $z_{m}=\frac{m}{i\omega}\left(-\omega^{2}+i\omega\frac{R_{m}}{m}+\omega_{r}^{2}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\begin_inset Formula $z_{m}=\frac{m}{i\omega}\left(-\omega^{2}+i\omega\frac{R_{m}\omega_{r}^{2}}{K_{m}}+\omega_{r}^{2}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Now, writing
|
||
\begin_inset Formula $R_{m}$
|
||
\end_inset
|
||
|
||
as:
|
||
\begin_inset Formula $R_{m}=2\zeta\sqrt{K_{m}m}$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula $\zeta=\frac{1}{2}\frac{r_{m}}{\sqrt{k_{m}m_{m}}}\Rightarrow\zeta=\frac{1}{2}\frac{r_{m}}{\sqrt{k_{m}m_{m}}}=\frac{1}{2}\frac{r_{m}}{\omega_{r}m_{m}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\begin_inset Formula $z_{m}=\frac{m}{i\omega}\left(\omega_{r}^{2}-\omega^{2}+2i\omega\zeta\omega_{r}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
z_{m}=m\left(i\omega+2\zeta\omega_{r}+\frac{\omega_{r}^{2}}{i\omega}\right),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\omega_{r}^{2}=\frac{k_{m}}{m_{m}}\qquad;\qquad\zeta=\frac{r_{m}}{2\sqrt{k_{m}m_{m}}}=\frac{r_{m}}{2\omega_{r}m_{m}}=\frac{\omega_{r}r_{m}}{2k_{m}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
After some algebraic manipulations we find:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $z_{m}u=\left(p_{l}-p_{r}\right)S+B\ell I$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{1}{Z_{\mathrm{el}}}\left(V_{\mathrm{in}}-V_{\mathrm{bemf}}\right)=I$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
where
|
||
\begin_inset Formula $V_{\mathrm{bemf}}=B\ell u$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Units of
|
||
\begin_inset Formula $\left[B\ell\right]=\frac{N}{A}=\frac{\mathrm{kg}\mathrm{m}s}{\mathrm{s}^{2}C}$
|
||
\end_inset
|
||
|
||
, knowing that
|
||
\begin_inset Formula $V=\frac{J}{C}$
|
||
\end_inset
|
||
|
||
, we can write this as:
|
||
\begin_inset Formula $\frac{\mathrm{kg}\mathrm{m}s}{\mathrm{s}^{2}C}=\frac{V\mathrm{kg}\mathrm{m}s}{\mathrm{s}^{2}J}=\frac{Vs}{\mathrm{m}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
And
|
||
\begin_inset Formula $\left[\frac{B\ell^{2}}{Z_{\mathrm{el}}}\right]=\left[\frac{Vs}{\mathrm{m}}\frac{N}{A}\frac{A}{V}\right]=\left[\frac{s}{\mathrm{m}}\frac{N}{A}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Results in:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $z_{m}u=\left(p_{l}-p_{r}\right)S+B\ell\frac{V_{\mathrm{in}}-V_{\mathrm{bemf}}}{Z_{\mathrm{el}}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{B\ell^{2}u}{Z_{\mathrm{el}}}+z_{m}u=\left(p_{l}-p_{r}\right)S+\frac{B\ell}{Z_{\mathrm{el}}}V_{\mathrm{in}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
To acoustic variables
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\frac{1}{S}\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}\right)U=\left(p_{l}-p_{r}\right)S+\frac{B\ell}{Z_{\mathrm{el}}}V_{\mathrm{in}}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
To transfer matrix notation:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p_{r}=p_{l}-\frac{1}{S^{2}}\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}\right)U+\frac{B\ell}{Z_{\mathrm{el}}S}V_{\mathrm{in}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\frac{1}{S_{l}}\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}\right)U_{l} & =p_{l}S_{l}-p_{r}S_{r}+\frac{B\ell}{Z_{\mathrm{el}}}V_{\mathrm{in}},\label{eq:U_vs_V}\\
|
||
U_{r}-U_{l} & =0,
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
which is in transfer matrix notation:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left\{ \begin{array}{c}
|
||
p_{r}\\
|
||
U_{r}
|
||
\end{array}\right\} =\boldsymbol{T}\left\{ \begin{array}{c}
|
||
p_{l}\\
|
||
U_{l}
|
||
\end{array}\right\} +\boldsymbol{s},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\boldsymbol{T}=\left[\begin{array}{cc}
|
||
1 & -\frac{1}{S^{2}}\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}\right)\\
|
||
0 & 1
|
||
\end{array}\right]\qquad;\qquad\boldsymbol{s}=\left\{ \begin{array}{c}
|
||
\frac{B\ell}{Z_{\mathrm{el}}S}V_{\mathrm{in}}\\
|
||
0
|
||
\end{array}\right\}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Computing the voltage input for given velocity
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Suppose we know the membrane velocity, and we want to know the corresponding
|
||
driving voltage.
|
||
For that we can rearrange Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:U_vs_V"
|
||
|
||
\end_inset
|
||
|
||
a bit:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{1}{S_{l}}\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}\right)U_{l}=p_{l}S_{l}-p_{r}S_{r}+\frac{B\ell}{Z_{\mathrm{el}}}V_{\mathrm{in}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Filling in
|
||
\begin_inset Formula $S_{l}$
|
||
\end_inset
|
||
|
||
is
|
||
\begin_inset Formula $S_{r}$
|
||
\end_inset
|
||
|
||
=
|
||
\begin_inset Formula $S_{d}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\frac{p_{r}-p_{l}}{U}=Z_{\mathrm{ac}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}+Z_{\mathrm{ac}}S\right)U=\frac{S_{d}B\ell}{Z_{\mathrm{el}}}V_{\mathrm{in}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Or:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(\frac{B\ell}{S_{d}}+\frac{Z_{\mathrm{el}}\left(Z_{\mathrm{ac}}+z_{m}/S_{d}\right)}{B\ell}\right)U=V_{\mathrm{in}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
V_{\mathrm{in}}=\left(\frac{B\ell}{S_{d}}+\frac{Z_{\mathrm{el}}\left(Z_{\mathrm{ac}}+z_{m}/S_{d}\right)}{B\ell}\right)U,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
or equivalently in terms of the mechanical velocity:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{B\ell^{2}+Z_{\mathrm{el}}\left(Z_{\mathrm{ac}}S_{d}+z_{m}\right)}{B\ell}u=V_{\mathrm{in}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
V_{\mathrm{in}}=\frac{B\ell^{2}+Z_{\mathrm{el}}\left(Z_{\mathrm{ac}}S_{d}+z_{m}\right)}{B\ell}u
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
For a COMSOL implementation, in terms of the computed acoustic pressure
|
||
and derivatives thereof (to create a linear system of equations):
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $V_{\mathrm{in}}=\frac{B\ell^{2}u+Z_{\mathrm{el}}\left(p+z_{m}u\right)}{B\ell}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $V_{\mathrm{in}}=\left(B\ell+\frac{Z_{\mathrm{el}}z_{m}}{B\ell}\right)u+\frac{Z_{\mathrm{el}}}{B\ell}p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
V_{\mathrm{in}}=\left(B\ell+\frac{Z_{\mathrm{el}}z_{m}}{B\ell}\right)u+\frac{Z_{\mathrm{el}}}{B\ell}F_{\mathrm{spk}},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $F_{\mathrm{spk}}$
|
||
\end_inset
|
||
|
||
is the net force the speaker exerts
|
||
\emph on
|
||
on the fluid
|
||
\emph default
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
As antireciprocal segment
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
As antireciprocal segment, a voltage controlled speaker has electrical connectio
|
||
ns on the left side, and acoustical connections on the right side:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left\{ \begin{array}{c}
|
||
p\\
|
||
U
|
||
\end{array}\right\} _{R}=\boldsymbol{T}_{\mathrm{spk}}\left\{ \begin{array}{c}
|
||
V\\
|
||
I
|
||
\end{array}\right\} _{L}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
A model us used for the back cavity pressure build-up which can be added
|
||
as an extra impedance, placed in series with the effective acoustic impedance
|
||
of the front side, hence the force balance reads:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
F_{\mathrm{emf}}=Z_{\mathrm{back}}U+Z_{\mathrm{front}}U
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
The transfer matrix reads:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\boldsymbol{T}_{\mathrm{spk}}=\left[\begin{array}{cc}
|
||
-\frac{S^{2}Z_{\mathrm{back}}+z_{m}}{SB\ell} & \frac{\left(B\ell\right)^{2}+Z_{\mathrm{el}}\left(z_{m}+S^{2}Z_{\mathrm{back}}\right)}{B\ell S}\\
|
||
\frac{S}{B\ell} & -\frac{SZ_{\mathrm{el}}}{B\ell}
|
||
\end{array}\right]
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Determinant:
|
||
\begin_inset Formula
|
||
\[
|
||
\frac{Z_{\mathrm{el}}\left(S^{2}Z_{\mathrm{back}}+z_{m}\right)}{B\ell^{2}}-\left(1+\frac{Z_{\mathrm{el}}\left(S^{2}Z_{\mathrm{back}}+z_{m}\right)}{B\ell^{2}}\right)=-1
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
For a closed back-cavity volume, the back-cavity is:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Then again:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Compute determinant:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\mathrm{det}=-S$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
(Micro)-perforated plate design
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Given
|
||
\begin_inset Formula $\beta$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\zeta$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\omega_{r}$
|
||
\end_inset
|
||
|
||
, a proper acoustic mass has to be chosen.
|
||
Given the resonator equations
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{align*}
|
||
C & =\frac{V}{\rho_{0}z_{0}},\\
|
||
m_{\mathrm{neck}} & =\frac{1}{\omega_{r}^{2}C}\\
|
||
\zeta & =\frac{1}{2}\omega_{r}CR_{v}.
|
||
\end{align*}
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula
|
||
\[
|
||
Z_{h}=\frac{1}{\omega_{r}C}\left(\frac{\omega_{r}}{i\omega}+2\zeta+\frac{i\omega}{\omega_{r}}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
, the viscous resistance and required acoustic mass can be determined.
|
||
This results in requirements for the (effective) acoustic mass and resistance
|
||
of the perforate.
|
||
For arbitrary hole sizes, the definition of the acoustic impedance of a
|
||
perforate is:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
z=\frac{\Delta p}{\overline{u}}.\label{eq:perforate_impedance_definition}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\overline{u}$
|
||
\end_inset
|
||
|
||
denotes the acoustic volume flow per unit of area through the perforate
|
||
(uncorrected yet for porosity), such that the area-averaged velocity
|
||
\emph on
|
||
in a hole
|
||
\emph default
|
||
is
|
||
\begin_inset Formula $u_{h}=\overline{u}/\phi$
|
||
\end_inset
|
||
|
||
, where
|
||
\begin_inset Formula $\phi$
|
||
\end_inset
|
||
|
||
denotes the porosity.
|
||
In Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:perforate_impedance_definition"
|
||
|
||
\end_inset
|
||
|
||
, it is assumed that the acoustic wavelength is typically much larger than
|
||
the length scale(s) of the perforate.
|
||
The model for the impedance of a perforate, in the linear range is
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
The COMSOL language, partially translated:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $z=-\rho_{0}c_{0}\frac{2i\sin\left(\frac{k_{c}t_{p}}{2}\right)}{\sqrt{\left(\gamma-\left(\gamma-1\right)\Psi_{h}\right)\Psi_{v}}}-\rho_{0}c_{0}\frac{i\omega}{c_{0}C_{D}\phi}\frac{2\delta}{\Psi_{v}}f_{\mathrm{int}},$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Using the fact that:
|
||
\begin_inset Formula $\Psi_{v}\equiv f_{\nu}-1$
|
||
\end_inset
|
||
|
||
and equivalently:
|
||
\begin_inset Formula $\Psi_{h}\equiv f_{\kappa}-1$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $z=\rho_{0}c_{0}\frac{2i\sin\left(\Gamma\frac{t_{w}}{2}\right)}{\sqrt{\left(\gamma-\left(\gamma-1\right)\left(f_{\kappa}-1\right)\right)\left(f_{\nu}-1\right)}}+\rho_{0}c_{0}\frac{i\omega}{c_{0}C_{D}\phi}\frac{2\delta}{1-f_{\nu}}f_{\mathrm{int}},$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
where
|
||
\begin_inset Formula $k_{c}$
|
||
\end_inset
|
||
|
||
is our
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $z=\rho_{0}c_{0}\frac{2i\sin\left(\Gamma\frac{t_{w}}{2}\right)}{\sqrt{\left(1+\left(\gamma-1\right)f_{\kappa}\right)\left(1-f_{\nu}\right)}}+\frac{i\omega}{c_{0}C_{D}\phi}\frac{2\delta}{1-f_{\nu}}f_{\mathrm{int}},$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\delta$
|
||
\end_inset
|
||
|
||
is the end correction length for one side:
|
||
\begin_inset Formula $\delta=4D/(3\pi)$
|
||
\end_inset
|
||
|
||
.
|
||
For small plate thicknesses:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $z=\rho_{0}c_{0}\frac{2i\sin\left(\Gamma\frac{t_{w}}{2}\right)}{\sqrt{\left(1+\left(\gamma-1\right)f_{\kappa}\right)\left(1-f_{\nu}\right)}}+\rho_{0}c_{0}\frac{i\omega}{c_{0}C_{D}\phi}\frac{2\delta}{1-f_{\nu}}f_{\mathrm{int}},$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
Z_{c}=\frac{kZ_{0}}{\left(1-f_{\nu}\right)\Gamma}.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
viscothermal wave number, i.e.
|
||
the wave number corrected for viscothermal losses:
|
||
\begin_inset Formula
|
||
\[
|
||
\Gamma=\frac{\omega}{c_{0}}\sqrt{\frac{1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\epsilon_{s}}}{1-f_{\nu}}}.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
For small plate thicknesses:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $z=i\omega\rho_{0}\frac{t_{w}+\frac{2\delta f_{\mathrm{int}}}{C_{D}}}{\left(1-f_{\nu}\right)},$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
z=\frac{i\omega\rho_{0}}{\phi}\left[\frac{t_{w}}{\left(1-f_{\nu}\right)}+2\delta f_{\mathrm{int}}\right]+\alpha\frac{\rho_{0}\omega\delta_{\nu}}{\phi},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $f_{\mathrm{int}}$
|
||
\end_inset
|
||
|
||
is the hole-hole interaction function which
|
||
\begin_inset Formula $\to1$
|
||
\end_inset
|
||
|
||
for
|
||
\begin_inset Formula $\phi\to0$
|
||
\end_inset
|
||
|
||
, and
|
||
\begin_inset Formula $\delta$
|
||
\end_inset
|
||
|
||
is the single-sided hole (therefore, the factor 2 in front) end correction
|
||
due to the added mass effect, for the situation of negligible hole-hole
|
||
interaction.
|
||
[Paper: Tayong, 2013].
