Week 9 - FVM Literature Review

FINITE VOLUME METHOD LITERATURE REVIEW:-

• OBJECTIVE:-

In this review we have disscussion about the finite difference method and finite volume method

Interpolation schemes in finite volume method.

Flux limiters in finite volume method.

IMPORTANT NOTES:-

1. FINITE ELEMENT METHOD .

2. FINITE VOLUME METHOD.

3. FINITE DIFFERENCE METHOD.

Finite element method:- A common technique for numerically resolving differential equations that appear in engineering and mathematical modelling is the finite element method (FEM). The conventional topics of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential are typical issue areas of interest. Using two or three spatial variables, the FEM is a general numerical method for solving partial differential equations (i.e., some boundary value problems). The FEM breaks down a complex system into smaller, more manageable pieces known as finite elements in order to solve an issue. This is accomplished using a specific space discretization in the spatial dimensions, which is carried out by creating a mesh of the object.

2. FINITE VOLUME METHOD:- The Navier-Stokes equations and other partial differential equations that can be expressed as algebraic equations can be solved using the finite volume method. Within the problem domain, discrete nodes surrounded by finite volumes serve as approximations for the physical parameters. A technique for expressing and analysing partial differential equations as algebraic equations is known as the finite volume method (FVM). The divergence theorem is used in the finite volume method to transform volume integrals in a partial differential equation containing a divergence term into surface integrals. The surfaces of each finite volume are then used to evaluate these terms as fluxes. These approaches are conservative because the flux entering a given volume is the same as the flux exiting the adjacent volume. The finite volume method also has the benefit of being simply formulated to support unstructured meshes. Many software for computational fluid dynamics use this technique. The small volume enclosing each node point on a mesh is referred described as a "finite volume."

3. Finite difference method:- The most straightforward way for discretizing partial differential equations is the finite-difference approach. Consider a location in space where a set of discrete equations known as finite-difference equations are used to substitute the continuum representation of the equation.

INTERPOLATION SCHEMES IN FINITE VOLUME METHOD:-

UPWIND INTERPOLATION SCHEME:-

• It satisfies boundedness criteria unconditionally
• However, UDS is numerically diffusive
• Diffusion is magnified if the flow is oblique to the grid in multi-dimensional problems
• Large variations or peaks will be smeared out
• It is only first-order accurate and hence very fine grids are required to obtain an accurate solution.
• UDS is an equivalent to backward or forward differencial approximation.
• the value of varibele

fe = fp if(v.n)e>0

fe = fE if(v.n)e<0.

Linear Interpolation (CDS)

• Approximates the value of the variable at CV face centre by linear interpolation of the values at two nearest nodes
• It is equivalent to the central differencing formula for the first-order derivative, and hence this is termed as the Central Differencing Scheme

fe=fE⋅λe+fP⋅(1−λe)

λe=xe−xPxE−xP

• CDS is second-order accurate on uniform grids, but formally of first order on a non-uniform grid
• Most widely used scheme as it results in simple approximations for evaluation of gradients required for the evaluation of diffusive fluxes
• However, it may produce an oscillatory solution

Hybrid Interpolation Scheme:

• Blend of CDS and UDS scheme based on local Peclet number

fe=fCDS⋅λ+fUDS⋅(1−λ)

Total Variation Diminishing Scheme (TVM):

fe=fP+12⋅ψ(r)⋅(fE−fP)

FLUX ELIMITERS IN FINITE VOLUME METHOD:-

1. Flux limiters are used in high resolution schemes. numerical schemes used to solve problems.

2. The flux limiters are used in fluid dynamics problems.

3. the flux limiters are used as partially differential equations.

consider 1D semi-discrete scheme.

dui/dt+1/deltax1[f(ui+1/2)-F(ui-1/2)] = 0

f(ui+1/2) and f(ui-1/2) represent edges of flux. the lowe and higher resolutions are represent flux schemes.

1D steady state equation:-

The governing equation for one-dimensional steady-state heat conduction equation with source term is given as

d/dx(alpha*dt/dx)+s = 0

where 'T' is the temperature of the rod. The boundary values of temperature at A and B are prescribed.

alpha is thermal diffusivity.

S is the source term.

CONCLUSION:- Finite volume method and finite element method , finite differential method are used as cfd problems.