Week 9 - FVM Literature Review

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.

Derivation:

• In FVM instead of writing the equation in a single point, equations are written in an entire volume. It means for a considered volume no assumptions are made over the shape of the volume. Inside the particular volume the flow is conserved, flow quantities are conserved, energy is conserved, mass is conserved, momentum is conserved and so on. This is called as Finite Volume Representation ,which begins with Integral form of governing equation.
• Once we have the integral form of equation.
• Apply Gauss Divergence theorem.
• Consider a steady state heat conduction equation with heat generation:

∂/∂x(α.∂T/∂x)+S =0
S is source term
α is thermal diffusivity term
∂T/∂x is temperature gradient along the x direction.
If small control volume (dv) is considered and it is multiplied with the governing equation and integrated it for entire volume then that should be also equal to zero.

Multiplying the equation by 'dv' and integrating it 

∫(∂/∂x (α.∂T/∂x)+S)dv=0

∫(∂/∂x (α.∂T/∂x)dv)+∫Sdv=0

∴dv=A.dx

∫(∂/∂x (α.∂T/∂x)A.dx)+∫Sdv=0

[αA(∂T/∂x) ]_w^(e )+S ̅dv=0

s ̅ is average value of source term(S)

α_e A ∂T/(∂x_e )-α_w A ∂T/(∂x_w )+S ̅dv=0

αA[(T_e-T_p)/∆x]-αA[(T_p-T_w)/∆x]+S ̅dv=0

[Heat flux]out –[Heat flux]in + Sourceterm =0

Differences between FDM & FVM:

• A finite difference method(FDM) discretization is based upon the differential form of the PDE to be solved. Each derivative is replaced with an approximate difference formula that can generally be derived from a Taylor series expansion. The computational domain is usually divided into hexahedral cells(grid) and the solution will be obtained at each nodal point.
• A finite volume method(FVM) discretization is based upon an integral form of PDE to be solved(eg. conservation of mass, momentum, or energy). The PDE is written in a form that can be solved for a given finite volume for cells. The computational domain is discretized into finite volumes and then for every volume, the governing equations are solved. The resulting system of the equation usually involves fluxes of the conserved variables, and thus the calculation of fluxes is very important in FVM
• 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 into a structured numerical grid internally is completely avoided

INTERPOLATION METHODS :-

• The approximation of surface and volume integrals require values of the variable at locations other than the computational nodes of the CV.
• Values at these locations are obtained using interpolation formulae.
• Various possibilities for interpolation. e.g.

1. Upwind interpolation
2. Linear interpolation
3. Quadratic upwind interpolation
4. Hybrid, TVD and ENO interpolation schemes.

FLUX LIMITER :-

• Flux limiters are used inhigh resolution schemes i.e. numerical schemes used to solve problem in science and engineering, paricularly fluid dynamics, described by PDE\'s. They are used in high resolution schemes, such as the MUSCL scheme, to avoid the spurious oscillations(wiggles) that would otherwise occure with high order spatial descritization schemes due to shocks, discontinuities or sharp changes in the solution domain. Use of flux limiters, together with an appropriate high resolution schemes, make the solution total variation diminishing (TVD).
• Note that flux limiters are also reffered to as slope limiters because they both have the same mathmatical form, and both have the effect of limiting the solution gradient near shocks or discontinuities. In general, the term flux limiter is used when the limiter acts on system stetes (like pressure, velocity...etc).

Need for Interpolation schemes and flux limiters in FVM :-

• The Guass or divergence theorem simply states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface.
• This theorem is fundamental in the FVM, it is used to convert the volume integrals appearing in the governing equation into surface integrals.
• The face values appearing in the convective and diffusive fluxes have tobe computed by some form of interpolation from the centroid values of the control volume at both sides of face.
• After spacial discretization, we can proceed with the temporal discretization. By proceeding in this way we are using the method of lines (MOL).
• The main advantage of the MOL method, is that it allows us to select numerical approximations of different accuracy for the spacial and temporal terms. Each term can be treated differently to yield to different accuracies.
• Nearly all flows in nature and industrial applications are unsteady (also known as transient or time dependent).
• Unsteadiness is due to-Instabilities, Non-equilibrium initial conditions, Time dependent boundary conditions, source term, chemical reactions, movieng or deforming bodies, turbulence, Buoyancy, Convection and Multiple phases