fvSchemes

Contains the instructions to discretize the problem’s equations. This settings can be changed on run time, however it is required to set the field controlDict.runTimeModifiable equal to ``yes``` in the “system/controlDict”. In fact, different discretization procedures suit different mesh refinement/quality and state of simulation.

This schemas defines how you case must be compueted.

gradScheme

This dictionary field tackles the gradient computation in 3 aspects, how to define the compute (scheme), its interpolation and its limitation, with this arrangement:

gradSchemes
{
    default         none;
    grad(p)         <optional limiter> <gradient scheme> <interpolation scheme>;
}

The gradients can be computed using two schemes.The keywords are:

• Gauss
• leastSquares

The leastSquares method is more accurate, however, it tends to be more oscillatory on tetrahedral meshes. The interpolation can be executed with the following methods:

• linear: cell-based linear
• pointLinear point-based linear
• leastSquares: Least squares

On the other side, gradient limiters will avoid over and under shoots on the gradient computations. They increase the stability of the method but add diffusion due to clipping. There are four available:

• cellMDLimited
• cellLimited
• faceMDLimited
• faceLimited

They are in descendent order of accuracy. faceLimited is the more diffusive one, however it provides stability. The gradient limiter implementation uses a blending factor

gradSchemes
{
    //         <limiter>     <gradient scheme> <interpolation scheme>
    default    cellMDLimited       Gauss            linear 1            // the blending factor is 1
}

where 0 means no blending hence the solution will be accurate but less stable. Of course 1 will introduce the maximum blending possible.

divSchemes

Relates to the evaluation of divergence across cell faces which usually transport a property Θ under the influence of velocity field φ (phi). The schemes are all based on Gauss integration, using the flux φ and the advected field being interpolated to the cell faces by one of the selected schemes.

These are the convective discretization schemes that you will use most of the times:

• Gauss upwind → first order accurate.

• Gauss linearUpwind grad(<Θ>) → second order accurate, bounded.

• Gauss limitedLinear → TVD scheme, second order accurate, bounded. Recommended for LES simulations.

• Gauss linear → linear interpolation, unbounded.

Gauss indicates that derivatives are evaluated via the Gauss’ theorem. While upwind indicates a standard 1st order upwind interpolation (bounded). First order methods are bounded and stable but diffusive. Second order methods are accurate, but they might become oscillatory, the most common method to use is ``linearUpwind```, which stands for standard 2nd order interpolation (unbounded). TVD methods such as superBee, minmod and vanLeer are second order accurate (bounded). TVD methods switch locally to upwind when they detect strong gradients. For the discretization of the diffusive terms instead, you can use a fully orthogonal scheme:

• linear corrected
• linear limited

0 for bad mesh and 1 for good quality mesh. An accurate and stable numerical scheme (second order accurate) would look like:

divSchemes
{
    div(phi,U)                      Gauss linearUpwind grad(U);
    div(phi,omega)                  Gauss linearUpwind grad(omega);
    div(phi,k)                      Gauss linearUpwind grad(omega);
    div((nuEff*dev2(T(grad(U)))))   Gauss linear;
}

laplacianSchemes

For the discretization of the diffusive terms, you can use a fully orthogonal scheme

laplacianSchemes
{
    default  Gauss linear  corrected;
}

You can also use a scheme that uses a blending between a fully orthogonal scheme and a non-orthogonal scheme:

laplacianSchemes
{
    default  Gauss linear limited 1
}

In summary, by setting the blending factor equal to 0 is equivalent to using the uncorrected method. You give up accuracy but gain stability. If you set the blending factor to 0.5, you get the best of both worlds. In this case, the non-orthogonal contribution does not exceed the orthogonal part. You give up accuracy but gain stability. For meshes with non-orthogonality less than 70, set the blending factor to 1. For meshes with non-orthogonality between 70 and 85, set the blending factor to 0.5. For meshes with non-orthogonality more than 85, it is better to get a better mesh. But if you definitely want to use that mesh, set the blending factor to 0.333, and increase the number of non-orthogonal corrections.

Discretization schemes example

Few examples are proposed here for different quality thresholds.

Non-orthogonality between 70 and 85

gradSchemes
{
    default     cellLimited Gauss linear 0.5;
    grad(U)     cellLimited Gauss linear 1.0;
}
divSchemes
{
    div(phi,U)                      Gauss linearUpwind grad(U);
    div(phi,omega)                  Gauss linearUpwind default;
    div(phi,k)                      Gauss linearUpwind default;
    div((nuEff*dev(T(grad(U)))))    Gauss linear;
}

laplacianSchemes
{
    default Gauss linear limited 0.5;
}
snGradSchemes
{
    default limited 0.5;
}

Non-orthogonality between 60 and 70

gradSchemes
{
    default cellMDLimited Gauss linear 0.5;
    grad(U) cellMDLimited Gauss linear 0.5;
}

divSchemes
{
    div(phi,U) Gauss linearUpwind grad(U);
    div(phi,omega) Gauss linearUpwind default;
    div(phi,k) Gauss linearUpwind default;
    div((nuEff*dev(T(grad(U))))) Gauss linear;
}

laplacianSchemes
{
    default Gauss linear limited 0.777;
}

snGradSchemes
{
    default limited 0.777;
}

Non-orthogonality between 40 and 60

gradSchemes
{
    default cellMDLimited Gauss linear 0;
    grad(U) cellMDLimited Gauss linear 0.333;
}

divSchemes
{
    div(phi,U) Gauss linearUpwind grad(U);
    div(phi,omega) Gauss linearUpwind default;
    div(phi,k) Gauss linearUpwind default;
    div((nuEff*dev(T(grad(U))))) Gauss linear;
}

laplacianSchemes
{
    default Gauss linear limited 1.0;
}

snGradSchemes
{
    default limited 1.0;
}