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Boundary conditions

The computational domain $\Omega$ is embedded in a 3D cuboid domain $\Omega_R$ . In the current version the following boundary conditions are implemented:

periodic boundary conditions
These conditions hold for ${\bf u},~\tilde{\bf u}$ , $p^{n+1}$ , and on opposite faces of $\Omega_{R}$ .
Dirichlet boundary conditions
On subsets $\Gamma \subset \partial \Omega$ Dirichlet conditions ${\bf u}={\bf u}_0$ may be defined. The same b.c. hold for $\tilde{\bf u}$ , i.e. $\tilde{\bf u}={\bf u}_0$ . These b.c. lead to homogeneous Neumann conditions for the pressure $p^{n+1}$ on $\Gamma$ . If or is calculated, Dirichlet b.c. for these quantities have to be specified on $\Gamma$ .
Slip conditions
Here, homogeneous Dirichlet conditions hold for the normal velocity component and homogeneous Neumann conditions for the tangential velocity components. The same b.c. hold for $\tilde{\bf u}$ . These b.c. lead to homogeneous Neumann conditions for the pressure $p^{n+1}$ .
Outflow boundary condition I
On faces $\Gamma$ of $\Omega_R$ homogeneous Neumann b.c. for the velocity may be defined as

$\displaystyle \frac{\partial {\bf u}}{\partial n}=0~.$
Outflow boundary condition II
Alternatively a boundary condition of convective type can be applied at outflow boundaries, i.e. it consists of a discretization of

$\displaystyle \frac{\partial {\bf u}}{\partial t} + {\bf u} \cdot \nabla {\bf u} = 0$
If no Dirichlet or periodic boundary conditions are specified, then homogeneous Neumann conditions are assumed for and .

Next: Computational grid and spatial Up: Numerical Method Previous: Time discretization Contents

Martin Engel 2004-03-15