Boundary Lyer Theory

222 X. Approximate methods for steady eqnations
Qnalitnt.ivcly it is at once possil)lc to mnkc the following nt~atcmcnt regarding the shape
of [,he potential velocity function U(x) which Icatls to no xcpnrntion. In viow of cqn. (10.41)
U" > 0
is a nrccnmry condition for n rctnrtlrd flow (IJ' < 0) to xrlhcro t,o the wall. In other words, t.11~
~nn.gnitude of the advcrsr pressure grntlicnt, n~r~st tlccrrnsr in t,lie flow direction. Ii'ig. 10.15.
, %
J hns scpnrntion will nlwiiys occllr il' the f~~r~t:l,ion I/(%) iu cx~rvctl tlownwnrtln 1)chinrl its mnxi~nnm
(11" < 0). In the opposit.e rase, whim tho vc1ocit.y function ci~rvrs i~pwnrds (U" > O), srjmration
tnny he ol>viatetl. 15vcn the limiting c,we of IJ" = 0,i. e. the rase of a velocity which tlccreanea
lir~carly with the length of arc, always Icatls to separation. Thin latter remark agrees with the
rmitlt fonnd in Src. IXd; it was conccrnrd with the boundary lnyer aaaoriatcd with a potentinl
flow vc1ocit.y which dccrcnwd linearly, and the solution of thc tlil~rret~tial cquntiona wm quokd
from a pnpcr by I,. Ilowarth. The su//icient condition for the absence of aepnration iu givcn by
Wo RIIRII now procrcd to cnlcnlntc tho potential flow and the varintion of hountlnry-lnycr
thickncus which are wnorinktl wiLh t,I~r 1imit.ing cane of o - I I, whrn tho bonntlnry lnycr re~nninu
on t,ho verge of sepraLiot~. Ikwr cqn. (10.41) we Imvo
. U" . .. - U'
u, -- I1 x -
or, npon intrgrathg: In U' -. 1 I In 11 -I- In (- C,'), i. r. IJ'/CJ1I = - C,', whero 6,' denotes
tho constant of integrat.ion. ltcpcntccl intagrnth~ ~ivcs
u
- 1
U-lo = C,' z + C, .
For z - 0 wo uhould hnve lJ(r) .- IJ,. no that C, = $6Nn-'O. Putting furthcr C,' UOl0 = C,,
we obtnin from cqn. (10.41)
w.
Eqnat,ion (10.43) reprcsrn1.s the pot.cnlia1 vslocit,y for wl~ich cloparation can jnst be nvoidwl.
Thr constm~t C, can IIC tlctnrminccl from the vnlnc of the bountlnry-layer I.hickness do at the
origin z = 0. We hnvc A - U' P/v = - 10 or d - 1/10 ;q/(--D7). From eqn. (10.43) we ohtain
and hrnrc
From 6 - r!, nt x = 0 we hxvr C, - 10 rl/lJ, doZ, which gives the final solution for thr potcntinl
flow nntl thr vnrinl,ion of bo~~ntlnry-lnyrr thirknrsu
It, in srrn that, t.hc n,ngnil.~~tlr of I.hr prrmissil)lr dcrrlornt.ion (tlcrrmsc in vclorit,~) is very small,
Irring ~)roporlion:il to .I: (1 1. Its vnl~~c is vrry nearly rcnlizctl for t,hc cnsc of constant vc1ocit.y
nlr~na tho IInh p1:ilr at. zrro iwitlr~~rr. Jn the prcsrnt. cnnr the incrrnsc in hound:wy-lnyrr thirltncsn,
0, is ~~roporlionnl to 3:I'.5" :his vnlnc also tli1h.m hnt lit,t,lc from lhe rase of n fln.1. plate nt zrro
in~~i~lr~~~~t-
I'cw whi~,Ii
0 - 2+5.
lly way of n further cxam~)le of retarded flow we shall ronnider the flow t.hrorrgh a
divergent chnnncl whose walls nro straight. This ca.w in corollary to the cum of the houndn.ry
layer in a divcrgmt chnnnol trentcd in Sec. IX b. The flow is nccn sketched in Fig. 10.16, where
x tlcnotrs tIw rndial tlislnncc frorn t.11~ nourrc a1 0. The wall is nsann~ctl to Iqin nt, x .- n \vhcrc
the entrnncc vcloril,y of the potrntinl strrarn is put cqnnl to U,. The poknlinl flow in givcn by
Computing thc vnhc of the qnantity a from cqn. (10.41), which is decisive for separation, we
obtnin here o = 2. Applying the criterion given in eqn. (10.4111) we cor~clude thnt, scpnration
occurs in :ill cnscn irrrnpcctivc of the megnit~~de of t.ho nnglo of divrrgence. This oxnmplc RIIOWR
very clrnrly thnt c lnminnr nt.rm~n has only n vcry limitctl cnpncity for nnppwting nn ntfvrrsn
prcrrsnrc grntlirnt without ncpration.
Acrording t,o a c:alculation pcrforrnctl hy K. Pohlhnuscn [In] tho point of ~cpnrnt.ion occurs
nt xr/rl = 1.21 nntl is sccn to be indcpentlcnt of tho anglc of divcrgence.
Fig. 10.16. J,nniinnr honnclnry layer in n tlivrrgent
chnnnc4. SrpnmLion occnrs at r,/n = 1.21 intlcpen-
dcnt,ly of the nnglc of tlivcrgence
-x :
p-- ,-, -I
rc----,ys ):
The prcrrding concl~~sions npply only n.s long as t,he displaccnwnt clTect of the Iioimdary
lu,yrr n~ny be nrglcrlccl. Ilo\vevcr, this is not the cnsc uhcn the angle of divcrgcnco in small.
Whcn thin nnglc is small, the boundary laycrs fill the whole channel cross-scction aflcr a certain
inlet length (r/. Scc. XI i) and the flow gorn over nsytnptoticnlly to that discussed in Scc. V 12
undrr the heatling of channel flow. When the included angle does not excced u certain valnc
which drprnds o~? the Reynolds number, there is no separation.
Ilcccntlg, S. N. l3rown nnd I(. Stewnrtnon [I] ~)~~l,linhrtl n nuninlnry rrvirw on nrpnrntior~ in
whirh the rnathematical qucst.ion renbrcd on thr ning~~larity which occllrn in 1.hr tlifl'~:r.rlll.ial
eqmt.ionn at tho critical point has been ernphnnized. Sccilso tho work of S. (:oldstein 141. 11 Inore
physicnlly inspircd review of thin ]~rob~cln nrm h ; rccc~~lly ~ been puh)inhrd by J. C. \villi:lln.
11 1 (291, n.nd by P. IC. Chang [2c].
45-72 (1969).
121 I3ussn1ann, I