Week 9 - FVM Literature Review

Finite Volume Method (FVM) is a numerical technique for solving partial differential equations governing the phenomena. The method involves discretizing the domain into smaller volumes. These control volumes are connected by common faces. The governing equation in integral form is applied for all the control volumes followed by appropriate integral transformations (like Gauss divergence theorem). The integral then leads to a set of linear equations of the quantities described by the differential equations. The linear equations are solved to obtain the solution.

In FDS, the domain is discretized into nodes and the properties are determined at the nodes. No conservation of quantity is used in FDM. In FVM, the domain is discretized into finite volumes with faces. The properties are usually stored at the cell centers. Conservation of quantity is applicable in FVM. 

Consider the 1-D steady-state heat diffusion equation. Also, consider the domain to be divided into smaller control volumes. Since the domain is 1-D, each control volume has two faces, left (West) and right (East) represented as w and e below. W and E refer to the neighboring volumes on left and right respectively. Consider the control volume named P of volume dV. The values of the properties to be determined are stored at the center of the control volumes. 

`d/dx(k(dT)/dx)=0.`

Apply the above equation in the integral form to the control volume P.

`intintintd/dx(k(dT)/dx)dV=0.`

dV is the volume of the finite volume element P. After applying the Gauss-divergence theorem it results as following 

`intintk(dT)/dxdA=0`. 

The surface integral is evaluated at all the faces (east and west for 1-D). 

`(kA(dT)/dx)_e-(kA(dT)/dx)_w=0` where subscript e and w mean east and west faces.

Here comes the use of interpolation schemes because the properties are known at only centers of volume. With interpolation schemes, values at faces are determined from neighboring centers. 

Such schemes include First order Upwind scheme, Power-law, Second-order Upwind scheme, QUICK, etc.

The first-order upwind scheme uses the upstream cell center value for a face. Depending on the flow, an appropriate interpolation scheme is used. Similarly, the derivatives at faces are also computed using neighboring cell center values. 

The purpose of flux limiter schemes is to restrict the derivatives to acceptable values like in cases of shocks, discontinuities in the domain. Usually, they are used in higher-order schemes. 

The gradients across a face (in higher-order schemes containing discontinuities) is determined using gradients obtained (if possible) by low and high-resolution schemes and an appropriate flux limiter function evaluated at that face.