Page 251 - Computational Fluid Dynamics for Engineers

240 7. Boundary-Layer Equations

y, that is, du/dx = 0. The momentum equation becomes an ordinary differential
equation and can be written as

V (P7 5 2)
-Ty = "d? - -
since from the continuity equation v(x, y) = v w = const.

(a) Show that the solution of Eq. (7.5.2) is

^ = 1 - exp( W ) (P7-5.3)

This velocity profile is known as the asymptotic suction profile.
(b) Using Eq. (7.5.3), show that

<S* = - — , 6 = - \ — , r(0) =r w = -gv wUoo (P7.5.4)
Z Vyj
V W
7-6. The idealized flow discussed in P7.5, does not exist close to the leading
edge of the plate, even though uniform suction starts at the leading edge. After
a certain distance, x downstream, however, the asymptotic suction will materi-
alize.
The distance x from the leading edge necessary to have the asymptotic-
suction flow has been determined by Iglisch [11] to be
w«ooxy/2 = __^ 1/2 _ 2

(a) Use the computer program of Section 7.4 and with appropriate changes ex-
amine the way in which the boundary-layer solutions approach the asymp-
totic values of
2
- - )
if = 2, Tw/tm oo = -— ( P7 6 2
u t oo
(b) Compare the calculated wall-shear parameter ^ with those given by Iglisch
/
for the region before the asymptotic suction profile is reached at the £-
stations shown in Table P7.1.
Hint: Since the boundary-layer equations are being solved in the transformed
plane (£,77), from Eq. (7.3.4)

U0 = -—(RL0 1/2 (P7.6.3)

From Eqs. (P7.6.1) and (P7.6.3), we see that when f w = 2, we have the
asymptotic suction profile according to Iglisch's expression (P7.6.1).

Figure P7.1 shows the computed values of H for a flat plate with uniform suction
and Table P7.2 shows the computed f£ values at the desired ^-stations.