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
f_{\mathrm{int}}(\phi) & =1-1.4092\sqrt{\phi}+0.33818\sqrt{\phi}^{3}+0.06793\sqrt{\phi}^{5}.\\
|
||
& -0.02287\sqrt{\phi}^{6}+0.063015\sqrt{\phi}^{7}-0.01614\sqrt{\phi}^{8}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
For square holes:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\xi^{2}=\frac{\pi D^{2}}{4P^{2}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{D}{P}=\sqrt{\frac{4\phi}{\pi}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For circular large holes with diameter
|
||
\begin_inset Formula $D$
|
||
\end_inset
|
||
|
||
, the end correction for both sides is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
2\delta=\frac{8}{3\pi}D\approx0.85D.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Here we use a more advanced model, which includes the shear wave number.
|
||
For unrounded edges and a perforate thickness of
|
||
\begin_inset Formula $t_{p}$
|
||
\end_inset
|
||
|
||
, the added mass end correction can be computed as:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Equation according to Temiz for added mass effect:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $2\delta=\frac{\delta_{\mathrm{temiz}}}{2}D$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Where:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\delta_{\mathrm{temiz}}=0.97\exp\left(-0.2S_{h}\right)+1.54-0.003\frac{D}{t_{p}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $S_{h}=\frac{D}{2}\sqrt{\frac{\rho_{0}\omega}{\mu_{0}}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
——–
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Ours:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $2\delta=\frac{8}{3\pi}D$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
2\delta=\frac{1}{2}\left[0.97\exp\left(-0.14\frac{D}{\delta_{\nu}}\right)+1.54-0.003\frac{D}{t_{p}}\right]D
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The factor
|
||
\begin_inset Formula $\alpha$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\alpha=5.08\left(\frac{D}{\sqrt{2}\delta_{\nu}}\right)^{-1.45}+1.70-0.002\frac{D}{t_{p}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Tuning the hole diameter for large holes and the negligible hole-hole interactio
|
||
n
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The coarse impedance of a Helmholtz resonator repeated here:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z(\omega)=\underbrace{i\omega m_{A}+R_{v}}_{Z_{h}}+\frac{\rho_{0}c_{0}^{2}}{i\omega V},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
The resistive and reacting part
|
||
\begin_inset Formula $i\omega m_{A}+R_{v}$
|
||
\end_inset
|
||
|
||
is due to the resonator holes,
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{h}=i\omega m_{A}+R_{v}\approx\frac{1}{S}\left[\frac{i\omega\rho_{0}}{\phi}\left[\frac{t_{w}}{\left(1-f_{\nu}\right)}+2\delta f_{\mathrm{int}}\right]+\frac{\alpha\rho_{0}\omega\delta_{\nu}}{\phi}\right].\label{eq:Zhole}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
In the large hole limit, or high shear wave number:
|
||
\begin_inset Formula
|
||
\[
|
||
\Re\left[i\omega m_{A}+R_{v}\right]\approx\frac{\rho_{0}\delta_{\nu}\omega}{\phi S}\left[\alpha+\frac{2t_{w}}{\left(D-4\delta_{\nu}\right)}\right]\underbrace{\propto}_{\mathrm{approx}.}\sqrt{\omega}.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In the large hole limit, without hole-hole interaction and
|
||
\begin_inset Formula $\delta_{\nu}\to0$
|
||
\end_inset
|
||
|
||
, we the resonance frequency of the system is:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\omega_{r,\mathrm{lh}}^{2}=\frac{\phi Sc_{0}^{2}}{V\left(1.54D+t_{w}\right)}\label{eq:omgr_largeholes}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{c_{0}^{2}\rho_{0}}{V\omega_{\mathrm{r,lh}}^{2}}\left[\frac{\omega_{\mathrm{r,lh}}^{2}}{i\omega}+\omega\left\{ \frac{\alpha\delta_{\nu}}{2\delta f_{\mathrm{int}}+t_{w}}+i\frac{Dt_{w}+2\delta f_{\mathrm{int}}\left(D-2\delta_{\nu}\left(1-i\right)\right)}{\left(D-2\delta_{\nu}\left(1-i\right)\right)\left(2\delta f_{\mathrm{int}}+t_{w}\right)}\right\} \right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{c_{0}^{2}\rho_{0}}{V\omega_{\mathrm{r,lh}}^{2}}\left[\frac{\omega_{\mathrm{r,lh}}^{2}}{i\omega}+\frac{\omega\alpha\delta_{\nu}}{2\delta f_{\mathrm{int}}+t_{w}}+i\omega\left(1+t_{w}\frac{2\delta_{\nu}\left(1-i\right)}{\left(D-2\delta_{\nu}\left(1-i\right)\right)\left(2\delta f_{\mathrm{int}}+t_{w}\right)}\right)\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{\mathrm{large\,holes},\mathrm{res}}(\omega)=\frac{c_{0}^{2}\rho_{0}}{V\omega_{r,\mathrm{lh}}^{2}}\left[\frac{\omega_{r,\mathrm{lh}}^{2}}{i\omega}+\frac{i\omega t_{w}}{\left\{ 1+2\frac{\delta_{\nu}\left(i-1\right)}{D}\right\} \left(2\delta f_{\mathrm{int}}+t_{w}\right)}+\frac{i\omega\left[2\delta f_{\mathrm{int}}-i\delta_{\nu}\alpha\right]}{2\delta f_{\mathrm{int}}+t_{w}}\right]\label{eq:Zlargeholes_forres}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
COMSOL boundary condition to useful
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
When using COMSOL to compute Helmholtz resonances, the added mass effect
|
||
is included just by solving the Helmholtz equation.
|
||
Therefore, to model the holes, only the final wall thickness part of the
|
||
added mass (and hole-hole interaction), and the resistive part of the impedance
|
||
should be added to the simulation.
|
||
If we look at Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:Zhole"
|
||
|
||
\end_inset
|
||
|
||
, it means only the following part:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
z_{\mathrm{bc,\,COMSOL}}=i\omega\rho_{0}\frac{t_{w}}{1-f_{\nu}}+\alpha\rho_{0}\omega\delta_{\nu}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Porosity estimator constraint
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
An estimation for the porosity is a good requirement, as a too large porosity
|
||
leads to too much hole-hole interaction and shift away from proper Helmholtz
|
||
resonators.
|
||
First of all, we set the surface area at the inner duct, which is available
|
||
for holes as
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
S=\Pi L_{h},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and we fix
|
||
\begin_inset Formula $L_{h}$
|
||
\end_inset
|
||
|
||
to
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
L_{h}=\lambda_{r}/20=\frac{2\pi c_{0}}{20\omega_{r,\mathrm{lh}}}=\frac{\pi c_{0}}{10\omega_{r,\mathrm{lh}}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Rewriting Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:omgr_largeholes"
|
||
|
||
\end_inset
|
||
|
||
to
|
||
\begin_inset Formula $\phi$
|
||
\end_inset
|
||
|
||
yields
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\phi\approx\frac{V\left(1.54D+t_{w}\right)\omega_{r,\mathrm{lh}}^{2}}{Sc_{0}^{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Fill in for
|
||
\begin_inset Formula $S=\Pi L_{h}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\phi\approx\frac{V\left(1.54D+t_{w}\right)\omega_{r,\mathrm{lh}}^{2}}{\Pi L_{h}c_{0}^{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
And for
|
||
\begin_inset Formula $L_{h}$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $L_{h}=\frac{\pi c_{0}}{10\omega_{r,\mathrm{lh}}}.$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\phi\approx\frac{10}{\pi}\frac{V\left(1.54D+t_{w}\right)\omega_{r,\mathrm{lh}}^{3}}{\Pi c_{0}^{3}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\phi_{\mathrm{estimation}}\approx\frac{10}{\pi}\frac{V\left(1.54D+t_{w}\right)\omega_{r,\mathrm{lh}}^{3}}{\Pi c_{0}^{3}}\leq0.1
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
See what this constraint does...*
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Large hole (boundary layer) limit
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\phi=\frac{S_{\mathrm{hole}}}{S_{\mathrm{tot}}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula $\delta_{\nu}\ll D$
|
||
\end_inset
|
||
|
||
.
|
||
Given
|
||
\begin_inset Formula $\zeta$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\omega_{r}$
|
||
\end_inset
|
||
|
||
.
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Now, the following substitutions are made:
|
||
\begin_inset Formula
|
||
\begin{align*}
|
||
C & =\frac{V}{\rho_{0}z_{0}},\\
|
||
m_{\mathrm{neck}} & =\frac{1}{\omega_{r}^{2}C}\\
|
||
\zeta & =\frac{1}{2}\omega_{r}CR_{v}.
|
||
\end{align*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $ $
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
such that we can write:
|
||
\begin_inset Formula
|
||
\[
|
||
Z_{h}=\frac{1}{\omega_{r}C}\left(\frac{\omega_{r}}{i\omega}+2\zeta+\frac{i\omega}{\omega_{r}}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
Z_{h}=\frac{1}{\omega_{r}C}\left(\frac{\omega_{r}}{i\omega}+2\zeta+\frac{i\omega}{\omega_{r}}\right)=\frac{1}{i\omega C}+\frac{2\zeta\omega_{r}}{C}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{2\zeta}{\omega_{r}C}=R_{v}.$
|
||
\end_inset
|
||
|
||
Or:
|
||
\begin_inset Formula
|
||
\[
|
||
\zeta=\frac{1}{2}\omega_{r}R_{v}C
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
But:
|
||
\begin_inset Formula $\frac{1}{Cm_{A}}=\omega_{r}^{2}$
|
||
\end_inset
|
||
|
||
Such that:
|
||
\begin_inset Formula $\frac{1}{C}=\omega_{r}^{2}m_{A}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\zeta=\frac{1}{2}\frac{R_{v}}{\omega_{r}m_{A}}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
Note that:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\zeta=\frac{1}{2}\frac{R}{m_{A}\omega_{r}}\approx\frac{1}{2}\frac{\Re\left[z\right]}{\Im\left[z\right]}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Procedure:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In the boundary layer limit:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
f_{\nu}=\frac{\left(1-i\right)\delta_{\nu}}{2r_{h}},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
such that:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
z_{\mathrm{perforate}}=\frac{i\omega\rho_{0}}{\phi}\frac{t_{w}+2\delta f_{\mathrm{int}}}{\left(1-\frac{\delta_{\nu}}{2r_{h}}+\frac{i\delta_{\nu}}{2r_{h}}\right)}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $z=\frac{i\omega\rho_{0}}{\phi}\frac{t_{w}+\frac{2\delta f_{\mathrm{int}}}{C_{D}}\left(1-\frac{\delta_{\nu}}{2r_{h}}+\frac{i\delta_{\nu}}{2r_{h}}\right)}{\left(1-\frac{\delta_{\nu}}{2r_{h}}+\frac{i\delta_{\nu}}{2r_{h}}\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Typical resistance: fill in
|
||
\begin_inset Formula $\omega=\omega_{r}$
|
||
\end_inset
|
||
|
||
.
|
||
Filling in:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\zeta\approx\frac{\delta_{\nu}}{D}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The real part of the perforate impedance is the resistive part.
|
||
In a 3D simulation, this impedance can be added to a surface of the hole,
|
||
to model the hole
|
||
\emph on
|
||
resistance
|
||
\emph default
|
||
in an otherwise inviscid simulation.
|
||
The real part is:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{}{}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Lots of holes
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Hereby, once we know the hole diameter, the required acoustic mass can be
|
||
tuned using the porosity:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
m_{A}\approx\frac{\Im\left[z(\omega=\omega_{r}\right]}{\omega S_{\mathrm{t}}}\approx\frac{1}{S_{\mathrm{tot}}\phi}\left(\frac{\rho_{0}8Df_{\mathrm{int}}(\phi)}{3\pi}+\rho_{0}t_{w}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
So that the porosity can be computed as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\phi\approx F(\phi)=\frac{D\rho_{0}\left(D-2\delta_{\nu}\right)\left(8Df_{\mathrm{int}}+3\pi t_{w}\right)}{3\pi S_{\mathrm{tot}}m_{A}\left(D^{2}-4D\delta_{\nu}+8\delta_{\nu}^{2}\right)}\approx\frac{\rho_{0}\left(8Df_{\mathrm{int}}(\phi)+3\pi t_{w}\right)}{3\pi S_{\mathrm{tot}}m_{A}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Note that this is a trancendental equation in
|
||
\begin_inset Formula $\phi$
|
||
\end_inset
|
||
|
||
, which can easily be solved by iterating
|
||
\begin_inset Formula $\phi$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\phi_{1} & =F(1)\\
|
||
\phi_{2} & =F(\phi_{1})\\
|
||
\phi_{3} & =F(\phi_{2})\\
|
||
\vdots & =\vdots
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Some holes
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For only
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
some holes
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
, far away from each other, we fill in for
|
||
\begin_inset Formula $\phi=\frac{1}{4}N_{\mathrm{hole}}\pi D^{2}/S_{\mathrm{tot}}$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
m_{A}\approx\frac{\rho_{0}}{3\pi N_{\mathrm{hole}}D}\left(\frac{32}{\pi}+\frac{12t_{w}}{D}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
So the number of holes can be chosen as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
N_{\mathrm{holes}}\approx\frac{4\rho_{0}\left(8Df_{\mathrm{int}}+3\pi t_{w}\right)}{3\pi^{2}D^{2}m_{A}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Small hole limit
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In the small hole limit,
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
f_{\nu}\approx1-\frac{iD^{2}}{16\delta_{\nu}^{2}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Such that:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\zeta=\frac{1}{2}\frac{R}{m_{A}\omega_{r}}\approx\frac{1}{2}\frac{\Re\left[z(\omega=\omega_{r}\right]}{\Im\left[z(\omega=\omega_{r}\right]}\approx\frac{3\pi\delta_{\nu}^{2}t_{w}}{D^{3}f_{\mathrm{int}}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Such that:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
D=\sqrt[3]{\frac{6\pi\delta_{\nu}^{2}t_{w}}{6\zeta}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
And:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
m_{A}=\rho_{0}\frac{8Df_{\mathrm{int}}}{3\pi S_{\mathrm{tot}}\phi}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Such that:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\phi\approx\rho_{0}\frac{8Df_{\mathrm{int}}}{3\pi S_{\mathrm{tot}}m_{A}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Geometry of hole patterns
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/hexagonal_pattern.pdf
|
||
width 50text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Geometry details of a hexagonal hole pattern
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:hexagonal_pitch"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For a square hole pattern, with hole-hole pitch
|
||
\begin_inset Formula $P$
|
||
\end_inset
|
||
|
||
, the overall surface of a unit cell
|
||
\begin_inset Formula $S_{\mathrm{unit}}=P^{2}$
|
||
\end_inset
|
||
|
||
.
