Week 9 - FVM Literature Review

Introduction :

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 (FM) 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 several 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.

FDM vs FVM:

- A finite difference method (FDM) discretization is based upon the differential form of the PDE to be solved. Each derivative is replaced with an approximate difference formula (that can generally be derived from a Taylor series expansion). The computational domain is usually divided into hexahedral cells (the grid), and the solution will be obtained at each nodal point. The FDM is easiest to understand when the physical grid is Cartesian, but through the use of curvilinear transforms the method can be extended to domains that are not easily represented by brick-shaped elements. The discretization results in a system of equations of the variable at nodal points, and once a solution is found, then we have a discrete representation of the solution.

- A finite volume method (FVM) discretization is based upon an integral form of the PDE to be solved (e.g. conservation of mass, momentum, or energy). The PDE is written in a form that can be solved for a given finite volume (or cell). The computational domain is discretized into finite volumes and then for every volume, the governing equations are solved. The resulting system of equations usually involves fluxes of the conserved variable, and thus the calculation of fluxes is very important in FVM.

The basic advantage of this method over FDM is it does not require the use of structured grids, and the effort to convert the given mesh into a structured numerical grid internally is completely avoided. As with FDM, the resulting approximate solution is discrete, but the variables are typically placed at cell centres rather than at nodal points.

Interpolation schemes in Finite Volume Method:

Interpolation is a process to estimate the values at unknown points by using the points with known values and sample points on either side of the unknown point. In FM; 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:

The upwind interpolation scheme is essentially used for convection-dominated problems. This interpolation scheme is essentially equivalent to using the forward difference method or backward difference method depending on the flow direction. Here, the direction of velocity at a given point determines the variable value to be used in an interpolation process. As the name suggests, we use the node upwind or upstream from the given node. The upwind interpolation (UDS) for approximating the value of variable "fat the east face of the control volume is given by:

`f_e = {(f_p,if (V.n)_e >0),(f_E,if(V.n)_e<0):}`

The values for the interface is given by

for V>0

`u_(i-1/2) cong u_(i-1)`

`u_(i+1/2) cong u_(i)`

`I_c = v(frac(u_(i)-u_(i-1))(Deltax))`

for V<0

`u_(i-1/2) cong u_(i)`

`u_(i+1/2) cong u_(i+1)`

`I_c = v(frac(u_(i+1)-u_(i))(Deltax))`

The resulting Tylor series expansion is

`v.u_(i-1/2) = vu_1-frac(vDeltax)(2)frac(delu)(delx)_(i+1/2)-frac(vDeltax^2)(8)frac(del^2u)(delx^2)_(i+1/2)+.....`

Upwind interpolation is the only approximation that satisfies the boundedness criterion unconditionally in the sense that irrespective of the velocity values and grid spacing, the solution will always remain bounded. This is the use of upwind interpolation which will never yield oscillatory solutions. Upwind Interpolation is numerically diffusive. If we have sharp peaks or oscillatory values which are present in the solution then those would be damped out because of the presence of numerical diffusion and hence numerical diffusion is magnified in multi-dimensional flows, especially in cases where the flow is oblique to the grid space. Peaks or rapid variations in the variables will be smeared out because of the introduction of an additional diffusive coefficient which is called a numerical diffusive coefficient when UDS is used. It is first-order accurate and hence grids are required to obtain accurate solutions.

2. Linear Interpolation(CDS):

The approximated value of the variable at control volume centroid by linear interpolation of the values at two nearest computational nodes.

`f_e = f_Elamda_e+f_p(1-lamda_e)`

where

`lamda_e=(x_e-x_p)/(x_E-x_P)`

The linear interpolation is equivalent to the use of the CDS formula of the first-order derivative, and hence this scheme is also termed as Central Differencing Scheme(CDS). This is 2nd-order accurate and may produce oscillatory results therefore small grids are preferred for this method. This is one of the most widely used interpolation methods.

The linear interplotation is equalient to the use of central differenceing formula of the first order derivative

