Week 9 - FVM Literature Review

Aim:Describe the finite volume method,and write down the major differences between FDM & FVM ,describe the need for interpolation schemes and flux limiters in FVM.

Procedure:

In computational fluid dynamics there are three methods for discretization for solving the governing navier stokes equation these are follows:

1) Finite Difference Method

2) Finite Element Method

3) Finite Volume Method

1) Finite Difference Method :

In finite difference method the grid is linear and aligned so that all grid points are lie in a straight line .In this method the properties are considered to concentrated at a single points.

2) Finite Element Method :

In this method the properties are valid over an elemental area . The properties are assumed to be concentrated at area centre .

3) Finite Volume Method :

The finite volume method is applied over a volume such that the governing equation is valid over entire volume.also the properties are assumed to be concentrated at the geometric centre of the volume concerned.

Drawbacks of Finite Difference Method :

• we cannot use finite difference method directly to a body whoes grid points are not aligned properly along the x,y,z direction ,this is may occure due to complex geometry,cuts in the structure
• Due to not aligned points we cannot calculate the properties at the next adjacent point exactly ,but we can make some correction factor and approximate it ,this method is called as 'variable transformation method '.which basically seek to create grid using different variable space which looks very similar to cartesian grid.
• The variable transformation method gives an approximate solution of property but it is time consuming and complex .for complex profile this method is less preferred due to that the Finite Volume Method is used.

Finite Volume Method :

The finite volume method is a method for representing and evaluating the partial differential equation in the form of algebric equations.In the finite volume method the volume integral in a partial differential equation that contain a divergence term are converted to surface integral,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 flux leaving from the adjacent control volume ,these method is conservative.Another advantage of finite volume method is that it is easily formulated to allow for unstructured meshes.Finite volume refers to the small volume surrounding each node point on a mesh.

In a finite volume method instead of equation writing for a single point,equations are written for a entire volume.It means for a given volume we cannot made a assumptions for a shape of the volume .Inside the particular volume the flow is conserved.

Main steps used to solve the 1D linear heat conduction equation using FVM method

1) Integrate the form of governing equation

2) Apply Gauss Divergence theorem

3) We get the descretized form which can be solved by using computer.

Consider a steday state heat conduction equation with heat generation ;

`del/(delx)(alpha.(delT)/(delx))+S=0`

where, S is source term

`alpha`=Thermal Diffusivity

`(delT)/(delx)`=Temperature gradient along the x direction

The governing equation `del/(delx)(alpha.(delT)/(delx))+S=0` is valid at every point in domain which means it is valid in entire control volume .

We consider small control volume and it is multiplied to governing equation and integrate it for entire volume then that should be also equal to zero ie the integral form of governing equation.

we take the conservation equation which is valid at a single point and if it is validated at a single point then it is validated at entire volume and in this entire volume the properties are conserved.

Multiplying the equation by dv and integrate it

`int(del/(delx)(alpha.(delT)/(delx))+S)dv=0`

`intdel/(delx)(alpha.(delT)/(delx))dv+intS dv=0`

since `dv=A.dx`

`intdel/(delx)(alpha.(delT)/(delx))A. dx+intS dv=0`

`[alpha.A (delT)/(delx)]_w^e+barS dv=0`

`barS` is average value of source term.

`alpha_e.A (delT)/(delx_e)-alpha_wA (delT)/(delx_w)+barS dv=0`

If we assume `alpha`as a constant and not a function of temperature ,

`alpha A[(T_e-T_p)/(Deltax)]-alpha A[(T_p-T_w)/(Deltax)]+barSdv=0`

The source term which is added in a cell, if it is a function of temperature we have to linearize it. so `barS rarr S_u+S_pT_p`

`alpha A[(T_e-T_p)/(Deltax)]-alpha A[(T_p-T_w)/(Deltax)]+barSdv=0`

[Heat flux]out-[Heat flux]in+Source term=0

The amount of heat that is removed from a particular box is conserved.

The different scheme used to solve the equation :

1.Gradient scheme

2.Interpolation scheme.

1) Gradient Scheme:

for calculating temperature gradient at east face we use neighbouring point temperature to numerically represent the temperature gradient and this is called as Gradient scheme.

2)Interpolation scheme:

In the finite volume formulation,evaluation of surface and volume integral sometimes requires the values of unknown variables at location other than the computational nodes of the control volume .These values are obtained by employing the certain interpolation schemes

for an example let us consider the case of steady one dimensional convection and diffusion.

`d/dx(rhouphi)=d/dx(Gamma(dphi)/dx)`

where the `phi`is our solution variable (scalar) ,`rho`is density,u is a velocity and `Gamma` is the diffusion coefficient

The above figure shows the three neighbouring control volumes, which we are going to derive the discretization equation.

integrating the governing differential equation over the control volume around the grid point P:

`(rhouphi)_e-(rhouphi)_w=(Gamma(dphi)/dx)_e-(Gamma(dphi)/dx)_w`

Evaluation of left hand side of the above equation requires the value of `phi` over the faces e and w these value are estimated by employing the certain interpolation schemes.

Types of Interpolation schemes:

1) Upwind Interpolation scheme

2) Linear Interpolation scheme

3) Quadratic Upwind Interpolation scheme

4) Hybrid scheme

5) Total Variation Diminishing (TVD)scheme

1) Upwind Interpolation scheme:

The upwind Interpolation scheme takes into account the direction of the fluid flow and based on it ,the value of solution variable at the face is approximated by its value at the upstream computational node.

If the fluid is moving from west to east (refer figure) then,

`phi_e~~phi_P`

and if the fluid is moving from east to west then,

`phi_e~~phi_E`

2) Linear Interpolation scheme :

It is the simplest amd most widely used interpolation. In this case we approximate the values of the variable at control volume face centre by interpolating the values lineraly at the two nearest computational nodes.

The linear interpolation is equivalent to the use of central difference formula of the first order derivative and hence it is also called as central difference scheme.It is second order accurate and produces oscilllatory solutions,the relation of linear interpolation is given below:

`f_e=f_Elambda_e+f_P(1-lambda_e)`

`lambda_e=(X_e-X_P)/(X_E-X_P`

• FLUX LIMITER:

They are used in high resolution schemes,to avoid the false oscillation that would otherwise occure with high order spatial discretization scheme due to shocks,discontinuities or sharp changes in the solution domain . Use of flux limiters,together with an appropriate high resolution scheme,makes the solutions total variation diminishing .The main idea behind the construction of flux limiter scheme is to limit the spatial derivatives to realistic values. The main purpose of flux limiter is to keep the values bounded at the cell faces. flux limiter do not let the solution value to overshoot.

consider the 1D semi discrete scheme

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

`F(u_(i+1/2))` and `F(u_(i-1/2))` represents the edge fluxes for the `i^(th` cell can be represented by low and high resolution scheme then the flux limiter can switch between these scheme depend upon the gradient close to the particular cell.

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

`F(u_(i-1/2))=f_(i-1/2)^(low)-phi(r_(i-1))(f_(i-1/2)^(low)-f_(i-1/2)^(high))`

where

`f_()^(low)`=Low resolution flux

`f_()^(high)`=High resolution flux

`phi(r)`=Flux limiter functio`

r represent the ratio of successive gradient on the solution mesh i.e `r_i=(u_i-u_(i-1))/(u_(i+1)-u_i`

`phi(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 positive gradient.