|
||
For a certain porosity, the pitch can then be computed as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
P=\sqrt{\frac{\pi}{4\phi}}D.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
For a hexagonal hole pattern (Fig.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:hexagonal_pitch"
|
||
|
||
\end_inset
|
||
|
||
) with hole-hole pitch
|
||
\begin_inset Formula $P$
|
||
\end_inset
|
||
|
||
, the overall surface of a unit cell
|
||
\begin_inset Formula $S_{\mathrm{unit}}=\frac{\sqrt{3}}{2}P^{2}$
|
||
\end_inset
|
||
|
||
.
|
||
Henceforth, the pitch can be computed from the porosity and the hole diameter
|
||
as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
P=\sqrt{\frac{\sqrt{3}\pi}{6\phi}}D.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The most important design parameters of a perforate are the porosity and
|
||
the hole diameter.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Addition of acoustic hole resistance in an otherwise inviscid simulation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
We assume that in a 3D FEM simulation, the imaginary acoustic impedance
|
||
of a single hole
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{\mathrm{hole}}=i\omega\rho_{0}\frac{4}{\pi D^{2}}\left[\frac{t_{w}}{\left(1-f_{\nu}\right)}+\frac{8Df_{\mathrm{int}}}{3\pi C_{D}}\right],
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Re[z_{\mathrm{hole}}]=\frac{2D\delta_{\nu}\omega\rho_{0}t_{w}}{\left(4\delta_{\nu}^{2}+\left(D-2\delta_{\nu}\right)^{2}\right)},\label{eq:Rv_hole}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Over-all transmission matrix
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
& & & & \left\{ \begin{array}{c}
|
||
p_{R}\\
|
||
U_{R}
|
||
\end{array}\right\} _{1} & = & \boldsymbol{T}_{1}\left\{ \begin{array}{c}
|
||
p_{L}\\
|
||
U_{L}
|
||
\end{array}\right\} _{1}\\
|
||
& & \left\{ \begin{array}{c}
|
||
p_{R}\\
|
||
U_{R}
|
||
\end{array}\right\} _{2} & & =\boldsymbol{T}_{2}\left\{ \begin{array}{c}
|
||
p_{R}\\
|
||
U_{R}
|
||
\end{array}\right\} _{1}\\
|
||
\left\{ \begin{array}{c}
|
||
p_{R}\\
|
||
U_{R}
|
||
\end{array}\right\} _{3} & =\boldsymbol{T}_{3} & \left\{ \begin{array}{c}
|
||
p_{R}\\
|
||
U_{R}
|
||
\end{array}\right\} _{2}\\
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
, hence
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left\{ \begin{array}{c}
|
||
p_{R}\\
|
||
U_{R}
|
||
\end{array}\right\} _{3}=\underbrace{\boldsymbol{T}_{3}\cdot\boldsymbol{T}_{2}\cdot\boldsymbol{T}_{1}}_{\boldsymbol{T}}\left\{ \begin{array}{c}
|
||
p_{L}\\
|
||
U_{L}
|
||
\end{array}\right\} _{1}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Miscellaneous models for acoustic components
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Acoustic impedance of small orifices
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Rectangular orifice
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Slit orifice
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
====================
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Lookup model
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
COMSOL model
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\align left
|
||
LRFTubes allows importing transfer matrix data from externally computed
|
||
sources (i.e.
|
||
finite element model results).
|
||
We focus on the use of COMSOL Multiphysics here.
|
||
The output data from COMSOL should be created using the
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
Port Sweep
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
functionality.
|
||
Implementation is only for 2 ports, as this is the only case for which
|
||
COMSOL is able to export data.
|
||
In COMSOL, the transfer matrix is defined as:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/comsol_transfermatrix.png
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\align left
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left\{ \begin{array}{c}
|
||
p_{i}\\
|
||
Q_{i}
|
||
\end{array}\right\} =\left[\begin{array}{cc}
|
||
T_{11} & T_{12}\\
|
||
T_{21} & T_{22}
|
||
\end{array}\right]\left\{ \begin{array}{c}
|
||
p_{o}\\
|
||
Q_{o}
|
||
\end{array}\right\} ,\label{eq:transfer_matrix_COMSOL}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
hence the transfer matrix definition of
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubes
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
is the
|
||
\emph on
|
||
inverse
|
||
\emph default
|
||
of the definition of COMSOL Multiphysics:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\boldsymbol{T}_{\mathrm{\lrftubes}}=\boldsymbol{T}_{\mathrm{COMSOL}}^{-1}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
To properly use the Lookup model, in COMSOL port 1 should be corresponding
|
||
to the LEFT side of a segment, and port 2 should be corresponding to the
|
||
RIGHT side of a segment.
|
||
Then, the data should be exported to a
|
||
\emph on
|
||
txt
|
||
\emph default
|
||
file with the columns in the following order: frequency, T11, T12, T21,
|
||
T22.
|
||
A file of this format, as exported by COMSOL can be passed to the constructor
|
||
of
|
||
\family typewriter
|
||
\emph on
|
||
LookupModel
|
||
\family default
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
SPICE model
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/two_port_probing.pdf
|
||
width 90text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Two-port model, probing the transfer matrix by computing the simulation
|
||
output.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:2-port-probing"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A SPICE model can be created from a COMSOL model, by performing a circuit
|
||
analysis of the system in two cases, one is the situation providing a voltage
|
||
source on one side, and measuring the current going in, and the current
|
||
going out on the other side, while the element is short-circuited.
|
||
The other is similar, only in this case the segment is
|
||
\emph on
|
||
open
|
||
\emph default
|
||
on the other side.
|
||
Fig.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:2-port-probing"
|
||
|
||
\end_inset
|
||
|
||
shows the schematic of the two cases that need to be computed.
|
||
If we assume:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left\{ \begin{array}{c}
|
||
p\\
|
||
U
|
||
\end{array}\right\} _{R}=\left[\begin{array}{cc}
|
||
A & B\\
|
||
C & D
|
||
\end{array}\right]\left\{ \begin{array}{c}
|
||
p\\
|
||
U
|
||
\end{array}\right\} _{L},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
for the components of the transfer matrix, we can set the following equations:
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
U_{R}^{(1)} & =C+DU_{L}^{(1)},\\
|
||
0 & =A+BU_{L}^{(1)},\\
|
||
0 & =C+DU_{L}^{(2)},\\
|
||
p_{R}^{(2)} & =A+BU_{L}^{(2)},
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
which gives four equations, for the four unknown transfer matrix coefficients.
|
||
We can directly perform this computation using the method
|
||
\family typewriter
|
||
LookupModel.from_pU
|
||
\family default
|
||
in
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
lrftubes
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Measuring the transmission matrix using the four microphone method
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Based on Brüel Kjaer - Transmission loss in impedance tube.pdf in /home/anne/next
|
||
cloud/wip_redusone/2021-Steegmuller/measurement_setup
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Modifications: volume flow U instead of velocity v; impedance Z instead
|
||
of characteristic impedance z; transfer functions Hir instead of cross
|
||
correlations (?).
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
TO DO:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
draw own image image
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
fix citation
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Transfer matrix according to our own definition instead of the definition
|
||
of Bruel & Kjaer = definition of COMSOL
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Consistently use Q or U for volume flow? Also in text above about COMSOL.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The transfer matrix of a device can be measured using a four microphone
|
||
setup as shown in figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:meas_transmatrix_4mic"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
The microphones record acoustic pressure and plane waves are assumed.
|
||
In the following equations, time dependency
|
||
\begin_inset Formula $\exp(+j*\omega*t)$
|
||
\end_inset
|
||
|
||
is not shown.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/Bruel_Kjaer_fig1.png
|
||
lyxscale 50
|
||
width 80text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Experimental setup to measure the transfer matrix, using the four microphone
|
||
method
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:meas_transmatrix_4mic"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The transfer matrix coefficients are calculated based on sound pressure
|
||
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
and volume velocity
|
||
\begin_inset Formula $U$
|
||
\end_inset
|
||
|
||
, as related by equation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:transfer_matrix_COMSOL"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Note that this definition is different than the definition used in LRFtubes
|
||
and therefore
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
should be inverted for further use.
|
||
Subscrips
|
||
\begin_inset Formula $i$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $d$
|
||
\end_inset
|
||
|
||
refer to
|
||
\begin_inset Formula $x=0$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $x=d$
|
||
\end_inset
|
||
|
||
respectively.
|
||
There are two equations and four unknowns, so two sets of measurements
|
||
are required.
|
||
The second set, indicated by superscript
|
||
\begin_inset Formula $*$
|
||
\end_inset
|
||
|
||
, must be performed with a different acoustic termination.
|
||
Together this results in four equations for four unknowns.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\align left
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left\{ \begin{array}{c}
|
||
p_{i}\\
|
||
Q_{i}
|
||
\end{array}\begin{array}{c}
|
||
p_{i}^{*}\\
|
||
Q_{i}^{*}
|
||
\end{array}\right\} =\left[\begin{array}{cc}
|
||
T_{11} & T_{12}\\
|
||
T_{21} & T_{22}
|
||
\end{array}\right]\left\{ \begin{array}{c}
|
||
p_{o}\\
|
||
Q_{o}
|
||
\end{array}\begin{array}{c}
|
||
p_{o}^{*}\\
|
||
Q_{o}^{*}
|
||
\end{array}\right\} ,\label{eq:transfer_matrix-double}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Solving for
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
yields:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left[\begin{array}{cc}
|
||
T_{11} & T_{12}\\
|
||
T_{21} & T_{22}
|
||
\end{array}\right]=\frac{1}{p_{d}Q_{d}^{*}-p_{d}^{*}Q_{d}}\left[\begin{array}{cc}
|
||
p_{i}Q_{d}^{*}-p_{i}^{*}Q_{d} & -p_{i}p_{d}^{*}+p_{i}^{*}p_{d}\\
|
||
Q_{i}Q_{d}^{*}-Q_{i}^{*}Q_{d} & -p_{d}^{*}Q_{i}+p_{d}Q_{i}^{*}
|
||
\end{array}\right]
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $Q$
|
||
\end_inset
|
||
|
||
at
|
||
\begin_inset Formula $x=0$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $x=d$
|
||
\end_inset
|
||
|
||
can be calculated from travelling
|
||
\begin_inset Formula $A$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $B$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $C$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $D$
|
||
\end_inset
|
||
|
||
.
|
||
The calculation of their second measurement counterparts
|
||
\begin_inset Formula $*$
|
||
\end_inset
|
||
|
||
goes analogously and uses
|
||
\begin_inset Formula $A^{*}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $B^{*}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $C^{*}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $D^{*}$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{i}=A+B
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Q_{i}=\frac{A-B}{Z_{0}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{d}=C\cdot e^{-jkd}+D\cdot e^{jkd}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Q_{d}=\frac{C\cdot e^{-jkd}-D\cdot e^{jkd}}{Z_{0}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
in which
|
||
\begin_inset Formula $Z_{0}=\frac{z_{0}}{S}$
|
||
\end_inset
|
||
|
||
is the impedance of an infinite duct, with
|
||
\begin_inset Formula $z_{0}$
|
||
\end_inset
|
||
|
||
the characteristic impedance and
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
the cross-sectional area,
|
||
\begin_inset Formula $j=\sqrt{-1}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $k$
|
||
\end_inset
|
||
|
||
the wavenumber.
|
||
Travelling waves
|
||
\begin_inset Formula $A$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $B$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $C$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $D$
|
||
\end_inset
|
||
|
||
can be calculated from transfer functions
|
||
\begin_inset Formula $H_{ir}$
|
||
\end_inset
|
||
|
||
from reference signal
|
||
\begin_inset Formula $r$
|
||
\end_inset
|
||
|
||
, as sent to the loudspeaker, to the recorded signal of microphone
|
||
\begin_inset Formula $i$
|
||
\end_inset
|
||
|
||
.
|
||
The calculation of their second measurement counterparts
|
||
\begin_inset Formula $*$
|
||
\end_inset
|
||
|
||
goes analogously and uses
|
||
\begin_inset Formula $H_{ir}^{*}$
|
||
\end_inset
|
||
|
||
.
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
A=\frac{j\left(H_{1r}\cdot e^{jkx_{2}}-H_{2r}\cdot e^{jkx_{1}}\right)}{2\sin\left(k\left(x_{1}-x_{2}\right)\right)}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
B=\frac{j\left(H_{2r}\cdot e^{-jkx_{1}}-H_{1r}\cdot e^{-jkx_{2}}\right)}{2\sin\left(k\left(x_{1}-x_{2}\right)\right)}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
C=\frac{j\left(H_{3r}\cdot e^{jkx_{4}}-H_{4r}\cdot e^{jkx_{3}}\right)}{2\sin\left(k\left(x_{3}-x_{4}\right)\right)}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
D=\frac{j\left(H_{4r}\cdot e^{-jkx_{3}}-H_{3r}\cdot e^{-jkx_{4}}\right)}{2\sin\left(k\left(x_{3}-x_{4}\right)\right)}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\sqrt{G_{rr}}$
|
||
\end_inset
|
||
|
||
has been removed from the equations because Caspers thinks that
|
||
\begin_inset Formula $H_{ir}$
|
||
\end_inset
|
||
|
||
refers to the cross spectrum instead of the transfer function.
|
||
If the transfer function is used, then
|
||
\begin_inset Formula $\sqrt{G_{rr}}$
|
||
\end_inset
|
||
|
||
shall be left out.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Note: if no reference signal has been recorded, the reference signal can
|
||
be set to the signal captured by microphone 1.
|
||
The equations have no way to figure out whether the loudspeaker really
|
||
was driven by such a signal.
|
||
Then a requirement is that all microphones are recorded simultaneously
|
||
and with synchronized ADC clocks.
