Boundary Lyer Theory

620 KVII. Origin of tr~rbrllonce li
lly M, -- 3.5, thc maxirnntn atnplificat,ion factmr of both t,llrec- and bwo-dimensionnl
dixt~~~rl~nnccs occrtrs at. R = oo. It, is in this second rrgion, whcrc the inst,abilit.y
nssrtmcs nn c:sscnt.ially frict,ionlrss ni~t.uiv, 1.lln.t. nn unst,al)lc bantl of frcqttrncics associakcd
with thc sccot~tl mode first appears for R < 2.26 x 100. In the third region,
M, .:- 5, t,l~c nmplilic~rt,ior~ fart.ors tlro.casc st.rdily in pt.oporl.ion to the incrcase in
(TI sllowrl in I'ig. 17.24.
For thr low snpctl flow of a gas, wc havc alrcatly disrusscd the tlcst,abilizing
cffcct, of ,z hcatrtl wnll and tltc stabilizing effect of a cooled wall. Lccs 1123, I241 cal-
cl~lat.etl similar rlrrcts for cornprcssihlc air 1)ountlary layers, antl, in addition, prrdictctl
the possit)ilit,y of co~nplct~cly st.ahilizing snpcrsonic bountlary layers by cooling. Alt,llouglr
t,his prrtliction anti s~lbscclnentcalcnlations of tllc cooling require(1for complete
st,aI~ili~n.t.iot~ by M. Tiloom [I I I alltl E. 12. van I)ricst [32, 331 were bascd on the asymptotic
t,l~eory of t.wo-ditncnsiond dist,~~rbances and took no account of the higher modes,
tJ~c more rrccnt, oomput.cr calcnIat,ions have verified t,l~a.t, suffioient cooling will indeed
~omplrt,~ly stal~ilizc, or nrn.rly so, bot,h two- and t,lircc-dimcnsiond first.-tnotlc tlist,ttrl):cnrrs
ovrr n witlc: hlac:l~ nnml)cr rnllgr. I'iq 17.27 shows, from t,llc: ft.ic1.ionlrs.q
tl~rory, t.llc rat.io of (/Ir),,,, to its vallro for thc adiabnt,ic wall, (/?t),nnz,ad as n ftlnetion
of 7',,./7',a, the rnt,io of t,l~c wall tempemt,tlrc to the adiabat,ir, wall t,empcrat,urc. The
st,nbilix:it.ion of t,l~rrc-dirnc~nsional first.-motlo distnrl~ancrs is clearly seen, with t1hc
st.at~ilizat~ion tlccrcasing with increasing Mach number for the same tcrnperat,ure ratio.
On t,hc contmry, sccontl-tnodc dist,urbnncrs, far from bcing st.abilizctl by coolil~g,
arc tlc-st,abilizrtl, The rrnson for t,l~is tlin'crent, behnviour is, once again, t,llat the
gcnrralizcd inllcxion point,, which is stmngly inflncnectl by cooling, has no importance
fi~r t.hc ~lnst~al)lo Iligl~rr motlcs. 'J'llc irnportnnt, q~tanl,it~y, t.1~ cxtcnt of the supersonic
~~c-lal.ivc-flow rrgion, is lil,t.lr inllucncrtl I)y cooling.
Simil:rr rrsult,s t,o (,l~osc showlt in I'ig. 17.27 a.rc ol)t.n.inctl from the viscous t.hcory,
exc.cpt, (.hat for M, > 3 Irss cooling is roquirrd for stabilim1,ion a(, any linit.c ky- noitls nnml1c.r than is given by the frict.ionltwi t,l~cory. In this corlncxion sec also
pa~wr 11y I