At 2Ax 4 (euvYT jut _ (puv)) ajo j-t Phas P, 2Ay Ax +n Wha. ~ 241) + 4 172,7 , Weyajet ~ Mia 12,7 t MLV 25-1 (Ax)? (Ay)? (6.91) or atl n At n n (Puy i,j = (Pu)} 6 1/2,; +A At Ay Pith Pi) (6.92) where, from Eq. (6.91), fae (Yi 370,5 (pu?)F_ Wi (PU) tyop44 (puv)i),71 2Ax 2Ay +a Ui 3/2,4 7 Mie 1/25 + 4-1/2) Mis aget ~ 12g t Mit 25| (Ax)? (Ay)? Equation (6.92) is a difference equation representing the x-momentum equation. Note that and v in Eqs. (6.91) and (6.92) are those values defined by Eqs. (6.90a and 5), i.e., and v use different grid points than those for wu. In like manner, a difference equation for the y-momentum equation is obtained. Here, we will difference Eq. (6.89) centered around point (i, j + 3) as shown in Fig. 6.17. We define average values of u at the points c and d on the left and right sides of the shaded cell in Fig. 6.17 as follows: At point c : u = 3 (uj1j.9 + Mi 1/2,541) At point d: = 4 (Ui41/2,7 + M4 1/2741)

Using a forward difference in time and central differences in space, Eq. (6.89) becomes n 1 At, n COMA = (OVi p12 +B At Ay Pit) Pi) (6.93) THE PRESSURE CORRECTION TECHNIQUE: APPLICATION TO INCOMPRESSIBLE VISCOUS FLOW 257 Ay FIG. 6.17 Computational module for the y-momentum equation. The filled-in area is an effective control volume. where B (pvt) 21 7 (pvt)? yj 412 ; (oY Vi jsp _ (pv); j-12 2Ax 2Ay Vijay 7 2Migjeie t-te | Yipes 7 Msi + Yj-1p (Ax)? ) (Ax)? +u Note that u and in Eq. (6.93) are those values defined by the average values at points c and d, ie., wu and w use different grid points than those for v. As outlined in Sec. 6.8.3, at the beginning of each new iteration, p = p*. For this situation, Eqs. (6.92) and (6.93) become, respectively, At, (pu ye 2,7 = (0M Yin ty,j tA At 7 i417 Pi) (6.94) / Ax 258 some siMPLe CFD TECHNIQUES: A BEGINNING

At and (pv Vijeap = (pv Jigen +B At Ry Piss PF ;) (6.95) Subtracting Eq. (6.94) from Eq. (6.92), we have At , (pu j = (pu')}, 257 A At A Pi Pi) (6.96) where n+l ntl xyntl (pu! Vi yj = (0uyf yj (pu Woap.3 (pul) 72,j = (PUyF iy _ (pu Vi. 1p2,; A =A-A* Phat. = Pit hij Pisty ij = Pig ~ Pi Subtracting Eq. (6.95) from Eq. (6.93), we obtain At 1 (pv isi = (ev) 1p +B At regeee Pi) (6.97) where 1 1 ayn (pv Pye = (every (pv ares (ev) je ap = (pr) ja 1p _ (Pv) peay2 Be=B-B Phat = Pig) Pij4t Pig = Pig Pi; Eqs. (6.96) and (6.97) are the xand y-momentum equations expressed in terms of the pressure and velocity corrections p, u, and v defined by Eqs. (6.86), (6.872), and (6.87b), respectively.

We are now in a position to obtain a formula for the pressure correction p by insisting that the velocity field must satisfy the continuity equation. However, we are reminded that the pressure correction method is an iterative approach, and therefore there is no inherent reason why the formula designed to predict p from one iteration to the next be physically correct; rather, we are concerned with only two aspects: (1) the formula for p must yield the values that ultimately lead to the proper, converged solution, and (2) in the limit of the converged solution, the formula for p must reduce to the physically correct continuity equation. That is, we are allowed to construct a formula for p which is simply a numerical artifice designed to expedite the convergence of the velocity field to a solution that satisfies the continuity equation. When this convergence is achieved, p 0, and the formula for p reduces to the physically correct continuity equation.