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
IEC Coupler impedances
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The Comsol model with which this data is gathered exports the input impedance
|
||
correctly, but the transfer impedance is actually the
|
||
\emph on
|
||
negative
|
||
\emph default
|
||
of the actual transfer impedance.
|
||
This is due to Comsol, which was only interested in the magnitude of the
|
||
impedance values, and due to us (sloppy work).
|
||
The input impedance is defined as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{\mathrm{in}}=\frac{p_{\mathrm{coupler,entrance}}}{U_{\mathrm{coupler,entrance}}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and the transfer impedance as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{\mathrm{tr}}=\frac{p_{\mathrm{DRP}}}{U_{\mathrm{coupler,entrance}}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Standard acoustic solutions
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Spherically symmetric breathing ball (monopole)
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
From Rienstra and Hirschberg:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{p}(r)=-z_{0}c_{0}k\frac{\hat{v}}{i\omega}\frac{k^{2}a_{0}^{2}}{1+ika_{0}}\frac{\exp\left(-i\left(kr-a_{0}\right)\right)}{kr}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
To our definitions and a bit of rewriting:
|
||
\begin_inset Formula
|
||
\[
|
||
\hat{p}(r)=\frac{i\rho_{0}c_{0}ka^{2}}{1+ika}\frac{\exp\left(-i\left(kr-a\right)\right)}{r}\hat{v}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
Radiation from a compact monopole with radius
|
||
\begin_inset Formula $a$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
breathing
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
velocity amplitude
|
||
\begin_inset Formula $\hat{v}$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{p}(r)=\frac{iz_{0}ka^{2}}{1+ika}\frac{\exp\left(-i\left(kr-a\right)\right)}{r}\hat{v}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Small source limit (
|
||
\begin_inset Formula $ka\ll1$
|
||
\end_inset
|
||
|
||
):
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{p}(r)\approx iz_{0}\frac{ka^{2}}{r}\left[\exp\left(-i\left(kr-a\right)\right)\right]\hat{v}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
In terms of the transfer impedance (
|
||
\begin_inset Formula $\hat{U}=4\pi a^{2}\hat{v}$
|
||
\end_inset
|
||
|
||
):
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\hat{p}(r)=\frac{i\rho_{0}c_{0}ka^{2}}{1+ika}\frac{\exp\left(-i\left(kr-a\right)\right)}{r}\frac{\hat{U}}{4\pi a^{2}}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\hat{p}(r)=\frac{iz_{0}k}{4\pi\left(1+ika\right)r}\left[\exp\left(-i\left(kr-a\right)\right)\right]\hat{U}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
which is also:
|
||
\begin_inset Formula
|
||
\[
|
||
\hat{p}(r)\approx\frac{iz_{0}}{2\lambda r}\left[\exp\left(-i\left(kr-a\right)\right)\right]\hat{U}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{p}(r)=\underbrace{\frac{iz_{0}k}{4\pi\left(1+ika\right)r}\left[\exp\left(-i\left(kr-a\right)\right)\right]}_{Z_{\mathrm{tr}}(r)}\hat{U},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
For easy estimations, in the small source (
|
||
\begin_inset Formula $ka\ll1$
|
||
\end_inset
|
||
|
||
) and far field limit (
|
||
\begin_inset Formula $kr\gg1$
|
||
\end_inset
|
||
|
||
):
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{p}(r)\approx\frac{iz_{0}}{2\lambda r}\hat{U}\left[\exp\left(-ikr\right)\right].
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Dipoles
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Translating sphere, exact solution
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula $\theta$
|
||
\end_inset
|
||
|
||
: pole angle.
|
||
Then the velocity follows:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{v}(\theta)=\hat{v}_{0}\cos\left(\theta\right).
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
After performing analysis, we find for the pressure:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{p}(r,\theta)=\frac{-i\omega\rho_{0}\hat{v}_{0}a^{3}\cos\theta}{2\left(1+ika\right)-\left(ka\right)^{2}}\frac{\partial}{\partial r}\left\{ \frac{\exp\left(-ik\left(r-a\right)\right)}{r}\right\} .
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
In the small source limit (
|
||
\begin_inset Formula $ka\ll1$
|
||
\end_inset
|
||
|
||
):
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\hat{p}(r,\theta)=-\hat{v}_{0}\frac{z_{0}k^{2}a^{3}\cos\theta}{2r}\left(1+\frac{1}{ikr}\right)e^{-ik\left(r-a\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{p}(r,\theta)\approx-\frac{z_{0}k^{2}a^{3}\cos\theta}{2r}\left(\frac{1+ikr}{ikr}\right)\left[\exp\left(-ik\left(r-a\right)\right)\right]\hat{v}_{0}.\label{eq:dipole_transl_sphere}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Small source limit, far field (
|
||
\begin_inset Formula $ka\ll1$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $kr\gg1$
|
||
\end_inset
|
||
|
||
):
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{p}(r,\theta)\approx-\hat{v}_{0}\frac{z_{0}k^{2}a^{3}\cos\theta}{2r}e^{-ikr}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Perfect dipole from two compact monopoles
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Distance between sources:
|
||
\begin_inset Formula $d\ll\lambda$
|
||
\end_inset
|
||
|
||
.
|
||
Volume flow from a single pole:
|
||
\begin_inset Formula $\hat{U}$
|
||
\end_inset
|
||
|
||
.
|
||
From the other source
|
||
\begin_inset Formula $-\hat{U}$
|
||
\end_inset
|
||
|
||
.
|
||
The angle
|
||
\begin_inset Formula $\theta$
|
||
\end_inset
|
||
|
||
is 0 at positions where the positive source is the closest to the listening
|
||
point.
|
||
Distance between the sources is
|
||
\begin_inset Formula $d$
|
||
\end_inset
|
||
|
||
.
|
||
Then the sound pressure is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{p}(r,\theta)\approx-k^{2}z_{0}\frac{\exp\left(-ikr\right)\cos\theta}{4\pi r}\left(\frac{1+ikr}{ikr}\right)\hat{U}d
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Comparing this equation to Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:dipole_transl_sphere"
|
||
|
||
\end_inset
|
||
|
||
, we find that for the same acoustic pressure of a perfect dipole vs.
|
||
a translating sphere:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
2\pi a^{2}\hat{v}_{0}a=\hat{U}d.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
So if we set the volume flow of a translating sphere equal to the frontal
|
||
area of
|
||
\begin_inset Formula $\pi a^{2}$
|
||
\end_inset
|
||
|
||
, the effective dipole distance is
|
||
\begin_inset Formula $2a$
|
||
\end_inset
|
||
|
||
, which corresponds to the diameter of the sphere!
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{a^{3}}{2}\hat{v}_{0}=\frac{1}{4\pi}\hat{U}d$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Hence: if we set
|
||
\begin_inset Formula $\hat{U}_{\mathrm{tr\,sphere}}=\pi a^{2}\hat{v}$
|
||
\end_inset
|
||
|
||
: the effective distance
|
||
\begin_inset Formula $d$
|
||
\end_inset
|
||
|
||
of a translating sphere is:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $2\pi a^{2}\hat{v}_{0}a=\hat{U}d$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Compact quadrupole
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/quadrupole.pdf
|
||
width 60text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Schematic of the quadrupole.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A compact square-shaped quadrupole with distances of
|
||
\begin_inset Formula $d$
|
||
\end_inset
|
||
|
||
between each pole, distance
|
||
\begin_inset Formula $kd\ll1$
|
||
\end_inset
|
||
|
||
.
|
||
Volume flow from a single pole:
|
||
\begin_inset Formula $\hat{U}$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{p}(x,y)=-ik^{3}z_{0}\hat{U}d^{2}\frac{xy\exp\left(-ikr\right)}{4\pi r^{3}}\left(1+\frac{3}{ikr}-\frac{3}{\left(kr\right)^{2}}\right).
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Optimized reactive silencers
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Parallel Helmholtz resonator transfer function and transmission loss
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Equations for a side branch Helmholtz resonator:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
p_{R} & =p_{L},\\
|
||
U_{R} & =U_{L}-p_{L}/Z_{h},
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $Z_{h}$
|
||
\end_inset
|
||
|
||
is the side branch impedance of the Helmholtz resonator, defined as
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{h}=\left(\frac{\rho_{0}z_{0}}{i\omega V}+R_{v}+i\omega m_{\mathrm{neck}}\right),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
m_{\mathrm{neck}}=\frac{\rho_{0}\ell_{\mathrm{eff},\mathrm{neck}}}{S_{\mathrm{neck}}},\label{eq:acoustic_mass_neck}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and for relatively large holes, air at STP, the resistance term can be estimated
|
||
as [SOURCE HERE!]:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
R_{v}\approx7.2\times10^{-3}z_{0}/S_{h},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Now, the following substitutions are made:
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
C & =\frac{V}{\rho_{0}z_{0}},\\
|
||
m_{\mathrm{neck}} & =\frac{1}{\omega_{r}^{2}C}\\
|
||
\zeta & =\frac{1}{2}\omega_{r}CR_{v}.
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{2\zeta}{\omega_{r}C}=R_{v}.$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
such that we can write:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Z_{h}=\frac{1}{\omega_{r}C}\left(\frac{\omega_{r}}{i\omega}+2\zeta+\frac{i\omega}{\omega_{r}}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The quality factor of the resonator is the ratio of the resonance frequency
|
||
to its bandwidth measure.
|
||
If we take
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Q\overset{\mathrm{def}}{=}\frac{f_{r}}{\Delta f},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\Delta f$
|
||
\end_inset
|
||
|
||
is the full width at half the maximum value, i.e.
|
||
the frequency distance between two points lying at
|
||
\begin_inset Formula $-3$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
dB w.r.t.
|
||
the maximum value.
|
||
The damping ratio
|
||
\begin_inset Formula $\zeta$
|
||
\end_inset
|
||
|
||
is related to
|
||
\begin_inset Formula $Q$
|
||
\end_inset
|
||
|
||
as:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\zeta=\frac{1}{2Q}=\frac{1}{2}\frac{\Delta f}{f_{r}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Assembling the transfer matrix
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left\{ \begin{array}{c}
|
||
p\\
|
||
U
|
||
\end{array}\right\} _{R}=\left[\begin{array}{cc}
|
||
T_{11} & T_{12}\\
|
||
T_{21} & T_{22}
|
||
\end{array}\right]\left\{ \begin{array}{c}
|
||
p\\
|
||
U
|
||
\end{array}\right\} _{L},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
T_{11} & =1\\
|
||
T_{12} & =0\\
|
||
T_{21} & =-Z_{h}^{-1}\\
|
||
T_{22} & =1
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Transmission loss
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The transmission coefficient can be computed as:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\tau=\frac{C}{A}=\frac{Z_{0}\left(T_{21}p_{L}+T_{22}U_{L}\right)}{\frac{1}{2}\left(p_{L}+Z_{0}U_{L}\right)},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
using
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
T_{11}p_{L}+T_{12}U_{L}=p_{R}=Z_{0}U_{R}=Z_{0}\left(T_{21}p_{L}+T_{22}U_{L}\right),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
we get
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $Z_{0}\left(T_{21}p_{L}+T_{22}U_{L}\right)=T_{11}p_{L}+T_{12}U_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
–
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $U_{L}=\frac{\left(T_{11}-Z_{0}T_{21}\right)}{\left(Z_{0}T_{22}-T_{12}\right)}p_{L}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
U_{L}=\frac{\left(T_{11}-Z_{0}T_{21}\right)}{\left(Z_{0}T_{22}-T_{12}\right)}p_{L},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
filling in:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\tau=\frac{2}{Z_{0}}\frac{T_{11}T_{22}-T_{12}T_{21}}{T_{11}-T_{12}/Z_{0}-T_{21}Z_{0}+T_{22}},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
assuming that the determinant of the transfer matrix be unity
|
||
\begin_inset Formula $(T_{11}T_{22}-T_{12}T_{21}\equiv1$
|
||
\end_inset
|
||
|
||
) [THIS IS TRUE, BUT WHERE DOES THIS ASSUMPTION COME FROM??], this can be
|
||
further simplified:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\tau=\frac{2}{T_{11}-T_{12}/Z_{0}-T_{21}Z_{0}+T_{22}},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For a Helmholtz resonator, this results in:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\tau=\frac{2}{T_{11}-T_{12}/Z_{0}-T_{21}Z_{0}+T_{22}},
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Filling in:
|
||
\begin_inset Formula $T_{11}=1$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $T_{12}=0$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $T_{21}=-1/Z_{h}$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $T_{22}=1$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\tau=\frac{2Z_{h}}{2Z_{h}+Z_{0}},
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\tau(\omega)=\frac{2Z_{h}(\omega)}{Z_{0}+2Z_{h}(\omega)},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Filling in the Helmholtz resonator equation:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\tau(\omega)=\frac{2\left(1+2\frac{\omega}{\omega_{r}}\zeta-\left(\frac{\omega}{\omega_{r}}\right)^{2}\right)}{2\left(1+2\frac{\omega}{\omega_{r}}\zeta-\left(\frac{\omega}{\omega_{r}}\right)^{2}\right)+i\frac{\omega}{\omega_{r}}\left(\frac{Cz_{0}\omega_{r}}{S}\right)}\label{eq:tau_hhres}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(\frac{Cz_{0}\omega_{r}}{S}\right)=\left(\frac{V\omega_{r}}{c_{0}S}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
The peak height, filling in for
|
||
\begin_inset Formula $\omega/\omega_{r}=1$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\tau=\frac{4\zeta}{4\zeta+\beta},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\beta$
|
||
\end_inset
|
||
|
||
is defined as the resonator strength:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\beta=\frac{V\omega_{r}}{Sc_{0}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
In terms of transmission loss:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathrm{TL}_{\omega=\omega_{r}}=20\log\left(\frac{\beta+4\zeta}{4\zeta}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In case of weak damping (
|
||
\begin_inset Formula $\zeta\ll1$
|
||
\end_inset
|
||
|
||
), Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:tau_hhres"
|
||
|
||
\end_inset
|
||
|
||
can be reduced to:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\tau(\omega)=\frac{1-\left(\frac{\omega}{\omega_{r}}\right)^{2}}{1-\left(\frac{\omega}{\omega_{r}}\right)^{2}+\frac{1}{2}i\frac{\omega}{\omega_{r}}\beta}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The width of the peak over which a certain transmission loss is higher than
|
||
a value of
|
||
\begin_inset Formula $\mathrm{TL_{\mathrm{min}}}$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
dB, can be computed as:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $|\tau(\omega_{r}+\Delta\omega)|=|\frac{1-\left(\frac{\omega_{r}+\Delta\omega}{\omega_{r}}\right)^{2}}{1-\left(\frac{\omega_{r}+\Delta\omega}{\omega_{r}}\right)^{2}+\frac{1}{2}i\frac{\omega_{r}+\Delta\omega}{\omega_{r}}\beta}|=10^{\frac{\mathrm{TL}_{\mathrm{min}}}{20}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $|\tau(\omega_{r}+\Delta\omega)|=|\frac{1-\left(\frac{\omega_{r}+\Delta\omega}{\omega_{r}}\right)^{2}}{1-\left(\frac{\omega_{r}+\Delta\omega}{\omega_{r}}\right)^{2}+\frac{1}{2}i\frac{\omega_{r}+\Delta\omega}{\omega_{r}}\beta}|=10^{\frac{\mathrm{TL}_{\mathrm{min}}}{20}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\beta=\frac{\Delta\omega}{\omega_{r}}4\sqrt{10^{^{\frac{\mathrm{TL_{\mathrm{min}}}}{10}}}-1}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The peak half width is the distance over which the transmission loss has
|
||
dropped 3
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
dB w.r.t.
