Boundary Lyer Theory

220
X. Approximate n~etlrods for steady equations e. Laminar flow wii.lr ndveruc presnure gmtlient; sepnrclt.ion 22 1
e. I,nmit~nr flow with nclverse pressure gradient ; sepnrntion
Flows with :dvcrso pressure gr:itlicni.s (rrtmdrd Ilows) arc of great practical inrportancc.
111 I.l~in connoxion it is always desired to nvoid ncpcr.m/ion from t,lw wall, hccanse thin phcnomenon
in associatccl with large energy IO~RCS. '1111~ flow al~wt an nrrofoil in a case in point,. Owing b
t.hc hct that on the nucLion side the pressure must, increase to it# free-stream val~~c at the txailing
edge, the flow is always likely 1.0 sc:parxtc. 'l'l~c flow in a divrrgcnt channcl (din'~~scr) nfTords
anollrcr cxan~plc. 'l'l~c objcd in using this sl~n.pc: of cl~nnncl is lo convcrt. kincl.ic cnrrgy i11t.o
prr:ssurc rncrgy, and if Lire angle of tlivcrgrnco is ~nndc t,oo large, srpamt,ion mrry oocnr.
l'hcoret~icd i~~vrst.igat.ions on the l)cl~nvio~~r of the I)ountlary Ia.ycr in the vieini1.y of I h
point of nrparntion hvr been carried out, hy S. Goldsbir~ pi] and 13. S. Strni.ford 121 :rl. C/. talm
rcvirw I)y S. N. Ibown a ~ 1C. d Stfiuvrrtnon I I).
Ol~scrvat.ions slrow thnt a Inlninnr 1)ounelnry Inyor whirlr separates from a wall frcqr~ent,ly
I)reolnes rmt.t.:drc~d lo it, having first hrron~c. t.~~rl)r~lcni.. Thin Irads t,o tho crent,ion of a laminar
separ:~t,ion bnbl)le. Fig. 10.13b, wl~iclr pl:~ccn it.sclf bet,\rrerr the separation point S ~hnd 1,110 rcaLt,nchtnent
~~oint, R. 'l'h flr~itl in the bul)l)le 1)erfornrs a rircrllntory motion. According t,o 1%.
10.1:h, the prrssure tlistrib~lt.ion nlong the wall can be represrrrtctl, in ~iml)lified fashion, by fi
ronstant, vnlne brtwcen thr point of separation S and point I' of largest thickncns followed by a
litrcnr inrroasc from I' t.o the point of reattachment n. Phenomena of this kind have been denrrilml
in tlchil l)g I. 'rani [2:3]. More recent experimental investigations into tl~e nnturc of
I;uninnr scpnrat.ion I~nhhIrs I I R V ~ I)rcn pcrfort~~rtl by A. 1). Young rL nl. [2R] as well an hy hf.
Gnstar 1481 and J. L. Van Ingen [GI. For theoret,ical contribr~tions see [Zb, 3a, 5~1.3.
It, will now bn shown wil,l~ the aid of srvrr:d cxntnplrs that, a laminar flow can only support
vrry sn~:~ll arlvrrsc: prcssorr gradin~~t.~ wiI.lro~~t srp:lrnt.ion. Adverse pressure gradients wlriclr exist
in practiral npplicxtions wonld, tlrorcrorc, nlmont nln.ays Ictul Lo separation if the flow were
laminar. 'l'hc: circ~~msl.ancc that real flows c:~n support consitlcr:~l)le rates of pressure increase
in n large nr~tnl)cr of rnsrs without scpnr:~t,ion is duo t.o the fact that the flow is mostly turbulent.
