Week 9 - FVM Literature Review

Finite volume method vs Finite difference method

• Finite volume method is a representation and evaluation of partial difference equation in the form of algebraic equation similarly Fiite difference method is calcuated at discreate places on a meshed geometry
• The important charecteristics of finite volume methos is its conservative proparties. The global conservation is also ensured by applyinfg conservative principle over each small control volume
• Finite volume method is fescribes as volume sorrunding each node point on a mesh means the grid points where variables are stored are defined at the centre of each control volume and the method startrd by deviding the flow domain into a number of small control volumes
• In finite volume method ,volume integrals in a partial differential equation that contains a divergence term are converted to surface integrals,using the divergence theorem.These terms are then evaluated as fluxes at the surface of each finite volume .because the flux entering a given volume is identical to that leaving the adjacent volume , these methods are conservative

The advantage of the finite volume method is thatit is easily formulated to alloe for unstructured meshes ,and this method is used in many CFD packages

Interpolation schemes in Finite Volume Method

Interpolation is theestiamtion of values at points by using the known points and sample points on either side of the unknown point. In FVM interpolation schemes are used to find values of volume integrals required at the points other than nodes

The differential 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

Flux Limiter

For conservative fluid flow , it is observed that the low order schemes are usually stable but quite dissipative in nature around the points of discontinuity or shockswhile thr higher order schemes are unstable in nature and show oscilation in the vicinity of discontinuity or shocks.Highly accurate and oscilation free schemes are known as High Resolution Schemes.

Flux limiter functions are used to fine tune the higher order and lower order schemes in such a way that the resulting scheme gives a higher 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 schemeis employed