Page 243 - Computational Fluid Dynamics for Engineers

232 7. Boundary-Layer Equations

V
^ ^ + ~ = 1 (7.5.2a)
cr b z
Inviscid-flow theory provides the external velocity distribution in the form

u e(s) = Uoo(l + t) cos(3 (7.5.2b)

where s is the surface distance, t the thickness ratio of the ellipse = b/a) and
(
(3 the angle between the line tangent to the body and the positive x-axis, that
is,
0 = an" 1 ^ (7.5.3)
t
ax
Since the computer program requires the specification of the dimensionless ex-
ternal velocity distribution as a function of dimensionless surface distance s/L,
the following expression is used to compute the dimensionless distance of a two-
dimensional body in which the dimensionless coordinates (x/L) and (y/L) are
given in tabular form
+
=
+
(i)* (iL v [Oi~ (Bi-J i^X" ® (7.5.4)
Here (s/L)i denotes the distance at a given surface location, (x/L)i, (y/L)i (i =
1,2,...,/) with the surface distance calculations starting from the stagnation
point.
If we shift the center of the ellipse to x/2a — 0.5 so that the reference length
L — 2a, then in an interval 0 < x/2a < 1, we can specify the total number of
^-stations to be, for example, 81 corresponding to a uniform Z\£-step length
of 0.0125. Next y/2a is computed from Eq. (7.5.2a) for each value of x/2a
specified at 1/8 intervals, and the distance is calculated from Eq. (7.5.4) and
the dimensionless external velocity Ue/uoQ obtained from Eq. (7.5.2b) for a
specified value of thickness ratio t.
The computer program also requires the specification of the onset of tran-
sition, which can be either input or computed. For this flow, it is specified at
different x/2a locations depending on the Reynolds number, which must also
be specified in the calculations.
Figure 7.7 shows the external velocity distribution over the ellipse for a thick-
ness ratio 1 to 4, and Fig. 7.8 shows the variation of the wall shear parameter
f/^ (= Cf/2y/R^) with x/2a for specified transition locations of x/2a) t r = 0.658
(
6
(
and 0.784 at Reynolds numbers i?2a = u 002a/u) of 10 7 and 10 , respectively.
It is seen from Fig. 7.7 that the flow starts as a stagnation point flow (m = 1)
with / ^ = 1.23259 and is relatively constant in the region where the flow ac-
celerates. If the slope of the u e and s curve in Fig. 7.7 were constant, then the
wall shear parameter f^ would have been constant and equal to its value at
m = 1. Thus, as m decreases from its value at the stagnation point, it reflects
the decrease in flow acceleration whic occurs around x/2a = 0.20 (see Fig. 7.7).