It will hc srrn later t.l~a.t. t.w+ulcnL flows arc r:~pal,le of overc~otning n~uclr larger adverse pressure
gr:~rlicnts \vil.l)out scpral.ion. 'l'lrc Iwst known rx:~mplcs inclutlc the cases of flow past circular
c~yIindrrs and s~II(.I-os, WIIOII srpnrat,ion ocr:urs ~nnc:lr f~~rt.lrcr r~psLrcam in laminar than in tur-
1)11lrnt. flow. In pri~cti(:t> wl~en a~lv~rsc prcss~~rc gr:dicnf.s exist, the flow is aln~ost always turbulent
l)rcn~~sc, in atlclilion, fl~o cxistcncr: of an nrlvorsc prrssure gradient favours the transition from
laminar 1.0 L11rh111c:nt. llow. 1 t is, ~~cvcrtlrcl~~~s, 11sef111 to clarify some of t,he f~mdamcntal relations
:wso~~i:~lcd o.it.11 tho provo~tlio~~ of srp;;rntin~~ on fl~c cxarnplc of I:m~itrnr flow, in particular,
0cw11sr I:IIII~II:I~ l10ns i~rc 11111~11 Inore rrarlily :I~II~II:II)IC 1.0 n~atlrcmai,icnl treatment than is tho
wsc wit11 I ~~rhlrnt, Ilnws.
'I'l~vrr arc. svvrr:tl ~nrt.l~ocls of prcvcnt,ing scpr:tl.ion. The simplcst of t,lrenr consists in
:rrr:~ngi~~g for IIIV ;~~lvrrsv 1)rrss11r(: gri~dir~~i.~ lo rrn~i~in hrlow the limit for wlrirl~ ncpnrat,i~n
I'j
s V R -- 5
I
Fig. 10.13. Scpar:~t,ion bul)l)le in a laminar bountlsry
li~yor nfiar I. 'l'nni 123). a) Shape of bubble (nchcnratir):
b) l'rcssl~ro distrib~~tion in hnbhle along tl~e wall (w!w
matic). 'l'hc nrc'snurc hetwoen S and V in tlro I)r~hhlo
docs occnr. 12 numcricnl example will serve to make thin idra clonr. Another possibility consisln
in ontrolling the bountlnry Iayer, e. g. by suction or by injecting fluid into it, or by addition
of -11 arrofoil at a poit~t where ils presence favoural)ly afictn t.lrc I~oundary Iayer in critical
regions. Thrsc mct,lrods will l)c discrlsscd rnorc fnlly in Chap. XI\'.
I'olloaing 1,. 1'randt.l [I61 we sllall show how it is possil)lo 1.0 cni.imaLc t.lw pcrn~issil,lc
rl~ngnilntlc of the :~dversc prcssure grn.dicnt for wl~irlr scjmrat.ion is jnst. prcvcnt.crl. 'l'llc arg~~mrnt
will he I):~.srd OII ~,III? von I ~&~~II~~II-I'~IIIII;LII~~II a~~~~roxi~~~:~tiot~
disc~~ssrd inSco. XI). 11, will hc:~w~~nir~l
ll~:it. tl~r I ~ o I I I I liiy~~
~ : ~ ~ in iwlwl 1111nt1 I1.y t.11~: 111~.xwrc: didril~ution (Irt(.r~ni~~c.(l I),y t110 11.(~.51 ~I.:IIII
poLr111i:rI flow 111) 1.0 :h point, which lirs wry clclsr in t,lie point, of scp:wation, sue11 as point8 0 in
Fig. 10.14. St:rr.Ling with this point, it will I)c assumed that i.hc pressure gratlicnt is srtoh that t.110
s11:1pc of IIIC vnlnrit,,y profiln TOIII:I~IIR IIIICIIRII~(~ ~)rocrc~ling (I~WIIR~~~:LIII, or trIu~t,, in oI,11rr \v~rds,
tho filq": C:~f:lor A rc:mainn ronst:~nt.; fiincc aL sc:lr:~ral.ion A .- - 12 a valnc of A -7 - 10 will
be cl~osrn. As seen from 'I':tl)le 10.2 this lcadn t.o a definite value for the second slrnpr factor,
narncly I( = - O.l:169, so L11:lt Il'(K) = 1.Tr23. Using tl~cse valurs it is sccn from rqns. (10.28)
and (10.29) that prevention of separation i~nplies the following relationsliip between the vclocity
U(z) of potential flow and the momentum thickness d,(x):
It follows that dZ/dz = 0,1369 U"/U'2, or
8,' 0.1369
- =z= -
v - V(x) .
Fig. 10.14. Devclopnrent of boundary Fig. 10.15. I'otcntial velocity fi~nction
I:~yer in thr! case when laminar separation for n laminar boundary layer with and
is prevented without separation
On the otlw hand the succeeding vclocity proflcn are given by tho ~non~entutn rquation (10.36)
for 3. =. 0, or
(le
U --- = F(I