FVM Literature Review

The finite volume method is a method of representing and evaluating the partial differential equation in the form of the algebraic equation. This method is applied over entire volume and properties are concentrated at the geometric centre of the volume concerned. In this method volume integrals in partial differential equation that contains a divergence term are converted to surface integrals, using the divergence theorem. This method is conservative form that the amount of flux entering the surface is equal to the amount of flux leaving the surface.The advantage of the this method is that it is easily formulated to allow unstructured meshes

METHOD TO SOLVE

Discretize the solution domain by a grid and define the computational node at which variables are to be evaluated.

Apply the integral form of conservative law to each finite volume.

Approximate the surface and volume integrals using appropriate quadratic formula in terms of the function values at computational nodes.

Collect the algebraic equations for all the volume to obtain a system of algebraic equation in terms of unknown values of the variable at the computational nodes .

Solve the resulting system of algebraic equation values of the variable at the each computational node

Example from the steady state heat conduction equation with a source term

`del/(delx)+(alpha(delt)/(delx))+S =0`

Where S is the source term

`alpha` is thermal diffusivity

multiplying the equation by dv and integrating it

`int((del)/(delx)(alpha(delT)/(delx))+S)dV=0`

`int((del)/(delx)(alpha(delT)/(delx))dV+intSdV=0`

dv=Adx

`int((del)/(delx)(alpha(delT)/(delx))Adx+intSdV=0`

`[alphaA(delT)/(delx)]_w^e+SdV`

S is the averge value source term which when multiplied by dv will be equal `intSdv`

`alpha_eA(delT)/(delx_e)-alpha _w(delT)/(delx_w)+SdV=0`

Assuming as constant and not function as temperature

`alpha A([T_e-T_p]/(trianglex))-alpha A([T_p-T_w]/(trianglex))+SdV=0`

(Heat flux)out-(Heat flux)in+source term=0

Hence the amount of heat removed from a particular box is conserved

Different schemes are used to solve the equation.

Gradient schemes

From the above figure when we calculate temperature gradient on the east face we basically use temperature from the neighbouring point trying to evaluate or numerically represent gradient of the temperature which is called gradient scheme.

Interpolation Scheme

Interpolation scheme is method of constructing new data points within the range of discrete set of known data points. It is often required to interpolate that is estimate the value of that function for an intermediate value of independent variable

Types of Interpolation scheme

Upwind interpolation

It is used to calculate the approximate value of variation at the east face of control volume it is first order accurate and is numerically diffusive with coefficient of numerical diffusions,it is equivalent to using backward or forward finite difference approximation .

Linear approximation the value of variable at the control volume face centre by two neatest computational nodes location .it is equivalent to the central difference schemes of the first order derivative and it is second order accurate and produces sporadic solution.

Quadratic Upwind interpolation

It is third order accurate but prone to oscilations it is derived from polynomial fitting

Flux limiters

Flux limiters are used in numerical schemes to solve problems in fluid dynamics described by te partial differential equation.They are used in numerical schemes to avoid wiggles that would otherwise occurs with high order spatial discretization scheme due to shocks discontinuities or sharp changes in the solution domain Use the flux limiters together with an appropriate high resolution resolution scheme make the solutions total variation diminishing.

Flux limiters are also refered to as slope limiters because they both have the same mathematical form and have the effect of limiting the solution gradient neat the shocks or discontinuities .In general flux limiters are used when the limiters acts on the system fluxes and slope limiters is used when the limiter is used when the limiter acts on system states

The main idea behind the construction of the flux limiters schemes is to limit the special derivatives to realistic values for scientific and engineering problems that usually means physicaly realisable and meaningful values.They are use in numerical schemes for solving problems described by PDE and only come into operation when sharp wave fronts are present