Week 9 - FVM Literature Review
objective-To describe the needs for interppolation schemes in FVM.
THEROY
In fluid dynamic there are three numerical method based solvers to solve th navier strokes equation
they are
1.Finite Difference Method.
2. Finite Volume Method.
3. Finite element Method.
In fDm the grids aree alligned linera along x y and z direction hence the flow properties are distrubted linearly. but the draw back of fdm is when grid are not alligned linealy along x,y and Zdirction its diffucult to calculate the flow properties at the adjacent points . bit we acn apporxiamte using variable tarnsformation method which is similar to the cartsiean method biu bthis method is not applicable for complex shaped objects . and thid ,ethod is non conseravative method.
Hence we go for Finte 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 mesh.
we apply finte volume method for one linear conduction eqyation
δ/δt( α*δT/δx)+s=0
δT/δx is the temperature gradient
α is the trhermal diffuisty
s is the source term
applying teun over the entire volume we get
∫(δ/δt( α*δT/δx)+s)*dv=0
∫δ/δt( α*δT/δx)*dv+∫s*dv=0
but dv=A*dx
∫δ/δt( α*δT/δx)*A*dx+∫s*dv
we use two methods in finite voluyme meyhod they are
1.Gradient scheme
2.Interpolation scheme
In the above figure when we calculate temperature gradient in the east face we basicallu use temperatute from neighbouring point trying to evalute or numerically resprent the gradient.
2.Interpollation scheme sice in the finite volume method the properties are distrubted at the center of the volume interpolation is needed to obtain the value of the properties at the quadrature points. we have differnt interpolation schemes
1.upwind interpollation scheme
2.linear upeind difference scheme
3.Quadaric upwind difference scheme
1.upwind interpolation scheme-
its mainly used for convection domiant problem .the velocity at the point the direction of that particular velocity determines variables uded in the interpolation process its udsed to calculate the apporxiamate value of the variable at east face of the control volume.
fe={fp if (vn)at east >o
or fe if (vn)e<0
}
whre fe is the point from left face to right face and fp is point right to left.
and vn is the velocity vector.
2.Linear interpolation scheme- Its method o9f generating new point with range of dicrete set of knoen data points its same central difference method its used to solve convection diffusion equation.
Quadaratic upwind interpolation scheme- This apporxiamtes the value of the variables at control volume fave center by quadartic interpolationb of the avlues at the tree nearest computational nodes one is doendtraem node and other two upward nodes .its third order accurate for both unifrom and non unifrom grids.
sai