Week 9 - FVM Literature Review

popular discretization approaches in CFD

• finite difference method
• finite volume method
• finite element method

finite volume method :-

finite it is the numerical tool that is borrowed from calculus of variation. there are lot of types of FEM like point collocation method etc. here they assume some trial function and multiply that trial function with weighting function. in gelerkins method the trial function itself weighting function. differnce method follows different ways in weighting. the this weighting function is multiplyed with trial function then integrate over the control volume and equate to zero. then we get the one set of algebraic equations. solving will give solution here we are wrking only in error and differential equation some times conservative law are violated. this method is more accurate then FDM and FVM. ideal for linear PDE's, expensive and complex for non linear PDE's here higher order is accuracy is achived by using higher order basis (i.e) shape functions. extending to higher order accuracy is relatively complex then FVM and FDM. higher order accurate calculations are explensive in computation and mathamatical formulation especially for no linear PDEs mostly sutable for heat transfer. structural mechanics, vibratinal analysis etc.

a FVM discretization is based upon an integral from of PDE,s is written in form which can be solved for a given finit volume. the computaion domain is dixcretized into finite volume and then for then for every volume the governing equations are solved. the resulting system of equations usually involes fulxes 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 effor t to convert the given mesh inti structured numarical grid internaly completly avoided. as with FDM, the the resulting approximate solution is a discrete, but the variables are typically placed at the cell centers rather then at nodal points, 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(eg. vertices) are obtained using inerpotaion.

major differnce between FVM and FDM

1) FDM:-

• mostly for defined geometries which could be represnted by structural grids
• i feel this method is subset of FEM as it works mainly for structured spatial discretizatin. which means, the problems solved by FDM could be solved using finite element, but converse is not necessarily true.
• one of the deviations in the case of FDM would come from approximating the higher order terms in taylor's series which is used to linearrize the PDE. this called as truncation error.
• other importent differrntiatinf factor is FDM compared to FEM and FVM is that it is based on the differential from of groverning equations. FEM and FVM are based on the integral from or weak fromulation

2)FVM:-

• the properties are calculated for every cell instead of nodes
• based in gthe integral form of conservation laws and can handle discontinuites in solution. in simple terms, what comes in, must go out.
• FVM approximate the value of the integral on the reference cell
• efficient in solving fuild flow problem
• like FEM,lot of established and theoretically proven subroutines are place for FVM which can be implemented in codes.

types of scheme

• gradient scheme
• interpolation scheme

here we are only talking about the interpolatuon scheme

in interpolation scheme the approximation of surface and volume integrals require values of the variables at location other than computaional nodes of the control volume. values of these locations are abtained using interpolation formulae

types

1. upwind interpolation
2. linear interpolation
3. quadratic interpolation
4. hybrid scheme
5. total variation diminish

1)upwind interploation scheme

here we would required upwind interplation scheme which is essentially used fpr convection dominated problems, and here the velocity at a given point direction of that which determines the variables values to be used in the interpluation process, the name upwind refers to that we are going to make use of the node which is upwind or upstream of the given node.

the upwind interploation for approximating the values of a varibale 'f' at the east face to the controlv volume is given as below. we want approximation at the eastern face 'e'

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

(v.n)e is velocity companent at point e in xdirection

this scheme satisfies the boundedness criterion, in the sense that irresprctive if what the velocity values are and irrespective of what is the grid spacing the solution will always remains bounded

2)hybrid scheme

this scheme is the combination of upwind interplation and cental distribution scheme based on the local peclet number

`fe=gammafCDS+(1-gamma)fcds`

3)total variation diminish

`fe=fp+0.5*psi*r*(fe-fp)`

`psi`= local flux limiter ,r= specified function whichbwill depend upon the loacl flow condition

flux limiter:-

``they are used in high resolution scheme , to avoid false oscillation that would otherwise occur with high order spatial discretization scheme due to shocks, discontinuites or shape change in the solution domain. use to flux limiters, togethor with a apropriate high resolution scheme make the solutions total variation

in TVD the main idea behind the construcation of flux limlter scheme is to limlit the spatial derivative to realsitic values the main purpose of flux limiter is to keep thevalues bounded at thecells faces, flux limiter dont let thye solution to overshoot.