F V M literature review

• The Finite Volume Method (FVM) is a numerical technique that transforms the partial differential equations representing conservation laws over differential volumes into discrete algebraic equations over finite volumes
• Following step involved in F. V. M

1) GRID GENERATION : In fvm method geometry devided in number of part that devided part called as control volume. At the centre of control volume partial differential solving being solved.

2). DISCRETISATION : Discretize control volume (Geometric Domain)

3). SOLVING PARTIAL DIFFERENTIAL EQUATION : The first step in the solution process is the Discretisation of the geometric domain.

• The partial differential equations are then discretizedtransformed into algebraic equations by integrating them over each discrete element. The system of algebraic equations is then solved to compute the values in the finite volume method.
• some of the terms in the conservation equation are turned into face fluxes and evaluated at the finite volume faces. Because the flux entering a given volume is identical to that leaving the adjacent volume, the FVM is strictly conservative.

• Let\'s apply F.V.M method to general convection diffusion equation :

• `(del rho.phi)/(delT) + grad(rho*phi*v) = grad (Gamma.grad(phi) + S`

for steady state and without heat generation equation written as , `grad(rho*phi*v) = grad (Gamma.grad(phi)`

`d((rho*phi*v) )/dx= d(Gamma.grad(phi))/dx`

flow must follow contunuity eqauion d(rho*u)/dx =0 .-------------------------------------------(1)

• Applying guass diverence equation we get :

`(rho*A*phi*v)e - (rho*A*phi*v)w = (Gamma.A*(d(phi)/(dx)))e -(Gamma.A*(d(phi)/(dx)))w`

`ubrace ((rho*A*phi*v)e - (rho*A*phi*v)w)` = convective flux across faces

• considering F=rho*u(convective flux per unit area ) & D= Gamma/dx (conductance flux at face )
• Fe = (rho*u)e & `De = ((Gamma)w )/ ((deltax)WP)` same as west face
• for Ae= Ap =A Equation is ,

• `Fe*(phi)e - Fw*(phi)w = De*((phi)E-(phi)P) - Dw*((phi)P-(phi)W)` --------------------------------(2)

• Before solving equation need to understand about peclet no :
• PECLET NO : Convective flux / Diffusive flux

• At different peclet no. different type of scheme is being used as per nature (value) of convective and diffusive flux.
• In order to solve eqution (2) we need to calculate the transported property phi at the faces e and w . To solve above equation different scheme being used .

1. CENTRAL DIFFERENCE SCHEME : When the flow direction is on the West to east it is equally distributed in the the both point.

• In central diffenrence scheme to solve phi at faces linear interpolation (phi at Point P & phi at point E or phi at point W) is use.

`(phi)w = ((phi) P + (phi)W)/2 & (phi)e = ((phi)P + (phi)E)/2`---------------------------(3)

• from (2) & (3) equation :

`AP*(phi)P = AW*(phi)W +AE*(phi)E` .

AW	AE	AP
Dw + (Fw/2)	De - (Fe/2)	AW + AE +(Fe-Fw)

• This central difference scheme is unstable for pe>2.

2. UPWIND DIFFERENCE SCHEME: Instability occured while using central difference scheme because it\'s inability to flow direction..

• When flow direct from West to east it\'s have influence on the West Side , which was not considered in Central difference scheme.
• Convective property phi has the same value as upsteam node value .

• for uw>0, ue>0 , (phi)W = (phi)w ,(phi)P = (phi)e -------------------------------------(4)
• from equation (2) & (4)
• `AP*(phi)P = AW*(phi)W +AE*(phi)E`
• for uw<0, ue<0 , (phi)E = (phi)e ,(phi)P = (phi)w -------------------------------------(5)
• from equation (2) & (5)
• `AP*(phi)P = AW*(phi)W +AE*(phi)E`

	AW	AE
Fw> 0 , Fe > 0	Dw + Fw	De
Fw< 0 , Fe < 0	Dw	De-Fe

• This method is stable for pe>2 .

• 3. QUADRATIC DIFFERENCING SCHEME : The face value at phi is obtained from interpolation by using two node value (on each side of faces ) and node on the upstream side .

`(phi) face = (6*(phi)( i-1) + 3 (phi)i -(phi) (i-2))/8`.

• Φ_w when u_w > 0 and u_e > 0 a quadratic fit through WW, W and P is used,
• Φ_e when u_w > 0 and u_e > 0 a quadratic fit through W, P and E is used,
• Φ_w when u_w < 0 and u_e < 0 values of W, P and E are used,
• Φ_e when u_w < 0 and u_e < 0 values of P, E and EE are used.
• Quick scheme is more accurate comapred to other scheme but shown overshoot and undershoot sometimes beacuse of this value fluctuates ..

• 4. HYBRID DIFFERENCING SCHEME : It is Combination of Central differencing & Upwind difference scheme .

• Discritize equation should have following characteristics. Because occured while using central difference scheme. .

1. Conservativeness. : Property must be conserved
2. Boundness :obtained value between boundary should bounded

• AP ,AW,AE must have positive sign.
• if source present (non linear ) like s=se + su*T then su must be nagavite .

3. Transportiveness : its purely depend on the value of peclet no . scheme should give accurate result for given peclet number.

FLUX LIMITERS :

• construction of flux limiter schemes is to limit the spatial derivatives to realistic value .
• As shown in quick scheme there is overshoot and undershoot meaning fluctuation of value between local max. to min value .
• Source of oscillation is actually is being interpolated when there is overshoot and undershoot.
• Need to figure out extapolate which doesnt create max. & min. value and sheme using is T.V.D (TOTAL VARIATION DIMINISHING SCHEME).

• First result got f.v.m method is converted into avarage value at cells . then with using interpolation value is obtained at respective side of face .

1. If extrapolated value is higher than avarage value then then fluctuation occured and we will require flux limiter.
2. if extapolated value is lower than avarage value at that face then extrapolation itself is used.

• DIFFERENT TYPE OF FLUX LIMITER:
• Mindmod
• superbee
• van leer

• HOW ITS WORK : Slope at i , i+1/2 cell being calculated and multiplity it with succesive gradient ratio.

• If r = 1 , then phi =1
• if r< 0 then phi =0.