|
||
the transmission loss at the resonance frequency.
|
||
This is an important design parameter.
|
||
We can compute it by setting:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
|\frac{\tau|_{\omega_{r}+\Delta\omega}}{\tau|_{\omega_{r}}}|=\sqrt{2},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
So given the -3
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
dB point, and the maximum required transmission loss, we can compute
|
||
\begin_inset Formula $\zeta$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\beta$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Eq 1:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\begin_inset Formula $\frac{\alpha_{-3\mathrm{dB}}-1}{\zeta}=\sqrt{2}\Rightarrow\zeta=\frac{\alpha_{-3\mathrm{dB}}-1}{\sqrt{2}}$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
Eq 2:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
\begin_inset Formula $\mathrm{TL}_{\mathrm{max}}=20\log\left(\frac{\beta+4\zeta}{4\zeta}\right)\Rightarrow\frac{\beta+4\zeta}{4\zeta}=10^{\frac{\mathrm{TL}_{\mathrm{max}}}{20}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\beta=4\zeta\left(10^{\frac{\mathrm{TL}_{\mathrm{max}}}{20}}-1\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $\zeta=\frac{\alpha_{-3\mathrm{dB}}-1}{\sqrt{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $\beta=4\zeta\left(10^{\frac{\mathrm{TL}_{\mathrm{max}}}{20}}-1\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Required volume in terms of resonator strength:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
V=\frac{Sc_{0}\beta}{\omega_{r}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Insertion loss
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For computation of the insertion loss, we require two more parameters:
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
The load impedance at the downstream end of the silencer
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
The output impedance of the source (
|
||
\begin_inset Formula $Z_{\mathrm{rad}}$
|
||
\end_inset
|
||
|
||
)
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Suppose the source strength is defined by
|
||
\begin_inset Formula $\mathcal{S}$
|
||
\end_inset
|
||
|
||
.
|
||
Situation without silencer:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
U_{L} & =\mathcal{S}/\left(Z_{s}+Z_{l}\right),\\
|
||
U_{R} & =U_{L},\\
|
||
p_{R} & =Z_{\mathrm{rad}}U_{R},
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $Z_{s}$
|
||
\end_inset
|
||
|
||
denotes the source output impedance, and
|
||
\begin_inset Formula $Z_{l}$
|
||
\end_inset
|
||
|
||
denotes the load impedance as felt by the source.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For the reference case, the load impedance equals the radiation impedance,
|
||
and the radiated acoustic power is:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[p_{R}U_{R}^{*}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[Z_{\mathrm{rad}}\left(\mathcal{S}/Z_{s}\right)\left(\mathcal{S}/Z_{s}\right)^{*}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}|\mathcal{S}/Z_{s}|^{2}\Re\left[Z_{\mathrm{rad}}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
P_{\mathrm{ref}}=\frac{1}{2}\frac{|\mathcal{S}|^{2}}{|Z_{\mathrm{rad}}+Z_{s}|^{2}}\Re\left[Z_{\mathrm{rad}}\right]
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Now, situation including silencer, with in general, transfer matrix
|
||
\begin_inset Formula $\boldsymbol{T}$
|
||
\end_inset
|
||
|
||
.
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[p_{R}U_{R}^{*}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[Z_{\mathrm{rad}}U_{R}U_{R}^{*}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Using:
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
P_{\mathrm{with\,silencer}}=\frac{1}{4}|\mathcal{S}|^{2}\frac{\Re\left[Z_{\mathrm{rad}}\right]}{|T_{22}Z_{\mathrm{rad}}-T_{12}+Z_{s}\left(T_{11}-T_{21}Z_{\mathrm{rad}}\right)|^{2}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
From that, computing the power ratio, that
|
||
\begin_inset Formula $\det\boldsymbol{T}\equiv1$
|
||
\end_inset
|
||
|
||
for a reciprocal system:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
R_{P}=\frac{P_{\mathrm{with\,silencer}}}{P_{\mathrm{ref}}}=\frac{|Z_{\mathrm{rad}}+Z_{s}|^{2}}{|T_{22}Z_{\mathrm{rad}}-T_{12}+Z_{s}\left(T_{11}-T_{21}Z_{\mathrm{rad}}\right)|^{2}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Insertion loss for a Helmholtz side branch resonator
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Filling in for a simple Helmholtz side branch resonator:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
R_{P,\mathrm{Helmholtz}}=\frac{|Z_{\mathrm{rad}}+Z_{s}|^{2}}{|Z_{\mathrm{rad}}+Z_{s}\left(1+\frac{Z_{\mathrm{rad}}}{Z_{h}}\right)|^{2}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Comparing this to the transmission loss curve:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
|\tau|_{\mathrm{Helmholtz}}^{2}=\frac{4|Z_{h}|^{2}}{|2Z_{h}+Z_{0}|^{2}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
High output impedance limit
|
||
\begin_inset Formula $(Z_{s}\gg Z_{\mathrm{rad}})$
|
||
\end_inset
|
||
|
||
, volume flow source
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
R_{P,\mathrm{Helmholtz}}=\frac{|Z_{h}|^{2}}{|Z_{h}+Z_{\mathrm{rad}}|^{2}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Low output impedance limit
|
||
\begin_inset Formula $(Z_{s}\ll Z_{\mathrm{rad}})$
|
||
\end_inset
|
||
|
||
, pressure source
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
R_{P,\mathrm{Helmholtz}}=\frac{|Z_{h}|^{2}}{|Z_{h}+Z_{s}|^{2}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Special case: barrier in an infinite space
|
||
\begin_inset Formula $(Z_{s}=Z_{\mathrm{rad}})$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
R_{P,\mathrm{Helmholtz}}=\frac{|Z_{h}|^{2}}{|Z_{h}+\frac{1}{2}Z_{\mathrm{rad}}|^{2}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Comparing limits to power transmission ratio
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
|\tau|^{2}=\frac{|Z_{h}|^{2}}{|Z_{h}+\frac{1}{2}Z_{0}|^{2}},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
So the transmission loss is the reduction in transmitted sound power for
|
||
the situation where the source output impedance equals the radiation impedance
|
||
on the other side of the silencer.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Multiple Helmholtz resonators at a single inlet
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In case multiple resonators are connected to the same inlet, the parallel
|
||
impedance can be computed by computing the equivalent parallel impedance:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{1}{Z_{h,\mathrm{tot}}}=\frac{1}{Z_{h,1}}+\frac{1}{Z_{h,2}}+\dots
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Transmission of the duct
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left\{ \begin{array}{c}
|
||
p_{R}\\
|
||
U_{R}
|
||
\end{array}\right\} =\left[\begin{array}{cc}
|
||
\cos\left(kL\right) & -iZ_{0}\sin\left(kL\right)\\
|
||
-iZ_{0}^{-1}\sin\left(kL\right) & \cos\left(kL\right)
|
||
\end{array}\right]\left\{ \begin{array}{c}
|
||
p_{L}\\
|
||
U_{L}
|
||
\end{array}\right\}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
3D (FEM) Models
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Apply equation of state:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{align*}
|
||
i\omega\rho+\rho_{0}\nabla\cdot\boldsymbol{u} & =0\\
|
||
i\omega\rho_{0}\boldsymbol{u} & =-\nabla p+\mu_{0}\nabla^{2}\boldsymbol{u}+\left(\frac{1}{3}\mu+\zeta\right)\nabla\left(\nabla\cdot\boldsymbol{u}\right)\\
|
||
i\omega\rho_{0}c_{p}T & =i\omega p+\kappa\nabla^{2}T\\
|
||
\frac{\rho}{\rho_{0}} & =\frac{p}{p_{0}}-\frac{T}{T_{0}}
|
||
\end{align*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Solving for
|
||
\begin_inset Formula $i\omega\rho_{0}c_{p}T=i\omega p+\kappa\nabla^{2}T$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
:
|
||
\begin_inset Formula $T=\frac{1}{\rho_{0}c_{p}}\left(1-h_{\kappa}\right)p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Where
|
||
\begin_inset Formula $\frac{i\delta_{\kappa}^{2}}{2}\nabla^{2}h_{\kappa}+h_{\kappa}=0$
|
||
\end_inset
|
||
|
||
and
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Same for velocity, negliging
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
bulk
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
viscosity terms:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $i\omega\rho_{0}\boldsymbol{u}=-\nabla p+\mu_{0}\nabla^{2}\boldsymbol{u}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
More or less solution:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\boldsymbol{u}=\frac{i}{\rho_{0}\omega}\left(1-h_{\nu}\right)\nabla p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Where
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{2i}{\delta_{\nu}^{2}}\nabla^{2}h_{\nu}+h_{\nu}=0$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $h_{\nu}|_{\mathrm{wall}}=1$
|
||
\end_inset
|
||
|
||
for a no-slip b.c.
|
||
and 0 for a slip b.c.
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Filling in the expression for eq of state,
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\rho=\frac{1}{c_{0}^{2}}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Substituting that one, for
|
||
\begin_inset Formula $\rho$
|
||
\end_inset
|
||
|
||
in continuity eq:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $i\omega\frac{1}{c_{0}^{2}}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p+\rho_{0}\nabla\cdot\boldsymbol{u}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\rho_{0}\nabla\cdot\boldsymbol{u}+i\frac{k}{c_{0}}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Fill in for momentum:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\rho_{0}\nabla\cdot\left(\frac{i}{\rho_{0}\omega}\left(1-h_{\nu}\right)\nabla p\right)+i\frac{k}{c_{0}}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\nabla\cdot\left(\left(1-h_{\nu}\right)\nabla p\right)+k^{2}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Multiplying with weight factor, applying greens theorem:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\int_{V}p_{w}k\left(1+\left(\gamma-1\right)h_{\kappa}\right)p-iz_{0}\nabla\cdot\boldsymbol{u}p_{w}\mathrm{d}V=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\int_{V}p_{w}k\left(1+\left(\gamma-1\right)h_{\kappa}\right)p+iz_{0}\nabla p_{w}\cdot\boldsymbol{u}\mathrm{d}V=iz_{0}\oint_{S}p_{w}\boldsymbol{u}\cdot\boldsymbol{n}\mathrm{d}S$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\int_{V}p_{w}k\left(1+\left(\gamma-1\right)h_{\kappa}\right)p+iz_{0}\nabla p_{w}\cdot\boldsymbol{u}\mathrm{d}V=iz_{0}\oint_{S}p_{w}\boldsymbol{u}\cdot\boldsymbol{n}\mathrm{d}S$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Filling in
|
||
\begin_inset Formula $\boldsymbol{u}$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{i}{\rho_{0}\omega}\nabla p\left(1-\psi_{v}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\int_{V}p_{w}k^{2}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p-\left(1-\psi_{v}\right)\nabla p_{w}\cdot\nabla p\mathrm{d}V=ikz_{0}\oint_{S}p_{w}\boldsymbol{u}\cdot\boldsymbol{n}\mathrm{d}S$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Axially symmetric:
|
||
\begin_inset Formula $\int_{z}\int_{r=0}^{a}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\int_{z}\int_{r=0}^{a}\left(p_{w}k^{2}\left(1+\left(\gamma-1\right)\psi_{T}\right)p-\left(1-\psi_{v}\right)\nabla p_{w}\cdot\nabla p\right)2\pi r\mathrm{d}r\mathrm{d}z=ikz_{0}\oint_{S}p_{w}\boldsymbol{u}\cdot\boldsymbol{n}\mathrm{d}S$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
—– Which
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $i\omega\frac{1}{c_{0}^{2}}p\left(1+\left(\gamma-1\right)h_{\kappa}\right)+\rho_{0}\nabla\cdot\left(\frac{i}{\rho_{0}\omega}\nabla p\left(1-\psi_{v}\right)\right)=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $k^{2}p\left(1+\left(\gamma-1\right)h_{\kappa}\right)+\rho_{0}\nabla\cdot\left(\nabla p\left(1-\psi_{v}\right)\right)=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
From
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
SLNS model
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\nabla^{2}h_{v}+\frac{2}{i\delta_{\nu}^{2}}h_{v} & =0,\\
|
||
\nabla^{2}h_{\kappa}+\frac{2}{i\delta_{\kappa}^{2}}h_{\kappa} & =0,\\
|
||
\frac{1}{k}\nabla\cdot\left(\left(1-h_{\nu}\right)\nabla p\right)+k\left(1+\left(\gamma-1\right)h_{\kappa}\right)p & =0\label{eq:slns}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The velocity is:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\boldsymbol{u}=\frac{i}{\rho_{0}\omega}\left(1-h_{\nu}\right)\nabla p
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Comsol writes for the effective density:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left(-\frac{1}{\rho_{c}}\nabla p\right)=i\omega\boldsymbol{u},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
such that
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{1}{\rho_{c}}=\frac{1-h_{\nu}}{\rho_{0}},$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\rho_{c}=\frac{\rho_{0}}{1-h_{\nu}},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
And:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\nabla\cdot\left(-\frac{1}{\rho_{c}}\nabla p_{t}\right)-\frac{\omega^{2}}{c^{2}\rho_{c}}p=Q_{m},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Filling in:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\nabla\cdot\left(-\frac{1}{\rho_{c}}\nabla p_{t}\right)-\frac{\omega^{2}}{c^{2}\rho_{c}}p=Q_{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\nabla\cdot\left(-\frac{\left(1-h_{\nu}\right)}{\rho_{m}}\nabla p\right)-\frac{k^{2}}{\rho_{m}}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Makes:
|
||
\begin_inset Formula $c^{2}\rho_{c}=\frac{c_{m}^{2}\rho_{m}}{1+\left(\gamma-1\right)h_{\kappa}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $c^{2}=\frac{c_{m}^{2}\left(1-h_{\nu}\right)}{1+\left(\gamma-1\right)h_{\kappa}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
c^{2}=\frac{c_{m}^{2}\left(1-h_{\nu}\right)}{1+\left(\gamma-1\right)h_{\kappa}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
With boundary conditions at isothermal no-slip wall:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
h_{\nu} & =1\qquad\mathrm{at\,the\,wall}\\
|
||
h_{\kappa} & =1\qquad\mathrm{at\,the\,wall}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Symmetry / inlet outlet:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
h_{\nu}=h_{\kappa}=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For pressure / velocity b.c.'s
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\boldsymbol{u}=\frac{i}{\rho_{0}\omega}\left(1-h_{\nu}\right)\nabla p
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Combine with pressure acoustics:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Weak form:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
(-acpr.gradpx*acpr.gradtestpx-acpr.gradpy*acpr.gradtestpy-acpr.gradpz*acpr.gradtestpz-
|
||
acpr.p_t*test(pac)*acpr.ik^2)*acpr.delta/acpr.rho_c
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
(-acpr.gradpx*acpr.gradtestpx-acpr.gradpy*acpr.gradtestpy-acpr.gradpz*acpr.gradtestpz-
|
||
acpr.p_t*test(pac)*acpr.ik^2)*acpr.delta/acpr.rho_c
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\int_{V}\left[-\nabla p_{t}\cdot\nabla p-p_{t}p\left(ik\right)\right]\frac{\delta}{\rho_{0}c_{0}}\mathrm{dV}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Weak form of SLNS:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\int_{V}p_{t}\left[\nabla\cdot\left(\left(1-h_{\nu}\right)\nabla p\right)+k^{2}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p\right]\frac{\delta}{\rho_{0}c_{0}}\mathrm{d}V$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\int_{V}\left[-\nabla p_{t}\cdot\left(\left(1-h_{\nu}\right)\nabla p\right)+p_{t}k^{2}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p\right]\frac{\delta}{\rho_{0}c_{0}}\mathrm{d}V$
|
||
\end_inset
|
||
|
||
+Boundary term.
