Week 9 - FVM Literature Review

What is Finite Volume Method(FVM), write down the major differences between FDM & FVM. Also, describe the need for interpolation schemes and flux limiters in FVM.

Ans

In computational fluid dynamics there are 3 numerical method based solvers to solve the governing navier strokes equations these are:

1. Finite difference method

2. Finite element method

3. Finite volume method

In finite difference method(FDM):

The grid is linear and aligned so that all grid points lie in a straight line . Main thing is that the properties are considered to concentrated at a single point.

In finite volume method(FVM):

The method is applied over a volume such that the governing equation is valid over the entire volume , aslo the properties are assumed to be concentrated at the geometric centre of the volume concerned.

In finite element method(FEM):

The properties are valid over an elemental area. the properties assumed to be concentrated at area centre.

Drawbacks of Finite Difference Method:

1. we cannot use FDM directly to a body whose grid points are not aligned properly along the x , y directions. these may occur due to complex folds , cuts in the structure.

2. as it is not aligned we cannot calculate the properties at the next adjacent point exactly, but we can make some correction factor and approximate it , this method is called as 'Variable Transformation Method' which basically seeks to create grid using different variable space which look very similar to cartesian grid.

3. variable transformation method gives an approx solution of property but it is time consuming and complex.

4. for complex profiles , this method is less prferred.

That's why Finite Volume Method (FVM) is used

Finite Volume Method

The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages. "Finite volume" refers to the small volume surrounding each node point on a mesh.

In FVM instead of writing the equation in a single point, equations are written in an entire volume. It means for a considered volume no assumptions are made over the shape of the volume. Inside the particular volume the flow is conserved, flow quantities are conserved, energy is conserved, mass is conserved, momentum is conserved and so on. This is called as 'Finite Volume Representation' which begins with 'Integral form of governing equation'.

Steps for FVM discretisation :

• We will write the required governing equation in its integral form.
• Apply Gauss Divergence Theorem.
• We obtain discretised form which can be solved using a computer ("Flux Terms").

Consider a steady state heat conduction equation with a source term (heat generation source) :

where

S is the Source term.

`alpha` is thermal diffusivity term.

`(delT)/(delx)` is Temperature gradient along x direction.

The above governing equation is valid at every point in the domain, meaning it is valid for the entire control volume.

If we consider a small control volume (dv) and it is multiplied with the governing equation and integrated for entire volume then that should also be equal to zero i.e. the integral form of governing equation.

Taking the conservation equation validated at single point and if it is validated at single point then it is validated atentire volume and in this entire volume the properties are conserved.

Multiplying the equation by 'dv' and integrating it, we obtain

`barS` is the average value of source term (S) which when multiplied by dv will be equal to ∫Sdv.

If we assume `alpha`as a constant and not as a function of temperature,

The source term which is to be added in a cell, if it is a function of temperature we have to linearize it. If we consider `alpha`as a function of temperature we have to linearize it as well.

The amount of heat that is removed from a particular box is conserved.

Interpolation Scheme :

Interpolations schemes contains terms that are interpolations of values typically from cell centres to face centres.

Here w and e are flux .

To find out the flux will use interpolation schemes i.e:

`T_w=(T_p+T_W)/(Deltax)`

`T_e=(T_P+T_E)/(Deltax)`

The interpolationSchemes sub-dictionary contains terms that are interpolations of values typically from cell centres to face centres. being divided into 4 categories: 1 category of general schemes; and, 3 categories of schemes used primarily in conjunction with Gaussian discretisation of convection (divergence) terms in fluid .It is highly unlikely that the user would adopt any of the convection-specific schemes for general field interpolations in the interpolationSchemes sub-dictionary, but, as valid interpolation schemes.

A general scheme is simply specified by quoting the keyword and entry, e.g. a linear scheme is specified as default by:

The convection-specific schemes calculate the interpolation based on the flux of the flow velocity. The specification of these schemes requires the name of the flux field on which the interpolation is based; in most OpenFOAM applications this is phi, the name commonly adopted for the surfaceScalarField velocity flux . The 3 categories of convection-specific schemes are referred to in this text as: general convection; normalised variable (NV); and, total variation diminishing (TVD). With the exception of the blended scheme, the general convection and TVD schemes are specified by the scheme and flux, e.g. an upwind scheme based on a flux phi is specified as default by:

Some TVD/NVD schemes require a coefficient where corresponds to TVD conformance, usually giving best convergence and corresponds to best accuracy. Running with is generally recommended. A limitedLinear scheme based on a flux phi with is specified as default by:

Apart Interpolation schemes there are something called Flux Limiters.

Flux limiter:

For convective fluid flow, it is ovservedthat low order schemes are usually stable but quite dissipative in nature around the points of discontinuity or shocks while the higher order schemes are unstable in nature and show oscillations in the vicinity of dinscontinuity or shocks. Highly accurate and oscillation free schemes are known as High Resolution Schemes.

In order to make the scheme TVD, we must constrain or limit the range of possible values of the additional convective flux

which was originally introduced to make the scheme higher-order. Hence, the function`varphi(r)` is called a flux limiter function.

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 scheme is employed.

`f_(low)`is low precision flux (first order)

`f_(high)`is high precision flux (higher order)