Week 9 - FVM Literature Review

FINITE VOLUME METHOD :-

• The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. Similar to the finite difference method (FDM) or finite element method (FEM), values are calculated at discrete places on a meshed geometry.
• One important feature of finite volume schemes is their conservation properties. Since they are based on applying conservation principles over each small control volume, global conservation is also ensured.
• \"Finite volume\" refers to the small volume surrounding each node point on a mesh means the grid points where variables are stored are typically defined as being at the centre of each control volume.
• The method starts by dividing the flow domain into a number of small control volumes.
• 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.
• The advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes, and this method is used in many computational fluid dynamic packages.

DIFFRENCES BETWEEN FDM AND FVM :-

FDM

Boundary conditions are needed to truncate the computational domain. They represent communication with the surrounding world, which is the part that we do not want included in our simulation. With the finite-difference method, we may easily run into problems handling boundaries for the purpose of defining the boundary conditions.

The finite-difference method is the most direct approach to discretizing based upon the differential form of the PDE to be solved. We consider a point in space where we take the continuum representation of the equations and replace it with a set of discrete equations, called finite-difference equations.

The finite-difference method is typically defined on a regular/structured grid (i.e. the computational domain is usually divided into hexahedral cells (the grid), and the solution will be obtained at each nodal point.) and this fact can be used for very efficient solution methods. Regular grids are useful for very-large-scale simulations on supercomputers.

The method is therefore not usually used for irregular CAD geometries, but more often for rectangular or block-shaped models.

Examples include Formula 1 car computational-fluid-dynamic (CFD) simulations and space-shuttle CFD simulations. The finite-difference method is more difficult to use for handling material discontinuities. In addition, it does not lend itself for local grid refinement or anything similar to “adaptive mesh refinement.” This may be needed to resolve local rapid variations in solutions such as around a corner of a complex shape, as described earlier.

But, if we can fit the simulation in a rectangular or box-shaped geometry using a regular grid, efficient implementations are much easier than for finite-volume methods.

If one can overcome the boundary-condition problem on curved boundaries, the method gives very efficient and high-quality results.

The finite-difference method is defined dimension per dimension; this makes it easy to increase the “element order” to get higher-order accuracy.

FVM

The finite-volume method is similar to the finite-differencing method in that the CAD model is first divided into very small but finite-sized elements of geometrically simple shapes. Apart from this, the finite-volume method is very different from the finite-differencing method, beginning with the concept of elements, which are instead referred to as cells.

A finite volume method (FVM) discretization is based upon an integral form of the PDE to be solved (e.g. conservation of mass, momentum, or energy) and can handle discontinuities in solutions. The PDE is written in a form which can be solved for a given finite volume (or cell). The computational domain is discretized into finite volumes and then for every volume the governing equations are solved. The resulting system of equations usually involves fluxes of the conserved variable, and thus the calculation of fluxes is very important in FVM.

The finite-volume method’s strength is that it only needs to do flux evaluation for the cell boundaries. This also holds for nonlinear problems, which makes it extra powerful for robust handling of (nonlinear) conservation laws appearing in transport problems.

The local accuracy of the finite-volume method, such as close to a corner of interest, can be increased by refining the mesh around that corner. However, the functions that approximate the solution when using the finite-volume method cannot be easily made of higher order. This is a disadvantage of the finite-volume method compared to the finite-difference method.

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 in to structured numerical grid internally is completely avoided. As with FDM, the resulting approximate solution is a discrete, but the variables are typically placed at cell centre’s rather than at nodal points. This is not always true, as there are also face-centered finite volume methods. In any case, the values of field variables at non-storage locations (e.g. vertices) are obtained using interpolation.

FVM can guarantee the local and global conservation of the system.

Interpolation schemes in Finite Volume Method:

Interpolation is a process to estimate the values at unknown points by using the points with known value and sample points on either sides of the unknown point. In FVM, interpolation schemes are used to find values of volume integrals required at the points other than nodes.

The different interpolation schemes are:

1. Upwind Interpolation Scheme (UDS)

2. Linear Interpolation Scheme

3. Quadratic Upwind Interpolation Scheme

FLUX LIMITERS:

For convective fluid flow, it is observed that the low order schemes are usually stable but quite dissipative in nature around the points of discontinuity or shocks while the higher order schemes are unstable in nature and show oscillations in the vicinity of discontinuity or shocks. Highly accurate and oscillation free schemes are known as High Resolution Schemes.

Flux Limiter functions are used to fine tune the higher order and lower order schemes in such a way that the resulting scheme gives a higher order accuracy in the smooth region of the flow and maintains first order accuracy in the vicinity of shocks and discontinuities. For such a scheme TVD scheme is employed.