Week 9 - FVM Literature Review

Week 9 - FVM Literature Review

Objective:

1. To explain about Finite Volume method
2. To write down the major differences between FDM & FVM.
3. To describe the need for interpolation schemes and Flux limiters in FVM.

Methods of CFD:

In fluid dynamics we know that there are three numerical-method based solvers which are used for the governing equation of Navier-Stokes. They are:

1. Finite Difference method
2. Finite Element method
3. Finite Volume method.

 Finite Difference method (FDM):

The finite difference method is an approximation method for solving partial differential equations. Basically it converts linear ordinary differential equation or non-linear partial differential equations into a system of equations that can be solved using matrix algebraic techniques. In this finite difference method the grid should be linear and must be aligned in such a way so that all the grid points lie in the straight line. In this method the governing equations properties are said to be concentrated at each and every point along the grid.This method solves the equations that are divided into single points or nodes.

 

Finite element method:

This method involves dividing the domain into elements of the desire. By solving the elements in the system the finite element analysis is a particular method for solving two or three dimensional problems. Here the computational domain is partitioned into smaller domains of finite element and the solution is built from the basic functions in each element.

Finite volume method:

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. The method is used in many computational fluid dynamics packages. "Finite volume" refers to the small volume surrounding each node point on a mesh.

In this method instead of writing the equation in a point, we write it for volume. Then we apply few mathematical techniques to help us solve these equations for any type of grids. Also the finite volume represents preserves the concept of conservation.

         

The commercial software is based on the FVM because of the unstructured mesh which made easy for solving the complex problems over the other methods.

Considering the 1-D steady state heat conduction equation with source term:

where `alpha`is the Thermal diffusivity, S is the source term.

and dho T by dho x is the temperature gradient with respect to x.

we know dv=A.dx

Differences between FDM & FVM:

Grid generation:

In the case of FDM, the domain is divided into numerous points that are referred to as grid points each of which represents the respective variable of interest.

In the case of FVM, the domain is divided into numerous control volumes mostly referred to as cells which consist of variables of interest that are located at the centroid of each control volume or cell.

Mode of discretisation:

In the case of FDM, the discretization is based on the derivatives since the differential form of governing equations is replaced by a system of algebraic equations at each grid point.

In the case of FVM, Volume integrals in a PDE that contains a divergence term are then converted into surface integrals by using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume.

Numerical Schemes:

Numerical Schemes which are applicable in the case of FDM are Forward differencing, Central differencing & Backward differencing.

Numerical Schemes which are applicable in the case of FVM are Upwind differencing, Hybrid differencing, Power-law which is used for giving a more accurate approximation to the one-dimensional exact solution.

Interpolation schemes in Finite Volume Method:

The interpolation or the approximation schemes are required in the FVM method for the conversion of the integral form of the governing equation to an algebraic equation which can then be solved using various numerical methods. These schemes also play a significant role in the calculation of the values of the flow quantities in the point of concern from the values that are already known from their neighboring faces/cells. Some of the interpolation schemes are listed as follows:

• Upwind Interpolation
• Linear Interpolation
• Quadratic Upwind Interpolation Scheme(QUICK)
• Hybrid, TVD(Total variation dimension) & ENO(Essentially non-oscillatory) interpolation scheme.

Upwind Interpolation:

The upwind interpolation for approximating the value of a variable f at the east face of the control volume is given by,

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.

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.

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.

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.

3. Quadratic Upwind Interpolation (QUICK):

The approximated values of the variable at the control volume's centroid by quadratic interpolation of the values at the three nearest computational nodes which consist of one downstream node 'D' and two upstream nodes UU.Quadratic Upwind Interpolation is third-order accurate in both uniform and non-uniform grids.The quick interpolation on a uniform cartesian grid is given by,

Where D, U, and UU denote the downstream, the first upstream and the second upstream node respectively. (E, P. and W or P. E, and EE) depending on the flow direction. Quick scheme is prone to oscillation because of the third order.

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,

Flux Limiters:

Flux limiters are used in the numerical schemes to solve problems of fluid dynamics, described by highly coupled non-linear Partial Differential Equations. If we tend to use low order numerical schemes for the solving of the governing equations then we might get a highly oscillatory and unstable solution near the discontinuity if present. Whereas if we tend to use higher-order numerical schemes to capture the phenomenon more accurately we tend to exploit the computational power unnecessarily. The main idea behind the use of the flux limiters is to limit the spatial derivative terms near the discontinuities to physical real values so as to maintain physical consistency. There are several types of flux/slope limiter functions. They come in handy when sharp wave-fronts/peaks are present. For smoothly changing waves/functions, the flux limiters do not operate and the spatial derivatives can be represented normally.

It is imperative to note that flux limiters are also referred to as slope limiters because they both have the same mathematical form, and both have the effect of limiting the solution gradient near shocks or discontinuities.

Here `f^(low)`= Low precision Flux

        `f_(high)`=High precision Flux

        `phi(r)=`Flux limiter function and `r=(u_i-u_(i-1))/(u_(i+1)-u_i`