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\int_{V}\left[\underbrace{-\nabla p_{t}\cdot\nabla p-p_{t}\left(ik\right)^{2}p}_{\mathrm{already\,there}}+\nabla p_{t}\cdot\left(h_{\nu}\nabla p\right)-p_{t}\left(ik\right)^{2}p\left(\left(\gamma-1\right)h_{\kappa}\right)\right]\frac{\delta}{\rho_{0}c_{0}}\mathrm{d}V$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Makes the weak contribution equal to:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\int_{V}\left[\nabla p_{t}\cdot\left(h_{\nu}\nabla p\right)+p_{t}\left(ik\right)^{2}p\left(\left(1-\gamma\right)h_{\kappa}\right)\right]\frac{\delta}{\rho_{0}c_{0}}\mathrm{d}V$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Written out:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
(hnu*(test(px)*px+test(py)*py+pz*test(pz))+test(p)*p*acpr.ik^2*(1-gamma)*hkappa)*
|
||
acpr.delta/acpr.rho_c
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
DEPRECATED, we doen het met de pressure acoustics interface en een enkele
|
||
weak contribution!
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Comsol implementation - General Form PDE
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Model in Comsol:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
e_{a}\frac{\partial^{2}p}{\partial t^{2}}+d_{a}\frac{\partial p}{\partial t}+\nabla\cdot\boldsymbol{\Gamma}=f
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Comparing with Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:slns"
|
||
|
||
\end_inset
|
||
|
||
results in:
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\boldsymbol{\Gamma} & =\frac{1}{k}\left(1-h_{\nu}\right)\nabla p\\
|
||
f & =-k\left(1+\left(\gamma-1\right)h_{\kappa}\right)p
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Comsol implementation - prescribed velocity
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Flux / source term form in Comsol:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
-\boldsymbol{n}\cdot\boldsymbol{\Gamma}=g-qp
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
From the mathematics, we find:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $k\boldsymbol{\Gamma}=\left(1-h_{\nu}\right)\nabla p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\boldsymbol{u}=\frac{i}{\rho_{0}\omega}\left(1-h_{\nu}\right)\nabla p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
– Combine:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\rho_{0}\omega}{i}\boldsymbol{u}=\left(1-h_{\nu}\right)\nabla p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
–
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\boldsymbol{\Gamma}=-iz_{0}\boldsymbol{u}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
-\boldsymbol{n}\cdot\boldsymbol{\Gamma}=iz_{0}\boldsymbol{u}\cdot\boldsymbol{n}\label{eq:Gam_vs_un}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Such that:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\boldsymbol{u}\cdot\boldsymbol{n}=\frac{i}{\rho_{0}\omega}\left(1-h_{\nu}\right)\nabla p\cdot\boldsymbol{n}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Note that:
|
||
\begin_inset Formula
|
||
\[
|
||
k\boldsymbol{\Gamma}=\left(1-h_{\nu}\right)\nabla p
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Fill in:
|
||
\begin_inset Formula $iz_{0}\boldsymbol{u}\cdot\boldsymbol{n}=-\boldsymbol{\Gamma}\cdot\boldsymbol{n}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
q & =0\\
|
||
g & =iz_{0}\boldsymbol{u}\cdot\boldsymbol{n}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Moreover, at such a boundary, we need to set
|
||
\begin_inset Formula $h_{\nu}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $h_{\kappa}$
|
||
\end_inset
|
||
|
||
to 0.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Normal impedance b.c.
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
We set
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
z\boldsymbol{u}\cdot\boldsymbol{n}=p
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Upon using Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:Gam_vs_un"
|
||
|
||
\end_inset
|
||
|
||
, we find:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Yields:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{i}{z_{0}}\boldsymbol{n}\cdot\boldsymbol{\Gamma}=\boldsymbol{u}\cdot\boldsymbol{n}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
-\boldsymbol{n}\cdot\boldsymbol{\Gamma}=-i\frac{z_{0}}{z}p
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Such that:
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
q & =i\frac{z_{0}}{z}\\
|
||
g & =0
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Interior impedance jump
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Equation:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{\mathrm{up}}-p_{\mathrm{down}}=z\boldsymbol{u}\cdot\boldsymbol{n}_{\mathrm{up}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
It should be implemented as a
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
weak contribution
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
.
|
||
For that we refer the the weak form equation:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Reverse engineering comsols weak contribution of such a split:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
-acpr.delta*acpr.iomega*(down(acpr.p_t)-up(acpr.p_t))*(down(test(acp))-up(test(acp))
|
||
)/acpr.Zi
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
waar: delta = 1/omega^2
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Leest:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
-i/omega*(down(p)-up(p))*(down(test(p))-up(test(p))) /z
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
We hebben altijd op een rand:
|
||
\begin_inset Formula
|
||
\[
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
We can write this as a weak contribution:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Weak contribution in pressure acoustics interface:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
|
||
\family typewriter
|
||
(hnu*(test(px)*px+test(py)*py+pz*test(pz))+test(p)*p*acpr.ik^2*(1-gamma)*hkappa)*
|
||
acpr.delta/acpr.rho_c
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Or we could write this with a custom density and speed of sound <— TODO!
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
2D Axisymmetric:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
|
||
\family typewriter
|
||
(hnu*(test(pr)*pr+pz*test(pz))+test(p)*p*acpr.ik^2*(1-gamma)*hkappa)*acpr.delta/ac
|
||
pr.rho_c
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset bibtex
|
||
LatexCommand bibtex
|
||
btprint "btPrintCited"
|
||
bibfiles "lrftubes"
|
||
options "plain"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
\start_of_appendix
|
||
Thermal relaxation in thick tubes
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "subsec:Thermal-relaxation-effect"
|
||
|
||
\end_inset
|
||
|
||
Thermal relaxation effect in thick tubes
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/prsduct_thermal_relax.pdf
|
||
width 80text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Schematic situation of a tube surrounded by a thick solid.
|
||
Note that the transverse acoustic temperature is drawn to be not zero at
|
||
the wall.
|
||
This happens in case of thermal interaction with a solid with finite thermal
|
||
effusivity.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:prsduct-heatwave-solid"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In this section, a formulation for
|
||
\begin_inset Formula $\epsilon_{s}$
|
||
\end_inset
|
||
|
||
is given for tubes where the temperature wave in the solid is present.
|
||
Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:prsduct-heatwave-solid"
|
||
|
||
\end_inset
|
||
|
||
shows a schematic overview of the situation.
|
||
As shown in the figure, the temperature wave accompanied with an acoustic
|
||
wave results in heat conduction to/from the wall of the tube.
|
||
To solve this interaction mathematically, the heat equation in the solid
|
||
has to be solved.
|
||
For constant thermal conductivity, density and heat capacity the heat equation
|
||
of the solid is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\rho_{s}c_{s}\frac{\partial\tilde{T}_{s}}{\partial t}=\kappa_{s}\nabla^{2}\tilde{T}_{s},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\rho_{s},c_{s},\tilde{T}_{s}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\kappa_{s}$
|
||
\end_inset
|
||
|
||
are the density, specific heat, temperature and thermal conductivity of
|
||
the solid, respectively.
|
||
In frequency domain and using cylindrical coordinates, assuming axial symmetry,
|
||
this can be written as
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$r$"
|
||
description "Radial position in cylindrical coordinates\\nomunit{\\si{\\m}}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left(r^{2}\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{\partial^{2}}{\partial x^{2}}\right)+r\frac{\partial}{\partial r}+\frac{2}{i\delta_{s}^{2}}r^{2}\right)T_{s}=0,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\delta_{s}$
|
||
\end_inset
|
||
|
||
is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\delta_{s}=\sqrt{\frac{2\kappa_{s}}{\rho_{s}c_{s}\omega}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Now, since
|
||
\begin_inset Formula $\partial T_{s}/\partial x\sim\frac{\delta_{s}}{\lambda}\frac{\partial T_{s}}{\partial r}$
|
||
\end_inset
|
||
|
||
, the second order derivative of the temperature in the axial direction
|
||
can be neglected.
|
||
In that case, the differential equation to solve for is
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\rho_{s}c_{s}i\omega T_{s}=\kappa_{s}\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\right)T_{s}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $-\kappa_{s}\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\right)T_{s}+\rho_{s}c_{s}i\omega T_{s}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\right)T_{s}+2\frac{\rho_{s}c_{s}\omega}{2\kappa_{s}i}T_{s}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\delta_{s}^{2}=\frac{2\kappa_{s}}{\rho_{s}c_{s}\omega}$
|
||
\end_inset
|
||
|
||
<<< subst
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\right)T_{s}+\frac{2}{i\delta_{s}^{2}}T_{s}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Multiply with
|
||
\begin_inset Formula $r^{2}$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(r^{2}\frac{\partial^{2}}{\partial r^{2}}+r\frac{\partial}{\partial r}+\frac{2}{i\delta_{s}^{2}}r^{2}\right)T_{s}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Say:
|
||
\begin_inset Formula $\xi^{2}=\frac{2}{i\delta_{s}^{2}}r^{2}\Rightarrow$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Then:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\partial^{2}}{\partial r^{2}}=$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left(r^{2}\frac{\partial^{2}}{\partial r^{2}}+r\frac{\partial}{\partial r}+\frac{2}{i\delta_{s}^{2}}r^{2}\right)T_{s}=0,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
which is a Bessel differential equation of the zero'th order in
|
||
\begin_inset Formula $T_{s}$
|
||
\end_inset
|
||
|
||
.
|
||
The solutions is sought in terms of traveling cylindrical waves:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\sqrt{\frac{2}{i}}=\sqrt{-2i}=\pm\left(i-1\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
T_{s}=C_{1}H_{0}^{(1)}\left(\left(i-1\right)\frac{r}{\delta_{s}}\right)+C_{2}H_{0}^{(2)}\left(\left(i-1\right)\frac{r}{\delta_{s}}\right),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $C_{1}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $C_{2}$
|
||
\end_inset
|
||
|
||
constants to be determined from the boundary conditions, and
|
||
\begin_inset Formula $H_{\alpha}^{(i)}$
|
||
\end_inset
|
||
|
||
is the cylindrical Hankel function of the
|
||
\begin_inset Formula $(i)^{\mathrm{th}}$
|
||
\end_inset
|
||
|
||
kind and order
|
||
\begin_inset Formula $\alpha$
|
||
\end_inset
|
||
|
||
.
|
||
If we require
|
||
\begin_inset Formula $T_{s}\to0$
|
||
\end_inset
|
||
|
||
as
|
||
\begin_inset Formula $r\to\infty$
|
||
\end_inset
|
||
|
||
, the constant
|
||
\begin_inset Formula $C_{2}$
|
||
\end_inset
|
||
|
||
is required to be
|
||
\begin_inset Formula $0$
|
||
\end_inset
|
||
|
||
.
|
||
From the acoustic energy equation, a similar differential equation can
|
||
be found for the acoustic temperature in the fluid:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\rho_{0}c_{p}i\omega T=i\omega\alpha_{P}T_{0}p+\kappa\nabla^{2}T$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(\nabla^{2}-2\frac{\omega\rho_{0}c_{p}}{2\kappa}i\right)T=-\frac{1}{\kappa}i\omega\alpha_{P}T_{0}p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left(\nabla^{2}+\frac{2}{i\delta_{\kappa}^{2}}\right)T=\frac{2}{i\delta_{s}^{2}}\frac{\alpha_{P}T_{0}}{\rho_{0}c_{p}}p$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\[
|
||
\left(r^{2}\frac{\partial^{2}}{\partial r^{2}}+r\frac{\partial}{\partial r}+\frac{2}{i\delta_{s}^{2}}r^{2}\right)T=\frac{2}{i\delta_{s}^{2}}\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p,
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
for which the (partial) solution is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
T=\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\left(1-C_{3}J_{0}\left(\left(i-1\right)\frac{r}{\delta_{\kappa}}\right)\right).\label{eq:temp_partial_sol}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
To attain at Eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:temp_partial_sol"
|
||
|
||
\end_inset
|
||
|
||
, use has been made of the fact that the temperature should be finite at
|
||
|
||
\begin_inset Formula $r=0$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\begin_inset Formula $C_{3}$
|
||
\end_inset
|
||
|
||
is a constant that is to be determined from the boundary conditions at
|
||
the solid-fluid interface.
|
||
These boundary conditions are:
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
T_{s}|_{r=a} & =T|_{r=a},\\
|
||
-\kappa_{s}\frac{\partial T_{s}}{\partial r}|_{r=a} & =-\kappa\frac{\partial T}{\partial r}|_{r=a},
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
i.e.
|
||
continuity of the temperature and the heat flux at the interface.
|
||
This yields two equations for two unknowns (
|
||
\begin_inset Formula $C_{1}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $C_{3}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $C_{2}$
|
||
\end_inset
|
||
|
||
is already argued to be
|
||
\begin_inset Formula $0$
|
||
\end_inset
|
||
|
||
).
|
||
Solving for the acoustic temperature yields:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $T|_{r=a}=T_{s}|_{r=a}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
–
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $C_{1}H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)=\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\left(1-C_{3}J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)\right)\Rightarrow C_{1}=\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\frac{\left(1-C_{3}J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)\right)}{H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}$
|
||
\end_inset
|
||
|
||
(1)
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Derivative b.c.
