Week 9 - FVM Literature Review

Major difference between finite volume and finite difference method:

There are four numerical methods:

1. Finite element method
2. Boundary element method
3. Finite volume method
4. Finite difference method

Finite volume method:

The Finite volume method (FVM) is a widely used numerical technique. The fundamental conservation property of the FVM makes it the preferable method in comparison to the other methods, i.e., FEM, and finite difference method (FDM). Also, the FVM’s approach is comparable to the known numerical methods like FEM and FDM, which means that its evaluation of volumes is at discrete places over a meshed geometry.

Furthermore, the FVM transforms the set of partial differential equations into a system of linear algebraic equations. Although the discrete approximation procedure in use in the FVM is distinctive, it also utilizes two basic steps. Firstly, it transforms and integrates the PDEs into balance equations over an element. The process incorporates the changing of the surface and volume integrals into discrete algebraic relations over elements as well as their surfaces using an integration quadrature of a specified order of accuracy. This will result in a set of semi-discrete equations.

Secondly, in this next step, the interpolation profiles are chosen to estimate the variation of the variables within the element and relate the surface values of the variables to their cell values and thus transform the algebraic relations into algebraic equations. In regards to the two steps in the FVM process, your approximation selection affects the overall accuracy of the subsequent numeric.

Finite difference method:

The finite-difference method is the most direct approach to discretizing partial differential equations. You consider a point in space where you take the continuum representation of the equations and replace it with a set of discrete equations, called finite-difference equations. The finite-difference method is typically defined on a regular grid and this fact can be used for very efficient solution methods. The method is therefore not usually used for irregular CAD geometries, but more often for rectangular or block-shaped models.

There is a connection with the finite-element method: Certain formulations of the finite-element method defined on a regular grid are identical to a finite-difference method on the same grid. Regular grids can often be used in meteorological, seismological, and astrophysical simulations,

With the finite-difference method, you may easily run into problems handling curved boundaries for the purpose of defining the boundary conditions. Boundary conditions are needed to truncate the computational domain. They represent communication with the surrounding world, which is the part that you do not want included in your simulation. If one can overcome the boundary-condition problem on curved boundaries, the method gives very efficient and high quality results.

Finite volume method use to solve the one dimensional heat conduction equation:

Diffusion coefficient

S = source term

Temperature gradient in x direction

dv = A*dx

α_w = αW + αP / 2

α_e = αP + αE / 2

Diffusion flux evaluated the

S ∆V = S_u + S_p φ_p

Adding three equation

- + S ∆V = S_u + S_p φ_p

Interpolation scheme:

Interpolation schemes are used to transform cell-centre quantities to face centres. The operation is used in many of the finite volume calculations, e.g. for the calculation of gradient, divergence, and Laplacian terms.

)

= Density

= velocity

Integration transport equation is

Some interpolation schemes used in CFD :

1. Up wind differencing scheme
2. Central differencing scheme
3. Hybrid differencing scheme

Up wind differencing scheme:

One of the major inadequacies of the central differencing scheme is its inability to identify flow direction. The value of property φ at a west cell face is always influenced by both φP and φW in central differencing. In a strongly convective flow from west to east, the above treatment is unsuitable because the west cell face should receive much stronger influencing from node W than from node P. The upwind differencing or ‘donor cell’ differencing scheme takes into account the flow direction when determining the value at a cell face: the convected value of φ at a cell face is taken to be equal to the value at the upstream node.

Φ_w = φ_W and φ_e = φ_p

The discretised equation becomes F_e φP – F_w φW = D_e(φE – φP ) – Dw(φP – φW)

Which can be rearranged as (D_w + D_e + F_e )φP = (D_w + F_w)φ_W + D_e φ_E

Central differencing scheme:

The central differencing approximation has been used to represent the diffusion terms which appear on the right hand side of equation and it seems logical to try linear interpolation to compute the cell face values for the convective terms on the left hand side of this equation. For a uniform grid we can write the cell face values of property φ as

Φ_e = (φ_p + φ_E )/2

Φ_w = (φ_W + φ_P )/2

F_e/2 (φ_p + φ_E ) – (φ_W + φ_P ) = D_e(φ_E – φ_P ) – D_W(φ_P – φ_W)

Hybrid differencing scheme:

The hybrid differencing scheme of Spalding (1972) is based on a combination of central and upwind differencing schemes. The central differencing scheme, which is second-order accurate, is employed for small Peclet numbers (Pe < 2) and the upwind scheme, which is first-order accurate but accounts for transportiveness, is employed for large Peclet numbers (Pe ≥ 2). As before, we develop the discretisation of the one-dimensional convection– diffusion equation without source terms. This equation can be interpreted as a flux balance equation. The hybrid differencing scheme uses piecewise formulae based on the local Peclet number to evaluate the net flux through each control volume face. The Peclet number is evaluated at the face of the control volume. The hybrid differencing formula for the net flux per unit area through the west face is as follows:

qw = Fw 1 + φW + 1 − φP for −2 < Pew < 2