Central differencing scheme

The central differencing approximation has been used to represent the diffusion terms which appear on the right hand side of convection-diffusion equation.In order to solve convection diffusion equation we need to calculate the transported property Φ at the e and w faces, the central differencing scheme is one of the process. It is logical to try linear interpolation to compute the cell face values for the convective terms on the left hand side of this equation. For a uniform grid we can write the cell face values of property Φ as

Steady state convection diffusion equation: div(ρuΦ)= div(ГgradΦ) +SΦ  ; here Г is diffusion coefficient & Φ is the property Formal integration over a control volume gives ʃAn.(ρuΦ)dA = ʃAn.(ГgradΦ) +ʃcvSΦdV → Equation 1. This equation represents flux balance in a control volume. The left hand side gives the net convective flux and the right hand side contains the net diffusive flux and the generation or destruction of the property within the control volume. In the absence of source term equation one becomes d/dx (ρuΦ) = d/dx(ГdΦ/dx) → Equation 2. Continuity equation: d/dx (ρu)=0 → Equation 3.

Assume a control volume:

Integration of equation 2 over control volume gives: (ρuΦA) e - (ρuΦA) w = (ГAdΦ/dx)e - (ГAdΦ/dx)w → integrated convection-diffusion equation Integration of equation 3 yields: (ρuA) e - (ρuA) w = 0 → integrated continuity equation It is convenient to define two variables F=ρu and D = Г/δx to represent the convective mass flux per unit area and diffusion conductance at cell faces. Assuming Ae = Aw, we can write integrated convection-diffusion equation as:

FeΦe - FwΦw = De (ΦE – ΦP) – Dw (ΦP – ΦW)

And integrated continuity equation as: Fe – Fw = 0 In central differencing scheme we try linear interpolation to compute cell face values for convection terms. For a uniform grid we can write cell face values of property Φ as Φe = (ΦE + ΦP) ̸ 2 Φw = (ΦW + ΦP) ̸̸ 2 On substituting this into integrated convection – diffusion equation we obtain Fe (ΦE + ΦP) ̸ 2 + Fw (ΦW + ΦP) ̸̸ 2 = De (ΦE – ΦP) – Dw (ΦP – ΦW) And on rearranging [ (Dw + Fw/2) + (De - Fe/2) + (Fe – Fw) ] ΦP = (Dw + Fw/2) ΦW + (De - Fe/2) ΦE aPΦP =aWΦW + aEΦE Different aspects of central differencing scheme:

    1. Conservativeness:

Conservation is ensured in central differencing scheme since overall flux balance is obtained by summing the net flux through each control volume taking into account the boundary fluxes for the control volumes around nodes 1 and 4.

Boundary flux for control volume around node 1 and 4

[ Гe1 (Φ2 – Φ1)/δx - qA ] + [ Гe2 (Φ3 – Φ2)/δx – Гw2(Φ2- Φ1)/δx ] + [Гe3(Φ4 – Φ3)/δx –

Гw3(Φ3 - Φ2)/δx ] + [ qB - Гw4(Φ4 – Φ3)/δx ] = qB - qA

Since Гe1 = Гw2 , Гe2 = Гw3 , Гe3 = Гw4

2. Boundedness: Central differencing scheme satisfies first condition of Boundedness

                     Since Fe – Fw = 0 from continuity equation, therefore  aP = aE+ aW

Another essential requirement for Boundedness is that all coefficients of the discretised equations should have the same sign (usually all positive). But this is only satisfied when (peclet number) Fe/De < 2 because for a unidirectional flow(Fe >0, Fw>0) aE = (De - Fe/2) is always positive if De > Fe/2

             3. Transportiveness:

It requires that Transportiveness changes according to magnitude of peclet number i.e. when pe is zero Φ is spreaded in all directions equally and as pe increases (convection>diffusion) Φ at a point largely depends on upstream value and less on downstream value. But central differencing scheme does not possess Transportiveness at higher pe since Φ at a point is average of neighbouring nodes for all pe.

              4. Accuracy:

The Taylor series truncation error of the central differencing scheme is second order. Central differencing scheme will be accurate only if Pe < 2. Owing to this limitation central differencing is not a suitable discretisation practice for general purpose flow calculations.