Week 9 - FVM Literature Review
The Finite Volume Method is a discretization method for the approximation of a single or a system of partial differential equations, expressing the conservation or balancing of one or more quantities.
In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface Integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative.
MAJOR DIFFERENCES BETWEEN FDM AND FVM:
1. The FDM discretization is based upon the differential form of the Partial Differential Equations or the Governing Equations to be solved whereas The FVM discretization is based upon the Integral form of the Partial Differential Equation or the Governing Equations.
2.The FDM the computational domain are converted into grid points called as nodes and the governing equations are solved to calculate the values at every node whereas in FVM the computational domain are converted into finite volumes and then the governing equations are solved for each and every volume.
3. In FDM we cannot use this method directly when the grids are not alligned along x,y or z direction whereas in FVM we can use this method for any geometry by defining the proper mesh element.
4. FVM conserves quantities better because also in discretization form we can tell that the equations are conserved inside the control volume while in FDM we cannot say that as there is no control volume.
5. In FDM it is not required to convert the geometry into mesh elements whereas in FVM we need to convert the geometry into mesh elements, as it is a necessary step in FVM.
6. Writing a FDM program is quite simpler than the FVM as in FVM it requires advance level of coding the program and it is a little bit complex also.
1 DIMENSIONAL HEAT CONDUCTION EQUATION:
The one dimensional heat conduction equation to solve using FVM is:
`(partialT)/(partialt) + alpha*((partial^2T)/(partialx^2)) =0`
which can be written as:
`(partial)/(partialx)(alpha*(partialT)/(partialx)) + S =0`
For FVM the governing equation must be written in Integral form;
`int((partial)/(partialx)(alpha*(partialT)/(partialx)) + S)dv =0`
After simplifying we will get:
`alphaA[(T_E - T_P)/dx] - alphaA[(T_P - T_W)/dx] + overline Sdv = 0`
where,
P = Present Volume to be calculated
W = Left side Volume
E = Right side Volume
alpha = Thermal Diffusivity
The first term can be called as 'Heat Flux Out' the second term can be called as 'Heat Flux In' and the third term is our source term.
From the above equation we can say that the quantities are conserved, In FVM the quantities are conserved inside the control volume also.
As Alpha and the Source term are function of Temperature so we need to linearize them.
Linearization can be used tp give important information about how the system behaves in the neighbourhood of equillibrium points or volumes and to find the local stability of an equillibrium point.
INTERPOLATION SCHEMES:
So in order to calculate the values of 'alpha' at different walls of the control volume and the we need to use Interpolation Schemes.
The interpolation schemes are used to transfer cell centre quantities to face centres.
The Interpolation schemes are used for the calculation of gradient, divergence and laplacian terms.
The approximations of Surface and Volume Integrals used in FVM requires values of the variables at locations other than the computational nodes of the control volume and the values at these locations are obtained by using Interpolation Schemes.
Flux Limiters are used while calculating gradients because we need to have thoose values bounded so that they cannot jump.
Basically the Flux limiters are used to avoid the spurious oscillations known as wiggles that would occur with high order spatial discretization schemes due to shock and discontinuities or sharp changes in the solution domain.