Interplotation ploynamial

`P_1(x) = u_Lfrac(x_R-x)(x_R-x_L) + u_Rfrac(x-x_L)(x_R-x_L)`

average Interface vales

`u_(i-1/2)congfrac(u_(i-1)+u_i)(2)`

`u_(i+1/2)congfrac(u_(i)+u_(i+1))(2)`

`I_c = vfrac(u_(i+1) -u_(i-1))(2Deltax)`

The tylor expansion is given by

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

3. Quadratic Upwind Interpolation(QUICK):

In computational fluid dynamics QUICK, which stands for Quadratic Upstream Interpolation for Convective Kinematics, is a higher-order differencing scheme that considers a three-point upstream weighted quadratic interpolation for the cell face values. In computational fluid dynamics, there are many solution methods for solving the steady convection-diffusion equation. Some of the used methods are the central differencing scheme, upwind scheme, hybrid scheme and QUICK scheme. To find the cell face value, a quadratic function passing through two bracketing or surrounding nodes and one node on the upstream side must be used. In the central differencing scheme and second-order upwind scheme, the first-order derivative is included and the second-order derivative is ignored. These schemes are therefore considered second-order accurate whereas QUICK does take the second-order derivative into account, but ignores the third-order derivative hence this is considered third-order accurate. This scheme is used to solve convection-diffusion equations using second-order central difference for the diffusion term and the convection term the scheme is third-order accurate in space and first-order accurate in time. QUICK is most appropriate for steady flow or quasi-steady highly convective elliptic flow.

Interplotation polunomial

`P_2(x) = u_Lfrac(x-x_M)(x_L-x_M)frac(x_R-x)(x_R-x_L) + u_Mfrac(x-x_L)(x_M-x_L)frac(x_R-x)(x_R-x_M)+u_Rfrac(x-x_L)(x_R-x_L)frac(x-x_R)(x_R-x_M)`

upwind biased interface value

if v>0,

`u_(i-1/2)congfrac(3u_i+6u_(i-1)-u_(i-2))(8)`

`u_(i+1/2)congfrac(3u_(i+1)+6u_(i)-u_(i-1))(8)`

`I_c=vfrac(3u_(i+1)+3u_i-7u_(i-1)-u_(i-2))(8Deltax)`

if v<0

`u_(i-1/2)congfrac(3u_(i-1)+6u_(i)-u_(i+1))(8)`

`u_(i+1/2)congfrac(3u_(i)+6u_(i+1)-u_(i+2))(8)`

`I_c=vfrac(3u_(i-1)+3u_i-7u_(i+1)-u_(i-2))(8Deltax)`

4. Hybrid Interpolation scheme:

This interpolation method is a blend of both central differencing scheme and Upwind differencing scheme which is based on local peclet number. It can be described by the general partial equation as follows:

`frac(del)(delt)(rhophi)+nabla(rhouphi) = (nabla(Gammafrac(delphi)(delx)))+S_(phi)`

Here, `rho` is density, 'u' is the velocity vector, `Gamma` is the diffusion coefficient & `S_(phi)` is the source term. In this equation property, `phi` can be temperature, internal energy, or component of velocity vector u in x, y, and z directions.

For one-dimensional analysis of the convection-diffusion problem in steady-state and without the source, the equation reduces to,

`frac(del)(delx)(rhouphi) = frac(del)(delx)(Gammafrac(delphi)(delx))`

where

0<x<L

Here, Peclet number(Pe) is a non-dimensional parameter used as a measure of the relative strengths of convection and diffusion given as:

`Pe = F/D = frac(rhou)(frac(Gamma)(delx))`

The grid used for discretization in Central Difference Scheme:

The grid used for discretization in Central Difference Scheme:

The grid used for discretization in Central Difference Scheme:

Flux limiter

Flux limiters are used in high-resolution schemes - numerical schemes used to solve problems in science and engineering, particularly fluid dynamics, described by partial differential equations (DE's). They are used in high-resolution schemes, such as the MUSCL scheme; to avoid the spurious oscillations (wiggles) that would otherwise occur with high order spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain. The use of flux limiters, together with an appropriate high-resolution scheme, makes the solutions Total variation diminishing (TVD). 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 realisable and meaningful values. They are used in high-resolution schemes for solving problems described by PDEs and only come into operation when sharp wavefronts 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 spurious oscillations. Consider the ID semi-discrete scheme below

`frac(du_i)(dt) + frac(1)(Deltax_i)[F(u_(i+1/2))-F(u_(i-1/2))] = 0`

Where `F(u_(i+1/2))` and `F(u_(i-1/2))` represents edge flux for the `i^(the)` cell . if these edge fluxes can be represented by low and high resolutions schemes then flux limiter can switch between these schemes depending upon the gradient close to the particular cell as follows

`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 , High resolution flux

`F^(high)` = high precision , low resolution flux

`phi(r)` = Flux limiter

and r represents the ratio of successive gradient on the solution mesh

`r_i = frac(u_i-u_(i-1))(u_(i+1)-u_i)`

The limiter function is constrained to be greater than or equal to zero, i.e.©(r) 2 0. Therefore, when the limiter is equal to zero (sharp gradient, opposite slopes or zero gradients). The flux is represented by a low-resolution scheme. Similarly, when the limiter is equal to 1 (smooth solution), it is represented by a high-resolution scheme. The various limiters have differing switching characteristics and are selected according to the particular problem and solution scheme. No particular limiter has been found to work well for all problems, and a particular choice is usually made on a trial and error basis.