Commit 723bddaf authored by Spencer Sherwin's avatar Spencer Sherwin
Browse files

Updated with details of moving body outflow modifications

parent 9b42e311
......@@ -451,3 +451,12 @@ year={2011}
publisher = {Springer London}
}
@Article{BaPlGrSh16,
author = {Bao, Y. and Palacios, R. and Graham, M. and Sherwin, S.J. },
title = {Generalized “thick” strip modelling for vortex-induced vibration of long flexible cylinders},
journal = {J. Comp. Phys},
year = {2016},
volume = {321},
pages = {1079-1097},
}
......@@ -210,12 +210,13 @@ $p=0$ which can be specified as
However when energetic vortices pass through an outflow region one can
experience instabilities as identified by the work of Dong, Karnidakis
and Chryssostomidis \cite{DoKa14}. In this work one impose a pressure
Dirichlet outflow condition of the form
and Chryssostomidis \cite{DoKa14}. In this paper they suggest to
impose a pressure Dirichlet outflow condition of the form
\begin{equation}
p^{n+1}= \nu \nabla\mathbf{u}^{*,n+1}\cdot \mathbf{n}-\frac{1}{2}
\mid \mathbf{u}^{*,n+1} \mid^2 S_o(\mathbf{n}\cdot \mathbf{u}^{*,n+1})-\mathbf{f}_b^{n+1}\cdot \mathbf{n}
\mid \mathbf{u}^{*,n+1} \mid^2 S_o(\mathbf{n}\cdot
\mathbf{u}^{*,n+1})+\mathbf{f}_b^{n+1}\cdot \mathbf{n}
\end{equation}
with a step function defined by $S_o(n\cdot
......@@ -225,13 +226,19 @@ is a non-dimensional positive constant chosen to be sufficiently
small. $\mathbf{f}_b$ is the forcing term in this case the analytical
conditions can be given but if these are not known explicitly, it is
set to zero, i.e. $\mathbf{f}_b=0$. (see the test
KovaFlow\_m8\_short\_HOBC.xml for a non-zero example). For the
velocity component one can specify
KovaFlow\_m8\_short\_HOBC.xml for a non-zero example). Note that in
the paper \cite{DoKa14} they define this term as the negative of what
is shown here so that it could be use used to impose a default
pressure values. This does however mean that the forcing term is
imposed through the velocity components $u,v$ by specifying the entry
\inltt{VALUE} (An example can be found in
ChanFlow\_m3\_VCSWeakPress\_ConOBC.xml). For the velocity component
one can specify
\begin{equation}
\nabla\mathbf{u}^{n+1}\cdot\mathbf{n}=\frac{1}{\nu}\Bigl[p^{n+1}\mathbf{n}+\frac{1}{2}
\mid \mathbf{u}^{*,n+1} \mid^2 S_o(\mathbf{n}\cdot
\mathbf{u}^{*,n+1})-\nu(\nabla\cdot\mathbf{u}^{*,n+1})\mathbf{n}+\mathbf{f}_b^{n+1}\Bigr]
\mathbf{u}^{*,n+1})-\nu(\nabla\cdot\mathbf{u}^{*,n+1})\mathbf{n}-\mathbf{f}_b^{n+1}\Bigr]
\end{equation}
This condition can be enforced using the \inltt{USERDEFINEDTYPE} ``HOutflow'', i.e.
......@@ -245,7 +252,28 @@ This condition can be enforced using the \inltt{USERDEFINEDTYPE} ``HOutflow'', i
</BOUNDARYCONDITIONS>
\end{lstlisting}
Dong has more also suggested convective like outflow conditions in \cite{Dong15} which can be enforced through a Robin type specification of the form
Note that in the moving body work of Bao et al. \cite{BaPlGrSh16}
some care must be made to identify when the flow over the boundary is
incoming or outgoing and so a modification of the term
\[
\frac{1}{2}
\mid \mathbf{u}^{*,n+1} \mid^2 S_o(\mathbf{n}\cdot
\mathbf{u}^{*,n+1})
\]
is replaced with
\[
\frac{1}{2} \left ( (\theta + \alpha_2)
\mid \mathbf{u}^{*,n+1} \mid^2 + (1-\theta + \alpha_1)
( \mathbf{u}^{*,n+1} \cdot \mathbf{n}) \mathbf{u}^{*,n+1} \right )
S_o(\mathbf{n}\cdot
\mathbf{u}^{*,n+1})
\]
where the default values are given by $\theta = 1,\alpha_1 = 0,\alpha_2 = 0$ and these values can be set through the parameters \inltt{OutflowBC\_theta},
\inltt{OutflowBC\_alpha1} and \inltt{OutflowBC\_alpha2}.
Dong has also suggested convective like outflow conditions in
\cite{Dong15} which can be enforced through a Robin type specification
of the form
\begin{equation}
\frac{\partial \mathbf{u}^{n+1}}{\partial n} + \frac{\gamma_0 D_0}{\Delta t} \mathbf{u}^{n+1} = \frac{1}{\nu}\Bigl[\mathbf{f}^{n+1} + \mathbf{E}(\mathbf{n},\mathbf{u}^{*,n+1}) + p^{n+1}\mathbf{n} -\nu(\nabla\cdot \mathbf{u}^{*,n+1}) \mathbf{n}
......
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