Boundary Lyer Theory

490 XVII. Origin ol t,urbulencc 11 a. I3hct of pressure gradient on tmnsitioti it1 bountlnry layer along ~moot.l~ wnlln 491
a. Effect of prennl~re grndierit on trnnsitian in boundary layer along ~n~ooth walls
r 7
J lw bountl:~.ry hyor on a llnt pl:lt,c a.t zero inridcncc whose stal)iliI,y was
investigated in Chap. XVI has the pcc~rlixr characteristic that its vclocity profiles
at tli&rcnt tlist,anccs from tho lending ctlgo are similar to each other (cf. Chap. VII).
In t,his case sirnilnrit.y results from the ahsencr of a pressllrc gm.tlicrit in the external
flow. On thc other h:md, in the case of a cylintlric:~,l htly of arbitmry shapc when the
pressurc gmdient along thc wall changes from point to point, t,llc rcsult.ing vcloc:it7y
profilrs arc not, grncrally speaking, similar to each other. In tho rangtrs where t,ho
pressure docrc.ascs tlownstmam, the v~locit~y profilns have no point3s of inllcxiol~
and are of t,llc typo shown in I'ig. 16.9~ wllcrcas in regions wl~crc t,llc: prossure inorcnscs
downstmam thoy arc of the type sli0~11 in Fig. 16.9~ ancl do posscss points of
inflexion. In t,hc caso of a flat plate all velocity profilcs have the same limit of
stability, namely R,,,, = ( fJ, d,/v),,,, - 520; in contrast with that,, in the case
of an arbitrary body sl~npc, the intliviclual velocity profiles have marlrrtlly tlifiront
limits of stabilit,~, I~ighcr than for a flat plate with favourable prcssure gratlicnt.s,
and lower with adverse prcssuro gradients. Consequrntly, in ortlcr to dotormino
the position of the point of instability for a body of a given, prescribed shape, it is
necessary to perform the following calculations:
1. Dctcrminat,ion of the pressure tlist,ribution along the contour of Lhe body
for frictionloss flow. 2. I)etcrminat,ion of the laminar boundary layer for tha.t pressure
distribution. 3. Dctcrmination of thc limits of stability for these indivitlual velociby
profiles. The problem of determining the prcssure distribution bclongs to potential
throry which supplies convenient met.hotJs of computation as, for example, tlescril)etl
by T. Theodorsen and J.1S. Garriclc 1242) and F. Riegels [193]. Convenient nlethods
for the calculat~ion of laminar boundary layers were given in Chap. X. The third step,
t,he st.al)ilit.y calculntion, will now be discussed in detail.
It is known from the theory of laminar boi~ndary layers, Chap. VIT, that,
generally speaking, the curvature of the wall has littlc influence on the development
of tho houndary layer on a cylindrical body; this is true as long as the radius of
curvahre of the wall is mnoh larger tjhan the boundary-lager thickness, which
amounts to saying that the effect of the centrifugal force may be neglected when
analyzing the formation of a boundary layer on such bodies. Hence the boundary
layer is seen to develop in the samc way as on a flat wall, but under t1he influence
of that, pressnrc gratlicnt which is tlctermined hy the potential flow pnst tfhe body.
The same applies t,o the tl~t~orminat~ion of the limit of stability of a boundary layer
with a pressure gradient which is different from zero.
Tn contrast with Ihc case of a flat plate, whrro the external flow is nnilbrin
at [J, 1 const,, wr now h:~ve Lo con0cncl with an ex1,ernal strram whose vrlority,
I/,(T), is :L f~lnrtion of the lrngth roordinatc The velocity Urn (z) is related to the
prrssuro gmtlirr~b tlp/tlr through tho Rrrnoulli orpation
to work witsli a nicnn llow whose ~olocit~y Il (y) rl~pcn(ls only 011 hllr 1 r:it1svrrsc
coortlir~~t~c y. 'l'ho inffl~oncc of Lhc I)~I:RSIICC gra(/irnt 011 ~l,al~ilit,y ~na~~if'rsl.~ iI,svlf
t.hro11g11 tltr form or tdic vclocit,y 1)rdiIc givv11 by fI(?y). \\'c IIILVI: nlrwtly wicl in
See. XVLb thnb the limit of st,al.)ilit,y of a vcloeil.y prolilo drpt:ntls shongly ou il,s
shape, profiles with a pint of hllcxion possessing colisitlrrably 1owc.r li~nitw of
stability than thosc without ono (poit~t-of-inflexion criterion). Now, since the pressure
grndjent cont,rols t,hc curvature of the velocity profile in accordance with ~(111. (7.15)
the shng tlcpentlencc of the limit of stability on the shapc of tho vrloni1.y profile
n.mount.s to a largo inflt~cnce of tJlc: ~)~CSRIIPO gmclicnt on sl,al)ilif~y. If, is, I,ltc~rc.li~ro,
true t,o say that accelerated flows (tlp/tlx < 0, clUrn/tl:c: > 0, f;~vour:lblc pressure
gradient) are considerd~ly morc stable thn clccclerat,ctl flows (tlp/clz>O, clIJ,,,/dz