Boundary Lyer Theory

In t,hc tr~rl~ulont region the pressure tlrop becomes approximately pr~port~ional
t,o the square of the mean flow velocity. In this case a consiclerably larger pressure
tliffcrencc is requirctl in ordcr to pnss a fixed quantit,y of fluid t.hrol1g11 the pipc,
ns corrlparocl with laminar flow. l'his follows from t,ho fact that t.ho plrcnomcnoll of
t.url)ltlrr~t mixing dissipat,cs a largc q~t:tt~t,it,y of' enorgy which c:~~~scs the rcsist,:tnc:c?
1.0 Ilow t.o incrcasc considcr:tl)ly. lrurl,llcr~norr, in Ihc casr? of Lurl~ulcrlt, llow t,hc volodistritlu(.ion
over the cross-scct,ior~al arca is much tnoro cvcn thrl in hminnr
flow. 'rhis circumst,ance is also t,o be explained by turbulent mixing which causes an
cxc:hangc of momcntum bctwecn the layers near the axis of the tube and those near
t,hc walls. Most pipc flows which are encountererl in engineering appliances occur at
such high Reynolds numbcrs that turbrllcrlt motion prevails as a rule. Thc laws of
turb~llent motion through pipes will be discrlssed in detail in Chap. XX.
111 a way which is similar to the motlion through a pipe, the flow in a boundary
laycr along a wall also becomes turbulent when the extcrnal velocity is sufficient,ly
largc. ISxpcrimental investigations into the transition from laminar to turbulent
flow in the I,ollntlnry Inyer were first carried out by J. M. Burgers [GI and I3. G.
vnll (lcr licgge Zijncrl 1171 as wcll as by M. IIansen [lG]. The t,ransit.iorl from
laminar to turbulent flow in the boundary layer becomes most clearly discernible
by a sutltlcn a.nd largc increase in the boundary-layer thiclrncss ant1 in the shearing
stress near the wall. According to eqn. (2.1), with 1 replaced by the current co-
ortlinatc s, the dimensionless boundary-layer thickness 6/1/1'27~; becomes constant
for laminar flow, and is, as seen from eqn. (2.la), approximately equal to 5. Fig. 2.23
contains a plot of this tlimcrlsiorllcss boundary-layer thickness agairlst the IZcynoltls
number IJ, z/v. At R, > 3-2 x 10" very sharp increase is clearly visil)le, and
Fig. 2.23. Boundnry-layer thickness plob-
tedr against the Reynolds number based
on'the current lcngth z along a plate in
pnrnllel flow at zero incidence, ~s mea-
sured by llanscn [I61
as sprn from rqn. (2 1 a). llr~lrc to thr rritiral Rrynoltls r~urnl~rr
there corrcspontls Rg crlt = 2800. The bountlary Inyrr or1 :I plate is Inr11in:cr near t.l~t:
leading edge and bcconles turbulent f~lrt.llcr tlowr~st,rca~n. 'I'llc nbscissn r,,,, of tl~t
point of lrn~lsit~ion can be clctcrminctl from L11c ktlow~~ v:~lric of R, .,,,. In t.llc caso
of n plate, as in the prcviot~sly discussed pipc flow, the nun~cricnl vaI11o of R,,,,
dcpcntls to a ~narkctl degree on the amount of' tlist.~lrl~ancc in tho nxt,crn:tl flow, :111tj
the value R, = 3.2 x 10%hot1lcl be regartlet1 ns a lower limit,. With oxccpt.iorl:~Ily
(list-rrrbnncc-frcc cxt.crnal flow, valrlcs of R, , - 10%rlrltl higllrr 11:~vc been :~tt.ail~rtl.
A 1):~rticul:trly rernarltable phcnorncnon connccld with the transit.ioll from
laminar to trlrbrllt:nt flow occurs in tJle casc of blunt llotlics, s11cl1 as circ~~lar cylintlers
or spheres. It will be seen from Figs. 1.4 ard 1.5 t,llaL the tlmg coef'ficierlt ofa circrtlar
cylintlcr or a sphcro suffcrs a sutltlcn :d consitlcral~le dccrcasc Ilr:lr Itcynoltls
n~iml~crs 1.' I)/v of bout 5 X lo5 or 3 x lo5 rcspccLive1~. This fact was first, obscrvrtl
on sphcrcs by G. 1I:iffrl 1141. It. is a conscquerlcc of t,ransition which causes t.he
point of separation to movc clownstacam, l)cca~rsc, in the case of a turbulcr~t 1)ountlary
laycr, the accelerating influence of the cxt.crn:d flow extmds furlhr due t,o t.t~rbulrr~t.
mixing. ~Tcncc the point of separation whicll lies near the equator for a laminar
I)o~rr~tlary I:~ycr nlovcs over a cor~sitlcml~lo tlislnr~cc in the downstream tlircct.ior~.
In t,urn, the tlcad arca decreases considcmbly, anti thc pressure di~t~ribution becomes
more like t,hat for frictionless motion (Fig. 1.11). The decrease in thc rlcad-wat,cr
region consitlcmbly reduces the prcssrlrc dmg, and that shows itself as a jump in
the curve G, .= f(R). L. Pmnrltl [26] provctl tl~e corrcctncss of t,hc prrcccling
reasoning 11y nlo~inl~ing n Ihiri wirc ring III; a ~Ilort, (li~Im(:c in fro~tt or IJIO ccl~i:ll,or
of a sphere. This car~scs the boundary laycr to bccome art,ificially turl)~llcrlt at n lower
Reynolds nl~mbcr and the tlccrcasc in t,hc drag cocfficicr~t taltes place carlicr Lllar~
would otherwise be the case. Figs. 2.24 and 2.26 reproduce photographs of flows
which have been made visible by smoke. They reprcscnt the subcritical pattern
with a large value of the drag coefficient and the supercritical pattern with a small
dead-water arca and a small value of the drag coefficient. The supercritical pat,tern
was achieved with Prandt,l's tripping wire. The preceding cxporimcnt shows in
a convincing manncr t,hat the jump in the drag curve of a rircular cylintlcr and
sphere can only be interprctcd as a borindary-layer phcnomcnor~. Othor bodies
with a blunt or rounded slcrn. (c. g. elliptic cyli~~tlcrs) display :I type of relationship
bctwcen drag coefficient and Rcynoltls number wllicl~ is s~~l)sta~~li:illy similar. \'Vit,h
increasing slcntlcrness the jump in t h curve bccomcs ~'iro~rcssivcl~ less pronor~nccd.
For a streamline body, such ns that shown it1 Fig. 1.12 t.h(:rc is rlo jump, I~nc:~usc
no :lpprrci:r.l)lc scp:~.rnt,io~~ occ~lrs; t,lw wry gmtlrr:~l Iyssrlrc ir~c!rr:lso on I,l~c Il;lclt