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# Computational grid and spatial discretization

In [1] only uniform grids were used. Therefore we present the difference stencils used for the staggered grid. A 2D example is shown in figure .
For convenience we denote the coordinates of the mesh lines by , and . These values completely define the computational grid. How to tell navcalc these values is explained in chapter . Here and in the following we denote by cells the rectangular subdomains . The computational domain is a union of cells.

Velocity components and pressure values are defined on the nodes:

 defined on    
, where .
The following stencils are used for the spatial discretization. We use the notation

and

The values , , , are defined analogously. To preserve the second order accuracy of the stencils a smooth distribution of the grid spaces ,.. is required. To obtain such smooth grids use the GridGen utility.

Diffusive terms:

The other diffusive terms are discretized in a similar fashion. Stencils similar to the one for are used for , and , where is either or .

Convective terms:
Five different discretizations of the convective terms are possible:
1. Donor-Cell (hybrid-scheme) (1st/2nd order)
2. Quadratic upwind interpolation for convective kinematics (QUICK) (2nd-Order)
3. Hybrid-Linear Parabolic Arppoximation (HLPA) (2nd-Order)
4. Sharp and Monotonic Algorithm for Realistic Transport (SMART) (2nd-Order)
5. Variable-Order Non-Oscillatory Scheme (VONOS) (2nd/3rd-Order) (default)
To select one of these schemes, you have to set the appropriate variable in the scene description file. How to do this is explained in section . By default the VONOS-scheme is used. In the following the Donor-Cell scheme is briefly described. More details about the other schemes can be found in [3].

Second order convective terms:

Stencils similar to the one for are used for the discretization of the convective terms in () and (), e.g. we have

The unknown values, e.g. are computed by linear interpolation.

First order upwind:
 where and where and

Stencils similar to the one for are used for the upwind discretization of the convective terms in () and (). First and second order terms can be blended using a parameter by, e.g.

first ordersecond order

The blending parameter is user definable (see chapter ) and may be chosen different for the equations of momentum and energy or transport of a scalar.

Laplacian for pressure:
We employ a conservative discretization which is simply the nested application of the centered difference for the pressure gradient and the centered difference for the natural discretization of the divergence, e.g.

Poisson solvers:

For the solution of the linear equation arising from discretization of the pressure poisson equation, the following numerical methods are implemented:

1. Successive Overrelaxation (SOR)
2. Symmetric SOR (forward/backward)
3. Red-Black scheme
4. 8-Color SOR
5. 8-Color Symmetric SOR (fw/bw)
6. BiCGStab

To select a method, select the corresponding option in the scene description file(see section  on how to do this). By default, the Poisson-equation is solved using the BiCGStab-method.

Next: Discretization of boundary conditions Up: Numerical Method Previous: Boundary conditions   Contents
Martin Engel 2004-03-15