Boundary Lyer Theory

240
XI. Axially symmetrical and three-dimensional boundary lnyera
The significance of r(x) may be inferred from Fig. 11.6. Retracing the steps of
Sec. X b we obtain the following differential equation for the quantity Z = cJ,~/v:
'l'he quantitirs Ii, fl(R), f2(K) have the same moaning as in tho two-dimensional
case, eqns. (10.27), (10.31) nnd (10.32). Introducing F(K) as before, cqn. (10.34),
we have
1 dr U
F(K)-2K--,I; K=ZUt. (1 1.40)
dz U r dz U
It is casy to see that the substitution
g=r2Z
transforms the prcrcding equation to the form
This form is preferable to that in cqn. (11.40) because it does not contain the
derivative drldx.
The point of separation is again at A = - 12, i. e. at Ii = - 04567, but
at the stagnation point the values of tfhe shape factors A and K are now different.
If the body of revolution has a blunt nose, we have at x = 0, i. e. at the upstreanl
stagnation point,
With this value the terms in the bracket in cqn. (11.40) reduce to F(K) - 2 Ii.
By following the same argnment as in tho two-dimensional case it is found that the
initial valw of Ii at the stagnation point is determined by the condition F(Ii) - 2 R =
-- 0 , or, explicitly
A, = -t 4.716 ; R, = 0.05708 .
Ilrnrr thr initial valurs of thc intcgml rurvc (11.40) at thc stagnation point l)cromc
K 0.05708
7 - -.!. -1
'0 - ur,, u', I
The initrial slope is zero for a body of revol~t~ion, because for reasons of symmetry
we must have (I,,'' = 0 at t.hc st:~grrat,ion pojnt. 'l'hc mcthotl of tlircct integration
tlrscribetl in Scc. X b can hc cxtendcd t,o the case of axially symnictrical bodies,
as shown by N. ltott and 1,. F. Crabtree [931. Equation (10.37) for the momcntuln
~t.hiclrness is now rcplacctl by
Some nunleriral examples have been calrulatrd by F. W. Sc!loll~emcier [I021
7--.- -8- ..
in llis pr;scrlkcd - - -.-- --
t-o tlij T-E"-----'--