Finite Volume Method Literature Review

There are three numerical methods based on solver to solve the governing equation (Navier stokes equation) in Computational fluid dynamics. Basically, we already discuss the finite difference method and here we discuss the finite volume method and the third one is a finite element method.

Finite Volume  Method:

The finite volume method is used for representing and evaluating partial differential equations in algebraic equations. It is quite similar to FDM or FEM. In this, values are calculated on discrete places on a meshed geometry. FVM refers to a small volume surrounding each node point on the mesh. In FVM, volume integrals in a PDE that contain a divergence term are converted into surface integrals using Gauss Divergence theorem. They are evaluated as fluxes at the surface of each finite volume. Because the flux entering a finite volume is identical o that leaving adjacent volume, it is conservative. This technique is versatile/flexible and can be applied to any complex geometry.

Interpolation Schemes:

The approximation of surface and volume integrals may require values of the variable at the locations other than the computational nodes of the CV. Values at these locations are obtained using the interpolation formula.

`frac{del}{delx}(alpha.frac{delT}{delx})+S =0`

where,

S is a source term

alpha is thermal diffusivity term

dt/dx is temperature gradient along the x-direction.

Example: In the 1D linear Heat conduction equation is steady-state, the source and alpha values are a function of temperature. We need to calculate them separately. According to the temperature of the face (shaded region). To calculate flow variables at that face (pressure, temperature, velocity), we use interpolation schemes which takes information from present and neighbor cell.

we know that the 1D Heat conduction equation is valid at every point in the domain, hence it is also valid in the entire control volume. 

If a small finite control volume (dv) is considered and it is multiplied with the governing equation and integrated for the entire volume, then that is also equal to zero. This is called as \'\'Integral form of the governing Equation\'\'.

`int(frac{del}{delx}(alpha.frac{delT}{delx})+S)dv=0`

`int(frac{del}{delx}(alpha.frac{delT}{delx}))dv+int(S)dv=0`

because of dv=A.dx

so `int(frac{del}{delx}(alpha.frac{delT}{delx}))A.dx+int(S)dv=0`

`[alphaA(frac{delT}{delx})]_w^ev+barSdv =0`

`barS` is the average value of source term (S) which when multiplied by dv is going to be equal to `intSdv`.

`alpha_eAfrac{delT}{delx_e) - alpha_wAfrac{delT}{delx_w} +barSdv=0`

If we assume alpha as a constant and not a function of temperature,

`alphaA[frac{T_e-T_p}{Deltax)] - alphaA[frac{T_p-T_w}{Deltax}] +barSdv=0`                    (`barS` linearization = `S_u+S_PT_p`)

|Heat flux|_out - |Heat flux|_in +Sourceterm =0

1) Upwind Differencing Scheme (UDS)

Approximation by its value the node upstream of \'e\'

2) Central Differencing Scheme (CDS)

Linear interpolation between nearest nodes 

3) Quadratic Upwind Interpolation (QUICK)

Interpolation through a parabola: three points necessary P, E, and point in the upstream side.

Flux Limiters

While calculating the variable gradients they tend to toggle and are not bounded to a particular value and keeps jumping, this occurs due to discontinuity due to shock occurring,  having a flux limiter will help to bound the gradient value thereby giving a stable solution.

Numerical approximation using higher-order spatial discretization schemes results in better accuracy but exhibits spurious oscillations near discontinuities such as shocks. On the other hand, lower-order spatial discretization schemes remain stable near discontinuities but dampen out the distributions because of their dissipative nature. Hence in order to capture discontinuities, high-resolution schemes are desired, which are highly accurate and stable throughout the domain.

Flux limiters operate functionally as a switch and tune the numerical fluxes obtained from lower-order schemes and higher-order schemes such that near the discontinuity lower order(stable) scheme is employed and away from it higher-order (accurate) scheme is employed.

`frac{du_i}{dt}+frac{1}{x_i}{F(u_(i+1/2))-F(u_(i-1/2))}`

where,

`F(u_(i+1/2)) and F(u_(i-1/2))` represents the edge fluxes for the ith cell. If this can be represented by low and high-resolution schemes, then flux limiter can switch between these schemes depending upon the gradients close to the particular cells as follow.

`F(u_(i+1/2) )= f_(i+1/2) ^ (low) - phi(r_i)(f_(i+1/2) ^ (low)-f_(i+1/2) ^ (high))`

`F(u_(i-1/2) )= f_(i-1/2) ^ (low) - phi(r_i)(f_(i-1/2) ^ (low)-f_(i-1/2) ^ (high))`

`f^(low)` = low precision, high resolution flux

`f^(high)` = High precision, low resolution flux

`phi(r)` = flux limiter function, r represents the ratio of the `phi(r)>=0`successive solution on the gradient mesh.

`r = u_i - frac{u_(i-1)}{u_(i+1)}-u_i`

this is the limiter function case.

if limiter function = 0, then there  will be a low resolution and sharp gradient

if limiter function >0, then there will be high resolution and +ve gradient