Week 9 - FVM Literature Review

FINITE VOLUME METHOD(FVM):

• The Finite volume method (FVM) is a method for representing and evaluating a partial differential equation. In the form of algebraic equations In the finite volume method, volume integrals in a partial differential equation 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 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.

Drawbacks of Finite Difference Method:

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 grid using different variable space which look very similar to 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 prferred.

That's why Finite Volume Method (FVM) is used

CONSIDERING 1D HEAT CONDUCTION EQUATION:

Governing equation is

`d/(dx)*alpha*((dT)/dx)+S=0`

here,

S- source term which is responsible for the heat equation

`alpha`- function of temperature.

consider a small volume from the control volume,then the governing equation is

`int(d/dx*alpha*(dT)/dx+S)=0`

to simplify the above equation consider it in the control volume w&e

to find the dT/dx value at the above highlighted points, we can use approximation methods (Taylor series)

on solving above equation we get,

`alpha*A*[(T_E-T_P)/(Deltax)]-alpha*A*[(T_E-T_W)/(Deltax)]+S*dv=0` ------->(1)

S- source term

A-face area

dv-control volume

For source term in CFD, the source term that is going to added in cell if it is actually the function of the temperature then linearisation is done to the source term.

on linearising this source term we get,

`S_u+S_P*T_P`

The above equation consist of a constant term and states that source term is the function of temperature.

equation (1) says that

[heat flux out]-[heat flux in]+[source term] = 0

hence FVM is said to conserve quantity better as in the derivation we are considering the equations for controlled volume and finally gretting results in the form that is conservative.

INTERPOLATION METHODS:

The approximation of surface and volume integrals require values of the variable at locations other than the computational nodes of the Control Volume.

Values at these locations are obtained using interpolation formulae.

Various possibilities for interpolation. e.g

• Upwind interpolation
• Linear interpolation
• Quadratic upwind interpolation
• Hybrid, TVD and ENO interpolation schemes.

1. Upwind Interpolation Scheme(UDS):

In this scheme we use the node in the upwind or upstream of the given node. It is generally used for convection dominated problems (i.e Pe>1). ,

`f_e = {(f_p,if,(v.n)_e,>,0),(f_E,if,(v.n)_e,<,0):}`

The values for this at the interface is given by,

for `v>0 ; u_(i-1/2)≈u_(i-1);u_(i+1/2)≈u_i`

`I_c ≈v(u_i-u_(i-1))/(Δx)`- (Backward differencing scheme)

for `v>0;u_(i-1/2)≈u_(i-1);u_(i+1/2)≈u_i`

`I_c ≈ v(u_(i+1)-u_i)/(Δx)`- (Forward differencing scheme)

Taylor series expansion is

`v.u_(i+1/2) = v.u_i - (v(Δx)/2).((del(u))/(del(x)))_(i+1/2)`where the second term has the truncation error which resembles a diffusive Flux term.

Upwind schemes are the approximations that satisfy the boundedness unconditionally and is First order accurate in space.

2.Central Differencing Approximation (CDS):

Approximate values of the variable at the control volume (CV) face centre by the linear interpolation of the values at the two nearest computational nodes.

`f_e =f_Eλ_e + f_p(1-λ_e)` where `λ_e = (x_e-x_p)/(x_E-x_P)`

The linear interpolation is equivalent to that of using central difference formula of the first order derivative and hence this scheme is also termed as Central Differencing Scheme(CDS).

3. Interpolation Polynomial:

`p1(x) = u_L.(x_R-x)/(x_r+x) + u_R.(x-x_L)/(x_R-x_L)`

The interface values are given by:

`u_(i-1/2)≈(u_(i-1)+u_i)/2` and `u_(i+1/2) ≈ (u_i+u_(i+1))/2`

`I_c ≈v.(u_(i+1)-u_(i-1))/(2Δx)`- (Central differencing scheme)

Taylor Series Approximation:

`u_(i+1/2) = (u_i+u_(i+1))/2 - ((Δx)^2/8).((del^2(u))/(del(x)^2))_(i+1/2) + ...`

FLUX LIMITERS:

Flux limiters are used in the numerical schemes to solve problems of fluid dynamics, described by highly coupled non-linear Partial Differential Equations. If we tend to use low order numerical schemes for the solving of the governing equations then we might get a highly oscillatory and unstable solution near the discontinuity if present. Whereas if we tend to use higher-order numerical schemes to capture the phenomenon more accurately we tend to exploit the computational power unnecessarily. The main idea behind the use of the flux limiters is to limit the spatial derivative terms near the discontinuities to physical real values so as to maintain physical consistency. There are several types of flux/slope limiter functions. They come in handy when sharp wave-fronts/peaks are present. For smoothly changing waves/functions, the flux limiters do not operate and the spatial derivatives can be represented normally.

It is imperative to 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.

`F(u_(i+1/2)) = f^(low)(i+1/2) - ϕ(r_i)(f^(low)(i+1/2)-f^(high)(i+1/2))`

`F(u_(i-1/2)) = f^(low)(i-1/2)- ϕ(r_i-1)(f^(low)(i-1/2)-f^(high)(i-1/2))`

Here`f^(low)`= Low precision flux (1st order precise)

`f^(high)`= High precision flux (Highest order precision)

`ϕ(r)`= Flux limiter function and `r = (u_i-u_(i-1))/(u_(i+1)-u_i)`