Boundary Lyer Theory

746 X XIV. I+cc turhttlrnt flown; jeta and wnkcn c. Examples 747
spectively, we may write
Consequently,
Fnrther, we put
' 8
s,= s, (:)' with E, = x1 0, U,
7 =a-Y,
where a denotes a free constant. The equation of continuity is integrated by the
use of a stream function tp, which i~ assumed to be of the form
Thus
y) = a-I Us 6"' z''~ F(q) .
On substituting into eqn. (24.42) we obtain the following differential equation for
F(v):
1. F' + 1
-. FF" + -EL a2F"' =O,
2 2 us
with the boundary conditions F = 0 and F' = 1 at TI = 0, and F' --; 0 at v = oo.
Since s, contain^ the free constant xl, we may put
This substitution simplifies the preceding differential equation which can now be
integrated twice, whence we obtain
FB+F'=l. (24.44)
This is exactly tlic same equation as that for the two-dimensional laminar jet,
eqn. (9.42). lksolution is F = t.anh v so that thevelocity is# = Us (~1.9)-lla(l -tanli2v).
'L'lie chamc~cristic velocity can be exprcsscd in terms of the constant momentum
-I m
per unit Icngth: .I -- p / UZ dy. Hence .I = ) p Us% s/a With J/p = R (kinematic
-03
momrnt,um), we obtain thc final form of the solution:
,rllr Vn~IIo
t,llr siligle cmpiricn.1 constant o was determined experimentally by
11. Rcicliardt [29] who found that a -1 7.67. Fig. 24.8 contains n compnrisou 1)ctwccn
the theoretical curve from eqn. (24.46) with the nicasurcmt:nts due to E. I'oertli-
mann, curve (2). The theoretical curve obtained by W. l'olltnicn [52] on the I)xsis of
Fig. 24.8. Vclocity tliatril)~~tion in a two-tlilncnnionnl, turbulent jot. hlctw~~rtmct~b
Foerthmann [ll]
Theory: rarrv (I) BIIC If* Tnlln~irn [Be]: curvr (2) lrom cqs. (24 45)
cllto Lo
Prandtl's mixing-length hypothesis, curve (I), hna also been shown for cornparinon
The first theoretical curve shows a slightly superior agreement with nieasurerncnt
as it is fuller near its maximum.
1.125 I
From the given nnmericnl value of a we obtain s,= - -
4n 112 J . or
E,= 0.037 bl12 U ,
whcrc hllr again denotes half the width at half depth.
A generalization of this problem consisting in a study of turbulent mixing under-
gone between a two-dimensional jet with a co-directional external stream was ex-
plored by S. Yamaguchi [GO]. See also S. Mohnmmadian [24n] nnd TI. Pfcil ct nl. [26a].
6. The circ~tlar jet. Experimental rcsnlb on circular jcts wcrc give11 11y W. Zitil~n
[61] and 1'. Ruder1 [33] as well as by IT. Reicliardt 1291 ant1 W. Wucst IFi!)] So&
results of measurements on circular jets are also contained in t h scrirs of rrl~orts
published by the Aerodynamic Institute in Gocttingen [GZ].
The first thcorctical treatment of a circular jct was givcn by W. 'l'ollrnit~n [52]
who based his study on Prandtl's mixing-length tlicory. In t.his cxsc, as ~cll as
in the preceding one, the assumption for shcaring stress given in eqn. (24.5) lcntls
to a considernbly simpler calculation. According to Table 24.1 (.lie witl(.li of t.11~