Week 9 - FVM Literature Review

Finite Volume Method (FVM) is method used in many computational fluid dynamic packages. FVM refer to small volume surrounding each node point on a mesh which means the grid points where variables are stored are typically defined as being at center of each control volume.

FVM is a method representing and evaluating partial diffrential equation in algebric equation. Similar to Finite Difference Method (FDM) or Finite Element Method (FEM) values are calculated at discrete places on meshed geometry. In finite volume method volume integral in partial differential equation contains a divergence term are converted to surface integral using divergence theorem these term are then evaluated as fluxes at surface of each finite volume because the flux entering given volume is identical to that leaving the adjacent volume these methods are conservative. Advantage of FVM is that it is easily formulated to allow for unstructured meshes.

Steps : -

• Integral form of equation
• Apply Gauss divergnce theorm
• We get discretize form which can be solved using computer it is in flux term

Equation :-

`alphaA[(T_e-T_p)/(Delta x)] - alpha A [(T_p-T_w)/(Deltax)] +overlineSdv = 0`

Where S is source term

`alphaA[(T_e-T_p)/(Delta x)]` is Heat flux out

`alpha A [(T_p-T_w)/(Deltax)]` is Heat flux in

It can be written as

`[Heat flux] _(out) - [Heat flux]_(in) +source term = 0`

Difference between FVM and FDM : -

In finite difference method the grid are linear and aligned so that all grid points lie in straight line. In FDM Properties are consided to be concentrated at single point.

In finite volume method which is applied over a volume such that governing equation is valid over entire volume. In FVM properties are assumed to be concentrated at geometric center of volume.

Interpolation schemes and flux limiters in FVM: -

Interpolation is a process to estimate the value at unknown points by using the points with known value and sample points on either sides of unknown points. In finite volume method (FVM) interpolation schems are used to find values of volume intergrals required at points other than nodes.

Different interpolation scheme are given below: -

1. Upwind Interpolation Scheme
2. Linear Interpolation Scheme
3. Quadratic Upwind Interpolation Scheme
4. Hybrid Interpolation Scheme
5. Total variation diminishing scheme
6. Essential Non-oscillatory Interpolation Scheme

Flux Limiters : -

Convective fluid flow it is observed that low order schemes are usually stable but quite dissipative in anture around the points of discontiunity or shocks while the higher order schemes are unstable in nature and show oscillation in the vicinity of discontinuty or shocks.Highly accurate and osccillation free schmes are known as High resolution Schemes. Flux Limiter function are used to fine tune the higher order and lower order schemes in such a way that resulting scheme gives a higher order accuracy in smooth region of the flow and maintains first order accuracy in the vicinity of shocks and discontinuties.