Week 9 - FVM Literature Review

Finite Volume Method (FVM):

Finite Volume Method is one of the numerical method to dicretise the governing equations of fluid flow,Heat Transfer etc.., Discretisation carried out by integrating the governing equation at a point on the domain over a control volume .Convective and diffusive fluxes at cell faces or CV faces can be interpolated with the help of few schemes from the control volume centered nodes.

For ex:Discretising the steady state 1D Heat conduction equation

k-conductivity,T-temperature,A-area,S-source term

1. `frac{d}{dx}(k*frac{dT}{dx})+S=0`

Integrate over the control volume

2.`int frac{d}{dx}(k*frac{dT}{dx}) dV+int S *dV=0`

by applying guass divergence theorem

3. `int frac{d}{dx}(k*frac{dT}{dx}) dV+int S *dV=int k*frac{dT}{dx} dA+int S*dV`

4. `int k*frac{dT}{dx} dA+int S*dV=underset(e)([k*frac{dT}{dx}*A])-underset(w)([k*frac{dT}{dx}*A])+underset(avg)(S)*Delta V`

interpolation described under the need for interpolation scheme and flux limiters in Fvm

5.`underset(e)([k*frac{dT}{dx}*A])=(frac{underset(P)(k)+underset(E)(k)}{2})*underset(e)(A)*frac{(underset(E)(T)-underset(P)(T))}{underset(PE)(deltax)}`

6.`underset(w)([k*frac{dT}{dx}*A])=(frac{underset(W)(k)+underset(P)(k)}{2})*underset(w)(A)*frac{(underset(P)(T)-underset(W)(T))}{underset(WP)(deltax)}`

source term has linearised for the dependent function

7.`underset(avg)(S)=underset(u)(S)+underset(P)(S)underset(P)(T)`

substituting and re-arranging above 5,6,7 on 4 equation ,we get

8.`(frac{underset(e)(k)}{underset(PE)(deltax)}*underset(e)(A)+frac{underset(w)(k)}{underset(WP)(deltax)}*underset(w)(A)-underset(P)(S))*underset(P)(T)=frac{underset(w)(k)}{underset(WP)(deltax)}*underset(w)(A)*underset(W)(T)+frac{underset(e)(k)}{underset(PE)(deltax)}*underset(e)(A)*underset(E)(T)+underset(u)(S)`

the above equation is the Discretised equation

Differences between Finite Volume Method (FVM) & Finite Difference Method (FDM) ?

Finite Volume Method (FVM)	Finite Difference Method (FDM)
1.properties calculated at control volume centered Nodes	1.Properties calculated at nodes
2.it works based on the integration and Guass divergence theorem	2.it works based on the taylor series
3.Conservation is depends on the stability	3.Conservation and non conservation form does not depend on stability
4.Naming the nodes by the cardinal directions	4.Naming the nodes by the numbers
5.It works on unstructured grid	5.It works on structured/rectangular Grid
6.convergence determined by series test such as ratio test	6.convergence determined by property of conservativeness,boundedness,transportiveness
7.TVD schemes is used for time marching	7.explicit and implicit approach is used for time marching
8.FVM would work with complex geometries	8.FDM work with simple geometries

Need for the interpolation schemes and flux limiters in FVM:

To determine the diffusion conductances at the cell faces of the control volume ,with the help of properties at the control volume node center by the interpolation schemes.

`underset(e)([k*frac{dT}{dx}*A])-underset(w)([k*frac{dT}{dx}*A])+underset(avg)(S)*Delta V=0`

parameters in the above formula defined at cell faces (with small letters as subscript,where e=east face of the primary cell (or) CV and w=west face of primary cell (or) CV)(CV indicates -Control Volume )

Linear interpolation scheme

`underset(e)(k)=frac{underset(P)(k)+underset(E)(k)}{2},underset(w)(k)=frac{underset(P)(k)+underset(W)(k)}{2}`

parameters with Subscript as capital letters indicates the centered node of the Control Volumes ,where E represents the East Control Volume to the Primary Control volume ,where W represents the West Control Volume to the Primary Control volume.For differencing the temperature used a central differencing scheme

`underset(e)([k*frac{dT}{dx}*A])=(frac{underset(P)(k)+underset(E)(k)}{2})*underset(e)(A)*frac{(underset(E)(T)-underset(P)(T))}{underset(PE)(deltax)}`

similarly

`underset(w)([k*frac{dT}{dx}*A])=(frac{underset(W)(k)+underset(P)(k)}{2})*underset(w)(A)*frac{(underset(P)(T)-underset(W)(T))}{underset(WP)(deltax)}`

Flux limiter is to limit the range of additional convective mass flux of different schemes to get converged and improve the order of the scheme

additional convective mass flux `F*psi(r)(underset(E)(phi)-underset(P)(phi))`

F-convective mass flux per unit area =`density * velocity`

`underset(E)(phi)-`parameter at centered node of east CV to the Primary CV

`underset(P)(phi)-`parameter at centered node of West CV to the Primary CV