Week 9 - FVM Literature Review

Finite Volume Method: it is a method of representing and evaluating the partial differential equation in the form of algebraic equations. by using the gauss divergence theorem, the volume integrals in a partial differential equation that contains the divergence of a vector field converts into a surface integral of a vector field. then these surface integral terms are evaluated as a flux at the surface of each finite volume. FVM is a conservative method because the amount of flux entering into a given volume is identical to leaving from an adjacent volume.

Advantage of FVM: it is easy to apply on an unstructured grid/mesh.

Note: Finite volume refers to a small volume that surrounds each node of the mesh.

The difference between FDM and FVM:

	FDM	FVM
1.	used only for structured mesh.	can also be used for unstructured mesh.
2.	compute the properties at every node	compute the properties at every cell instead of node
3.	use differential form `(delf)/(delx)=(f_(i+1)-f_i)/(Deltax)`	use integral form `intint(delf)/(delx)dxdy=intfdy=c*Area`
4.	Use Taylor series	Use Divergence theorem
5.

Consider 1-D heat conduction equation.

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

now integrate over the finite volume dV

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

`int_w^edel/(delx)(alpha(delT)/(delx)).Adx+int_w^es.Adx=0`

`alpha(T_E-T_P)/(Deltax)-alpha(T_P-T_W)/(Deltax)+S_u+S_PT_P=0`

Heat flux out-Heat flux in+Source term=0

Need for interpolation scheme:

we are finding the temperature gradients `((delT)/(delx))_w and ((delT)/(delx))_e`at west face and east face with the help of temperatures at grid pint W and E by using liner interpolation method.

Interpolation schemes are:

1. linear interpolation scheme

2. Quadratic interpolation scheme

3. Hybrid interpolation scheme

4. Upwind interpolation scheme