Boundary Lyer Theory

184 IX. Exact solutions of the steady-skate boundary-layer equations
by int'roduring t,hc dimensionless transverse coordinat,e 9 = y 1/ lJl/v z and tlte
stream fur~ct.ios y~ = 1/ v V1 z /. Assuming t,l~nt IL/U = /I, we are led to t,Iic boundnry
contlit ions
IICCXIISC Y) =- 0 t,l~cre. The sol~lt~ion of the dilTerential equation (9.49) subject to the
boundary contlitior~s (9.50) and (9.51) cannot be obtained in closed form, and a
numerical mcthd nimt be employed. It is possible to obtain exact numerical solut
h s I)y the IISC of asymptotic expansions for 77 + - co and 17 -+ -1- cro togetfher
wit.11 a series expansion about r] = 0; several such solutions were provid~d by R. C.
1,oc:Iz 1451. 'f'hc prthlcrn was first, solved by n~lmerical integration by M. 1,essen [44a]
st,art.ing with an nsymptot.ic expansion for r] -+ -00.
. Jlw . tlia.gr:~rn in Pig. 9.14 prc.scnt,s volo~il~y profiles for I = U,/U1 = 0 and 0.5.
An irnprovcd ~~umerical solution was p~~hlishctl by W. J. Christian [lo]. This special
cnsc of the int.eract,ion l~et~ween n wide, l~olnogericous jet ancl an adjoining mass of
quiescent, air is oftm tlescribcd by tho term "plane half-jet".
Fig. 9.14. Velocity distrihut,ion in tl~c
zone hetrvecn two int,crnct.ing parallel
streams, after R. C. Lock [45]
11. Flow in the irlet lengt.h of n strnigl~t chnn~~rl 185
R. C. J.oclr [45l studied, in atltlit.ion, the case wl1e11 t,hr t.wo half-jets differ in
their clensit.ics ancl viscosit,ies, and riot only in tllcir velocities. An exanlplc of stlch :I
case is t,lrc flow of air over a wnt,cr srlrf:~.cc. The solution now tlrpcntls on t,l~c p:lr:Ltnctcr
x -- I,, p2/p1 p1 in atltlit.ion t.o I. Lock provided sevcrnl cxnct solut.iotls ns wcll
as solutions which were l)ascd 011 the rnoment.um int.cgral rquat~ion. An approsim;~lc
mc.t.llotl was also conccivc.tl 1)y 0. I