Week 9 - FVM Literature Review

Definition of the finite volume method(FVM)- It is a method for representing and evaluating the partial differential differencing in the form of algebraic equations. In the FVM volume integrals in a partial differential equation that contains a divergence term are converted to surface integrals using the divergence theorem. These terms are then evaluated as fluxes at the surface at each finite volume. FVM is conservative in its inherent conservative.

#) Difference between the FVM and the FDM-

FVM- 1)It evaluates the exact expressions for the average value of solutions over some volume and uses the data to construct an approximation of the solution within cells.

2) FVM discretization is based upon an integral form of the pde to be solved for example the conservation of the mass, momentum and the energy equation.

3) The PDE is written in a form that is solved for a given finite volume or the cell..

4) The computational domain is discretized into the finite volume and then for every volume, the governing equations are solved.

5)The resulting system of equation usually involves fluxes of the conserved variables and thus the calculation of the fluxes is very important in the FVM.

6)It does not require the structured mesh or grids and the effort to convert the given mesh into a structured numerical grid is completely avoided.

7)Since the FVM is an integral scheme since we are integrating the across the surface area the chance of error is minimized.

8) Integral schemes are slower but score over the FDM when the mesh is irregular.

9) FVM based discretization schemes are being used in the commercial software packages like the Ansys fluent.

The points for the FDM-

1)  It approximates the derivative using the nodal values.

2) Discretization is based upon the differential form of the pde to be solved. Each derivative is replaced with an approximate difference formula of the pde to be solved. Each derivative is replaced with an approximate difference formula taht is generally derived by the Taylor series expansion.

3) The computational domain is divided into the hexahedral cells called as the grid, the solution will be obtained at each nodes or nodal point.

4) The FDM is easiest to understand when the physical grid is cartesian but through the use of the curvilinear transformation the method can be extended.

5) The discretization results in a system of equations of the variable at the nodal points and once the solution is found then we have a discrete representation of the solution.

6) Since the FDM  method uses the differential form of the governing equation the rate of convergence is faster than the FVM .

7) FDM-based discretization software is mainly used in academic roles or academic circles.

8) The concept of conservation which is important for the fluid flows equations are not inherently built into the FDM codes but are built in the FVM codes.

#) Some of the  interpolation schemes are as follows-

1)Central differencing schemes(CDS)- Approximate value of the variable at the control volume face center for linear interpolation

2) Upwind interpolation approximation scheme(uds)- The name upwind refers to the assumption that we use the node in the upwind direction of the given node. It is used for the high convection nodes or the high peclet numbers flows.

forward differencing= (u(i+1)-ui)x/delta 

backward differencing = (ui-ui-1)x/delta

Taylor series expansion- vu_(i+1/2) = vu_i-v(deltax)/2*((delu)/(delx))-(i+1/2).

3) Quadratic upwind differencing scheme(QUICK)- It is used for the convective kinematics . It is a third-order flux approximation. Although it has second order overall accuracy. It approximates the value of the variable at cv face by quadratic interpolation of the values of the variable at cv face by centre by quadratic interpolation of the values at 3 nearest computational nodes.

#) Flux limiters basically used in high-resolution schemes, to avoid the random and violent oscillation that can occur with high order spatial discretization schemes due to shock and discontinuities or sharp change in the solution domain which is not really much appreciated. The logic behind the construction of the flux limiters scheme is to limit the spatial derivatives to values that are closer to real-world data.