Week 9 - FVM Literature Review

OBJECTIVES:

1. DEFINE WHAT IS FVM(FINITE VOLUME METHOD). 

2. DIFFERENTIATE BETWEEN FDM AND FVM. 

3. DESCRIBE THE NEED FOR INTERPOLATION SCHEMES & FLUX LIMITERS IN FVM. 

THEORY: 

 We know that in Fluid Dynamics there are three numerical-method based solvers which are used to solve for the governing equation of Navier-Stokes: 

• Finite Difference Method
• Finite Element Method
• Finite Volume Method

Finite Difference Method(FDM): The finite difference method (FDM) is an approximate method for solving partial differential equations. Basically, it converts linear ordinary differential equations(ODEs) or non-linear partial differential equations(PDEs) into a system of equations that can be solved using matrix algebraic techniques. Here, the grid should be linear & must be aligned in such a way so that all the grid points lie in a straight line. Here, the governing equation's properties are said to be concentrated at each and every point along the grid. 

Finite Element Method(FEM): This method consists of the pictorial representation of the solutions in terms of defined basis functions. Here, the computational domain is partitioned into smaller domains of finite elements & the solution is built from the basis functions in each element. The actual equations are usually founded by restating the conservation equations in a weak form. Here, the field variables should be stated in terms of the basis functions, then the equation is multiplied by the appropriate test functions & the equation is then integrated over an element. Since the finite element method's solution is expressed in terms of specific basis functions hence far more information is available when compared to FDM or FVM. 

Finite Volume Method(FVM): FVM is a discretization method using which we can approximate a single PDE or even a system of PDEs that expresses the conservation of one or more quantities. These PDEs are often regarded as conservation laws that can be of various nature such as elliptic, parabolic, hyperbolic, etc. It is used to describe the relations between partial derivatives of unknown fields such as temperature, pressure, molar fraction, concentration, the density of electrons, etc with respect to the variables within the domain(time, space, etc) under consideration.

Here in FVM, a mesh is being generated that consists of a partition of the domain where the space variables reside. Elements of those meshes are termed as control volumes where the integration of those PDEs over each control volume results in a balanced equation. Then, those sets of balanced equations are then discretized with respect to the set of discrete unknowns. The issue here is that the discretization of those fluxes at the boundaries of each control volume as in order for the FVM to be efficient, the numerical fluxes are generally conservative and consistent. When the numerical fluxes are conservative then the flux entering a control volume from its neighbor should be the opposite of the one entering the neighbor from the control volume. When the numerical fluxes are consistent then the numerical flux of a regular function interpolation tends to the continuous flux as the mesh size vanishes. 

Sometimes it is possible to discretize the fluxes at the boundaries of the control volume by the FDM and then it is referred to as the conservative finite difference method. Specifically, in the case of FVM with respect to FDM, the discretization can be performed on the local balance equations, rather than on the PDEs, and hence the fluxes on the boundaries of the control volumes are discretized, rather than the continuous differential operator. 

CONSIDERING 1D LINEAR HEAT CONDUCTION EQUATION: 

$\frac{\partial /}{\partial x}=(\alpha \cdot \frac{\partial \mathrm{T/}}{\partial x})+S=0$

Here, 

$\alpha =$Thermal Diffusivity

S = Source Term

$\frac{\partial \mathrm{T/}}{\partial \mathrm{x=}}$ Temperature gradient with respect to x

Considering the Volume Integral:

$\int (\frac{\partial /}{\partial x}(\alpha \cdot \frac{\partial \mathrm{T/}}{\partial x})+S)\cdot dV=0$

$\int \frac{\partial /}{\partial x}(\alpha \cdot \frac{\partial \mathrm{T/}}{\partial x})dV+\int S\cdot dV=0$

We know, $dV=A\cdot dx$

DIFFERENTIAL FORM OF FDM:

DIFFERENTIAL FORM OF FVM: 

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. 

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 'f' at the east face of the control volume is given by: 

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 actually 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. 

$f\underset{e}{}$ = $fE\lambda \underset{e}{}$+$f\underset{p}{}$$\lambda \underset{e}{}$)

where $\lambda \underset{e}{}$= $x\underset{e}{}$-/-

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'.

$f\underset{e}{}$=$f\underset{U}{}$+$g1$$f\underset{D}{}$-$f\underset{U}{}$) + $g$($f\underset{UU}{}$-)

  = ($x\underset{e}{}$-)(-$x\underset{UU}{}$)/(-)(-

$g\underset{2}{}$ = (-$x\underset{U}{}$)($x\underset{e}{}$-$x\underset{D}{}$)/($x\underset{UU}{}$-)(-$x\underset{D}{}$)

Quadratic Upwind Interpolation is third-order accurate in both uniform and non-uniform grids. 

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: 

Here, $\rho$ is density, 'u' is the velocity vector,  is the diffusion coefficient & $S$is the source term. In this equation property, $\varphi$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, 

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

The grid used for discretization in Central Difference Scheme:

The grid used for discretization in Upwind Difference Scheme for positive Peclet number (Pe>0):

The grid used for discretization in Upwind Difference Scheme for negative Peclet number (Pe < 0):

FLUX LIMITERS:

For convective fluid flow, it has been observed that low-order schemes are usually stable but quite dissipative in nature around points of discontinuity or shocks while the higher-order schemes are unstable in nature and show oscillations in the vicinity of discontinuity. Highly accurate oscillation-free schemes are known as high-resolution schemes. Flux limiters are used to tune higher 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 total variation diminishing(TVD) criteria is being maintained of which basic form is as follows: 

fhigh-  low precision , high resolution flux

flow-  high precidion , low resolution flux

ϕ(r)-  flux limiter function

ri=(ui−ui−1)/(ui+1−ui)

\'r\'  reperesents the ratio of successive gradients on the solution mesh.

 ϕ(r)≥0

this is the limiter function case

if limiter function = 0  , then there will be a low resolution and sharp gradient.

if limiter function > 0 , then there will be a high resolution and +ve gradient.