Week 9 - FVM Literature Review

Interpolation schemes and flux limiters in FVM:

According to mathematics, interpolation is a called as a method of estimating new data points from a discrete set with in a certain range say (a,b). This method is widely used in computational analysis to find the field variables at unknow grid point.There are various procedure to perform these calculations and are called as interpolation schemes. But this word interpolation gives slightly a different meaning when considered in CFD softwares (eg: OpenFOAM). Here interpolation schemes are the procedure to convert or transform cell centered quantities to face centered quantities. Based on the geomery, physics of the component certain schemes are adapted.

Certain schemes mentioned in OpenFOAM are given below:

1) Gradient schemes - a) cell based linar scheme

b) point based linear

c) least squares

2) Divergence Schemes - a) linear divergence b) Upwind divergence c) QUICK divergence etc

3) Laplacian Schemes - It is a combination of divergence and gradient operators.

For example: consider 1D heat conduction equation

`del/(delx) (alpha (delT)/(delx)) + S =0`

Integrating this equation gives the field variable T with respect to x.

But here `alpha` α is a function of T. So we need to find alpha using interpolation schemes.

Flux limiters:

In CFD, when the complex real time scenarios like supersonic or hypersonic flows or the problems with complex geometries are dealt, flux variations are high. To limit these variations beyond a certain limit, fluxes are limited by using flux limiters. These limit the spatial derivatives when the solution at discontiuities are going beyond limit. Spurious oscillations in solutions are created by shocks, discontinuities and sharp edges in higher order schemes. These limiters help in the convergence of solution.