Week 9 - FVM Literature Review
AIM:
To explain Finite Volume Method(FVM) and the major differences between FDM & FVM. Describe the need for interpolation schemes and flux limiters in FVM.
DESCRIPTION:
In CFD there are three numerical method based solvers to solve the governing navier strokes equations.They are
1.Finite difference method: The grid is linear and aligned so that all grid points lie in a straight line.Main thing is that the properties are considered to concentrated at a single point.
2.Finite element method: The properties are valid over an elemental area.The properties assumed to be concentrated at area centre.
3.Finite volume method: The method is applied over a volume such that the governing equation is valid over the entire volume, also the properties are assumed to be concentrated at the geometric centre of the volume concentrated.
DRAWBACKS OF FDM:
1.We cannot use FDM directly to a body whose grid points are not aligned properly along the x,y directions.This may occure due to complex folds and cuts in the structure.
2.As it is not aligned we cannot calculate the properties at the next adjacent point exactly, but we can make some correction factor and approximate it, this method is called as 'Variable transformation method' which is basically seeks to create grid using different variable space which look very similar to cartesian grid.
3.Variable transformation method gives an approximate solution of property but it is time consuming and complex.
4.For complex profiles, this method is less preferred.So, FVM method is used.
FINITE VOLUME METHOD
The FVM method is used for representing and evaluating PDEs in the form of algebraic equations.In FVM method, volume integrals in a PDE that contain a divergence term are converted to surface integrals, using the gauss 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 FVM is that it is easily formulated to allow for unstructured meshes.The method is used in many CFD dynamic packages.'Finite volume' refers to the small volume surrounding each node point on a mesh.
In FVM instead of writing the equation in a single point, equations are written in an entire volume.It means for a considered volume no assumptions are made over the shape of the volume. Inside the particular volume the flow is conserved, flow quantities are conserved, energy is conserved, mass is conserved, momentum is conserved and so on.This is called as 'Finite volume representation' which begins with 'Integral form of governing equation.'
STEPS FOR FVM:
1.Once we form the integral form of equation apply Gauss divergence theorem.
2.We get discretized form which can be solved using computer (flux terms).
Consider a steady state heat conduction equation with heat generation;
`(del)/(delx)(alpha.(delT)/(delx))+S=0`
Here S is the source term
`alpha`is the thermal diffusivity term
`(delT)/(delx)`is temperature gradient along x-direction.
Governing equation `(del)/(delx)(alpha.(delT)/(delx))+S=0` is valid at every point in domain, which means it is also valid in entire control volume.
If small control volume (dv) is considered and it is multiplied with the governing equation and integrated it for the entire volume then that should be also equal to zero i.e the 'integral form of governing equation.'
Taking the conservation equation validated at single point and if it is validated at single point then it is validated at entire volume and in this entire volume properties are conserved.
Multiplying the equation by 'dv' and integrating it
`int((del)/(delx)(alpha.(delT)/(delx))+S)dv=0`
`int((del)/(delx)(alpha.(delT)/(delx))dv+intSdv=0`
since dv=A.dx
`int(del)/(delx)(alpha.(delT)/(delx))A.dx+intSdv=0`
`barS`is the average value of sourve term S which when multiplied by dv is going to be equal to `intSdv`
`alphac.A.(delT)/(delxe)-alphaw.A.(delT)/(delxw) + barSdv=0`
If we assume `alpha`as a constant and not a function of termperature
`alpha.A[(Te-Tp)/(Deltax)]-alpha.A[(Tp-Tw)/(Deltax)]+barSdv=0`
The source term which is to be added in a cell, if it is a function of temperature we have to linearize it. If `alpha` is a function of temperature we have to linearize this also.
`(bars)`(linearization)=`Su+SpTp`
`alpha.A[(Te-Tp)/(Deltax)]-alpha.A[(Tp-Tw)/(Deltax)]+barSdv=0`
`underset(out)(Heat flux)-underset(IN)(Heat flux)`+source term =0
The amount of heat that is removed from a particular box is conserved.
SCHEMES USED TO SOLVE EQUATION
1.Gradient scheme
From the above figure when we calculate temperature gradient in the east face, we basically use temperature from neighbouring point trying to evaluate or numerically represent gradient of temperature is called as 'Gradient scheme.'
