Week 9 - FVM Literature Review

Finite Volume Method: -

The finite volume method (FVM) is a discretization technique like FEM and FDM used in CFD. FVM evaluates the variables at discrete locations (center of the cell) of the geometry under consideration. FVM divides the geometry into a finite number of volumes (by forming mesh/grid). This mesh can be structured or unstructured, forming a finite number of control volumes in which the governing equations are conserved. These control volumes are known as cells in FVM and each cell has one center. The Gauss divergance therom is the foundation of the finite volume method.

Interpolation Schemes in FVM: -

Approximation at the surface and volume integral needs values of the variables at locations other than defined nodes at the control volume and values at these nodes are obtained by interpolation. These interpolation methods are as follows.

Interpolation Methods: -

1.Upwind interpolation scheme (UDS): - Upwind interpolation scheme depends on the direction of the flow and same as forward differencing or backward differencing finite difference scheme. It is first-order accurate.

2.Central differencing approximation scheme (CDS): - This scheme is same as the central differencing scheme in the finite difference method. It is second-order accurate and may give oscillatory solutions.

3.Linear upwind difference scheme (LUDS): - This scheme is same as a one-sided three-point differencing scheme in finite difference method. It is second-order accurate.

4.Quadratic upwind interpolation (QUS): - This scheme is third-order accurate. This scheme is derived using polynomial fitting and may have oscillations.

Flux Limiters: -

Flux limiters are used in high resolution schemes. Flux limiters are used to tune the numerical flux of high order and low order scheme using the flux limiter function in such a way that the resulting scheme gives a high order accuracy in the smooth region of flow and sticks with the first order of accuracy in the vicinity of socks/discontinuities. Some of the common flux limiters in FVM are as follows:

1.Van Leer

2.Min-Mod

3.SuperBee

4.Van Albada

5.UMIST

6.Sweby

Differences in FDM and FVM: -

1.In FEM, the dependent values are stored at the element nodes, but in FVM, the dependent values are stored in the center of the Finite volume.

2.It is much easier to implement FVM to unstructured grids.

3.The finite volume method(FVM) has an advantage over FDM, of the conservation of mass and momentum and energy is exactly satisfied over the entire computational domain.