Boundary Lyer Theory

490 XVII. Origin of turbrllence I1 b. Drt.crminntion of the ponition of thc point of inshbility for prrscril)rtl body shnl~c 407
On sevcral oc:cnoions wc have sttrcsscti thc fact that n pressure increme along
a 1)oundary lnycr sttrongly favours tmnsition to t,url)ulent flow in it. Conversely,
a st,rong pressure tlecrease, such ns may bc crcat8cct bchi~id sharp cdgcs in supersonic
flow, cnn car~sc a turbulent boundary laycr to become laminar. Interesting observations
of this kind were made by J. Sternberg 12151 who employed a cylinder
providctl wit,li e conical forc-body. Figure 17.5 shows a shadogmpll of the flow
along the conics1 fore-body at a Mach number M = 3. The boundary laycr turns
turbulentr at thc tripping wire provided for the purpose. Further downstream,
hohintl t.11~ corner formed at the jnnct.ion of thc two bodies, the t~~rhulent bountlary
1:ry-r hrns lamirrar again, Fig. 17.0. This phrnorncnon is explainctl by the circunnstnncc
t11:lt t11c lnrgc f:tvo~~rnhlc pressure gradient at the shonlder impresses a very
strong acrolrr:tl,ion 011 the flow and this, in turn, ~xt~inguishcs the t.nrl)ulrncc, in
a w:iy rcniinisc:rnt of the eKccl of a strong contmction placed ahead of the test
scct,iori OF a wind tunnt:l. Q~lalit~ative indications on this process can bc found in a
~a1)or I)y W. 1'. ,Jones nntl E. 15. Immtlcr 1 low. According to thcsc nuthors, rclnminnrixa.Iion
(nxt,inc:tion of t,rlrbrtlcncc) occurs in incomprossil~lc st.rcnrns w11c:n t,l~c
tli~nc~nsionlws acc:rlcr:~t,ion paramctcr sat,ixfics the incq~ralil.~
Introtltrring I'ol~lhnusen's shape fartor A from cqn. (10.21), anti using eqn. (17.3),
wc ran tra~~slntc the prrcrding contlit.ion to rcari
'I'hc: t.ransit.ion fro~n n t,ctrl)~tlrnt. to n laminar flow pat,t,crn in a tnbo of cirrular
cross-scot,ion was invcsbigat.cil in dctail cxpcrimrntxlly by M. Sibnlltin ns cnrly as
1962. In partirr~lar, this investigation cxtcntlcd to a study of the attenuation of
longit,ntlinnl t.r~rl~rrlont flnctuat.ions and discovered that t.llis is st,rongcr ncnr the wall
t.l~nn in thc ccnt.c:r of the pipe.
. I . he prrwtli~~g rrs111t.s will cnal~lo us 1.0 ralrulat~c in thc following secfiot~ the
posit ion of l.I~i, pint of in~t~ability for the casc of two-tlirncnsional flow past a body
of arbit mry .;hapc.
11. I)c.lc.rniinnlio~~ of the position of the point of instnbilily for prescribed hotly shape
.,
I 11c: tlct.crmina.l.inr~ of tl~c position of tJtc pdint ol tmnsit,ion for prescribed
I)otlj~ sl~:~pcs (in t,wo-tli~r~c:r~sior~al flow) becomes very emy if IISC is made of the rcsults
c:o~~l.:ririrrl it\ Figs. 17.3 nntl 17.4.'I'lrc essc:nt.i:d atlvnnt:r.gc of t,hc tnrt.Iiotl t.o \)ctlcscri\)ctl
II(*I.~ consisl.~ in I.lt(: f:~:l, 1~1r:iL no ft~rtl~cr l:iI~orio~is ~nlc~tl:it.ions arc rcquirotl, t,he Lctlious
j)arI, of the work I~nvil~g 1)ocn con~plctctl once nntl for all whcn compnt,ing the diagran~sii~
Fig 17.3.
We 1)cgin wit,ll t,he evalrlnLion of t8hc laminar 0onntla.ry 1:hyc.r from thr pot,cwtinl
velovit,y tlist,rilrnl.ion ll,,(x)/U,, whicl~ is regnrdrtl :i.s known, I)y t.11~ IISV 01' 1'0111-
II:IIISCII'S :I~)~)I~~X~III:I~,