Boundary Lyer Theory

XX. T~lrbrrlcnt flow tl~ro~rgh pipcs
Fig. 20.15. Srconclnry flows in piprs of triancross-section
(urllcn~atic)
g~tlnr R I TCC~II~II~R~
~
Fig. 20.16. (hrvcu of constant vrlocit,y for n.
rcrtnng~~lor opcn rliannol, nfkr Nikt~ratlsc (431
*
tli;igrams of sccontln.ry flows in t,riangul:ir and roct.angular pipcs are shown inXFig. 20.15.
It is swn that, the sccondnry flow in (.he rectangular oross-section which proccetls
from t.ho wall inwnrtls in Clic neiglibourhood of the cnds of the larger sidcs and
of t.lic mitltlle of the shorbr sitlcs creates zoncs of low velocity. They appear vcry
clcnrly in (.he pic:t,urc of curvcs of constant velocity in Pig. 20.13. Such secondary
flows romc int,o play also in opcn channels, as cviclcnccd by the pattcrn of curves
of const,ant, vrlooity in Via. 20.16. Y'hc maximum velocihy docs not occur near thc
free surfiicc I,nt, at. about, one fifth of 1.11~ tlcpth down, and the flow in the frcc surface
is not, at n.ll two-clirnensionnl a.s might have been expected. When the cross-section
of the cllannol cont.ains a narrow region, transition does not occur simulbanoously
ovrr t.licr wholo or the Ilow. I'or cxnmplc, in the rrgior~ within an acute angle of a
tfrinngulnr cross-section, the flow remains laminar to very large Reynolds numk)crs,
whcrc.~ in thc bulk il 1i:t.d tnrrlctl turhrlcnt long ago. Such a state of affairs is seen
illustrntc~l wil,li tlic nit1 of Pig, 20.17 which represents the results of mcasuremenLq
Fig. 20.17. Bollndnry betwrrn laminar
nl tl t.llrlmlrnt flow in nn n(:utr, t.rinw
gular 1 cl~annel, dct~rniincd visually by tho
we of smoke injection, after E. 1L G.
Eckcrt nnd l'. E. Irvinc [13]
R, - h,drrulir. rndius = dh/2
pcrfornlctl by E. R. G. J3cltcrt ant1 T. E. Irvinc [13]. At a Itcynoltls nilnil)cr of
R == 1000, the flow remains Inrninar ovcr 40 per cant. of [.he hcight of 1,hc triangle:,
t.lic region of lamirmr flow dccrcnsing :is the Itcynoltls ri~trrtl)cr is iric:rc::lsc:tl.
13. Meycr 1381 invcst.ignt,ctl the prcssilrc antl vclociljy tlist,riI~irtion iri a Ilow
through a stmight channrl wit11 :i cross-sc:ct.ion whose sllapo varictl but whosc crossscc:t.ional
nrca rcmainctl const.nnt,. IIc 11sc:tl a chnnr~cl in wllic:li ;I cir~i~li~r (:rmsscvtion
was gmc1u:illy trxnsf'ornictl int,o n roc:t.n.nglc: wilJi ills sitlos in t.lrc r:il,io 1 : 2.
'I'rn.rlsition was clli:ctctl in hot11 tlirt:c:t.ions over two rlilTc:rcrit~ I~:ngths, :rntl it, w:is
tlisc:ovort:tl I,hnl. I,hc prcssnrc: loss in l11t: ~~ortioii with Im.nsit,ion from c:irc:lc: to rc:c:l.:lriglc:
co~isitlcr:hly cxc:cctlctl t.Imt in the oplmsito tlirc:c:t.iori.
Most, pipcs llsctl ill cngir~coring sl.ruc:tirrt:s c:nrlnot, IN: rc:g:lrtlctl :w hoing I~~tlr:ir~lic:n.lly
smoot,h, ;it, Ic:lslf att highor Itc:ynoltls rll~nil)crs. 'I'll(: rc:sisL;~nc:c: 1.0 Ilow olli:rc:tl
I)y rongh walls is larger than tl1:~t irnplictl I)y thc prc:c:t:tling oclllnlions for smoot.ll
pipes. Conscq~icnt~ly, the laws of fric:tion in rough pips ;rrc of grc:;lt pr:ic:tic::rl importmwc,
awl cxp(:rimcnlnI work on thcrn I)c:gan vory wrly. 'l'llc tl(sirt: Lo c:xl~lorc:
the laws of frictionof rough pipcs in a systc:matic way is frr~st~r;it~(d l&.t.llc: funtl;~rnc:nt.~~.l
. - . - . . cliffi~~~t,.~~.that the numbcr .of parn.nict,cr~ tlcscribirig roilgllncss is cxtmortliri;lrily
large owing to t11c grcnt tlivcrsity of gcornc:t~ic forms. If wo consiclcr, for cxnrnplc,
a w:dI with . .~. itlcntjcal prot.r~~sior~s ~~~~~~nt:i~~~t~Ilccor~clr~siv~i.bI~:it
its tlr:tg clt:l)t:ritls
on .. lhc . ,, tlcnsitty -. -. - of ... :-.-. tIist.riOnt+t~ .. of siic:li rougl~nrsscs, i. c. on their nllrnl)t:r IFr unil, :Lra:L
as we n.s on thr shapc antlhnight nr~tl, fin:llly, also on t.hc way in whtt:h they :m:
tiist,ribtit,&l over i!s surface, It took, ~hcroforo, ;a i o r t.irnc ~ ~ to fortnnlat,~ oIn:w ant]
- - - -. . .
sirnik ~ W which S tIcscril~c the flow of flnitls tl~ro~gll rough pipes. I,. Tlopf [25]
rn:do :i comy)rol~cnsivc rrvicw of 1.110 nllnicw)ils cnrlicr cxocrirnont,:d rtwults :~ntl
found two l.ypcs of ro~lghncss in rthtiori to 1.l11: rcsist~in(:(: fnrmuh for rough pipes
and open c:ltnnncls. 'l'hc first, ltilltl of rongllrloss c::ll~sc:s :,. rcsisl,:irlc:~~ whic:ll is proj)orlSion:d
to tljc.glunrc .of the vclocit.y,; !,I~is~~l~cans t,h;~t, t,ll(: c:ocfficiorrt of rc:sist,:l.nc-c:
. . .
Is itdnpt:ntlcrlt~ of 1,hc ltcynoltls ni~rnhcr ant1 corrosl)orltls to rcIat,ively coarse n.ntl
tightly sp:icccl roughness clcrncnb SIIC~ as for cxninplc c:o:srsc s:lnrl gr:tins gluctl on tJl(:
surf:lcc, cerncnl, or ro11g11 cast iron. In st~nlt cxscs lllc: nalirrc of Lllc ro~lglltlt:ss ~:;III
bc cxprcssccl with tlic aid of n sitlglc rougllncw p:~r:imc:l.cr k/l