The gradient of a scalar property \(\phi\) is represented using the notation:

\[\grad \phi = \vec{e}_1 \frac{\partial}{\partial x_1} \phi + \vec{e}_2 \frac{\partial}{\partial x_2} \phi + \vec{e}_3 \frac{\partial}{\partial x_3} \phi\]

where the \(\vec{e}\) vectors represent the unit vectors of the 3-D space.

Usage🔗

Gradient schemes are specified in the fvSchemes file under the gradSchemes sub-dictionary using the syntax:

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

Options🔗

Gradient schemes🔗

• Gauss gradient scheme
• Least-squares gradient scheme

Interpolation schemes🔗

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

Gradient limiters🔗

The limited gradient schemes attempt to preserve the monotonicity condition by limiting the gradient to ensure that the extrapolated face value is bounded by the neighbouring cell values.

• Cell-limited gradient scheme
• Face-limited gradient scheme
• Multi-directional cell-limited gradient scheme

• Multi-directional face-limited gradient scheme

• clippedLinear: limits linear scheme according to a hypothetical cell size ratio

Example🔗

See this example to see the relative performance of the schemes.

Further information🔗

Source code:

• $FOAM_SRC/finiteVolume/finiteVolume/gradSchemes/gradScheme

API:

• Foam::fv::gradScheme