Week 9 - FVM Literature Review

AIM:-  FVM Literature Review

OBJECTIVE:-What is the Finite Volume Method(FVM), write down the major differences between FDM & FVM. Also, describe the need for interpolation schemes and flux limiters in FVM.

You can use the 1D linear heat conduction equation as an example.

Finite Volume Method (FVM)

In the finite volume method, the entire fluid domain is divided into a finite number of cells each associated with a finite-sized volume

Governing equations are solved at the cell controls,

i.e. the cell centers act as nodes and store the volume averaged values of field variables.

The mathematical foundation of the FVM Method because of the unstructured mesh Easy for solving complex problems over the other methods.

consider the 1D Heat Conduction Equation:

`∂/∂x(α .∂T/∂x)+S = 0

Where S = Source term

α = Thermal diffusivity

Let us consider the Control volume p and the centroid at the p as shown above.

The above figure has a set of control volumes with the centroids located at p, E(EAST), W(WEST), N(NORTH),S(SOUTH POINT), and the further point with the representation.

The faces center is denoted as n, e,s, and w.

Come back to the Heat conduction equation, because of the 1D, it doesn't have the North and south control volume.

Considering the small volume 'dv' in the Control volume P the integral form of the equation for the small volume is given by.

Advantages:-

volume averaged quantities to make much more physical meaning than point-like ones.

The FV formulation sources built-in conservation laws which is an extremely useful property for numerical implementations

Working of FVM

The governing equation is usually available in the PDE form

Convert the governing equation into their finite volume integral form

Apply Gauss- Divergence theorem to convert volume integrals to surface integrals and approximate / distance those integrals

Apply the boundary conditions suitably and solve the algebraic discrete equation using the appropriate solution method.

FDM VS FVM

 FINITE DIFFERENCE METHOD                                                                                                        FINITE VOLUME METHOD

     (FDM)                                                                                                                                              (FVM)

.Governing equations are in the differential form                                                            . Governing equations are in the integral form

.Solution computed at each node i.e. intersection                                                            . Solution computed at the cell centroids

 on a mesh line.                                                                                                            . Can handle unstructured grid and hence complex 

. Requires the use of a structured grid difficult to implement                                              geometries with the case.

  for complex geometry                                                                                                  . Driving schemes higher than second order is 

 .It is Easy to obtain higher order schemes, so preferable for                                             next to impossible due to the complexity involved

  in applications that require very high accuracy                                                                . Can handle discontinuities due to the nature of 

. All flow variables need to be differentiable and continuous                                              integral equations

  over the entire domain unsuitable for handling discontinuities                                       .imposes and ensures conservation qat both cell and     

  .Does not ensure conservation                                                                                    domain levels

Gradient schemes:-

Gradient schemes are used to approximate gradient terms in the governing equation (like the pressure gradient term).

The gradient is evaluated at cell centroids

Gauss gradient scheme.

Approximates the gradient slope using Gauss divergence theorem

Gradients are computed from face value which in turn requires interpolation.

Last squares gradient scheme.

Approximate the gradient at the cell center Grid using the neighbor cell centroid values (not face values)

The method is designed to compute the gradients such that the sum of squares of the errors is minimized.

.It results in more equations than the number of unknowns (one for each face )

Interpolation Schemes:-

interpolation schemes are used to evaluate if force values that arise as part of the convective flux terms in the discretization of its equation in the integral form

upwind interpolation scheme.

This is a first-order accurate interpolation scheme.

The scheme is bounded (non-oscillatory) and has a diffusive character based on the direction of flow

The face value is considered the same as the cell centroid of the neighboring in the upwind direction 

We compute the face values by the equation.

           `ϕf =  ϕp for F>0

ϕF =

          ϕf =  ϕN for F<0

. Linear interpolation Differencing scheme

.This is a second-order accurate interpolation scheme which unbounded and might generate oscially solutions

The face values are computed as a linear interpolation from the owner and neighbor cell

 . The face value is computed as

ϕF = fxϕp + (1-fx)ϕN

fx = fX/px = |xf - xN|/|d|

.Linear  upward Differencing 

.This is a second-order accurate interpolation scheme.

.it is employed for high convection (strong gradient flows.

.The scheme is unbounded and prone to oscillations

.The face values are calculated

           ϕ p+1/2(ϕp  - ϕpp = 2/3ϕp -1/2ϕp - 1/2ϕpp for F>0

 ϕ F =  

           ϕN +1/2 (ϕN - ϕNN) = 2/3ϕN - 1/2ϕNN for F<0

. The quadratic upwind interpolation scheme (Quick )

. This is a third-order accurate interpolation scheme 

. The face values were approximated using a three-point quadratic function with two surrounding nodes.

. The scheme is highly oscillatory 

. The face values are computed as

ϕ face = 6/8ϕi - 1 + 3/8 ϕi - 1/8 ϕi - 2

Flux limiters

. for the second and higher order interpolations (high resolution), which predict face values based on a gradient at cell centroids in the presence of strong gradients due to overshooting the face might exceed the local minimum/maximum.

. This will create new maxima and minima in the solution and might lead to the solution blowing up 

. This issue can be overcome by using a Flux Limiter function.

. The flux limiter function keeps monitoring the ratio of successive gradients and locally switches to the first order scheme low resolution.

. When it detects strong gradients.

. Adding a flux limiter of new local minima and maxima.

. Such higher resolution and bounded schemes are called total variation Diminishing (TVD) schemes.

. Commonly used flux limiters are super Bee, Minmod, and vanleer.