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
–
|
||
\begin_inset Formula $-\frac{\partial T}{\partial r}|_{r=a}=-\frac{\kappa_{s}}{\kappa}\frac{\partial T_{s}}{\partial r}|_{r=a}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
where
|
||
\begin_inset Formula $-\frac{\partial T}{\partial r}|_{r=a}=\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\frac{\left(i-1\right)}{\delta_{\kappa}}C_{3}J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
using
|
||
\begin_inset Formula $\frac{\partial H_{0}^{(1)}(z)}{\partial z}=-H_{1}^{(1)}(z)$
|
||
\end_inset
|
||
|
||
==>
|
||
\begin_inset Formula $-\frac{\kappa}{\kappa_{s}}\frac{\partial T_{s}}{\partial r}|_{r=a}=\frac{\kappa}{\kappa_{s}}C_{1}\frac{\left(i-1\right)}{\delta_{s}}H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Such that:
|
||
\begin_inset Formula $\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\frac{\left(i-1\right)}{\delta_{\kappa}}C_{3}J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)=\frac{\kappa_{s}}{\kappa}C_{1}\frac{\left(i-1\right)}{\delta_{s}}H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Filling in
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\frac{\left(i-1\right)}{\delta_{\kappa}}C_{3}J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)=\frac{\kappa_{s}}{\kappa}\left(\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\frac{\left(1-C_{3}J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)\right)}{H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}\right)\frac{\left(i-1\right)}{\delta_{s}}H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Solving for
|
||
\begin_inset Formula $C_{3}$
|
||
\end_inset
|
||
|
||
gives:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $C_{3}=\frac{1}{\left[\frac{J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}{\frac{\kappa_{s}}{\kappa}\frac{\delta_{\kappa}}{\delta_{s}}H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}+J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)\right]}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
or:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $C_{3}=\frac{1}{\left[\left(1+\epsilon_{s}\right)J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)\right]}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\epsilon_{s}=\frac{\kappa\delta_{s}}{\delta_{\kappa}\kappa_{s}}\frac{H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)}{H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\kappa\delta_{s}}{\delta_{\kappa}\kappa_{s}}=\sqrt{\frac{\kappa^{2}\delta_{s}^{2}}{\kappa_{s}^{2}\delta_{\kappa}^{2}}}=\sqrt{\frac{\kappa\rho_{0}c_{p}}{\kappa\rho_{s}c_{s}}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\[
|
||
T=\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}\left(1-\frac{1}{\left(1+\epsilon_{s}\right)}\frac{J_{0}\left(\left(i-1\right)\frac{r}{\delta_{\kappa}}\right)}{J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)}\right)p,
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\epsilon_{s}=\frac{e_{f}}{e_{s}}\frac{J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}{J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
-
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
-Asymptotic form of the Hankel function for large argument, and
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $-\pi<\arg(z)<2\pi$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $H_{\alpha}^{(1)}(z)\sim\sqrt{\frac{2}{\pi z}}e^{i\left(z-\pi\frac{1+2\alpha}{4}\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
And for
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $J_{\alpha}(z)\sim\sqrt{\frac{2}{\pi z}}\cos\left(z-\pi\frac{1+2\alpha}{4}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Filling this in into
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{e_{f}}{e_{s}}\cdot-ii=\frac{e_{f}}{e_{s}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $e_{f}$
|
||
\end_inset
|
||
|
||
is the thermal effusivity
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset nomenclature
|
||
LatexCommand nomenclature
|
||
prefix "A"
|
||
symbol "$e$"
|
||
description "Thermal effusivity\\nomunit{\\si{\\joule\\per\\square\\metre\\kelvin\\second\\tothe{ \\frac{1}{2} } }}"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
of the fluid, and
|
||
\begin_inset Formula $e_{s}$
|
||
\end_inset
|
||
|
||
the thermal effusivity of the solid, such that the ratio is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{e_{f}}{e_{s}}=\sqrt{\frac{\kappa\rho_{0}c_{p}}{\kappa_{s}\rho_{s}c_{s}}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Note that for large
|
||
\begin_inset Formula $a/\delta_{\kappa}$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)}{J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)}\to i,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and for large
|
||
\begin_inset Formula $a/\delta_{s}$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}{H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}\to-i,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
such that for both numbers large
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\epsilon_{s}\to\frac{e_{f}}{e_{s}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Derivation of Karal's discontinuity factor
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "chap:Derivation-of-Karal's"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
|
||
\series bold
|
||
Note: this documentation is incomplete.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename img/discontinuity_appendix.pdf
|
||
width 60text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Schematic of a discontinuity at the interface between two tubes with different
|
||
radius.
|
||
Domain B is the smaller tube and domain C is the larger tube.
|
||
The radius of the tube in domain B is
|
||
\begin_inset Formula $b$
|
||
\end_inset
|
||
|
||
, and the radius of the tube in domain C is
|
||
\begin_inset Formula $c$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:karal-1"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
This appendix describes the derivation of Karal's discontinuity factor.
|
||
The following assumptions underlie the model:
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $z=0$
|
||
\end_inset
|
||
|
||
: position of the discontinuity
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Assume
|
||
\begin_inset Formula $f\ll f_{c}$
|
||
\end_inset
|
||
|
||
, such that far away from the discontinuity, only propagating modes exist.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Assume axial symmetry, so dependence of
|
||
\begin_inset Formula $\theta$
|
||
\end_inset
|
||
|
||
is dropped
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In cylindrical coordinates, the solution of the Helmholtz equation can be
|
||
written in terms of cylindrical harmonics
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "blackstock_fundamentals_2000"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Assuming axial symmetrySuch that the acoustic pressure in for example tube
|
||
|
||
\begin_inset Formula $B$
|
||
\end_inset
|
||
|
||
can be written as:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{B}=\left\{ \begin{array}{c}
|
||
J_{m}\left(k_{r}r\right)\\
|
||
N_{m}\left(k_{r}r\right)
|
||
\end{array}\right\} \left\{ \begin{array}{c}
|
||
e^{im\phi}\\
|
||
e^{-im\phi}
|
||
\end{array}\right\} \left\{ \begin{array}{c}
|
||
e^{\beta z}\\
|
||
e^{-\beta z}
|
||
\end{array}\right\}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $J_{m}$
|
||
\end_inset
|
||
|
||
is the cylindrical Bessel function of order
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
k_{r}^{2}-\beta^{2}=k^{2}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Using the boundary condition that
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial p_{B}}{\partial r}|_{r=b}=0,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and assuming axial symmetry (only the
|
||
\begin_inset Formula $m=0$
|
||
\end_inset
|
||
|
||
modes) it follows that
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial J_{0}}{\partial r}\left(k_{r}b\right)|_{r=b}=0.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Assuming that
|
||
\begin_inset Formula $\alpha_{k}$
|
||
\end_inset
|
||
|
||
is the
|
||
\begin_inset Formula $k^{\mathrm{th}}$
|
||
\end_inset
|
||
|
||
zero of
|
||
\begin_inset Formula $J_{0}^{'}(x)$
|
||
\end_inset
|
||
|
||
, we can write for
|
||
\begin_inset Formula $k_{r,k}$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
k_{r,k}=\frac{\alpha_{k}}{b}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Hence we find the following reduced expression for the pressure in tube
|
||
|
||
\begin_inset Formula $B$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{B}=B_{0}^{0}\exp\left(ikz\right)+B_{0}^{1}\exp\left(-ikz\right)+\sum_{n=1}^{\infty}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)\left\{ \begin{array}{c}
|
||
e^{\beta_{n}z}\\
|
||
e^{-\beta_{n}z}
|
||
\end{array}\right\} ,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where accordingly,
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\beta_{k}^{2}=\left(\frac{\alpha_{k}}{b}\right)^{2}-k^{2}\label{eq:beta_k}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
For
|
||
\begin_inset Formula $k^{2}<\left(\alpha_{k}/b\right)^{2}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\beta_{k}^{2}>0$
|
||
\end_inset
|
||
|
||
, the modes are evanescent.
|
||
And since we only allow finite solutions for
|
||
\begin_inset Formula $z\leq0$
|
||
\end_inset
|
||
|
||
, the final results for
|
||
\begin_inset Formula $p_{B}$
|
||
\end_inset
|
||
|
||
is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{B}=B_{0}^{0}\exp\left(ikz\right)+B_{0}^{1}\exp\left(-ikz\right)+\sum_{n=1}^{\infty}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)e^{\beta_{n}z},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\beta_{n}$
|
||
\end_inset
|
||
|
||
is defined as the positive root of the r.h.s.
|
||
of Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:beta_k"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
We simplify this relation to:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{B}(z)=p_{B}^{0}(z)+\sum_{n=1}^{\infty}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)e^{\beta_{n}z}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
For the velocity we find
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $u=\frac{i}{\omega\rho_{0}}\frac{\partial p_{B}}{\partial z}=u_{B}^{0}(z)+\sum_{n=1}^{\infty}\frac{i\beta_{n}}{\omega\rho_{0}}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)e^{\beta_{n}z}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
u_{B}(z)=u_{B}^{0}(z)+\sum_{n=1}^{\infty}Y_{B,n}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)e^{\beta_{n}z},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Y_{B,n}=\frac{i\beta_{n}}{\omega\rho_{0}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Similarly, for the positive
|
||
\begin_inset Formula $z$
|
||
\end_inset
|
||
|
||
we find
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{C}(z)=P_{C}^{0}(z)+\sum_{m=1}^{\infty}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)e^{-\gamma_{m}z},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\gamma_{m}=\sqrt{\left(\frac{\alpha_{m}}{c}\right)^{2}-k^{2}}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
u_{C}(z)=u_{C}^{0}(z)+\sum_{m=1}^{\infty}Y_{C,m}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)e^{-\gamma_{m}z},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Y_{C,m}=-\frac{i\gamma_{m}}{\omega\rho_{0}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Boundary conditions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
At the interface (
|
||
\begin_inset Formula $z=0$
|
||
\end_inset
|
||
|
||
), the following boundary conditions are valid:
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
u_{B}|_{z=0} & =u_{C}|_{z=0} & 0\leq r\leq b\label{eq:derivative1bc}\\
|
||
u_{C}|_{z=0} & =0 & b\leq r\leq c\label{eq:derivative2bc}\\
|
||
p_{B} & =p_{C} & 0\leq r\leq b\label{eq:continuitybc}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
Taking Eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:derivative1bc"
|
||
|
||
\end_inset
|
||
|
||
, multiply by
|
||
\begin_inset Formula $r$
|
||
\end_inset
|
||
|
||
and integrating from
|
||
\begin_inset Formula $0$
|
||
\end_inset
|
||
|
||
to
|
||
\begin_inset Formula $c$
|
||
\end_inset
|
||
|
||
, taking into account Eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:derivative2bc"
|
||
|
||
\end_inset
|
||
|
||
yields:
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $u_{B}(z)=u_{B}^{0}(z)+\sum_{n=1}^{\infty}\zeta_{B,n}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)e^{\beta_{n}z}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Integrating from 0 to
|
||
\begin_inset Formula $b$
|
||
\end_inset
|
||
|
||
for
|
||
\begin_inset Formula $u_{B}$
|
||
\end_inset
|
||
|
||
and integrating from 0 to
|
||
\begin_inset Formula $c$
|
||
\end_inset
|
||
|
||
for
|
||
\begin_inset Formula $u_{C}$
|
||
\end_inset
|
||
|
||
cancels out the Bessel functions, as the primitive of
|
||
\begin_inset Formula $J_{0}(x)x$
|
||
\end_inset
|
||
|
||
is
|
||
\begin_inset Formula $J_{1}(x)x$
|
||
\end_inset
|
||
|
||
, for which due to the no-slip b.c.
|
||
the resulting integral is zero, and at
|
||
\begin_inset Formula $r=0$
|
||
\end_inset
|
||
|
||
, the integral is zero as well.
|
||
Hence we obtain only the propagating mode contribution to the volume flow.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
b^{2}u_{B}^{0}=c^{2}u_{C}^{0}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
We require one more equation at the interface, which is found from the continuit
|
||
y boundary conditions as well.
|
||
Multiplying Eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:derivative1bc"
|
||
|
||
\end_inset
|
||
|
||
with
|
||
\begin_inset Formula $J_{0}(\alpha_{q}\frac{r}{c})r$
|
||
\end_inset
|
||
|
||
and integrating setting
|
||
\begin_inset Formula $q=m$
|
||
\end_inset
|
||
|
||
and dividing by
|
||
\begin_inset Formula $bc$
|
||
\end_inset
|
||
|
||
yields:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $u_{B}=u_{B}^{0}+\sum_{n=1}^{\infty}\zeta_{B,n}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $u_{C}=u_{C}^{0}+\sum_{m=1}^{\infty}\zeta_{C,m}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
–
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int\limits _{0}^{b}u_{B}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int\limits _{0}^{c}u_{C}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
– Work out stuff, first line:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
- Using the rule:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
\int J_{0}(C_{1}x)J_{0}(C_{2}x)x\mathrm{d}x=x\frac{C_{1}J_{1}(C_{1}x)J_{0}(C_{2}x)-C_{2}J_{0}\left(C_{1}x\right)J_{1}(C_{2}x)}{C_{1}^{2}-C_{2}^{2}}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
–>
|
||
\begin_inset Formula $C_{1}=\frac{\alpha_{q}}{c}$
|
||
\end_inset
|
||
|
||
;
|
||
\begin_inset Formula $C_{2}=\frac{\alpha_{n}}{b}$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $x=b$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int\limits _{0}^{b}u_{B}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=u_{B}^{0}J_{1}(\alpha_{q}\frac{b}{c})\frac{bc}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}b\frac{\frac{\alpha_{q}}{c}J_{1}(\frac{\alpha_{q}}{c}b)J_{0}(\frac{\alpha_{n}}{b}b)-\frac{\alpha_{n}}{b}J_{0}\left(\frac{\alpha_{q}}{c}b\right)J_{1}(\frac{\alpha_{n}}{b}b)}{\left(\frac{\alpha_{q}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}=$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Using:
|
||
\begin_inset Formula $J_{1}\left(\alpha_{i}\right)=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int\limits _{0}^{b}u_{B}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=u_{B}^{0}J_{1}(\alpha_{q}\frac{b}{c})\frac{bc}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}\frac{b}{\left(\frac{\alpha_{q}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}\frac{\alpha_{q}}{c}J_{1}(\frac{\alpha_{q}}{c}b)J_{0}(\frac{\alpha_{n}}{b}b)=$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Using:
|
||
\begin_inset Formula $\rho=\frac{b}{c}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int\limits _{0}^{b}u_{B}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=u_{B}^{0}J_{1}(\alpha_{q}\frac{b}{c})\frac{bc}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}\frac{\alpha_{q}\rho}{\left(\frac{\alpha_{q}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}J_{1}(\frac{\alpha_{q}}{c}b)J_{0}(\frac{\alpha_{n}}{b}b)=$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Setting:
|
||
\begin_inset Formula $q=m$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int\limits _{0}^{b}u_{B}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{bc}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}\frac{\alpha_{m}\rho}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n})$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
———————————————————————
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
And the rhs:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int\limits _{0}^{c}u_{C}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=\int\limits _{0}^{c}\left[u_{C}^{0}J_{0}(\alpha_{q}\frac{r}{c})r+\sum_{m=1}^{\infty}Y_{C,m}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)J_{0}(\alpha_{q}\frac{r}{c})r\right]\mathrm{d}r$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int\limits _{0}^{c}u_{C}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=\int\limits _{0}^{c}\left[\sum_{m=1}^{\infty}Y_{C,m}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)J_{0}(\alpha_{q}\frac{r}{c})r\right]\mathrm{d}r$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Setting:
|
||
\begin_inset Formula $q=m$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int\limits _{0}^{c}u_{C}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=\int\limits _{0}^{c}\left[\sum_{m=1}^{\infty}Y_{C,m}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)J_{0}(\alpha_{m}\frac{r}{c})r\right]\mathrm{d}r$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Using the rule:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
\int J_{0}(C_{1}x)^{2}x\mathrm{d}x=\frac{1}{2}x^{2}\left(J_{0}(C_{1}x)^{2}+J_{1}(C_{1}x)^{2}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $C_{1}=\alpha_{m}\frac{r}{c}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $x=c$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int\limits _{0}^{c}u_{C}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=Y_{C,m}C_{m}\frac{1}{2}c^{2}\left(J_{0}(\alpha_{m}\frac{c}{c})^{2}+J_{1}(\alpha_{m}\frac{c}{c})^{2}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int\limits _{0}^{c}u_{C}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=Y_{C,m}C_{m}\frac{1}{2}c^{2}J_{0}(\alpha_{m})^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
— OR:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{bc}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}\frac{\alpha_{m}\rho}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n})=Y_{C,m}C_{m}\frac{1}{2}c^{2}J_{0}(\alpha_{m})^{2}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
– Divide by bc:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{1}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}\frac{\alpha_{m}\rho}{\left[\rho\alpha_{m}^{2}-\rho^{-1}\alpha_{n}^{2}\right]}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n})=Y_{C,m}C_{m}\frac{1}{2}\rho^{-1}J_{0}(\alpha_{m})^{2}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
- Deel teller en noemer in breuk door
|
||
\begin_inset Formula $\rho$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{1}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}\frac{\alpha_{m}}{\alpha_{m}^{2}-\frac{\alpha_{n}^{2}}{\rho^{2}}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n})=Y_{C,m}C_{m}\frac{1}{2}\rho^{-1}J_{0}(\alpha_{m})^{2}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{1}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}T_{mn}B_{n}=Y_{C,m}\frac{1}{2}\rho^{-1}J_{0}(\alpha_{m})^{2}C_{m},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
T_{mn}=\frac{\alpha_{m}}{\alpha_{m}^{2}-\frac{\alpha_{n}^{2}}{\rho^{2}}}J_{0}\left(\alpha_{n}\right)J_{1}\left(\alpha_{m}\rho\right).
