Week 9 - FVM Literature Review
FINITE VOLUME METHOD:
Its also a discritization approach,where weconsider a finite volume,find the required parameters for that volume rather than at a point dislike FEM,then we use gradient and interpolation,extrapolation schemes to fine the parameters for our finite volume.its a whole conserved approach where each parameter is conserved rather than any errorical losses.
its useful when our geometry along a directioon is not linear or where we dont have gradients in a line.
major difference between finite volume and finite element method.
FVM FEM
*in we analyse for a parameter over a volume. *we analyse our parameters at a respective point.
*all the values or parameters are conserved i.e., *there are loss in the results due to approximations.
no losses related to like approximations.
*we employ surface and volume integrals to solve *we use difference and at max single integration to solve for
our criterions. parameters.
*we consider the parametrical value at the centre *all are subjected at a point.
of the respective volume.
*also called as gradient method. *reffered as variable transformation method.
*we deal with equations max. *we deal with formulas.
advantage of FVM:
accurate,can use interchangable volume and surface integrals to solve our governing equation.no approximated valuees so no error in us.
using the vector and integral analysys,we solve for the flux at each of our volume.
we generally use gradient and interpolation schemes to solve for our values for a govrning equation.
so lets consider one dimensional linear heat conduction equation
∂( α*∂T/∂x)/ ∂x +s=0
s-source* might be many types.
consider a volume
|-----------dx-------------|
| * | * | * |
|---dx/2-----|
W w P e E
consider a finite volume hold the middle or present value,marked W,P,E at the centre of each volume.
let crosss secctional area be A
upon applying inegration to 1D LINEAR heat eqn,
∫[∂( α*∂T/∂x)/ ∂x +s]dv=0
for the present volume P mid of w,e let us find gradient
`int_w^e`[∂( α*∂T/∂x)/ ∂x +s]dv=`
dv=A*dx
`int_w^e`[∂( α*∂T/∂x)/ ∂x *a*dx+s*dv
its in the form of `int f'(x)dx=f(x)`
`implies``[alpha*A*partial(T)/(partialx)]_w^e+``bar s`*dv=0
where `bar s` is average of source s in its domain.
upon substituting and normally,`alpha` is also function of derivative terms so we consider at each gradient.
((underset(e)(α)*A*partialT/partial(`underset (e)(x)`))-((underset(w)(α)*A*partialT/partial(`underset (w)(x)`)`+`bar s`*dv=0
underbrace(`(α*A[`underset(E)(T)`)`-underset(P)(T)/dx])_("heat flux"))-underbrace((α*A[`underset(P)(T)`-underset(W)(T)`/dx])_("heat flux))+`bar s`*dv=0
we do linearise `bar s` to be underset(u)(S)+underset(P)(T)*underset(P)(T).
need of interpolation schemes:if we want the gradiend value preceeding alike cds srhscheme,skewed left handed scheme,we required interpolation scheme which frames our domain.
we have newtons and eulers interpolation schemes to find next values or next gradient or next cell.
FLUX LIMITERS:flux limiters used in fvm to avoid the random and violent oscillations or jumps that can occur during higher order schemes which are unsatable in nature.
common forms of flux limiters are CHARM,HCUS,HQUICK,korean,minmod.(from a source)