Week 9 - FVM Literature Review

ANSWER:

For Understanding the need of Interpolation schemes and flux limiters, first we would try to do the Finite Volume Discretiztion of the Steady State One Dimensional Heat Conduction Equation with a Source term as shown below:

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

where

`T implies` Temperature

`alphaimplies`thermal diffusivity

`S implies`source term which represents the heat generation

Now let us consider the FV domain having FV elements and their centroids as shown in figure :

The above figure has 3 Finite volume elements with their respective centroids located at P (current point), W (west point) and E (east point). Also faces 'w' and 'e' denote the west and east faces respectively. Consider the small volume element 'dv' as shown in the middle finite volume element, the integral form of equation for the middle volume element can be written as:

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

`implies intdel/(delx)(alpha(delT)/(delx))dV+intSdV=0`

`implies intdel/(delx)(alpha(delT)/(delx))Adx+intSdV=0`

`implies [alpha A(delT)/(delx)]_e- [alpha A(delT)/(delx)]_w + barSdV = 0`

`implies alpha_e A((delT)/(delx))_e- alpha_w A((delT)/(delx))_w + barSdV = 0`

Now the method in which Temperature gradients in the above equations are described, defines the gradient schemes chosen for the given FVM setup. In our case we will be chooosing the gradient scheme which takes into account the temperature values at point 'P' and point 'E' for east face and the temperature values at point 'P' and point 'W' for west face repectively. Moreover, if the source term is a function of temperature itself (like the `alpha`), then it becomes necessary to linearize the source term. After selecting the gradient schemes and the linearization of source term, the equation becomes as shown below

`implies alpha_e A((T_E-T_P)/(Deltax))- alpha_w A((T_P-T_W)/(Deltax))_w + S_U+S_PT_P = 0`

`implies` Heat Flux at east face - Heat flux at west face + Source terms = 0

This is the FVM discretization. But we need to take care of the values of thermal diffusivities which are a function of temperature.

Need for interpolation schemes :

To calculate `alpha_e` we must know the value of `T_e` i.e. Temperature value at east face. Since we have a value available for `T_P,T_W and T_E` by interpolating these values we can get the value of `T_e and T_w` i.e. the temperatures at east and west faces respectively and thus the values of `alpha_e and alpha_w`can be calculated. This is the need of interpolation schemes.

Need for Flux Limiters :

The gradient schemes (of temperature in this case) may have very large values in case of discontinuities (or in case of say shock waves) and might give a vague results. So, in order to prevent the gradients to take large values, these gradients' values are bounded by what is known as the 'Flux Limiters'. These provided bounded ness to the vlaues of the gradients.