Week 9 - FVM Literature Review
FVM Literature Review
AIM: To study and understand the Finite Volume Method(FVM)
OBJECTIVE:
- What is the Finite Volume Method(FVM)
- Write down the major differences between FDM & FVM.
- Also, describe the need for interpolation schemes and flux limiters in FVM.
- You can use the 1D linear heat conduction equation as an example.
LITERATURE REVIEW:
Discretization is the process of converting partial Differential Equations into Algebraic form. In computational fluid dynamics there are 3 numerical method based solvers to solve the governing Navier strokes equations these are:
Finite Difference Method: The grid is linear and aligned so that all grids points lie in a straight line. The main thing is that the properties are considered to concentrate at a single point.
Finite Element Method: The properties are valid over an elemental area. The properties assumed to be concentrated at the area center.
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 center of the volume concerned.
Finite Volume Method:
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 meshes. The method is used in many computational fluid dynamics 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 'Finite Volume Representation' which begins with 'Integral form of governing equation'.
Steps:-
- once we have this integral form of equation.
- Apply Gauss divergence theorem
- We get descretized form which can be solved using computer ("Flux Terms").
consider a steady state heat conduction equation with heat generation:
`del /(del x) (alpha. (delT)/(delx)) + S = 0`
S is source term
`alpha`is thermal diffusivity term
`(delT)/(delx)` is temperature gradient along the x direction.
Governing equation - `del /(del x) (alpha. (delT)/(delx)) + S = 0` is valid at every point in domain, which means it is also valid in entire control volume.
If a small control volume (dv) is considered and it is multiplied with the governing equation and integrated 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 a single point and if it is validated at a 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+ intS dv = 0`
`because dv = A .dx`
`intdel/(delx)(alpha.(delT)/(delx))A.dx+ intS dv = 0`
`[alphaA((delT)/(delx))]_w^e +overlineS dv = 0`
`overlineS` is average value of source term (S) which when multiplied by dv is going to be equal to`intSdv`.
`alpha_eA(delT)/(delx_e)-alpha_wA(delT)/(delx_w) +overlineS dv = 0`
If we assume `alpha` as a constant and not a function of temperature,
`alpha A[(T_e-T_p)/(Deltax)]-alphaA[(T_p-T_w)/(Deltax)] + overlineSdv =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.
`(overline S underline("linearization") to S_u+S_pT_p)`
`alpha A[(T_e-T_p)/(Deltax)]-alphaA[(T_p-T_w)/(Deltax)] + overlineSdv =0`
`[Heat flux]_("out")-[Heat flux]_("in")+Source term =0`
The amount of heat that is removed from a particular box is conserved.
Different schemes used to solve the equation:
1. Gradient scheme
2. Interpolation scheme
Gradient Scheme: we calculate temperature gradient in the east face, we basically use temperature from neighboring point trying to evaluate or numerically represent a gradient of temperature is called as Gradient scheme.
Interpolation scheme:
n the 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 one-dimensional convection and diffusion.
`d/(dx)(rho u phi)=d/(dx)(Gamma(dphi)/(dx))`
where `phi`is our solution variable (a scalar), `rho` is the density, u is the velocity and `Gamma` is the diffusion coefficient. As we know that the three neighboring control volumes, which we are going to use for deriving the discretization equation.
Integration of the governing differential equation over the control volume around grid point P yields:
`(rho u phi)_e - (rho u phi)_w = (Gamma(dphi)/(dx))_e - (Gamma(dphi)/(dx))_w`
Evaluation of the left-hand side of the above equation requires the values of `phi` over faces `e and w`. These values are estimated by employing certain interpolation schemes.
DIFFERENCE BETWEEN FDM & FVM:
1. we cannot use FDM directly to a body whose grid points are not aligned properly along the x, y directions. these may occur due to complex folds, 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 basically seeks to create a grid using different variable space which look very similar to a cartesian grid.
3. variable transformation method gives an approx solution of property but it is time-consuming and complex.
4. for complex profiles, this method is less preferred.
That's why the Finite Volume Method (FVM) is used.
Flux Limiters
While calulating 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 occurring, having a flux limiter will help to bound the gradient value thereby giving a stable solution.
Numerical approximation using higher-order spatial discretization schemes results in better accuracy but exhibits spurious oscillations near discontinuities such as shocks. On the other hand, lower-order spatial 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 the lower-order schemes and higher-order schemes such that near the discontinuity lower order (stable) scheme is employed and away from its higher-order (accurate) scheme is employed.
Suppose represents flux at face e calculated using high-resolution scheme, `L_e` represents flux at e calculated using lower order scheme and `H_e` represents flux at e calculated using higher-order scheme. Then `psi(r)` represents the flux limiter function such that:
`F_e = L_e + psi(r)(H_e-L_e)`
Flux limiter function depends upon the smoothness parameter r which is defined as:
`r_P = (phiP-phiW)/(phiE-phiP)`
The flux limiter function is constrained to be greater than or equal to zero. It assumes a value of 0 near discontinuities and a value of 1 away from them. Hence it switches between a higher-order scheme and a lower order scheme for numerical representation of flux, thereby keeping the scheme stable and accurate throughout.
Note that flux limiters are also referred to as slope limiters because they both have the same mathematical form, and both have 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:
`(du_i)/(dt)+1/(Deltax_i)[F(u_(i+1/2)) - F(u_(i-1/2))] = 0`
`F(u_(i+1/2)) & F(u_(i-1/2))` these are edge fluxes for the ith cell
`F(u_(i+1/2)) = f_(i+1/2)^(low) - phi(ri)(f_(i+1/2)^(low)-f_(i+1/2)^(high))`
`F(u_(i-1/2)) = f_(i-1/2)^(low) - phi(ri)(f_(i-1/2)^(low)-f_(i-1/2)^(high))`
`f^(high)` - low precision , high resolution flux
`f^(low)`- high precidion , low resolution flux
`phi(r)`- flux limiter function
`r_i = (u_i-u_(i-1))/(u_(i+1)-u_i)`
'r' represents the ratio of successive gradients on the solution mesh.
`phi(r)ge 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.