Added description of boundary condition: incident plane wave on port in infinite wall.

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Casper Jansen 2021-07-19 10:58:27 +02:00
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5 changed files with 411 additions and 24 deletions

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@ -1570,6 +1570,12 @@ To model absorption of sound, a one-dimensional porous material model should
This work has been postponed to a later stage. This work has been postponed to a later stage.
\end_layout \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 \begin_layout Section
Overview of this documentation Overview of this documentation
\end_layout \end_layout
@ -7075,7 +7081,7 @@ A membrane is a mechanical
\end_layout \end_layout
\begin_layout Section \begin_layout Section
Hole Holes in plate
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard
@ -7083,16 +7089,73 @@ series_impedance.py/class CircHoleNeck(SeriesImpedance)
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard
Behaves like an acoustic mass with losses. A plate with several holes can be modelled using CircHoleNeck.
It represents holes in sheet material, which can form the neck of a Helmholtz It behaves like an acoustic mass with losses and can represent the neck
resonator. of a Helmholtz resonator.
Hole-hole interaction is neglected. Typical uses are to connect volumes to eachother or volumes to ducts, to
The resistance term is an approximation. form Helmholtz resonators.
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard
Usable for connecting volumes to eachother or volumes to ducts, to form Limitations are that hole-hole interaction is neglected and that the resistance
Helmholtz resonators. 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 \end_layout
\begin_layout Section \begin_layout Section
@ -7278,7 +7341,7 @@ literal "true"
\begin_inset Formula \begin_inset Formula
\begin{equation} \begin{equation}
M_{A}=\chi(\alpha,k)\frac{8\rho_{0}}{3\pi^{2}a_{L}}, M_{A}=\chi(\alpha,k)\frac{8\rho_{0}}{3\pi^{2}a_{L}},\label{eq:acoustic_mass}
\end{equation} \end{equation}
\end_inset \end_inset
@ -7904,12 +7967,347 @@ Filling this in, we obtain the following low-frequency approximation to
\begin_layout Standard \begin_layout Standard
\begin_inset Formula \begin_inset Formula
\begin{equation} \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] 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{equation}
\end_inset \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 'Sound absorbing materials' (1949) Zwikker
et al., pp.
132-134.
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 and
\begin_inset Formula $p_{s}$
\end_inset
the scattered pressure field.
All depend on both position and time.
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 and its
reflection can be described as:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
p_{i}(x,t)=\begin{cases}
P_{i}\cdot\cos(kx) & x<0\\
undefined & 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 half the amplitude of the incident plane wave (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 wavenumber 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
no longer is 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 als 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_{rad}U
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
in which
\begin_inset Formula $Z_{rad}$
\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_{rad}v\label{eq:bc-planewave-port-pressure}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
in which
\begin_inset Formula $z_{rad}$
\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 'measured' 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_{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 'measured' by averaging it over the port's boundary.
\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 \end_layout
\begin_layout Chapter \begin_layout Chapter
@ -8542,7 +8940,7 @@ Z_{h}=\left(\frac{\rho_{0}z_{0}}{i\omega V}+R_{v}+i\omega m_{\mathrm{neck}}\righ
where where
\begin_inset Formula \begin_inset Formula
\begin{equation} \begin{equation}
m_{\mathrm{neck}}=\frac{\rho_{0}\ell_{\mathrm{eff},\mathrm{neck}}}{S_{\mathrm{neck}}}, m_{\mathrm{neck}}=\frac{\rho_{0}\ell_{\mathrm{eff},\mathrm{neck}}}{S_{\mathrm{neck}}},\label{eq:acoustic_mass_neck}
\end{equation} \end{equation}
\end_inset \end_inset
@ -9580,18 +9978,6 @@ For circular large holes with diameter
\end_layout \end_layout
\begin_layout Standard \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
Here we use a more advanced model, which includes the shear wave number. Here we use a more advanced model, which includes the shear wave number.
For unrounded edges and a perforate thickness of For unrounded edges and a perforate thickness of
\begin_inset Formula $t_{p}$ \begin_inset Formula $t_{p}$
@ -10388,7 +10774,7 @@ Z_{\mathrm{hole}}=i\omega\rho_{0}\frac{4}{\pi D^{2}}\left[\frac{t_{w}}{\left(1-f
\begin_layout Standard \begin_layout Standard
\begin_inset Formula \begin_inset Formula
\begin{equation} \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)}, \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{equation}
\end_inset \end_inset

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