Boundary Lyer Theory

328 X I1 1. J,nn~i~~nr bo~~ntl:~ry Inyrrs in comprcusihle flow
or1)itnl velocity of a satellite of w, = 8 lzmlsec, the temperature rise even in a real
gas is still of t,l~c order of 10,000 tlcg C. Tho mngc of Mach numbers M, 1 6, in which
thrrc cxisl, Inrgo cIiKcrrnncs bctwcon tho bchavionr of a real as opposed to a perfect
,gas, is givrn t.Itt: tmme of hypersoltic flow. 'l'llc occtlrrence of chemical reactions
(ioniznl.io~l, tlissocint.ion) wl~ich sc:t in behind a shock wave or in t.he Imundary layer
OII :I. solitl Imly ill :I Itypt?rsonic sl.rt::~m by virt,no of tho cxistcncc: ol:t high f,c:tnj)c:r:~.t,trrc,
c.otwitlcr;tl)ly c.otnldic:xtcs t,llc I.;~slr of annlyzing t,ho flow. For this reason, wc sl~i~ll
rt,s(.rit:t, our consitlcrat.ions to that range of Mach numbers in which tho fluid can
still 1)c :~ssl~mctl t.o obry 1.11~ perfect-gas law; ir~ air, this corrcspontls to a range
of M.,, -r: 6. In motlcrn t~itncs ni11(:11 nttcnt.ion has I)ccn giver: t,o tllcstntly of l)onntJary-
Inyrr Ilows at hyprrsonic vclocit.irs ant1 in 1,hc presence of c1iemic;tl reactions. For
drtails, tllc rrntlrr is rcfcrrctl to t.he hole by 1%'. 11. J)ormnce [20].
Fig. 13. I. Tcmpcrnt,uro rise in air in ternis
ol thc flight velocit.y, w,, and t,lle Mach
w,Rm/secJ
n~~nil~cr, M,. 'Vhc curve Iabcllcd "l)erI'ect
gas" mas calculst~d wit.11 the nit1 of eqns.
(13.1) nntl (13.2). Thc velocity 111s = 7.0
km/sec in that of nn nrtificinl satellite in
orbit,, and lo,< = 11.2 km/scc roprcsenh
the e-9cnpc ve~ocit~y of n satalIite from
I
0
~
6 I2
I
8
I
24
I
36
I
42
, the earth ~
Him
Even in I.llc mngc of snpcrsonic Mach numbers ( M, < 6 in air), the t,cmperature
rise irt thr gascww stream is high enough to force us to talrc int,o a.ccount the effect
of t,cmpcrehrc: on the proportics of tJllc gas, in particular, on ils viscosity. The lrinematic
viscosit,y of most gases, and of air wnong Lhem, incrcascs cor~sitlcrably as the
t.ernl)craturc is incronsctl.
In t,llc caso of air, as sltown I)y E. R. van Driest [30], it is possible to use an interpoI:~t.ion
fbrmuln l):~.sctl on I). M. Sntdlcrland's theory of viscosi1,y. This can be written
wllrrc /I,, clcnotcs the viscosity at the reference Lrnjperaturc To, and Sl is a constant
whit:l~ for air assumes the value
S1=llOK. '
l'ltc lm?twli~tg rel:~t.ion I)ct~weon tlle viscosiby /I of air and the temperature, T, is
scerl plottd a.s curve (I) in Fig. 13.2. Sinco t,hc relation (13.3) is still too complicated,
it is c:nst.omnry l,o npproxirn:~.l,c i(( in thcorc~t.icnl calculat.ions by tho simpler power law
where 1,llc constant h sc3rvcs to nchicve a better apl)rosim:tl,ion 1.0 thc more cx:~c%
Sut,hrrIn.nd formt~la (13.3) in 1.h~ nciglll)onrl~ootl of a tlcsirotl l,rt~ll)c~r:l.l.lr~~c r:111p
(cf. Scc. XIlTtl).
Fig. 13.2. The dynamic vis-
coaity, 11, of air in tcr~ns
of the temperaturc T
Curvc(1) ?dras~~rc~ncnls
and inler-
pnlalion forlaula (13.3) hased on
Sntherlnntl's rquntlnn. Ourvcn (2).
(3). rind (4) pow~r lacs. cqn.
(13.0, wit11 difirnnt values of
thc exponent ro