Boundary Lyer Theory

528 XVII. Origin of turbulence I1
IIICIII, \vitJ~ tncn~~~rctr~r~~l~~.
This (::~n l)c infcrrotl very clearly from the pictures of snch
'l':~.ylor vorl,icc,s ol)t:ainotl I J I?. ~ Schultz-(lrnnow and 11. IIein [204], several of whicll
hnvo I1t:t:n rq~rotlnrcti in I'ig. 17.333. In their cxperimcntal arrangement in which the
pip 11:t.tI I.lw tlimc~~sion or d .= 4 mm, and the inner radius was R, -- 21 mm, the
vcwl.it~r.s :i.pp~:~~:d ;LI> :L ~wril)hcrnl vclority, I/(, which corrcspontls to n lboynoltls
n~~tnlwr R (Ii d/v : .: !)4.5, Fig. 17.33:~. It, is t~ol~cworl,l~y t,h;~I, t,l~t: flnw rctn:iint:tI
lihtni~t:ir :I(, L11t' IIIII(:~ I~igl~t:r ~bt:ynohIs I I I I I I I I ) ~ of ~ ~ R r--- 322 (T, --= 141) a11t1 R -:= 8(i8
(T, =- M7), IGgs. 17.33 IJ, c. 'l'urbulcnt flow did not bccomc tlevelopctl until a Rcynoltls
rli~mlm R -= :l!)fiO (T, = 1715) had bcct~ mncl~ctl, Pig. 17.33tl. It, should I)c sl,rcsscd
rn~pl~n~.ir:~lly LhnL tshc first nppcnmncc of ncutral vortices n.t thc limit of stnl~ility
in nocortlnncc wit,h cqn. (17.20) and thc pcrsistcnce of arnpliIict1 vorticos at higher
Tnylor numl)rrs tlors not in any way imply that thc flow has bccome turbulent. On
t.lw contrary, cvcn if the limit of st,ahility is exceeded by a large margin, the flow
rcmains wt,Il ortlrrcd nntl Inminnr. Turbulent flow does not bccome developed until
'I'nylor, ant1 t.l~rrcfort:, ltryr~oltls numbers vastly cxccctling the limit of st.al)ilit,y
:~rr nl hit~ctl.
.I. '1'. Sh~nrl, 12181 s~tc:c:twlcd in con~puting thc flow pattern of the unstal)lc
Intnin:~r Ilow in l.11~ prcscnrc of Taylor vorticcs nntl with thc non-lincnr terms in
(.llr r(luat.iot~ of' mo(,ion rctninctl. Ilc disrovnretl the sxist,once of equilibrium bctwccn
Kg. 17.34. I'low hot\rsc:cn t.wo conoentxic rylintlrr.q: tor+io cocflicicnt for inner cylinder in t,rrms
of t.hr 'I'nglor nnml)cr, T,,.
f. Stability of a bo~inclary layer in Lho proscnm of tl~rcc-di~ncnniotlrbl tlisl,t~rl~:inws 52!)
the transfcr of energy from the base flow to the sccondnry flow ant1 t.11~ viscous
energy dissipation in t>hc secondary flow. The t,ransfcr of cncrgy fron~ the Onsc
flow t,o thc secondary flow causes a Inrgc incrcnsc in thn torqrtc rcq~~irrd to roht,c
t.l~(! inner cylinder. 7'hc diagram in Fig. 17.34 cotitair~s n compnrison I)ct,wrcn Ihc
t.l~rorrt.icn.lly dcrivctl :inti thr cxpcrirntwt.ally mcns~trrcl v:~l~~rs of 1 h t.orcj~tc, roc*f'L
ricnl, C,,. '1'11t- I:~l.tvr is clcfinwl as
Xi
C, = ------- - . - . . (17.21)
R,~ 1' '
-4- n ~ ( 2
wit,l~ 1~ as tthc l~cight, of I,hc cylintlcr. '1'11~ 1inc:tr l,l~(:or~l wil,l~ SLII:LII rrhtivt: g;111s,
(//I(,, yields
In :tcldit.ion to the? txtrvc wllic:h (:orrt~spot~l~ 10 IJiis lint.:t.r t.lt(!ory, IIII(I wl1i(41 I(-:I.(IH
lo :L Ior(11tc t;ot:l'fit~it:t~l, (,'M . - 047/T,, Sor d/11', O.OW, lht* (li~~pxtn (:o~~t,;~it~:+
t,Iit:
cllrvc provitlc:d 11y .J. '1'. SI,~~nrt,'s ~IOII-lirtwr t1~:or.y as wcll ILS on(: givt:tt l),y ;L I.Iwory
lor turl)ulcnt flow; thc Intber leads to tho formul;~ tlial. Cnr - T,,-".2. 111 all, wt: may
tlisccrn t,hrcc rcgirncs of Ilow, cnch circ:umscril)c:tl I)y 1.h~ 'l':~ylor r~~~rnltc:r in tho
li)llowing way:
T, < 41.3: laminar Coucttc flow,
41.3 < T, < 400 : laminar llow with 'I'nylor vorl,ic:os,
T, > 400 : lurl~tllor~t Ilow.
Agrccmcnt bctwccn theory and cx~wrimrnt is cxecllrnt in thc first I.wo rangost.
An extension of Taylor's thoory can bc found in a study hy Ii. Iiirchgarssner [IOG].
A detailed experimental investigation of Couettc Row, particularly in transition,
was carried out in 1965 by 1) Colcs [291
Ekct of an axinl velocity: The preceding stability calculations have been
extended by 13. Ludwieg [I 32, 1331 to includc the case when the two ryl~ntlct s arc
also axially displaced with respect to each other. Let u(r) denote the tangential
velocity, and let w(r) denotc the axial velocity. If we now introduce the dimensionless
velocity gradients
- r dzl r dw
u=-- and GI=--,
u dr u dr
wc can writc the stability criterion for n non-viscous fl~tid in the form
t l'lic cxperirner~tnl msulkq displnycd in Fig. 17.34 dernonstrntc furtl~or that an increase in the
Taylor number, that is, that an increase in tho lteynolrls number at a constant value of d/.R,,
cansen n trflnnition from cellular to tnrhulcnt flow. Whcn thc flnw is tc~rI)nlcnt (1, > 400),
wo have CM - Td-0.2. and I~cncc, nt constnnt d/Rt al~o CM w ((I, dlv)-o.Z - R (1 2. 'l'lm sarno
~(:RIIIL WIW discov(:red IIY {I. It~it:J~nrtlI, ((201 in Cl~np. XI X) WIICII 110 ~t.ndi(:~I I h I:ILH(: or I~III::~~
Couette flow between flat parallol walls. It is remarkable that ll~c same dependence of the
torqm coefficient on Iteynolds number exists for a disk rotating in a llnid at rcst, eqn. (21 30).