Week 9 - FVM Literature Review

Finite volume method (FVM):- iv evaluating partial differential eqn in the form of algebraic eqn similar to fea or fdm values are calculated at mesh grid point

In FVM the entire fluid volume is divided into number of cells each associated with finite volume.

The governing eqn are solved at the centroid of each cell.

Advantages of FVM

• Volume avrg quantities makes more physical meaning than the point like ones.
• Valid for polyhedral cells.
• Unstructured mesh can be solved.
• Analysis is valid for hanging node.
• Can handle discontinuities

Difference b/w FVM and FDM

• Governing eqn are in integral form in FVM and eqn in partial form in FDM
• In FDM solution is calculated at mesh node point where as in FVM the solution is calculated at centroid of the cell.
• In FDM is difficult to make grid for complex geometry in FVM can handle unstructured mesh so here it is possible.
• FDM is preferable for higher order accuracy. In FVM its difficult
• In FDM flow should be continuous at all point
• FDM does not ensure conservation. FVM ensure conservation at both cell and domain level.

1D linear heat conduction

`d/dx(alpha*(dT)/dx)+s=0`

`int (d/dx(alpha*(dT)/dx)+s)*dV=0`

`int (d/dx(alpha*(dT)/dx))dV+int sdV=0`

`int (d/dx(alpha*(dT)/dx))*A*dx+int sdV=0`

`(alpha*A*(dT)/dx)+S*dV=0`

`(alpha*A*(dT)/dx)_e-(alpha*A*(dT)/dx)_w+S_(avg)*dV=0`

`heat flux_(out)-heat flux_(enter)+Source term=0`

source term is linearised

`S_u+S_p*T_p`

different schemes used to solve the eqn

1. gradient scheme
2. interpolation scheme

gradient scheme

gradient term used to approximate gradient tem in governing eqn.

two gredient scheme are

• guase gradient scheme
• least square gradient scheme

Interpolation schemes

Interpolation schemes are used to evaluate the face values that arises as part of the heat flux in the discretization of NS eqn in integral form.

Different types of interpolation

1. Upwind interpolation scheme
2. Linear interpolation / central differencing scheme
3. Quadratic upwind interpolation

1. upwind inter polation:-

upwind interpolation depends on the direction f the flow o`phi_f=F_f*phi_n+(1+F_f)*phi_p`f mass flux

`F_f=rho_f*A_f*(U_f*n^(cap))`

if `F_f>0` ->mass flow out

if `F_f<0` -> mass flow in

• upwind diffrencing no longer varies linear across the cetroide face.
• phi is constant b/w cell centroid & cell face.
• upwind diffrencing is a first order accurete diffrenceing.
• the solution is is bounded and diffusive charecter based on the direction of fow.
• the face value is considered same as the nabouring cell centroide value.

the face value campute using this eqn

• upwind diffrencing is not accurate
• most suitable for convectional dominante flow
• we can use it to generate an intial , & stable solution

2.linear interpolation or centeral diffrencing scheme

• this is second order accurate interpolation scheme but unbounded might generate the oscilatery solutions
• the variation b/w centroide is linear
• used for diffusion terms never be used for convective terms in RANS solutions
• it can be used for convective terms in LES for greater accurecy is required

`phi_f={(phi_f)=(phi_p) for F_f>0) or (phi_f=(phi_p) for F_f<0)`

`F_f=(x_f-x_p)/(x_n-x_p)`

Linear upwind diffrencing:-

• This is second order accurate interpolation scheme. It employed for highly convective flows
• The scheme is unbounded and prone to oscillations

3. Quadratic upwind interpolation scheme

• The face value approximated using a three point quadratic function with surrounding nodes and one upstream node
• The scheme is highly oscilatory

• Third order accurate interpolation

`phi_f=6/8*pho_(i-1)+3/8*phi_i-1/8*phi_(i-2)`

Flux limiter: -

for second order or higher order interpolation which predict face value based on gradients at cell centroids in the presence of strong gradients due to overshooting the face values might exceed the local minima/ maxima