Page 174 - Computational Fluid Dynamics for Engineers

5.6 Finite-Volume Methods 161

scheme, central scheme, or the upwind scheme discussed in Section 5.5. For
a central cell-centered finite-volume method there are several choices for eval-
uating the flux terms as discussed by Hirsch [1]. Consider here the choice of
averaging the fluxes; for example, write eAB a s

e
e
CAB = 2(^+1 J + ij) = i+i/2j (5.6.8a)
and / B C as
/BC = g (fij+i + fid) = fi,j+i/2 (5.6.8b)

Eq. (5.6.7) can then be written as

Yl F ' $ = (/iJ+i/2 - fij-i/2)k + e i + i/ 2 j - ei-i/2,j)h (5.6.9)
(
sides
Apply Eq. (5.6.Id) to the x- and y-components of the momentum equa-
tions since, for the continuity equation, Q s = 0. For the x-component of the
momentum equation, write Eq. (5.6.Id) for the control volume cell ABCD as

^ Q SS = -(PAB ~PCD)h = -^(Pi-ij -Pi+ij)h (5.6.10a)
sides
and for the ^/-component

-
] T Q SS = -(PBC ~ PDA)h = (Pij-i ~ Pij+i)k (5.6.10b)
sides
Similarly for the energy equation, write

^2 Qs s = ~[(PU)AB - (pu)cD}h - [(PV)BC - (pv)DA]k
sides
= ^[(P^i-ij ~ (pu)i+ij]h+ ^[{pv)ij-i - (pv)ij+i]k(5.6.U)

Inserting the expression given by Eq. (5.6.9) into Eq. (5.6.2) and noting that
fiij = kh, write the left-hand side of the generic form of the conservation integral
equation (2.2.24) as
e
dUjj e i+l/2,j ~ i-l/2,j fij+1/2 ~ fi,j-l/2
dt k h
or
dUjj | ej+ij — Cj-i,j , fij+i — fij-i (Kaii\

Equation (5.6.2) with its right-hand side given by the expressions in Eq.
(5.6.10) leads to second-order accurate space discretizations on Cartesian meshes.
It should be noted that Uij does not represent the quantity at a specific grid
point; it represents an average value of the quantity for cell (i, j). Similarly,