Boundary Lyer Theory

450
XVI. Origin of tnrhulence I
pying trnnsil.ion from I:~niinn,r 1.o h~rh~rlcrit flow is ol' fctntlamc?nl~a.l iniport,anc:c
for I.hc whole snirncc of flnid ~ncchnnics. 'rho iricidcnrc of t,url)ulcnc:c was first, rocngnizcd
in rclt~tion to flows tl~rorrglr strn.iglit pipes n.ncl elrn.ti~iols. In n flow at vcry
low ltcynoltls rinrnlwr t.l~rongh n straight pip(: of uniform cross-sccttion ant1 smoot.11
wn.lls, every llnitl p:irf.icln tnovrs wit,lt :I uniform vrlo~it~.~ long a straight path.
Viscous forces slow down the p:~rl.iclrs nwr t.11~ w:d1 in relation to tliosn in t,l~c cxtcrnnl
core. 'l'lic flow is wcllortlcrctl and pnrtic:las tmvcl alo~lg noighboirring lnycrs (1n.niiriar
llow), Pig. 2.22a. Ilowcvcr, obscrvation shows t,l~at this onlorly pttcrn of Ilow
(:oases to cxisL nt higher Itcynoltls r~ntnl~crs, Vig. 2.22b, :d t11:i.t sl,rong mixing
of all l.lic pnrtiolcs occurs. 'l'liis rnising ~)rot:oss can ho mn.tlc visil)lc in a flow t,hrongh
a pipc, as first shown hy 0. ItcynoI~I~ (711, hy feeding into it R thin thrci~cl of li(\uid
dye. As long as the Ilow is 1arninn.r the tlrrcatl maint~ins sharply drfincd Im~n~lnrics
all nlong tho sl,ro:i,~n. As soon a.7 the flow 1)ccomes turl)ulont the tfl~rcacl diffuses into
bhc stream ancl thc flnict appcnrs uniformly colonrctl at, a short tlistnnce clownstmatn.
In this ease thcrc is supcrimposctl on thc main motion in thc cliroct,ion of the axis
of t,hc pipo a snt)sitliary motion at, right nnglcs to it wl~inh clli?c:l,s mixing. 'l'hc pattc.rn
of strcnmlincs at a fixed point bccomcs snl)jcotccl to continnorts fluctuations ant1 thc
sul)sidiary motion causcs an exchnngc of momcntnm in a tmnsvcrse direction beeausc
each particlc su1)stantinlly robins its forward momcnt,tim while mixing is taldng
placc. As a conscqiicncc, the vclocity tlistrihntion ovcr the cross-section is considcrably
morc uniform in turbulcnt than in laminar flow. The mc~urcd velocity distribution
for these two types of flow is shown in Fig. 16.1, where the mass flow is the
samc for both cases. 111 laminar flow, according to the Hagen-Poiseuille solution
given in Chap. I, the velocity tlist,ribut.ion ovcr the cross-section is parabolic (see
also Fig. 1.2), bnt in turbulcnt, Ilow, owing to thc transfer of momcntnm in tho tmnsverse
direction, it becomes considcrnbly more uniform. On closer investigntion it
appears that thc most, essential fcature of a turbulcnt flow is the fact that at a given
point in it, thc vclocity and tlic pressure arc not constant in time but exhibit very
irregnlar, high-frcqucncy flnctuations, Fig. 16.17. The velocity at a given point
can only be consitlcrcd constant on the average and ovcr a longer ~eriod of timc
(q~~asi-steady flow).
Thc first syst,cmatio investigation into thcsc two f~~ndan~cntnlly tliffcr~rit~ patkerns
of flow wcre conducted by 0, Rcynolds [71]. 0. Rcyr~olds was also the first to
investigate in greater tlctail thc circnmstn.r~cos of the transition from laminar to
tnrbnIrnt flow. l'hc provionsly mcntionctl tlyc oxperimcnt was nscd by him in t,his
coru~cxion, antl he discoverctl the law of similarity which now bears his name, and
which states that, trnnsition from laminar to turbulent flow nlways occurs at nearly
tho samc Reynolds nnniher 171 dlv, wlierc t3 = (;)/A is thc mean flow-velocity
(Q -1 volrlme m1.c of Ilow, A = cross-sectional area). 'rhc numerical value of tho
R.c~ynoltls n~inibcr at, which t,rn.nsition occurs (critical ltcynoltls number) was
Fig. 16.1. Vrlority di~t,rihtll.ion it1 pipc; n) Inminnr: h) turhulrnt
a. Some ~xperin~rnl.al rrrrtrltn otl transit ion from lnrninar to turl~r~lrnt flow 45 1
~stablishcd as bring approximately
Accortlingly, flows for which thc Itcyrioltls nnm1)cr R < R,,,,, arc snppost~l t.o he
laminar, nncl flows Ihr which R > R , nrn cxl)cctctl t,o IIC tt~rhrrl~~t~t~. 'l'li(: t~i~tii(~i(d
value of the criCical Itcynolds nurnt)cr tlc~~cnt1s vcry st,rongly on th: vo~~tlit~iotin
which prevail in Ihc inil.inl pipe lcrigtl~ ns wrll as in t,lic n.l~l)roac-h to il.. 1':vc~n It~~~t~oltls
tlionght that the cril.ionl ltcynoltls ri~~ttil~cr inc:rn:~w.s as thc tlisI,itrl~:~nc:vs ill t.Iit:
flow bcforc thc pipe arc tlccreasctl. 'l'liis fact was confirmed csprrimc~ltally I)y
11. T. 12arnes and 15. G. Colcer [I b], antl latcr by L. Schillcr 1801 who rc-nrhctl
critical vn.lurs of the Itcynoltls numl)cr of up to 20,000. V. W. 1Slrman [24] succ~:ctlcd
in mai~it~nir~ing laminar flow up to a critical Itcynoltls nun~bcr of 40,000 by providing
an irilct which was mnclc cxeept~ionnlly frcc from tliat urh:~nccs. 'I%(: 111)pt:r
limit to which tho critical lteynoltls nl~mhnr can IK: tlrivcn if cxtrcmc: cnro is !,:II