Updated documentation
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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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@ -46,16 +66,25 @@
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newcommand{
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articletitle}{Numerical solution of the (Stokes) viscous boundary layer}
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Boundary layer flow
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filename /home/anne/bin/doctemplates/ascee_beeldmerk_withacr.eps
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ASCEE, Máximastraat 1, 7442 NW Nijverdal, info@ascee.nl
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1
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@ -152,9 +549,260 @@ and
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% Optionally: set this document to confidential
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||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Differential equation
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The Stokes layer of the velocity satisfies the following differential equation:
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\]
|
||||
\begin{equation}
|
||||
\frac{\partial\hat{u}}{\partial t}-\frac{\mu}{\rho_{0}}\nabla_{\mathrm{T}}^{2}\hat{u}=\hat{K}(t),\label{eq:diffeq}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $t$
|
||||
\end_inset
|
||||
|
||||
is time,
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $\nabla_{\mathrm{T}}^{2}$
|
||||
\end_inset
|
||||
|
||||
is the Laplacian in
|
||||
\emph on
|
||||
transverse
|
||||
\emph default
|
||||
direction.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $\hat{u}$
|
||||
\end_inset
|
||||
|
||||
is the
|
||||
\emph on
|
||||
axial
|
||||
\emph default
|
||||
velocity,
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $\hat{K}(t)$
|
||||
\end_inset
|
||||
|
||||
is the forcing function as a function of time, which in acoustics equals
|
||||
minus the gradient of the pressure,
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $\mu$
|
||||
\end_inset
|
||||
|
||||
is the dynamic viscosity,
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $\rho_{0}$
|
||||
\end_inset
|
||||
|
||||
the density.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Harmonic solution
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
If we assume harmonic motion, we may write
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\hat{u}=\Re\left(ue^{i\omega t}\right)\quad;\quad K=\Re\left(\hat{K}e^{i\omega t}\right)\quad\mathrm{etc}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $\Re$
|
||||
\end_inset
|
||||
|
||||
is the operator taking the real part of its argument,
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $i=\sqrt{-1}$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $\omega$
|
||||
\end_inset
|
||||
|
||||
is the frequency in rad/s.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
For flow close to a plate and unbounded in the
|
||||
\begin_inset Formula $+y$
|
||||
\end_inset
|
||||
|
||||
-direction, the solution for
|
||||
\begin_inset Formula $u$
|
||||
\end_inset
|
||||
|
||||
yields:
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
u=\frac{1}{i\omega\rho_{0}}K\left(1-\exp\left(-\left(1+i\right)y/\delta_{\nu}\right)\right),
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $\delta_{\nu}=\sqrt{\frac{2\mu}{\rho_{0}\omega}}$
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
For oscillating flow between two parallel plates, separated at distances
|
||||
|
||||
\begin_inset Formula $2y_{0}$
|
||||
\end_inset
|
||||
|
||||
, and
|
||||
\begin_inset Formula $y=0$
|
||||
\end_inset
|
||||
|
||||
at the center between the two plates:
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
u=\frac{1}{i\omega\rho_{0}}\frac{1-h_{\nu}}{1-f_{\nu}}K,
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $h_{\nu}=\frac{\cosh\left(\left(1+i\right)y/\delta_{\nu}\right)}{\cosh\left(\left(1+i\right)y_{0}/\delta_{\nu}\right)},$
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $f_{\nu}=\frac{\tanh\left(\left(1+i\right)y_{0}/\delta_{\nu}\right)}{\left(1+i\right)y_{0}/\delta_{\nu}}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Forward Euler time, central in space
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
For
|
||||
\begin_inset Formula $\nabla_{\mathrm{T}}^{2}\equiv\frac{\partial^{2}}{\partial y^{2}}$
|
||||
\end_inset
|
||||
|
||||
, the forward Euler time, central in space (uniform grid) formulation of
|
||||
Eq.
|
||||
\begin_inset space ~
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:diffeq"
|
||||
|
||||
\end_inset
|
||||
|
||||
is
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}-\frac{\mu}{\rho_{0}}\frac{u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}}{\Delta y^{2}}=K^{n},
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
where upper index
|
||||
\begin_inset Formula $n$
|
||||
\end_inset
|
||||
|
||||
denotes a discrete time instance and lower index
|
||||
\begin_inset Formula $i$
|
||||
\end_inset
|
||||
|
||||
denotes a discrete position index.
|
||||
This is an explicit form for the velocity at the next time index
|
||||
\begin_inset Formula $n+1$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Solving for
|
||||
\begin_inset Formula $u_{i}^{n+1}$
|
||||
\end_inset
|
||||
|
||||
yields:
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
u_{i}^{n+1}=u_{i}^{n}+\Delta tK^{n}+\frac{\mu\Delta t}{\rho_{0}\Delta y^{2}}\left(u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}\right)
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
BIN
doc/blflow.pdf
Normal file
BIN
doc/blflow.pdf
Normal file
Binary file not shown.
Loading…
Reference in New Issue
Block a user