Week 9 - FVM Literature Review
Finite Volume Method(FVM):
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. 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. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. "Finite volume" refers to the small volume surrounding each node point on a mesh.
Divergence theorem:
The divergence theorem states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface.
`intint_s(F.n)ds=intintint_vDiv(F)dv`
where F is the force vector and n is the normal vector to the surface.
Steps in Finite Volume discretization,
1. Write the governing equations in its integral form.
2. Apply Gauss divergence theorem.
3. Come up with suitable methods to calculate flux terms.
Difference between FDM and FVM:
Finite volume methods can be compared and contrasted with the finite different methods, which approximate derivatives using nodal values, or finite element methods , which create local approximations of a solution using local data, and construct a global approximation by stitching them together. In contrast a finite volume method evaluates exact expressions for the average value of the solution over some volume, and uses this data to construct approximations of the solution within cells.
In FDM the grid is linear and aligned so that all the grid points align in a straight line.The properties are considered to be concentrated at a single point.
In FVM the method the governing equations are valid throughout the entire volume.The properties are concentrated at the geometric centre of the volume.
FDM cannot be applied for complex geometries whereas FVM can be used for complex geometries.
One Dimensional steady state Heat conduction:
`del/(delx).alpha(delT)/(delx)+S=0`
Where,
S is the source term
`alpha` is the thermal diffusivity
`(delT)/(delx)`is the temperature gradient along x direction
Multiplying the above equation by dv and integrqting it we get,
`int(del/(delx).alpha(delT)/(delx)+S).dv=0`
`int(del/(delx).alpha(delT)/(delx))dv+intS.dv=0`
`dv = A.dx`
`int(del/(delx).alpha(delT)/(delx))A.dx+intS.dv=0`
`[alphaA((delT)/(delx))]_w^e+barS.dv=0`
`barS` is the average value of source term when multiplied by dv will be equal to `intS.dv`
`alpha_eA(delT)/(delx_e)+a_wA(delT)/(delx_w)+barS.dv=0`
If we assume `alpha` as a constant and not a function of temperature
`alphaA[(T_e-T_p)/(Deltax)]-alphaA[(T_p-T_w)/(Deltax)]+barS.dv=0`
If the source term is a function of temperature we have to lineraize it.
The equation can also be written as
`[Heat flux]_(Out)-[Heat flux]_(In)+Source Term=0`
Interpolation schemes:
In the finite volume formulation, evaluation of surface and volume integrals sometimes requires the values of unknown variables at locations other than the computational nodes of the control volume.These values are obtained by employing certain interpolation schemes.
Some interpolation scheme used in CFD are as follows
1. Upwind interpolation scheme
2. Linear interpolation scheme
3. Quadratic upwind interpolation for convective dynamics (QUICK) scheme
4. Hybrid scheme
5. Total Variation Diminishing (TVD) scheme
Upwind interpolation scheme:
The Upwind interpolation scheme takes into account the direction of the fluid flow and based on it the value of solution variable at the upstream computational node.For instance ,if the fluid is moving from west to east, then:
`phi_eapproxphiP`
And if the fluid is moving from east to west then,
`phi_eapproxphiE`
Linear interpolation Scheme:
It is the simplest and most widely used interpolation. In this case we approximate the values of the variable at control volume face centre by interpolating the values linearly at the two nearest computational nodes.The linear interpolation is equivalent to the use of the central difference scheme.
Quadratic upwind interpolation:
In this case we have to approximate the value of the variable at control volume face centre by quadratic interpolation at three nearest computational nodes.
Hybrid scheme:
This scheme is the combination of upwind interpolation scheme and the linear interpolation based on the local Peclet number.
`f_e =yf_(CDS)+(1-y)f_(CDS)`
Flux limiters:
The main idea behind the construction of flux limiter schemes is to limit the spatial derivatives to realistic values—for scientific and engineering problems, this usually means physically realizable values. They are used in high-resolution schemes for solving problems described by PDE's and only come into operation when sharp wave fronts are present. For smoothly changing waves, the flux limiters do not operate and the spatial derivatives can be represented by higher-order approximations without introducing nonreal oscillations.