Boundary Lyer Theory

518 XVI I. Origin of turhr~lcncc 11 c. Effects duc to heat transfer and cornprcsnihilil.y 510
rtl 11y n~lrncrirnl rnlrulnt,ions thnt with n region in tho boundary layer whcrc lk > 1
tI1rr.c- nrc- :In inlinitc nrlmbcr of nculml wnvc numbers, or modes, with the sarnc plrasc
vrlocity c,. 'l'hr mult,iplc modes arc n result. of the change in the govcrning tlilPcrc:nt.inl
rquation for, say, thc: ~mwurc oscillntictn from rlliptic whon MZ < 1 to hyperbolic
\vhrn M" I. 'l'l~e first motlo is t.lx snmn as in incomprrssiblc flow, nntl was first
rotnl~~trtl I'IW c~otnpr(~ssil~lc flow 1)y Ir. IATS tu10 1':. Itc:sllot.ko 11421. Thc ntltlit.ionn1,
or I~ighrr. rnotlw hnvo no incornprc~ssil~lc ronnt,crpxrts. c, = c,, MZ, first, reaches
11nit.y at M, 2.2, nntl the uppcr I~ountlary of t,hc rcgion of supersonic relative flow
is nt ?I/(> -- 0.16, 0.43, 0.50, Lor f& =- 3, 5, 10, rcsprctivcly.
..
lhc multiple 11tmt.rn1 di~t~url~nnrcs wit,lr phnsc vdocit,y c8 are not tho only ones
possiblo wllcw ME, > 1. Thcrc nrr also rnult.iplc ncut,rnl dist,urbances wit.h U, < c,
< Il, -t a,. Tllrse dist,nrbnnccs (lo not. tlcpond on the boundary layer having a
gc.nrrnlizcd inllrxion point,. I~urt.llcrmore, there nre always adjacent amplified tlist~rlrlmnrrs
of t IIC .w~rre t?/pr idh, plrass 11e1ocitie.s c, < I/,. Co~tacquentl?/, the co~rr.prossil)le
Ooi~?tdar?/ /fl?/rr is isnslable lo frictioltkss dislicrba~~ces rrgardloss of any other f~aturen o/
thc idoci/?/ rr~rtl hrg~cr(r/rtrr profilrs (1,s lonq n.p th,rre in a rqlion idtrre M2 > 1.
A limiting fn.rt,or in t,he amplifim.t,im of first,-motln tlisturbnnccs is t.hnt c, must
lir I)c:t.wcvw ro n.11tl r,. Any lhing t,llnb incrcnsrs the diffcrrnoo cr - co nlso incrc-asps thc
nmpIilic:nt~ion f:~.c:Ior /?,. As shown Iiy Iqig. 17.24, t,llis tlifkroncc can be cl11it.c stnnll.
Tho co~lst.rn.int inlposrtl I)y co, which tlnlilrc c, is nnrclnkcl to the boundary-layer
profile, call only 1)o rrnlovctl 11y consirlcring n moro general form of disturbancc than
has I)crn usrtl up to t.llis point,. With
As n rcsult of c, - co increasing wit,ll {I t.ho ~naxirrlr~tn nmpliBcnt,ion fnet,or of
t