FVM Literature Review

 FVM Literature Review

Given problem:

What is 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. 

• What is Finite Volume Method(FVM)?

In order to solve the real-world problem, the people discover three simples schemes for the numerical solution are 

• Finite Difference Method
• Finite Element Method
• Finite volume Method

Finite Difference Method:This method solves the equation, that is divide into single points or nodes. This method involves solving the discretizations using the differencing equation.

Finite Element Method:This method involves dividing the domain into elements of the desire. By solving the element in the system. The finite element analysis is a particular method for solving two or three-dimensional problems.

Finite Volume Method: The finite volume methods (FVM) is a method for representing and evaluating partial differential equation in the form of algebric equations. In the finite volume method, volume integral in a PDE\'s that cantain a divergence theorem. These term are then evaluated as fluxes at the surface of each finite volume. Because the flux entering a given volume is a identical to that leaving the adjecent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unconstructed meshes. The method is used in many CFD\'s packages. \"Finite Volume\" reffers to the small volume surrounding each node point on a mesh.

 

 

The Commercial software is based on the FVM Method because of the Unstructured mesh, Easy for solving the complex problems over the other methods.

Application of finite volume method to 1-D steady-state heat conduction problem:

The governing equation for one-dimensional steady-state heat conduction equation with source term is given as 

where 'T' is the temperature of the rod. The boundary values of temperature at A and B are prescribed.

α  is thermal diffusivity.

S is the source term.

 

 

STEPS:

 

• Grid generation

The first step in the finite control volume is to discretize the domain into discrete control volumes as shown in figure

The number of nodal points is placed between A & B. The boundary of the control volume is positioned midway between the two adjacent nodes, thus each node is surrounded by the control volume. The control volume should be set up near the edges of the domain such that the physical boundary and the control volume boundary should coincide with each other.

A general nodal point is identified as P and its neighboring nodes to the west and east are identified as W and E respectively. The west side of the control volume is identified as w whereas the east side is identified as e. The distance between the nodes W and P, and between nodes P and E are identified asδxWP and δxPE respectively. Similarly between w and P, and between P and e are identified as δxwP and δxPe. The control volume width 

• Discretization

The key step in the finite volume method is to integrate the governing equation over a control volume to yield a discretize equation at its nodal point P. For the control volume defined above this gives 

The equation becomes

A is the cross sectional area of the control volume 

$△V$ is the control volume.

S is the average value of the source S over the control volume.

The above equation states that the diffusive flux of temperature leaving the east face of the control volume minus the diffusive flux of temperature entering the west face is equal to the generation of temperature. It constitutes a balance equation of temperature over a control volume.

In order to derive useful forms of the discretized equation, the interface diffusion cofficient

α   and temperature gradient dT/dx at east and west face are required`. Linear approximations seem to be the obvious and simplest way of calculating interface values and gradients. This practice is called a central differencing scheme.

 

And the diffusive flux terms are evaluated as 

The finite volume method approximates the source term as

The final equation can be rearranged as 

This can be arranged as 

Identifying the coefficient of as  and rearranging the equation as under 

  where

               

• Differences between FDM and FVM

S No

Finite Difference Method

Finite Volume Method

 1        It is evaluated at the nodes                                            It is evaluated  at the

                                                                                               volume surrounding

                                                                                                the nodes.           

 2        It is well suited for structured mesh                          The process is much                                                                                                       more efficient for the                                                                                                     unstructure mesh                                                                                            (compex geometry,arbitary shape)  

                                                                                                                                                                             

 3       It can be applied to the unstructured                The discretization process             

        mesh but the process becomes tedious.              looks really simplier and

                                                                               easier when FVM comes to

                                                                                   complex geometry shapes.

4.       It is computationally more expensive              It is computationally less expensive

          for the unstructured mesh.                             for the unstructured mesh.                             

 5.It has failed to preserve the concept of                   It preserves the concept of conservation.                                                             conservation.

