Week 9 - FVM Literature Review

Finite volume Method(FVM) Literature Review

Aim:-

Define the Finite Volume Method(FVM)
Write down the Difference Between FDM and FVM.
Describe the need for interpolation schemes and also describe Flux limiters in FVM, use 1D linear heat conduction equation as an example.

Objective:-

Define the finite volume method (FVM). Also describe the difference between FDM, FVM, and FEM.
Consider the example of a 1D heat conduction equation to describe how to solve the FVM problem.
Explain about grid generation process in FVM
Describe how to discretize the equation by using various interpolation methods.
Write down how flux limiter is important to reduce shocks and distortion is the solution.
Calculate the solutions for discretizing equations.
Lastly, compare calculated numerical results with actual results.

Finite Volume Method(FVM):

• The Finite volume method(FVM) is a method for representing and evaluating partial differential equations in form of algebraic equations.
• In this method, volume integrates into a partial differential equation that contains a divergence term and is converted into 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.
• 'Finite volume' refers to the small volume surrounding each node point on a mesh.
• The advantage of the finite volume method is that it is easily formulated to allow unstructured meshes and this method is used in many computational fluid dynamic packages.

Differences between FDM & FVM:

• A finite difference method(FDM) discretization is based upon the differential form of the PDE to be solved.
• Each derivative is replaced with an approximate difference formula that can generally be derived from a Taylor series expansion.
• The computational domain is usually divided into hexahedral cells(grid) and the solution will be obtained at each nodal point.
  A finite volume method(FVM) discretization is based upon an integral form of PDE to be solved(eg. conservation of mass, momentum, or energy).
• The PDE is written in a form that can be solved for a given finite volume for cells. The computational domain is discretized into finite volumes and then for every volume, the governing equations are solved.
• The resulting system of the equation usually involves fluxes of the conserved variables, and thus the calculation of fluxes is very important in FVM
  The basic advantage of this method over FDM is it does not require the use of structured grids, and the effort to convert the given mesh into a structured numerical grid internally is completely avoided

EXAMPLE:

1D Heat conduction equation:

Governing equation or the conservative equation is given by;

Here, the S- the source term is responsible for heat generation.

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

alpha - a function of temperature.

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

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

Simplify the above equation and consider (w,e) in the control volume;

To find the dT/dx value at the above-hilighted points, we can use approximation methods (Taylor series).

On solving the above equation we get,

(alpha*A[(T_E-T_P)/(Deltax)])-(alpha*A[(T_P-T_W)/(dx)])+S*dV=0 -main equation

Here,

¯¯¯SS¯- source term ; A- face area; dV= volume.

For the source term in CFD, the source term that is going to added in a 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 consists of a constant term and states that the source term is the function of temperature.

main equation says that [Heat flux out ] - [Heat flux inn] + 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 getting results in the form that is conservative.

Interpolation Schemes in FVM:

• Upwind Interpolation Scheme
• Linear Interpolation Scheme
• Quadratic upwind Interpolation Scheme
• Hybrid, TVD and ENO Interpolation Scheme

Upwind Interpolation Scheme -

> It is mainly used for the convection dominant problems. Velocity for a given point is the direction of that particular velocity that determines the values of the variable used in the interpolation process. In an upwind scheme, it makes use of nodes that are upstream of the given node.

QUADRATIC UPWIND INTERPOLATION (QUICK):
A quadratic upwind interpolation (QUICK) scheme can be derived using polynomial fitting. The QUICK interpolation on a uniform Cartesian grid is given by;

phi_e=6/8 phi_U+3/8phi_D-1/8phi_(U)

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). The QUICK scheme is third-order accurate, but prone to oscillations.

Flux limiters:

Flux limiters are used in the high-resolution scheme used to solve the problem in science and engineering, particularly fluid dynamics described by PDE. They are used in a high-resolution scheme such as the MUSCL scheme to avoid the spurious oscillation that would otherwise occur with a high-order spatial discretization scheme due to shocks, discontinuities, or sharp changes in the solution domain. The use of flux limiters together with an appropriate high-resolution scheme makes the solution's total variation diminishing (TVD).

The main idea behind the construction of flux limiters schemes is to limit the spatial derivatives to realistic values for scientific and engineering problems this usually means physically realizable and meaningful value. They are used in the high-resolution scheme for solving problems described by PDE's and only come into operation when sharp wavefronts are present. For smoothly changing waves, the flux limiters do not operate, and the spatial derivatives can be represented by high-order approximation without introducing spurious oscillations.

Consider the 1D semi-discrete scheme below;

where;

F(u_(i+1/2)) and F(u_(i-1/2))
represents low and high-resolution schemes, then the flux limiters can be switched between these schemes depending upon the gradient close to a particular cell as follows;

where,

f(low)= low-resolution flux

F(high)=high-resolution flux

r_i=(Uinitail-Uintial+1)/ (Uintial+1-Uintial)

The limiter function is constrained to be greater than or equal to zero. i.e ϕ(r)≥0ϕ(r)≥0. Therefore, when the limiter is equal to zero the flux is represented by a low-resolution scheme. Similarly, when the limiter is equal to 1 it is represented by a high-resolution scheme. The various limiters have differing switching characteristics and are selected according to the particular problem and solution scheme.