3.13 Limiting multiple components

The calculation and application of a limiter is introduced in Sec. 3.11 for advection of a single scalar property. The figure below examines when the advected property is a vector, e.g. eqn in the momentum conservation Eq. (2.47 ).

PICT\relax \special {t4ht=

It shows a 2D cut along the eqn-eqn plane through a selection of cells in a regular mesh with the velocity vector eqn displayed for each cell. When eqn is interpolated to faces along the eqn-direction using a limited scheme, the simplest approach is to calculate a limiter for each vector component, e.g. eqn, and interpolate the component with that limiter.

In the example above, the profiles of eqn and eqn in the eqn-direction are quite different, so the calculated limiters will be different for eqn and eqn components at each face. The limiting will not be invariant under a rotation of the co-ordinate axes, leading to a different solution depending on the initial orientation of the geometry with respect to the axes.

The limiting can be invariant using a single limiter calculated from the magnitude eqn, which is applied to all components of eqn. The strength of the limiting corresponds to an average across the components, which is usually insufficient for the component which requires strongest limiting. This can cause instability.

Instead, the ‘V’ scheme calculates the limiter based on the ‘worst-case’ direction, i.e. the direction of steepest gradient in eqn at the cell face. It uses the following expression for eqn for a vector eqn, replacing Eq. (3.12 ) for a scalar:

 rn f d r r = 2 j-djr-- --- -r-- --- 1; for vector : n f n f \relax \special {t4ht=

PICT\relax \special {t4ht=

While the V scheme ensures invariance, it also provides greater stability than component limiting. It can remove oscillations in solutions, e.g. in the example above of supersonic flow over a step showing the effect in velocity in the cells adjacent to the step corner.

Multivariate limiting

Multivariate limiting applies the same limiter for advection discretisation across a set of 2 or more equations. It works by calculating the limiter eqn for each solution variable in the equation set and applying the lowest eqn to all equations.

It can be used in order to maintain consistency in the transport of individual fluid species, such as eqn, eqn, e.g. in the propagation of laminar flame (which is beyond the scope of this book).

Notes on CFD: General Principles - 3.13 Limiting multiple components