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\author{C.-H. LAI\footnote{School of Computing \& Mathematical Sciences,
University of Greenwich, Wellington Street, Woolwich, London SE18 6PF, U.K.
email: C.H.Lai@grn.ac.uk},
A.M. CUFFE\footnote{Address same as above.  This author is supported by
a Bursary of the University.}, \&
K.A. PERICLEOUS\footnote{Address same as above.}}
\authorhead{LAI, CUFFE, \& PERICLEOUS}
\chapter{A Domain Decomposition Technique for Viscous/Inviscid Coupling}
\chaptitle{VISCOUS/INVISCID COUPLING}
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\section*{INTRODUCTION}

The numerical simulation of viscous flows past realistic configurations
using time-dependent compressible Navier-Stokes for high Reynolds numbers 
requires a large amount of computing time \cite{cfdcc94}.  Although
steady-state solutions of a compressible inviscid flow past over a
complete aircraft described by the Euler equation can be obtained in just a few
minutes time, modelling 3-D unsteady flows to understand the aerodynamics of
a complete aircraft is still a grand challenge.
With the limits of computing resources, it is important to develop efficient
numerical technqiues which are suitable for implementation on parallel
computers and for engineering design purposes.

One such technique relies upon a problem
partitioning concept.  The concept in relation to viscous flows past
over obstacles was originally observed by Prandtl \cite{prandtl04},
of which the following two observations are involved.  When a
viscous fluid flow past a solid boundary, tangential and normal components
of the velocity must both equal to the components of the velocity of the
surface, and fluid sticks to the surface.  For an ideal inviscid fluid,
only the normal components are equal and there is no restriction on tangential
slip of fluid relative to the boundary.  The fundamental differences
lead to the idea of a boundary layer first proposed by Prandtl.  It is
a thin layer of fluid adjacent to the solid surface within which viscous
effects are dominant and that such effects become negligible outside
the thin layer.  Hence it is valid to divide the flow region into two
parts, i.e. the boundary layer and the inviscid flow outside the layer.

\section*{TRUNCATION TECHNIQUES}

Truncation methods have been around for a long time.  It has been
used, in a different form, for the determination of local hyperbolic and
elliptic regions in a plane transonic flows for shock location
\cite{murman71} or local
adaptive mesh refinement for shock capturing.  This Section describes two
frequently used truncation techniques related to flows past an obstacle.

\subsection{Prandtl's truncation}
Prandtl introduced the idea of boundary
layer by neglecting the effect of viscosity outside the thin layer next to a
surface.  One assumption to implement the idea in 2-D incompressible
viscous flows past an obstacle described by
\begin{equation}
\nabla\cdot\underline{u} = 0
\end{equation}
\begin{equation}
\frac{\partial\phi}{\partial t} + \nabla\cdot (\underline{u}\phi -
\nu\nabla\phi ) = - \frac{1}{\rho}\frac{\partial p}{\partial\underline{n}}
\label{mom}
\end{equation}
\noindent where $\phi = u, v$ and $\underline{u} = (u , v)^T$ is
$u \gg v$.  The assumption leads to the boundary layer equations as
described by
\begin{equation}
\nabla\cdot\underline{u} = 0
\end{equation}
\begin{equation}
\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x}
+ v\frac{\partial u}{\partial y} = - \frac{1}{\rho}
\frac{\partial p}{\partial x} + \nu\frac{\partial^2u}{\partial y^2}
\end{equation}
\begin{equation}
\frac{1}{\rho}\frac{\partial p}{\partial y} \approx 0
\end{equation}
\noindent There are other assumptions which reduces the 2-D
incompressible viscous flow model to other turbulent boundary layer models
\cite{knott86} or models suitable for regions with separations and wakes 
\cite{green73} \cite{raudkivi75}.

The upstream Reynolds number is chosen to be 1000.  Figure \ref{fig4} shows the
velocity profiles against the distance along $\eta$-axis, obtained at
$\xi = 0.46$ and $\xi = 0.66$.  Figure \ref{fig4} shows the effect of
$||u_{\epsilon ,\sigma} - u_{fv}||_\infty$ for the aerfoil problem.

\section*{CONCLUSIONS}

The scalar truncation method is 
extended to a vector truncation method and is embedded in a cell-centred finite 
volume method.  The effect of the threshold parameter $\sigma$ is insignificant 
to the accuracy of the solution.  The choice of the threshold parameters namely, $\sigma$ and $\epsilon$, is effectively equivalent to the determination of a 
minimal inner region defined by the thickness function $t$.  The present 
approach shows that the computation of velocity gradients in the finite volume 
method can be used in both the evaluation of the line integrals as well as for 
viscous effect comparison.

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\begin{figure}[tbh]
\begin{center}
\leavevmode
\hbox{\epsfysize=8cm\epsffile{aies1f5.eps}} \\
\end{center}
\caption{$||u_{\epsilon , \sigma} - u_{fv}||_\infty$ for some values of
$\epsilon$ and $\sigma$.}
\label{fig4}
\end{figure}

