Week 9 - FVM Literature Review

Objective -

To describe the need for interpolation schemes and flux limiters in FVM.

Basics

In computational fluid dynamics there are 3 numerical method based solvers to solve the governing navier strokes equations these are:

1. Finite difference method

2. Finite element method

3. Finite volume method

In finite difference method:

The grid is linear and aligned so that all grid points lie in a straight line . Main thing is that the properties are considered to concentrated at a single point.

In finite volume method:

The method is applied over a volume such that the governing equation is valid over the entire volume , aslo the properties are assumed to be concentrated at the geometric centre of the volume concerned.

In finite element method:

The properties are valid over an elemental area. the properties assumed to be concentrated at area centre.

Interpolation schemes in Finite Volume Method:

Interpolation is a process to estimate the values at unknown points by using the points with known value and sample points on either sides of the unknown point. In FVM, interpolation schemes are used to find values of volume integrals required at the points other than nodes.

The different interpolation schemes are:

1. Upwind Interpolation Scheme (UDS)

2. Linear Interpolation Scheme

3. Quadratic Upwind Interpolation Scheme

4. Hybrid Interpolation Scheme

5. Total variation Diminishing Scheme

6. Essentially Non-oscillatory interpolation Scheme

Some interpolation schemes used in CFD are as follows:

1. Upwind interpolation scheme

2. Linear interpolation scheme

3. Quadratic Upwind Interpolation for Convective Kinematics (QUICK) scheme

4. Hybrid Scheme

5. Total Variation Diminishing (TVD) scheme

Explaining the Upwind interpolation scheme for an example

The Upwind interpolation scheme takes into account the direction of fluid flow and based on it, the value of solution variable at the face is approximated by its value at the upstream computational node. For instance, if the fluid is moving from west to east (refer figure), then:

ϕe≈ϕPϕe≈ϕP

And if the fluid is moving from east to west (refer figure), then:

ϕe≈ϕE

Flux Limiters

While calulating the variable gradients they tend to toggle and are not bounded to a particular value and keeps jumping , this occurs due to discontinuity due to schock occuring , having a flux limiter will help to bound the gradient value there by giving a stable solution