 6.It is highly undesirable in the case of                      It is desirable in the case of moving moving meshes.                                                                    meshes.

pic of FDM and FVM:

FVM:

FDM:

 

•  The need for interpolation schemes and flux limiters in FVM

Consider the 1D Heat Conduction Equation:

Where S= Soure term

Thermal diffusivity 

 Let us consider the Control volume P and the centroid at the P as shown in the above figure. The above figure has a set of control volume 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. Consider the small volume 'dv' in the Control volume P, the integral form of the equation for the small volume is given by,

now dV=A*dx

On solving the equation,

are the function of the temperature we need to linearize it.

Assuming the  is constant,

(Heatflux out)- (Heatflux in)+(source term)=0.

1. To calculate the Tw and Te Temperature at the west and east faces

the value at the TP,TE and TW by interpolating the values, we can get

the temperature at the east and west faces. By using them we can

calculate the thermal diffusivity of the faces are  

• Upwind interpolation scheme.
• Linear Interpolation scheme.
• Quadratic Interpolation scheme.
• Hybrid Interpolation Scheme.

Upwind Interpolation scheme:

The upwind interpolation scheme (UDS) for approximating the value of a variable at either front or black face of a control volume is given by 

The interpolation scheme is equivalent to using a backward or forward finite difference approximation(depend on the flow direction). It is the first order accurate scheme and is numerically diffusive with a coefficient of numerical diffusion

 

The scheme is developed from the convective flows with the suppressed diffusion effects.

Linear Interpolation scheme:

The value of the variable at CV face center approximate from the values near two computational nodes. This location at 'e' on a cartesian grid, the variable value is approximated by,

The linear interpolation is equivalent to the use of the central differencing formula of the first-order derivative, and hence, this is also termed as the central differencing scheme(CDS). This scheme is the second derivative accurate and may produce an oscillatory solution.

Hybrid interpolation:

This companion of both the upwind and linear interpolation schemes. The cartesian grid is given  by

where

Quadratic Upwind interpolation(QUICK)

A quadratic upwind interpolation is the three order scheme, it is derived from the polynomial fitting. It is also represented as the QUICK Interpolation.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.

2.Flux limiters:

In solving the convective flow fields, it is observed that the discontinuity or shocks in results. For the High order, it is observed that the Oscillation is more and the solution became unstable due to this reason.

To obtain the highly accurate and oscillation free results we need to use the flux limiter function. The function provides the smooth region over the flow and maintains the first-order accuracy. Flux limiters are mainly used to get the total variation diminishing [TVD].

Their simple forms for 1D Equation are given by 

 

This terms represent edge fluxes for the ith cell.

Where, fluxes can be represented by low and high-resolution schemes.

where

where r represents the ratio of successive gradients on the solution mesh,

 

the limiter function is constrainted to be greater than or equal to zero 

When the limiter equals to zero, the flux represents the low resolution scheme.

When the limiter equals to 1, the flux represents the high-resolution scheme.

There are many limiters having distinguished characters and are selected according to the problem. There is no particular limiter found to work well for all and it is selected by the trial and error method.

Some of the limiters are

Above all limiters are symmetric, exhibits the following condition,

This factor determines the limiting actions for forward and backward gradients operate in 

the same way.

 

 

3.The Guass or divergence theorem simply states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface.This theorem is fundamental in the FVM, it is used to convert the volume integrals appearing in the governing equation into surface integrals.The face values appearing in the convective and diffusive fluxes have tobe computed by some form of interpolation from the centroid values of the control volume at both sides of face.After spacial discretization, we can proceed with the temporal discretization. By proceeding in this way we are using the method of lines (MOL).The main advantage of the MOL method, is that it allows us to select numerical approximations of different accuracy for the spacial and temporal terms. Each term can be treated differently to yield to different accuracies.Nearly all flows in nature and industrial applications are unsteady (also known as transient or time dependent).

   In this way I desribe the need for interpolation schemes and flux limiters in FVM by explaining detailed terms.