Week 9 - FVM Literature Review

Finite Volume Method(FVM):

The finite volume method(FVM)is a method to for representating and evaluating partial differential equation in form of algebric equation.

In this method volume integrates into a partial differential equation that contain a divergence term and converted into surface integrals, using the divergene theoram. These terms are then evaluated as a flux at surface of each finite volume.

Finite volume refers to the small volume sorrounding each node point on a mesh.

The advantage of the finite volume method is that it is easily formulated to allow unconstructed meshes and this method used in many computational fluid dynamics packages.

Difference between FDM & FVM:

The finite volume method is a method for representing and evaluating partial differential equationsin the form of algebric 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 features of finite volume schemes is their conservations properties. Since they are based on applying conservation principles over each nodes.

'Finite Volume' refers to the small volume surrounding each nodes point on a mesh means the grid points where variables are stored are typically defined as being at the center of each control volume.

We cannot used FDM when the grids are not aligned along x,y & z direction. Instead of writing the equation in a point, we write it for a volume.

Also the FVM representation preserves the concept of conservation.

Interpolation :

interpolation is the process of using pointwith known values to estimate value at other unknown points. In FVM interpolation schemes are used to find the value of volume integrals required at the points other than than nodes.

Some of the used methods are the central differencing scheme,hybrid scheme,upwid scheme and QUICK scheme.

d/dx(α.∂T/∂x) + S = 0

where T = temperature

S = source term

α = thermal diffusivity

∂T/∂x = temperature gradient x direction

multiplying the governing equation by 'dv' and integrating it

∫(d/dx(aplha.∂T/∂x) + S) dv = 0

here , dv = A*dx

∫d/dx(α. ∂T/∂x) A dx + ∫ S dv = 0

After soving we got

[Heat flux ] out - [ Heat flux] in +sourseterm = 0

Flux limiters :

The flux limiters are used in high resolution schemes - numerical schemes used to solve problems in science and engineering,particularly fluid dynamics, described by partial differential equation(PDE). They are used in high resolution schemes such as MUSCL scheme to avoid the oscillations.