Challenge

Week 9 - FVM Literature Review

AAjaykumar Mst26 Jun 2021

AIM:

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

Methods of Discretization :

1. Finite difference method

2. Finite element method

3. Finite volume method

Infinite difference method:

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

Infinite 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 center of the volume concerned.

Infinite element method:

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

Drawbacks of Finite Difference Method:

1. We cannot use FDM directly to a body whose grid points are not aligned properly along the x, y directions. these may occur due to complex folds, cuts in the structure.

2. As it is not aligned we cannot calculate the properties at the next adjacent point exactly, but we can make some correction factor and approximate it, this method is called as 'Variable Transformation Method '.

3. Variable transformation method gives an approx solution of the property but it is time-consuming and complex.

4. For complex profiles, this method is not preferred.

Interpolation schemes:

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

Upwind interpolation scheme:

The upwind interpolation for approximating the value of a variable 'f' at the east face of the control volume is given as below. We want approximation at the eastern face 'e'

(v.n)_e is the velocity component at point e in the x-direction

Linear interpolation scheme:

The linear interpolation is equivalent to using the central difference formula of the first-order derivative; hence, it is also called the Central Difference Scheme(CDS). It is second-order accurate whose accuracy is very good but we might have one small problem for advection-dominated situations, that this scheme may produce oscillatory solutions, This scheme leads to simple approximation for evaluation of gradients required for evaluation of diffusive fluxes.

Flux limiters

Flux limiters are employed in high-resolution schemes, which are numerical systems for solving problems in science and engineering, especially fluid dynamics, that are described by partial differential equations (PDEs). They are employed in high-resolution schemes, such as the MUSCL scheme, to eliminate spurious oscillations (wiggles) caused by shocks, discontinuities, or rapid changes in the solution domain that would otherwise occur with high order spatial discretization schemes. The use of flux limiters, together with a suitable high-resolution scheme, reduces the total variation of the solutions (TVD).