Finite Volume Method Literature review

Finite Volume Method Literature review

Finite Volume Method

Finite Volume method provides a robust way of discretization the equations to solve for the certain class of heat transfer & fluid flow problem .Similar to the FDM technique wherein derivatives are replaced/discretized in form of differences that is been applied on the points/nodes,FVM applies the same concept but the derivatives which are discretized are applied to the volumes (with centers).In short, FVM computes solutions at each finite volume where FDM computes solutions at each point. Also, FVM seems to conserve the properties in a better way than the FDM because of the fact that the governing equations are solved in an integral form. On the other hand, the finite element method technique is also a popular technique that is been used in many structural simulation codes, its usage in CFD is limited because it could not able to model the turbulence.

Interpolation methods

The approximation of surface and volume integrals require the values of the variables at a location other than the computational nodes of the control volume.

Now the various possibilities for interpolation include upwind interpolation, linear interpolation, Quadratic upwind interpolation and recently developed hybrid schemes, TVD (Total Variation Dimensions Schemes) & ENO (Essentially Non-oscillatory interpolation schemes)

Interpolation schemes of finite volume method

• We will have to look at why we need interpolation schemes and finite volume method
• Few most commonly used interpolation schemes namely,
  1. Upwind interpolation
  2. Linear interpolation
  3. Quadratic upwind interpolation

Let’s have look one by one.

Upwind interpolation Scheme:

The name upwind refers to the assumption that we are going to use the node that is upwind (or) upstream of the given node. It is generally used for convection dominated problems.

For the east face of a quantity, It is given as:-

Where the second-order term is the leading truncation error, which resembles a diffusive flux (Numerical diffusion).

Upwind Differencing Scheme is the only approximation of the satisfies boundedness criterion unconditionally. It is first-order accurate in space.

Linear interpolation

The approximate value of the variables at the control volume face center by linear interpolation of the values at two nearest computational nodes:-

The linear interpolation is equivalent to the use of the Central Difference Scheme (CDS) formula of the first-order derivative & hence this scheme is also termed as CDS.

The approximation is second-order accurate. It is the simplest and most widely used interpolation for the evaluation of diffusive fluxes.

Quadratic Upwind Differencing Scheme: (QUICK)

QUICK stands for Quadratic Upwind Interpolation Scheme for Convective Kinematics. It is a third-order flux approximation. Although it has second-order overall accuracy, It approximates the values of the variable at CV face center by Quadratic interpolation of the values at three nearest computational nodes.

Other interpolation Schemes:

1.Hybrid Interpolation Schemes: (CDS and UDS)

2.Total Variation Diminishing: (TVD Scheme)

Flux Limiters:

For Convective fluid flow, it is 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 the first-order accuracy in the vicinity of shocks and discontinuities. For such a scheme TVD criteria are maintained. The basic forms are shown below:

Some of the other forms of flux limiters are: