Week 9 - FVM Literature Review

Introduction:- To study the Finite Volume Method and its importance in CFD problems.

Finite Volume Method(FVM):-

• FVM transforms the set of partial differential equations into a system of linear algebraic equations.
  Although the discrete approximation procedure is use in the FVM is distinctive, it also utilizes two basic steps.
• Firstly, it transforms and integrates the PDEs into balance equations over an element.
• The process incorporates the changing of the surface and volume integrals into discrete algebraic relations over
  elements as well as their surfaces using an integration quadrature of a specified order of accuracy.
  This will result in a set of semi-discrete equations.
• Secondly, in this next step, the interpolation profiles are chosen to estimate the variation of the variables
  within the element and relate the surface values of the variables to their cell values and thus transform the
  algebraic relations into algebraic equations.
• In regards to the two steps in the FVM process, your approximation selection affects the overall accuracy of the subsequent numeric.
• The FVM is a natural choice for solving CFD issues because the PDEs we have to resolve for CFD are conservation laws.
• The FVM’s most significant advantage is that it only needs to do flux evaluation for the cell boundaries.
  Also, this is true for nonlinear issues as well, which, in turn, makes it an excellent choice for the handling of (nonlinear) conservation laws.
• The accuracy of the FVM, for example, close to a corner of interest, can be increased by refining the mesh around that corner, similar to the FEM.
• However, unlike the FEM, the functions to estimate the solution is not easily made for higher-order.

Major differences between FDM & FVM:-

1. 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).
2. The computational domain is usually divided into hexahedral cells (the grid), and the solution will be obtained at each nodal point.
3. The FDM is easiest to understand when the physical grid is Cartesian, but through the use of curvilinear transforms the method can be extended to domains that are not easily represented by brick-shaped elements.
4. The discretization results in a system of equation of the variable at nodal points, and once a solution is found, then we have a discrete representation of the solution.

1. A finite volume method (FVM) discretization is based upon an integral form of the PDE to be solved (e.g. conservation of mass, momentum, or energy).
2. The PDE is written in a form which can be solved for a given finite volume (or cell). The computational domain is discretized into finite volumes and then for every volume the governing equations are solved.
3. The resulting system of equations usually involves fluxes of the conserved variable,
   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 in to structured numerical grid internally is completely avoided.
4. As with FDM, the resulting approximate solution is a discrete, but the variables are typically placed at cell centers rather than at nodal points.
5. This is not always true, as there are also face-centered finite volume methods. In any case, the values of field variables at non-storage locations.(e.g. vertices) are obtained using interpolation.

Need for interpolation schemes and flux limiters in FVM:-

Interpolation Methods:-
The approximations of surface and volume integrals require values of the variable at locations other than the variable at locations other than the computational nodes of the CV.

Values at these locations are obtained using interpolation formulae.

There are various types of interpolation methods:-

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

• Upwind Interpolation(UDS):-

The upwind interpolation (UDS) for approximating the value of the variable f at the east face of a control volume is given by,

`f_e={(f_p if (v.n)_e >0); (f_e if (v.n)_e <0)`

UDS is equivalent to using a backward or forward finite difference approximation (Depending upon the flow direction)

Advantage:- Upwind interpolation (UDS) is the only approximation that satisfies boundedness criterion unconditionally.

Disadvantage:-

1. UDS is numerically diffusive.
2. Numerical diffusionis magnified in multidimensional problems if flow is obliqueto the grid.
3. Peaks and rapid variations in the variables will be smeared out.
4. Only first order accurate. Hence, very fine grids are accurate solutions.

• Linear Interpolation (CDS):-

Approximate the values of the variable at CV face centre by linear interpolation of the values at two nearest computational nodes.

`f_e= f_Elambda_e + f_P(1-lambda_e),`

`lambda_e=(x_e-x_p)/(x_E-x_p)`

The linear interpolation is equivalent to the use of central difference formula of the first order derivative, and hence, this scheme is also termed as central difference scheme(CDS).

Characteristics:-

• CDS scheme is second order accurate
• It may produce oscillatory solutions.
• Simplest and most widely used interpolation.
• Leads to very simple approximation for evaluation of diffusive fluxes.
• Approximation of gradients has second order accuracy on uniform grids.
• It's formally of first order on non-uniform grids.

Quadratic Upewind Interpolation(QUICK):-

QUICK- Quadratic Upewind Interpolation for Convective

Kinematics

Approxiamate the value of the variable at CV face centre by quadratic interpolation of the values at three nearest computation nodes(One downstream node D and two upstream nodes U, UU):

`f_e=f_U+ g_1(f_D-f_U)+g_2(f_UU-f_U)`

`g_1=[(x_e-x_U)(x_e-x_(UU))]/[(x_D-x_U)(x_D-x_(UU))]`

`g_2=[(x_e-x_U)(x_e-x_(D))]/[(x_(UU)-x_U)(x_(UU)-x_(D))]`

It is third order accurate on both uniform and non-uniform grids.

Hybrid Interpolation Methods:-

Blend of CDS and UDS scheme based on local peclet number.

`f_e= gammaf_(CDS)+(1+gamma)f_(UDS)`

Total variation diminishing (TVD) schemes:-

`f_e=f_P+1/2psi(r)(f_E-f_P)`

`psi "is called a flux limiter function"`

Conclusion:- FVM is the one of the best way to solve computational problems and the with help of proper interpolation scheme and accurate flux limiter, one can solve wide range of CFD problems. FVM is more suitable method as compare to FDM and FEM.