Boundary Lyer Theory

334 XIIT. Lnrninnr 1)oundary laycra in nompressihlc flow
with that for an irlromprcssiblc fluitl from eqn. (12.80) provicled that in thc former
rase P = I. IT. W. I'Gnmons and J. G. Brained [34) have shown t,hat, it1 the rasc
of T'mntltl 11ntn11crs which differ from unity the deviations in wall tempcraturr
caused by comprrssibility effects, as compared with the incon~pressihlc cqt~ation
(12 SO), arc- only very slight2. TIIIIS Ihc atIinbat,ic-\vnll trmpcmtnre cqnatiorl
remains vnlitl for rompressil~lo flows with a vrry gootl tlcgrrc of npproxirn:lfio~l
For nir, with y -- 1.4 :mi P -- 0.7 1, wt. ol)t,nill
Thv rrs~~lling tlrprntlrnce of thr ntlinbatic-wall trmpcraturc on thc Mar11 nun~ber has
Iwrn rrl~rrsrr~trti gmpl~icwlly by tho plot in Pig. 13.4. For rxamplc, at, a Mnrli
nl~nil)rr M,, -= I thr wall 1)ccornrs Ilc~atrtl by 4.5O C (or 80° F) in roirnrl fig~rrrs.
A[. M,, - - 3, t llc tr~nprratl~rr inorc:rso brc~ornca ns l~igll as 400' C (or 720° P), ;i~ltl
nl M,, = 5, it is as rnlrrll as 1200° C (or 2200° 1').
c. Tllc flnt plntc at. zrro incitlrnct!
The recovery /actor, r, then represcths tl~c ratio of the frictional lnmpcraturr inc.rcnsr
of tllc plntc, (T, - T,), to that due to adiabat,ic con~prrssion,
urnz
AT, = -- - >
2 c,,
from cqn. (12.14). 011 cornprrirtg rqns. (13.lH) ant1 (13.19) il. is scwl l.I~:,t, t,l~o mcovc:r,y
factor has the val~rc
Ilcncc: for air
:wi
r = dF- (Inminnr) , (lXl!):~)
r -- d0.71 = 0.84 (Inniinnr) . (l:!.I!)l))
Fig. 13.5. Mrnunrrrl rrcovrry Z't 10 IZ
factors, r, for laminar boundary
layen on conra nt sqwsonic
veloriticq lor difkrcnt
Mnrh n11111brm nnd Ilrynoltln
v %I0
A 60'
0 Boo
br? TO 33
019 to 25
%I lo 18
numbcra, al1r.r C. 11. I ~ rJ2]; Y r
ronipnrison \\ it11 theorel icnl
vnlncs lrotn rqn. (I 3.19%)
'.rhc diagrams in Fig. 13.5 rcprcscnt the rcsults of ~nnn.suronirnls on t,llc rccovcry
factor in the cast of laminar boundary layers on conos in supcrsonie strca.rns, porformed
by G.R. Eber !32J. The numcrirnl valnc r = is swn to be ~onfi~~n~t:~I
lly t,llcse men.surernents. Similar results follow from ~nc:~surcnlcr~l,s p(di)rtllt:~l on
various cones and a paraboloid pcrformcd by B. dcs Clcrs ant1 J. St,ert~l~crg 1271 :d
It. Scl~crrcr [89].
Velneity nncl tctnpernturc distributions in thc nbnetlce nl lwnt trmslcr: 'I'wo
ppt:rs by W. Ilant.zscht? ant1 11. Wcntlt. [44, 461 ant1 :L p:l.prr by 1,. (Irorxo 121 1
coll(,nill cxplici(, formll]ac for the c:rlc~rln.l.ion or Ll~c vvle~cil~y :1,11tl I,c:ln~)c~l~:li.llt~t* tlish'i-
~)llt,iorl ill Illlmbcr of spceific cnscs. J'ignrc 13.0 conI.:iilw 11lot,s of lSllt: vtdor.il,y rlisl.l.ihlt.ion
in t,lle l~ol~ntlary 1;~ycr for sovcral M ;~I numlmrs. It, reprrst!~lt,s Cl-oc:c:o's c:~l(:i~-
Intions for a boulldary hycr on an arlirrhcttic /kt plnlc on tho :~,ssnn~l)l ion of ;I. vise-11sil.y
law = 1 and for P = 1. The distance, y, from the wall has 11cc11 rnntle (litncnsiolllcss
\vit,ll rcfcrcnco 1.0 ~GTu, where 11, tlrnotcs 1,lle Itinrrn:rl,ic visc80sit,y in
t,llo oxt,crrln,l flow. It is seen l,Il:~t for incrcasirlg I\l:rcl~ tl~tml)rw l,llc:ro is :I. c:r~l~sitl~:~.:~.l)le:
t~lickcrlillg of tllo Ilollntln.ry laycr and tll;~I, for very ~:LI.~I: hl:11.11 n11t11111.t.s (.)I(: v~.lor,i(y
clist,rihllt,ion is approximately lincar over ib W~IO~C thicloless.