Boundary Lyer Theory

688 XXJI. The incon~prcssible t,urbulent boundary lnycr
the Inrgcst local advcrse prcssurc gradicnta occur in that region when the angles of incidence arc
large. The required mir~imuni suction mta, ns described by the suction cocfficicnb co. ,{, are
of the order of 0.002 to 0,004. A. R.wpet [78] performed flight mensure~nonts on wings provided
with suction at the nosc.
AnoI.hcr clli?cl.ivc? mcl~hotl lo incrrrr.qe wrrrin~lo~r li/l
*! )arl,icnlarly in wings with a large flapdcflcct.ion
:~nglc, consists in Ll~c injccliou of :t I.l~in jet of atr of largc vclority closc to the nosc of
the flap, I'ig. 22.11. 'J'his dcvicc inll)arl~ a considamhlc arnonnt of enrrgy to the turbulcnt bounrl.
nry lrtyor and causes it lo adhere lo 1.hc wing. 'Ulc pin in lift nchicvcd by this method can be
esbin~at~tl by comparing t,hc pressnrc distril)utiorm of the flap wing with nnd witl~ot~t ~eparixtion,
rwpecthly. According to J. Willinrns [122], t~l~ceffcotivencssof (.he jet can be judged with reference
to t.lm tlin~rnsionlr~a ~nomcnlnn~ cocfficicnt
wl~cre 11, dcnolcs t.hc vclociby of tShc jet and n rcprcnolta ib witltl~. F. 'l'l~o~itns [log, 1101
~~wfortnod rxlcnsivc nic:murrn~cnt.~ on t,hc rlli:ct.ivc~mn of injcction Tor tlrc incrcnne in the lift
of Ilnp wings. 110 W.ZS RIRO able to formul:rh n procedure which allows us to calculnte t,hc value
of thc motncntnni c~ocflicicnt rcquircd to avoid separation h injcction through a slit into a
turbulent boundary Iaycr. In addition, I?. Thomas [I001 perLrmed detailed measuremente in
Lhc turl~nlcnt 1)ound:rry Inycr I~chind an injection slit. Sirnilnrly, iuvratigations were performed
by P. Cnrrik nnd 15. A. Eichclbrenner[lC,J on t.11~ q~wkion of the rct,urn of a ~epsrakd boundary
Iaycr in a large ildvcrsc presstire gradient throngh t,hc application of a tangential jet.
H. Srhlirht.ing rgl] gnvc n ~hort snmn~nry of irivc~t.igal~ion~ into tho pro1)lcnl of increasing
the n~aximum lift of wings by snit,nl~ly controlling the boundary layer.
If a dilfcrctit gw is injccLcd into :r turbulent I)OIIII~I~~,V I~rycr, wc nrc ngail~ fnccd witell a
Oinnr!y lnyer, .w wns the cwc with laminar flow (Scc. XIVc), in which the concc~~tratiotl varies
throughout tlw flow field. Various physical hypotheses have bcen proyoscd in order to bc in s
position to annlyssc the procrss of injection into a turbnlcnt boundary layer. 11. L. Turcotte [115]
amumcs tlint the proccss of mixing is c~scn1i:rlly complete in the laminar sublayer and derives
in this manner an approxin~ate formula for t,he shcnring atreas at the wall for the case of an inc:on~prenail~lc
Iluirl. The formnln wm extended to include compressible boundary Iaycn; ih form in:
111 this cquntion, the sobacript tu rcfcrs to the wall, thc subscript 0 relaks Lo the ewe without
injection and thc subscript 1 dracribcs tllc frce ~trenn~. Tl~c validity of the preceding equation
haa been confirmd by n~enn~~rclncnta lwrformcd by several authors on plaks and cones at Mach
numbers ranging from 0 to 4.3.
Extensive mcamrementa on the c(fect of the injection of an other gas on the shearing strew
at the wall in boundarv layera formed on conea in wmpresaible flow have been reported upon
by C. C. I'appna and A: I?. Okuno [731.
M. W. Jtnb&n aud C.C. Pappna [UU] propo~cd a mixing-length theory for tho calculation
of the effects of the injection of a foreign gas into a turbulent boundary layer. This waa applied
to the calculation of thc rate of hwit transferred from the wall, and the corresponding results
for the injcction of hclium and hydrogen are shown in Fig. 22.12; they have been plotted along
with experimental results for comparison. The latter show an even larger decrease in heat transfer
mka than predicted by the theory. By contrast, the dcovery factor seems to be affected but
little by the injection of a lighter gtw, in a t>wbuIcnt as w&II aa in a laminar boundary layer.
Expcrin~enb in which a heavy gnn (freon) was blown into a tarbuicnt boundary layer of
air yiclded npproximnlcly idrnticn.1 velocity profiles ns those in which air was discharged, even
though t.lie deosit,.y ratio of the gasos between t,he wall and outer edge of the boundary layer wna
ns hig11 as 4. &x(:ept for t,l~c cnec of an adverse prcwure gmdicnt or of very vigorous blowing,
the phcnomenn rnn Im described quite well with the aid of Prandtl's mixing-lengt,ll theory.
b. Tho cxlcnlntion of two-dimensional turbulent bonndnry ltrycrs
1) without Injection
gain in lit1
"due lo injeclion
separated llow
dl with injeclion
Fig. 22.1 1. Flnt wing wit,h injection thrortgl~ n ulit at thc nose of Lhc 11111) for 1.ho IIII~~OHV ol'i~~~witsing
maximum lift; a) separated flow, without injection; b) adhering flow with injeclion; c) prcsstm
dist,ribution: d) vclocit,y distribution in tho boundary layer(
Fig. 22.12. Heat-transfer rates for a binary bouiidary layer on a flat plnte at zero incidence with
the injection of hydrogen or helium int,o air in a turbulent bonndnry Iaycr, after M. W. ltubcsin
and C. C. Pappna [UU]. Comparison between theory and mennnren1cnt for tho St.anlon number
S = q/el ti1 cpl (T, - T w)
689