2.Interpolation scheme
In this finite volume formulation, evaluation of surface and volume integrals sometimes requires the values of unknown variables at locations other than the computational nodes of the control volume.These values are obtained by employing certain interpolation schemes.
As an example let us consider the simple case of steady !D convection and diffusion.
`del/(delx)(rhouphi)=d/(delx)(gamma(delphi)/(delx))`
where `phi`is our solution variable (scalar), `rho`is the density, u is the velocity and `gamma` is the diffusion coefficient.
The above figure shows the three neighbouring control volumes, which we are going to use for deriving the discretization equation.
Integration of the governing differential equation over the control volume around the grid point P yields
`(rhouphi)c-(rhouphir)w=(gamma(delphi)/(delx))e-(gamma(delphi)/(delx))w`
Evaluation of left hand side of the above equation required the values of `phi` over faces e and w.These values are estimated by employing certain interpolation schemes.
Some interpolation schemes used in CFD are:
1.Upwind interpolation scheme
2.Liner interpolation scheme
3.Quadratic upwind interpolation for convective kinematics (QUICK) scheme
4.Hybrid scheme
5.Total variation diminishing (TVD) scheme
Explaining the Upwind interpolation scheme for an example:
The upwind interpolation scheme takes into account the direction of fluid flow and based on it, the value of solution variable at the face is approximated by its value at the upstream computational node.For instance, if the fluid is moving from west to east (like in figure) then
`phie approx phiP`
And if the fluid is moving from east to west then
`phie approx phiE`
In addition to these Gradient scheme and interpolation schemes there are flux limiters used.
FLUX LIMITERS
While calculating the variable gradients they tend to toggle and are not bounded to a particular value and keeps jumping, this occurs due to discontinuity due to shock occuring, having a flux limiter will help to bound the gradient value there by giving a stable solution.
Numerical approximation using higher order spacial discretization schemes results in better accuracy but exhibits spurious oscillations near discontinuities such as shocks. On the other hand, lower order spacial discretization schemes remain stable near discontinuities but dampen out the distributions because of their dissipative nature.Hence in order to capture discontinuities, high resolution schemes are desired which are highly accurate and stable throughout the domain.
Flux limiters operate functionally as a switch and tune the numerical fluxes obtained from lower order scheme and higher order scheme such that near the discontinuity lower order (stable) scheme is employed and away from it higher order (accurate) scheme is employed.
Suppose flux at e calculated using higher resolution scheme `He`and flux at e is calculated using lower resolution `Le`Then `varphi(r)` represents the flux limiter function such that
`Fe=Le+varphi(r)(He-Le)`
Flux limiter function depends upon the smoothness parameter r which is defined as
`rp=(phiP-phiW)/(phiE-phiP)`
The flux limiter function is constrained to be greater than or equal to zero.If assumed a value of 0 near discontinuities and a value of 1 away from them.Hence it switches between higher order scheme and lower order scheme for numerical representation of flux,thereby keeping the scheme table and accurate throughout
Note that flux limiters are also referred to as slope limiters because they both have the same mathematical form, and both the effect of limiting the solution gradient near shocks or discontinuities.In general , the term flux limiter is used when the limiter acts on system fluxes and slope limiter is used when the limiter acts on system states (like pressure, velocity etc).
Consider 1D semi discrete scheme
`(delui)/(delt)+(1)/(Deltax(i))[F(ui+1/2)-F(ui-1/2)]=0`
`F(ui+1/2)` and `F(ui-1/2)` these are edge fluxes for the ith cell.
`F(ui+1/2)=f^(low)(i+1/2)-phi(ri)(f^(low)(i+1/2)-f^(high)(i+1/2))`
`F(ui-1/2)=f^(low)(i-1/2)-phi(ri)(f^(low)(i-1/2)-f^(high)(i-1/2))`
`f^(high)`=low precision, high resolution flux
`f^(low)`=high precision, low resolution flux
`phi(r)`=flux limiter function
`ri=(ui-u(i-1))/(u(i+1)-ui`
'r' represents the ration of successive gradients on the solution mesh.
`phi(r)>=0` this is the limiter function case
If limiter function =0, then there will be a low resolution and sharp gradient.
If limiter function>0, then there will be a high resolution and +ve gradient.