Week 9 - FVM Literature Review

Objective: Finite Volume Method literature review and significant differences between FDM and FVM.

First, We will walk through an understanding of CFD as follows.

CFD:

CFD is a branch of Fluid Mechanics that involves numerical analysis and data structures to analyze and solve problems involving fluid flows in surfaces. Generally, It uses computers to calculate and simulate the free stream flow of fluids defined by boundary conditions.

CFD is widely used in industries as it is more cost-effective than physical testing. On the contrary, Complex flow simulations are challenging and error-prone, and it takes profound engineering expertise to obtain stabilized solutions. The strategy of CFD is to replace the continuous problem domain with the Discrete domain using a grid. In a CFD solution, one would directly solve the relevant flow variables only at the grid points. The values at other locations are determined by interpolating the values at the grid points.

There are many methods to solve the Discritize domain; some are as follows

1. Finite Difference method
2. Finite Volume method

Finite Difference method:

finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. The method of deriving the discrete equation using Taylor’s series expansions is called the finite-difference method. FDM solves Partial differential equations into a system of linear equations that can be solved by matrix algebra techniques. FDM is simple to code and is used in a few specialized codes. It could handle complex geometries with high accuracy and efficiency by using embedded boundaries or overlapping grids.

Finite Volume method:

The finite volume method (FVM) is a method for representing and evaluating partial differential equations 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 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. "Finite volume" refers to the small volume surrounding each node point on a mesh.

Major differences between FDM and FVM:

• FDM is easier to code than FVM.
• FDM may easily run into the problem over curved boundaries.
• On the other hand, The finite-volume method is a natural choice for CFD problems, since the partial differential equations you have to solve for CFD are conservation laws.
• FVM has a feature like adaptive mesh refinement which could handle complex shapes, unlike FDM.
• However refining the mesh over the corners, the functions that approximate the solution when using the finite-volume method cannot be easily made of higher-order.
• This is a disadvantage on FVM when compared to FDM.

Interpolation Schemes in FVM:

The approximation of surface and volume integrals may require values of the variable at the location other than the computational nodes of the control volume. These values are obtained by employing certain interpolation schemes.

The interpolation schemes used are as follows:

1. Upwind interpolation scheme
2. Linear interpolation scheme
3. Quadratic Upwind interpolation for Convective Kinematics scheme (QUICK)
4. Hybrid interpolation scheme
5. Total Variation Diminishing Scheme (TVD)
6. Essentially Non-oscillatory interpolation scheme

For example, let us consider the simple case of steady one dimensional convection and diffusion.

`(drhouphi)/dx = d/dx(Gamma (dphi)/dx)`

Where `phi` is solution variable,

`rho`is the density,

u is the velocity, and

`Gamma`is the diffusion coeffecient

If there is a 3 control volumes linearly, where middle control volume is Point p and surrounded by West point and east point.

Integration of the governing differential equation over the control volume aorund the grid point P yields:

`(rhouphi)_e - (rhouphi)_w = (Gamma(dphi)/dx)_e - (Gamma(dphi)/dx)_w`

Evalution of left hand side of the above equation requires the values of `phi` over faces e and w. These values are estimated by emoploying the above interpolation schemes.

Flux limiters:

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

`(delu_i) /(delt) + 1/x_i [F(u_(i +1/2)) - F(u_(i-1/2))] = 0``

Where `F(u_(i +1/2))` and `F(u_(i-1/2))` represents edge fluxes for the ith cell.

The edge fluxes represented by high and low resolution schemes, Then a flux limiter can switch between these schemes depending upon the gradients close to the particular cell.

`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,

`phi(r)` = flux limiter function,

and r represents the ratio of successive gradients on the solution mesh

`r_i = (u_i - u_(i-1))/(u_(i+1)-u_i)`

This is the limiter case

if the limiter function = 0, then there will be a low resolution and sharp gradient.

if the limiter function>0, then there will be a high resolution and positive gradient.