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Setting
|
||
\begin_inset Formula $p_{B}=p_{C}$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int_{0}^{b}p_{B}(z=0)r\mathrm{d}r=\int_{0}^{b}\left[p_{B}^{0}+\sum_{n=1}^{\infty}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)\right]r\mathrm{d}r$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int_{0}^{b}p_{B}(z=0)r\mathrm{d}r=\frac{b^{2}}{2}p_{B}^{0}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
———————————————–
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)r\mathrm{d}r=\int_{0}^{b}\left[p_{C}^{0}+\sum_{m=1}^{\infty}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)\right]r\mathrm{d}r$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)r\mathrm{d}r=\frac{b^{2}}{2}p_{C}^{0}+\sum_{m=1}^{\infty}\frac{bc}{\alpha_{m}}C_{m}J_{1}\left(\alpha_{m}\rho\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Such that
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
\frac{b^{2}}{2}p_{B}^{0}=\frac{b^{2}}{2}p_{C}^{0}+\sum_{m=1}^{\infty}\frac{bc}{\alpha_{m}}C_{m}J_{1}\left(\alpha_{m}\rho\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Divide by
|
||
\begin_inset Formula $\frac{b^{2}}{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
p_{B}^{0}=p_{C}^{0}+2\sum_{m=1}^{\infty}\frac{J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}C_{m}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{B}^{0}=p_{C}^{0}+2\sum_{m=1}^{\infty}\frac{J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}C_{m}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int_{0}^{b}p_{B}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\int_{0}^{b}\left[p_{B}^{0}J_{0}\left(\alpha_{p}\frac{r}{b}\right)r+\sum_{n=1}^{\infty}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\right]\mathrm{d}r$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int_{0}^{b}p_{B}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\sum_{n=1}^{\infty}B_{n}\int_{0}^{b}J_{0}\left(\alpha_{n}\frac{r}{b}\right)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Setting
|
||
\begin_inset Formula $p=n$
|
||
\end_inset
|
||
|
||
en
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
\int J_{0}(C_{1}x)^{2}x\mathrm{d}x=\frac{1}{2}x^{2}\left(J_{0}(C_{1}x)^{2}+J_{1}(C_{1}x)^{2}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $C_{1}=\frac{\alpha_{n}}{b}$
|
||
\end_inset
|
||
|
||
en
|
||
\begin_inset Formula $x=b$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int_{0}^{b}p_{B}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=B_{n}\frac{1}{2}b^{2}J_{0}(\alpha_{n})^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
– Zelfde voor integraal voor
|
||
\begin_inset Formula $p_{C}$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\int_{0}^{b}\left[P_{C}^{0}+\sum_{m=1}^{\infty}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)\right]J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\sum_{m=1}^{\infty}C_{m}\int_{0}^{b}J_{0}\left(\alpha_{m}\frac{r}{c}\right)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Gebruik de regel:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
\int J_{0}(C_{1}x)J_{0}(C_{2}x)x\mathrm{d}x=x\frac{C_{1}J_{1}(C_{1}x)J_{0}(C_{2}x)-C_{2}J_{0}\left(C_{1}x\right)J_{1}(C_{2}x)}{C_{1}^{2}-C_{2}^{2}}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Waarbij:
|
||
\begin_inset Formula $C_{1}=\frac{\alpha_{m}}{c}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $C_{2}=\frac{\alpha_{p}}{b}$
|
||
\end_inset
|
||
|
||
;
|
||
\begin_inset Formula $x=b$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\sum_{m=1}^{\infty}C_{m}b\frac{\frac{\alpha_{m}}{c}J_{1}(\frac{\alpha_{m}}{c}b)J_{0}(\frac{\alpha_{p}}{b}b)-\frac{\alpha_{p}}{b}J_{0}\left(\frac{\alpha_{m}}{c}x\right)J_{1}(\frac{\alpha_{p}}{b}b)}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{p}}{b}\right)^{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\sum_{m=1}^{\infty}C_{m}\frac{\rho\alpha_{m}}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{p}}{b}\right)^{2}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{p})$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Zet
|
||
\begin_inset Formula $p=n$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\sum_{m=1}^{\infty}C_{m}\frac{\rho\alpha_{m}}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n})$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Zodat:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
B_{n}\frac{1}{2}b^{2}J_{0}(\alpha_{n})^{2}=\sum_{m=1}^{\infty}C_{m}\frac{\rho\alpha_{m}}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n})
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
B_{n}\frac{1}{2}b^{2}J_{0}(\alpha_{n})^{2}=\sum_{m=1}^{\infty}C_{m}\frac{\rho\alpha_{m}}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n})
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Deel linker en rechterzijde door
|
||
\begin_inset Formula $\frac{1}{2}b^{2}$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
B_{n}J_{0}(\alpha_{n})^{2}=2\sum_{m=1}^{\infty}\rho^{-1}C_{m}\frac{\alpha_{m}}{\alpha_{m}^{2}-\frac{\alpha_{n}^{2}}{\rho^{2}}}J_{0}(\alpha_{n})J_{1}(\alpha_{m}\rho)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Oftewel:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula
|
||
\[
|
||
B_{n}J_{0}(\alpha_{n})^{2}=\frac{2}{\rho}\sum_{m=1}^{\infty}T_{mn}C_{m}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
B_{n}J_{0}(\alpha_{n})^{2}=\frac{2}{\rho}\sum_{m=1}^{\infty}T_{mn}C_{m}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $B_{n}=\frac{2}{\rho J_{0}(\alpha_{n})^{2}}\sum_{q=1}^{\infty}T_{qn}C_{q}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{1}{\alpha_{m}}+\sum_{n=1}^{\infty}Y_{B,n}T_{mn}\frac{2}{\rho J_{0}(\alpha_{n})^{2}}\sum_{q=1}^{\infty}T_{qn}C_{q}=Y_{C,m}\frac{1}{2\rho}J_{0}(\alpha_{m})^{2}C_{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\sum_{n=1}^{\infty}\frac{2Y_{B,n}}{J_{0}(\alpha_{n})^{2}}T_{mn}\sum_{q=1}^{\infty}T_{qn}C_{q}-\frac{1}{2}Y_{C,m}J_{0}(\alpha_{m})^{2}C_{m}=-u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{\rho}{\alpha_{m}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
—————Setting ——-
|
||
\begin_inset Formula $C_{m}=ikbu_{B}^{0}z_{0}D_{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\sum_{n=1}^{\infty}\frac{2Y_{B,n}}{J_{0}(\alpha_{n})^{2}}ikbu_{B}^{0}z_{0}T_{mn}\sum_{q=1}^{\infty}T_{qn}D_{q}-\frac{1}{2}Y_{C,m}ikbD_{m}u_{B}^{0}z_{0}J_{0}(\alpha_{m})^{2}D_{m}=-u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{\rho}{\alpha_{q}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Using:
|
||
\begin_inset Formula $z_{0}Y_{B,n}=\frac{i\beta_{n}}{k}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $z_{0}Y_{C,m}=-\frac{i\gamma_{m}}{k}$
|
||
\end_inset
|
||
|
||
and ,
|
||
\begin_inset Formula $\gamma_{m}=\sqrt{\left(\frac{\alpha_{m}}{c}\right)^{2}-k^{2}}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\beta_{n}=\sqrt{\left(\frac{\alpha_{n}}{b}\right)^{2}-k^{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\sum_{n=1}^{\infty}\frac{2}{J_{0}(\alpha_{n})^{2}}\sqrt{\left(\frac{\alpha_{n}}{bk}\right)^{2}-1}kbT_{mn}\sum_{q=1}^{\infty}T_{qn}D_{q}+\sqrt{\left(\frac{\alpha_{m}}{kc}\right)^{2}-1}\frac{1}{2}kbD_{m}J_{0}(\alpha_{m})^{2}D_{m}=+J_{1}(\alpha_{m}\rho)\frac{\rho}{\alpha_{m}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
When
|
||
\begin_inset Formula $kc\sim kb\ll1$
|
||
\end_inset
|
||
|
||
, this can be rewritten to:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\sum_{n=1}^{\infty}\frac{2\alpha_{n}}{J_{0}(\alpha_{n})^{2}}T_{mn}\sum_{q=1}^{\infty}T_{qn}D_{q}+\frac{\alpha_{m}\rho}{2}D_{m}J_{0}(\alpha_{m})^{2}D_{m}=J_{1}(\alpha_{m}\rho)\frac{\rho}{\alpha_{m}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\sum_{n=1}^{\infty}\frac{2\alpha_{n}}{J_{0}(\alpha_{n})^{2}}T_{mn}\sum_{q=1}^{\infty}T_{qn}D_{q}+\frac{1}{2}\rho\alpha_{m}J_{0}(\alpha_{m})^{2}D_{m}=J_{1}(\alpha_{m}\rho)\frac{\rho}{\alpha_{m}},\label{eq:D_meq}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
D_{m}=\frac{C_{m}}{ikbu_{B}^{0}z_{0}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:D_meq"
|
||
|
||
\end_inset
|
||
|
||
is a set of infinite equations in terms of an infinite number of unknowns
|
||
for
|
||
\begin_inset Formula $D_{m}$
|
||
\end_inset
|
||
|
||
.
|
||
In matrix algebra for a finite set, this can be written as
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
(\boldsymbol{M}_{1}\cdot\boldsymbol{M}_{2}+\boldsymbol{K})\cdot\boldsymbol{D}=\boldsymbol{R}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
M_{1,ij} & =\frac{2\alpha_{j}}{J_{0}(\alpha_{j})^{2}}T_{ij}\\
|
||
M_{2,ij} & =T_{ji}\\
|
||
K_{ij} & =\frac{1}{2}\rho\alpha_{j}J_{0}(\alpha_{j})^{2} & ;\quad i=j\\
|
||
K_{ij} & =0 & ;\quad i\neq j\\
|
||
R_{i} & =J_{1}(\alpha_{i}\rho)\frac{\rho}{\alpha_{q}}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Finally, the added acoustic mass,
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{C}^{0}=p_{B}^{0}-i\omega M_{A}U_{B},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
can be computed as
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{B}^{0}=p_{C}^{0}+\sum_{m=1}^{\infty}\frac{2J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}C_{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{B}^{0}=p_{C}^{0}+ikbu_{B}^{0}z_{0}\sum_{m=1}^{\infty}\frac{2J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}D_{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Filling in:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{C}^{0}=p_{B}^{0}-i\omega M_{A}U_{B}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
Then:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $p_{B}^{0}=p_{C}^{0}+i\omega M_{A}U_{B}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
or:
|
||
\begin_inset Formula $i\omega M_{A}U_{B}=ikbu_{B}^{0}z_{0}\sum_{m=1}^{\infty}\frac{2J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}D_{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
And since:
|
||
\begin_inset Formula $M_{A}=\chi(\alpha)\frac{8\rho_{0}}{3\pi^{2}a_{L}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $\chi(\alpha)=\frac{3\pi}{4}\sum_{m=1}^{\infty}\frac{J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}D_{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\rho_{0}\sum_{m=1}^{\infty}\frac{2}{\pi b}\frac{J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}D_{m}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For a given velocity
|
||
\begin_inset Formula $u_{C,0}$
|
||
\end_inset
|
||
|
||
the velocity profile at
|
||
\begin_inset Formula $z=0$
|
||
\end_inset
|
||
|
||
is
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $u_{C}(z)=u_{C}^{0}(z)+\sum_{m=1}^{\infty}Y_{C,m}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)e^{-\gamma_{m}z}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\lang english
|
||
\begin_inset Formula $u_{C}=u_{C}^{0}+u_{B}^{0}\sum_{m=1}^{\infty}\gamma_{m}bD_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
u_{C}=u_{C}^{0}+bu_{B}^{0}\sum_{m=1}^{\infty}\gamma_{m}D_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|