Aim: FVM Literature Review.

Objective

• What is Finite Volume Method(FVM)
• Write down the major differences between FDM & FVM.
• Describe the need for interpolation schemes and flux limiters in FVM.

Description (reference Versteeg)

What is the finite volume method?

The finite volume method is one of three distinct numerical solution techniques: Finite Difference, Finite Element and Spectral methods to which FVM is a special case of the former.

This algorithm consists of

• Integrating the governing equations over all finite control volumes of the domain
• Converting the integral equations into their respective algebraic equations.
• Solution of those algebraic equations by iterative methods.

The control volume integration distinguishes the FVM from all other CFD techniques in that the resulting statements express the exact conservation of the properties for each finite cell thus providing a clear relationship between the numerical algorithm and the physics. This makes concepts more easily understood compared to the FEM and spectral methods. The numerical method/physics can be expressed in the following way

Discretisation techniques are available to model the physics of transport phenomena, convection/diffusion and external/internal sources such as heat generation etc. Iterative methods are required due to the non-linearity of the fluid flow.

The governing equations for the flow of an incompressible Newtonian fluid is given below.

Due to commonality of the various terms in each of the above equations, it can be conveniently condensed into the following form via a general variable phi.

The above equation is called the transport equation for property phi. The key step in finite volume methods is to provide an integral form of the above equation over a three-dimensional control volume, given below.

After a suitable reordering of integrating/differentiation and using Gauss Theroem which states

The resulting integral form can then be re-cast as below

This can be read as follows

Major Difference between FDM and FVM

• The form of the equations to be solved between FDM and FVM are different in that the former consist of algebraic equations approximating the derivative and the latter consists of algebraic equations approximating the integrals/fluxes.
• Finite difference methods are generally restricted to structured grids. This can be attributed to the fact that the Taylor expansion methods require that 2 of 3 spatial components are held constant while constructing the Taylor expansion on the 3rd component.
• A workaround to enable the analysis of unstructured grids using the FDM requires the use of transformation/mapping methods for coordinate transformations.
• In FVM, unstructured grids are conveniently handled because the integral form of the equations are used which allows the physical quantities to be conserved (and thus tracked) at both the finite cell level and over the complete domain.

Interpolation Schemes

In the analysis of fluid flow, we must account for the effects of convection on the diffusion behaviour of the transported material. As an example, we can consider the steady state convection -diffusion equation which is derived from the transport equation shown earlier.

The LHS gives the net convective flux and the RHS provides the net diffusive flux with its source term.

The need for interpolation schemes arises because of the fact that the convective terms present discretisation problems for the following reasons

• Convection process imparts influence on the transport material only in the direction of flow whereas
• Diffusion imparts influence along its gradients in all directions.

This crucial difference presents problems in calculating the convective fluxes across the boundary control faces.

It has been shown that using the central differing scheme is inadequate for describing the flow of convective-diffusive phenomena. For example, one of the major disadvantages of this scheme is the ability to identify flow direction.

Therefore, the properties of discretisation schemes are evaluated to ascertain their suitability via the following:

• Conservativeness
• Boundedness
• Transportability

Thus, alternative discretisation schemes have been developed to circumvent the weakness of central differencing schemes using the above mentioned properties.

• Upwind differencing schemes
• Hybrid differencing schemes
• Power law scheme

The disadvantage of the above 3 schemes is that although they are stable and obey transportability requirements, they are only 1^st order accurate thus making them prone to numerical diffusion errors. Consequently, the concept of higher order differencing schemes are considered including but not limited to

• QUICK (quadratic upstream interpolation for convective kinetics)
• TVD (total variation diminishing)

The QUICK scheme although they can minimise false diffusion errors, is known to give overshoot/undershoot or wiggle behaviour in certain conditions and so the TVD was developed to overcome this spurious oscillation behaviour.

To make a scheme TVD it must follow the criteria of the flux limiter. This is a concept used in higher order schemes to avoid erroneous oscillations that would otherwise occur due to real physical phenomena such as shockwaves. The idea is to limit the spatial derivative to realistic values.

Thus, TVD schemes which follow the flux limiter can give 2^nd order accurate solutions free of non-physical wiggles making it suitable for CFD applications.