Week 9 - FVM Literature Review

AIM: To describe the need for interpolation schemes and flux limiters in FVM.

THOERY: Discretization is the process of converting partial Differential Equation in to Algebraic form. The whole idea behind discretization is to get the solution at descrete points.

Discretization methods are used to chop a continuous function (i.e., the real solution to a system of differential equations in CFD) into a discrete function, where the solution values are defined at each point in space and time. Discretization simply refers to the spacing between each point in your solution space.

Methods of Discretization 

1. Finite Difference Methiod: Simple, Straightforward to apply and Structured Grids. 

2. Finite Element Method: applied to any Geometries.

3. Finite Volume Method: Conservative and applied to any Geometries.

 FINITE VOLUME METHOD: 

• The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. Similar to the finite difference method (FDM) or finite element method (FEM), values are calculated at discrete places on a meshed geometry.
• One important feature of finite volume schemes is their conservation properties. Since they are based on applying conservation principles over each small control volume, global conservation is also ensured.
• "Finite volume" refers to the small volume surrounding each node point on a mesh means the grid points where variables are stored are typically defined as being at the centre of each control volume. 
• The method starts by dividing the flow domain into a number of small control volumes.
• 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.
• The advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes, and this method is used in many computational fluid dynamic packages.

Consider 2 particular finite volumes A and B as shown above. The flux is going in and out of the volume  To get the complete set of domain or the grid all the individual volumes like A and B  are combined 

Features of FVM 

1. FVM can accomodate any type of grid and hence it is naturally suitable for any type of grid 

2. Unstructured grid generation methods and mesh partitioning techniques developed for Finite element  Method can be directly used in FVM.

Interpolation schemes in Finite Volume Method:

Interpolation is a process to estimate the values at unknown points by using the points with known value and sample points on either sides of the unknown point.  In FVM, interpolation schemes are used to find values of volume integrals required at the points other than nodes. 

The different interpolation schemes are:

1. Upwind Interpolation Scheme (UDS)

2. Linear Interpolation Scheme

3. Quadratic Upwind Interpolation Scheme

4. Hybrid Interpolation Scheme

5. Total variation Diminishing Scheme

6. Essentially Non-oscillatory interpolation Scheme 

1. Upwind Interpolation Scheme (UDS):

The upwind interpolation scheme (UDS) for approximating the value of a variable f  at the east face of a control volume is given by

        `f_e= {(f_(p ) if (v.n)_e>0),(f_E if (v.n)_e<0):}`

It is generally used convection dominated problems (i.e pe >1). 

The values of the interface is given by:

for v > 0, we have

`u_(i-1/2) approx u_(i-1) , u_(i+1/2) approx u_(i)`

`I_c approx v {u_i - u_(i-1)}/{Deltax}`- Backward difference

for v < 0, we have

`u_(i-1/2) approx u_(i) , u_(i+1/2) approx u_(i+1)`

`I_c approx v {u_(i+1) - u_i}/{Deltax}` - Forward difference

Taylor series expansion

`vu_(i+1/2) = vu_i - {vDeltax}/2({delu}/{delx})_(i+1/2) - v(Deltax)^2/8({del^2u}/{delx^2})_(i+1/2) +.....`

a first order accurate flux approximation, the leading truncation error resembles a diffusive flux `d{delu}/{delx}` with `d = {vDeltax}/2` being numerical diffusion coefficient.

2. Linear Interpolation:

To approximate the values of variables at CV face centre by linear interpolation of the values at two nearest computational nodes. This is equivalent to the use of the central difference scheme of first order derivative, and hence it is also known as central difference scheme.  This also leads to very simple approximation of evaluation of gradients required for evaluation of diffusive fluxes. The relation is given as follows:

`f_e = f_E lambda_e + f_p(1-lambda_e)` where `lambda_e = {x_e - x_p}/{x_E-x_p}`

Interpolation polynomial:

`p_1(x) = u_L.{x_R-x}/{x_R-x_L} + u_R.{x-x_L}/{x_R-x_L}`

Average interface values are given by,

`u_(i-1/2) approx {u_(i-1) + u_i}/2`and `u_(i+1/2) approx {u_(i) + u_(i+1)}/2`

`I_c approx v.{u_(i+1) - u_(i-1)}/{2.Deltax}` - Central differencing

Taylor series approximation

`u_(i+1) = u_(i+1/2) + {Deltax}/2 .({delu}/{delx})_(i+1/2) +(Deltax)^2/8.({del^2u}/{delx^2})_(i+1/2) + .....`

`u_i = u_(i+1/2) - {Deltax}/2 .({delu}/{delx})_(i+1/2) +(Deltax)^2/8.({del^2u}/{delx^2})_(i+1/2) + .....`

From the above expression we get,

`u_(i+1/2) = {u_i + u_(i+1)}/2 - (Deltax)^2/8.({del^2u}/{delx^2})_(i+1/2)+...`

The approximation is second order accurate. It is the simplest and most widely used interpolation for evaluation of gradients reqired for evaluation of diffusive fluxes.  

3. Quadratic Upwind Interpolation Scheme (QUICK): This approximates the value of variable at CV face centre by quadratic interpolation of the values at three nearest computational nodes (one downstream node, D and two upstream nodes, U). It is third order accurate for both uniform and nonuniform grids.

`f_e = f_U + g_1(f_D - f_U) + g_2(f_(UU) -f_u)`

`g_1 = {(x_e - x_U)(x_e - x_(UU))}/{(x_D - x_U)(x_D-x_(UU))`

`g_2 = {(x_e - x_U)(x_e - x_D)}/{(x_(UU) - x_D)(x_(UU)-x_(U))`

The relation obtained as:

`phi_e = 6/8phi_c + 3/8phi_D - 1/8phi_(UU)`

4. Hybrid Interpolation Scheme: It is the blend of CDS and UDS scheme, based on Peclet number

`f_e = gammaf_(CDS) + (1-gamma)f_(UDS)`

5. Total variation Diminishing scheme:

`f_e = f_p + 1/2 psi(r)(f_E - f_p)`

where `psi` is the flux limiter.

FLUX LIMITERS:

For convective fluid flow, it is observed that the low order schemes are usually stable but quite dissipative in nature around the points of discontinuity or shocks while the higher order  schemes are unstable in nature and show oscillations in the vicinity of discontinuity or shocks. Highly accurate and oscillation free schemes are known as High Resolution Schemes.

Flux Limiter functions are used to fine tune the higher order and lower order schemes in such a way that the resulting scheme gives a higher order accuracy in the smooth region of the flow and maintains first order accuracy in the vicinity of shocks and discontinuities. For such a scheme TVD scheme is employed.

`F(u_(i+1/2)) = f^(low) (i+1/2) - phi(r_i)(f^(low) (i+1/2)-f^(high) (i+1/2))`

`F(u_(i+1/2)) = f^(low) (i-1/2) - phi(r_i -1)(f^(low) (i-1/2)-f^(high) (i-1/2))`

where `f^(low)` = Low precision flux (1st order accurate)

           `f^(high)` = High precision flux (Highest order accurate)

             `phi(r)`   = Flux limiter function where, r = `{u_i-u_(i-1)}/{u_(i+1) - u_i`