Week 9 - FVM Literature Review

* Finite Volume Method:

A discretization technique for partial differential equations, especially those that arise from physical conservation

laws. FVM uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the

equations.

- FDM is a differential scheme, that is approximation of a taylor series expansion.

- We use first order, second order differential in terms of difference between two point values.

- This may bring in some approximation error.

- Finite Volume method is an integral scheme.

- Since integrating across the area, chance of error is minimized.

- Finite Element Method is also an integral scheme.

- So between a differential and an integral scheme (integral scheme scores in terms of error).-

- Differential schemes are faster and error can be minimized if you have regular grids (squares and rectangles).

- Integral schemes are slightly slower but score over FDM when the mesh is irregular.

- Most of the commercial software like FLUENT (FVM) are either FVM or FEM (Deform)

- FDM are very popular methods in academics.

* Differences between FDM & FVM:

FDM

1. Governing equations are in the differential form.

2. Solution computed at each node i.e., intersection of mesh lines.

3. Requires the use of structured grid, difficult to implement for complex geometries.

4. Easy to obtain higher order schemes, so preferable for applications that require very high accuracy.

5. All flow variable need to be differentiable and continuous over the entire domain, unsuitable for handling discontinuities.

6. Does not ensure conservation.

FVM

1. Governing equations are in the integral form.

2. Solution computed at the cell centroids.

3. Can handle unstructured grid and hence complex geometries with ease.

4. Deriving schemes higher than second order is next to impossible due to the complexity involved.

5. Can handle discontinuities, due to the nature of integral equations.

6. Imposes and ensures conservation at both cell and domain levels.

Working of FVM

* Need for interpolation schemes and flux limiters in FVM:

Flux limiters basically used in high resolution schemes, to avoid the random and violent oscillations that can occur with high

order spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain which is not really

much appreciated.

The logic behind the construction of flux limiter schemes is to limit the spatial derivatives to to values which are close to the

real world datas.

FLUX LIMITERS:

For convective fluid flow it has been observed that low - order schemes are usually stable but quite dissipative in nature

around points of discontinuity or shocks while the higher order schemes are unstable in nature and show oscillations in the

vicinity of discontinuity. Highly accurate and Oscillation free schemes are known as High Resolution Schemes.

Flux limiter functions are used to tune high order and low order schemes in such a way that the resulting scheme gives a high

order accuracy in the smooth region of the flow and maintains first order accuracy in the vicinity of shocks and discontinuities.

For such a scheme TVD (Total Variation Diminishing) criteria is maintained.

Interpolation Schemes in Finite Volume Method:

The approximation of surface and volume integrals may require values of the variable at locations other than the computational nodes of the CV. Values at these locations are obtained using interpolation formulae.

Some commonly used interpolation schemes are: as follows,

1. Upwind Interpolation Scheme: (UDS)

2. Central Differencing Approximation: (CDS)

3. Quadratic Upwind Differencing Scheme: (QUICK)

4. Hybrid Interpolation Schemes: (Both CDS and UDS)