Week 9 - FVM Literature Review
In CFD there are three known methods to solve the governing equation. In the last challenges we have already completed and worked with Finite difference methods.
The Finite Volume Method is used for representing and evaluating the governing equations in the terms of surface integrals. Similar to FDM in FVM the values are calculated in the small volume surrounded by the node points of the mesh. In this method the volume integrals in the PDE are converted into the surface integrals using the gauss divergence theorem. The surface integrals can be evaluated as the fluxes. As the flux entering the system is equal to the flux leaving the system this method is conservative. Unlike FDM this method can be applied to the complex geometries. As surface and volume integrals are used in this method, thus the variables require the descretisation at the nodes. Thus they require an interpolation formula.
Considering 1D Linear Heat Conduction Equation
`frac{del}{delx}(alphafrac{delT}{delx})+s=0`
Where,
S is the source term
alpha is the thermal diffusivity
Considering a small control volume dv and multiplying it to the governing equation and then integrating it over the entire volume, then it becomes equal to zero as net flux is zero.
`int(frac{del}{delx}(alphafrac{delT}{delx})+s)dV=0`
`int(frac{del}{delx}(alphafrac{delT}{delx}))dV+intsdV=0`
We know that
`dV=Adx`
`int(frac{del}{delx}(alphafrac{delT}{delx}))Adx+intsdV=0`
`(alphaAfrac{delT}{delx})^eunderset(w)( )V+S'dV=0`
where S' is theaverage value which is multiplied by dV gives the value
`alphaunderset(e)( )Afrac{delT}{delxunderset(e)( )}-alphaunderset(w)( )Afrac{delT}{delxunderset(w)( )}+S'dV=0`
Considering alpha as the constant
`alphaA[frac{Tunderset(e)( )-Tunderset(p)( )}{Deltax}]-alphaA[frac{Tunderset(p)( )-Tunderset(w)( )}{Deltax}]+S'dV=0`
`|Heat Flux|underset(out)()-|Heat Flux|underset(text(in))()+Source=0`
1.) Upwind Differencing Scheme
Approximation is done by the upper or next node of 'e'.
2.) Central Differencing Scheme
Interpolation is done with two nearest nodes
3.)Quadratic Upwind Interpolation
It is done through parabola. It is done through the next three points P,E and next node of 'e'.
Flux Limiters
When calculating the numerical gradientsthey tend to jump rather than sticking to a particular value, this occours due to the discontinuities and the shocks. Flux Limiters help to bind the values thus giving a stable solution. Methods using higher order schemes result in better accuracy but shows larger oscillationsnear shocks or discontinuitiesthus leading to an unstable solution. Whereas low order schemes show the stable solution near shocks or discontinuities because of their dissipative nature. Thus to capture discontinuities higher oredre schemes are required as they are highly accurate.
Flux Limiters work as a switch to tune the numerical fluxes obtained from lower order schemes or higher order schemes such that when the discontinuities occour the lower order scheme is employed else the higher order scheme is employed.
`frac{du underset(i)()}{dt}+frac{1}{x underset(i)()}(F(Uunderset(i+1/2)())-F(Uunderset(i-1/2)())`
where `F(Uunderset(i+1/2)())` and `F(Uunderset(i-1/2)())`represent the fluxes at the edge points of the ith element of the mesh. We will represent these fluxes with higher resolution and low resoltion schemes then the flux limiter will switch between these schemes depending upon the gradient close to the particular cells.
`F(Uunderset(i+1/2)())=f^(low)underset(i+1/2)()-phirunderset(i)()f^(low)underset(i+1/2)()-f^(high)underset(i+1/2)()`
`F(Uunderset(i-1/2)())=f^(low)underset(i-1/2)()-phirunderset(i)()f^(low)underset(i-1/2)()-f^(high)underset(i-1/2)()`
where
f^low= low precision high resolution flux
f^high= high precision low resolution flux
phi=flux limiter function
`r=u underset(i)()-frac{u underset(i-1)()}{u underset(i+1)()}-u underset(i)()`
this is the limiter function case
if r=0 then the low resolution sharp gradient
if r>0 then high resoltion positive gradient