Boundary Lyer Theory

720 XXIII. Turbulent boundary layers in con~prcasible flow
Fig. 23.0. Meanurcrncnts on
vclociLy tIistril~uLio~~ in
turi~ulcnt boundary layer
on flnL plate at zero incidcncontsupcrsonic
velocity,
ahr It. M. O'I)onn~ll [20]
M, = 2.4; d, - nlomcnlun~
lluieknrss rrorn cqn. (13.76);
TW - "'a
vclociljy profile in n aom~~rcssiblc Ifnrl~~rlcnt Iwundary laycr is given in Fig. 23.0 which
contains a plot of u/U, in terms of y/02 for M, = 2.4 as mcasurctl by It. M. O'llonncll
[26]. Ilcrc, d, rcpmscnls thc momcnt~rm thickness dcfincd in eqn. (13.75).
In t4hc atloptctl systcnl of coortlinatt?~, thc points for tliffcrcr~t, Rcynoltls numbcrs
arrange tli~rnsclvcs well on a single curve. 'l'llc t,l~corctia~l curvc shown on thc graph
tlcviates from the corrrspontling curves for incompressible flow much Icss than was
tl~c cnsc with laminar flow, Fig. 13.10. As cxpccted, thc 11ountlar.y-layer tl~it:ltncss
incrcn.scs with Mach nnmbcr ; this is I~rongl~t, into evidcncc in Fig. 23.10 which
clisplnys velocity profiles up to M, = 0.0. It is worth noting in this conncxion t,l~at
turbulent boundary layer on n. flat plate
in'supcrsonic flow at various Mnch num-
bh, r29 memured'hy I!. W. Matting, D.
It. Clqman, J. R. Nyholm, and A. C..
T1iornn.q [58]
c. lnfl~lence of Mach number; lawe of friction 72 1
t.hc momentnm thickness from eqn. (13.75) bccomes smaller comparccl with the
boundary-layer thickness, 8, as the Mach numl>cr is incrcascrl, bccausc the clcnsity
tlccreascs in t,hc direction of tho wall.
The tliagrarn in Fig. 23.1 1 contains a logarithmic plot of the vclocitty ratio 'M/IJ,
against 71 - y v*/vW of thc type er~countercd in Fig. 20.4, in wliiclr thc valucs of
clcnsity, Q, ant1 kinematic viscosity, v, havc bccn taken at thc wall tcmpcratnre. It is
noted that the characteristic shapc familiar from incornprcssiblc flow persists at!
lligl~cr Rlnch numbers, but qnantitativc tlepxrturcs makc their appcamncc. 'I'his
follows from I,l~c ex~~crimcntal rcs~~ltn plot,l.ctl in lhc figtlro ant1 due tm It. I