Week 9 - FVM Literature Review
TO EXPLAIN THE NEED FOR THE INTERPOLATION SCHEMES AND FLUX LIMITER IN THE FVM(FINITE VOLUME METHOD).
Aim:
To describe the need for the interpolation schemes in the FVM and flux limiter.
METHOD OF THE CFD:
To solve the real-world problem, the people discovery three simples schemes for the numerical solution are
- Finite Difference Method
- Finite Element Method
- Finite volume Method
Finite Difference Method:
This method solves the equation, that is divide into single points or nodes. This method involves solving the discretizations using the differencing equation.
Finite Element Method:
This method involves dividing the domain into elements of the desire. By solving the element in the system. The finite element analysis is a particular method for solving two or three-dimensional problems.
Finite Volume Method:
In the FVM Method the system to divide into an unstructured mesh which is conservative. In the algebraic form, the partial equation represents in the volume integral, which converted into the surface integral by the Gauss divergence theorem. These terms are solved as a flux in each surface. Because the flux entering the surface is identical to the adjacent surface, this is a conservative principle.
The Commercial software is based on the FVM Method because of the Unstructured mesh, Easy for solving the complex problems over the other methods.
Consider the 1D Heat Conduction Equation:
`del/(delx)*(alpha*(delT)/(delx))+S = 0`
Where
S= Soure term
`alpha`=thermal diffusivity
Let us consider the Control volume P and the centroid at the P as shown in the above figure. The above figure has a set of control volume with the centroids located at P, E(EAST), W(WEST), N(NORTH), S(SOUTH POINT), and the further point with the representation. The faces center is denoted as n, e, s, and w. Come back to the Heat conduction equation, because of the 1D, it doesn't have the North and south control volume. Consider the small volume 'dv' in the Control volume P, the integral form of the equation for the small volume is given by,
`int[del/(delx)*(alpha*(delT)/(delx)) + S]dV = 0`
`intdel/(delx)*(alpha*(delT)/(delx))dV + intSdV = 0`
Now, dV = A.dx
`int_w^e del/(delx)*(alpha*(delT)/(delx))Adx + intSdV =0`
On solving the equation,
`alpha_eA(delT)/(delx_e) - alpha_wA(delT)/(delx_w) + S_(avg)dv = 0`
S_(avg),`alpha` are the function of the temperature we need to linearize it.
Assuming the αα is constant,
`alphaA((T_E-T_P)/(Deltax))-alphaA((T_P-T_W)/(Deltax)) +S_u + S_PT_P = 0`
(Heatflux out)- (Heatflux in)+(source term)=0.
Need for interpolation schemes:
To calculate the Tw andTe, Temperature at the west and east faces, we have the value at the TP,TE,andTW by interpolating the values, we can get the temperature at the east and west faces. By using them we can calculate the thermal diffusivity of the faces are αeandαwαeandαw.There are different ways of Interpolation. Some of the ways are given by
- Upwind interpolation scheme.
- Linear Interpolation scheme.
- Quadratic Interpolation scheme.
- Hybrid Interpolation Scheme.
Upwind Interpolation scheme:
The upwind interpolation scheme (UDS) for approximating the value of a variable at either front or black face of a control volume is given by
The interpolation scheme is equivalent to using a backward or forward finite difference approximation(depend on the flow direction). It is the first order accurate scheme and is numerically diffusive with a coefficient of numerical diffusion
`Gamma_e^n = ((rhou)_eDeltax)/2`
The scheme is developed from the convective flows with the suppressed diffusion effects.
Linear Interpolation scheme:
The value of the variable at CV face center approximate from the values near two computational nodes. This location at 'e' on a cartesian grid, the variable value is approximated by,
`f_e = f_E(lamda_e) + f_p(1-lamda_e)`
`lamda_e = (x_e - x_p)/(x_E - x_p)`
The linear interpolation is equivalent to the use of the central differencing formula of the first-order derivative, and hence, this is also termed as the central differencing scheme(CDS). This scheme is the second derivative accurate and may produce an oscillatory solution.
Hybrid interpolation:
This companion of both the upwind and linear interpolation schemes. The cartesian grid is given by
Φf=ψΦ+(1−ψ)⋅Φcd
where
Φf=Value of the variable at the face(phi).
Φud= value of the variable at the face(by the upwind differencing technique).
`phi_(cd)`=
Value of the variable by Central differencing method at the face.
`psi`=blending function to switch between schemes.
Quadratic Upwind interpolation(QUICK)
A quadratic upwind interpolation is the three order scheme, it is derived from the polynomial fitting. It is also represented as the QUICK Interpolation.the quick interpolation on a uniform cartesian grid is given by,
`phi_e = 6/8*phi_U + 3/8*phi_D - 1/8*phi_(UU)`
Where D, U, and UU denote the downstream, the first upstream and the second upstream node respectively. (E, P, and W or P, E, and EE) depending on the flow direction. Quick scheme is prone to oscillation because of the third order.
Flux limiters:
In solving the convective flow fields, it is observed that the discontinuity or shocks in results. For the High order, it is observed that the Oscillation is more and the solution became unstable due to this reason.
To obtain the highly accurate and oscillation free results we need to use the flux limiter function. The function provides the smooth region over the flow and maintains the first-order accuracy. Flux limiters are mainly used to get the total variation diminishing [TVD].
Their simple forms for 1D Equation are given by
`(delu_i)/(delt) + 1/(Deltax_i)[F(u_(i+1/2))-F(u_(i-1/2))] = 0`
This terms represent edge fluxes for the ith cell.
Where, fluxes can be represented by low and high-resolution schemes.
`F(u_(i+1/2)) = f_(i+1/2)^(low) - phi(r_i)(f_(i+1/2)^(low) - f_(i+1/2)^(high))`
`F(u_(i-1/2)) = f_(i-1/2)^(low) - phi(r_(i-1))(f_(i-1/2)^(low) - f_(i-1/2)^(high))`
Where
`f^(low)`=low resolution flux
`f^(high)`=high resolution flux,
ϕ(r)=flux limiter function.
where r represents the ratio of successive gradients on the solution mesh,
`r_i = (u_i-u_(i-1))/(u_(i+1)-u_i)`
the limiter function is constrainted to be greater than or equal to zero
ϕ(r)≥0
When the limiter equals to zero, the flux represents the low resolution scheme.
When the limiter equals to 1, the flux represents the high-resolution scheme.
There are many limiters having distinguished characters and are selected according to the problem. There is no particular limiter found to work well for all and it is selected by the trail and error method.
Some of the limiters are
Above all limiters are symmetric, exhibits the following condition,
`(phi(r))/r = phi(1/r)`
This factor determines the limiting actions for forward and backward gradients operate in the same way.
Reference
1.NPTEL
2.WIKIPEDIA