Week 9 - FVM Literature Review
Finite difference method
It is based on Tylors series of expansion.
Finite volume method is bases on integral form of diffential equations.
For FDM mothod
`T(i+1,j)=T(i,j)+( (d /dx)T)*(delta x)+(d^2/dx^2)T)*(deltax^2)/2` (1)
`T(i-1,j)=T(i,j)-( (d /dx)T)*(delta x)+(d^2/dx^2)T)*(deltax^2)/2` (2)
By adding equations we get the
`(T(i+1,j)+T(i-1,j)-2*T(i,j))/(deltax^2)`=`(d^2/dx^2)T`
So this equaion we get by dropping higher order deriveatives of 3 rd order and so on.
So this equations can be solved using Tridiagobal Matrix Formulation to get required results.
FVM
Now take the considerations of heat conduction for plane slab.
`int[K*(d^2/dx^2)T+K*(d^2/dy^2)T]+intSg*dv=(1/(rho*cp))(d/dt)T` (A)
Now considering steady state and 1 dimensional heat transfer.
Then we get.
`(k/(deltax))*((dT)/dx)=0` (B)
Now we have 1 as West point and 3 is East point and 2 is our center point
Now equation B reduces to
`k/(deltax)*((T3-T2)-(T2-T1))=0`
now further the equation reduces to
`(a1+a2)T2=a1T1+a3T3`
`a1=(2*K)/(deltax)`
`a3=K/(deltax)`
`deltax=(Dimension X direction)/(Total no of control volume)`
Therefore all equations are written for all grid points say we have 7 equations and 7 unknowns can be solved using TDMA approach to get the solution.
Numerical Schemes
- Upwind scheme
- Central diffference scheme
- Second order upwind
- Quadric Interpolation scheme
Now depending on the flow direction the upwind schemes are used as it takes 100% weightage from upstreame point.
`Fephie-Fwphiw=De(phiE-phiP)-Dw(phiP-phiW)`
Now `phie=phip` for flow from right to left and giving 100% weightage
Now for CDS scheme
`phie=(phiP+phiE)/2`
and
`phiw=(phiP+phiW)/2`
These linear schemes helps to calculate the approximate values near the faces of finite volumes.
SOU `ax+b=0`
Quick `ax^2+bx+c=0`
Now F= convection term=`rho*U`
D=diffusion term=`gamma/(deltax)`
So depending on Peclet Number we can decide the schemes to be used as Peclet Number greater than 2 leads CDS schemes unstable.
SOU and QUICK are considers higher order terms which increases the computational cost.
Flux Limiters
This is useful to limit spatial derivatives to yield realistic solution in case of shcok wave problem.