Boundary Lyer Theory

560
XVIII. P~tndan~cnlal~ of turbulent flow
lorm topctllrr a c*omplrte ~trtss tensor oi l~~rbulenl /low. Eql~ationst (1 8.6) were first
tlctlncrtl 1)y 0. ltcy~oltls 1431 from the equations of motion of fluid dynamics (see
also the next srct ion).
It, is easy 1.0 vis~~alizc - that f,he time-averages of thc mixed products of velocity
Il~~ct.u:sl.ions, SII(~I as C. g. TI,' 11' - do, in fact,, diffcr from zero. The stmss component
a,,' - T,,' -- - - Q 11' 11' c:~n he int,erprct,cd n.s the t,ransport of z-morncntfnm through
a surfacc 11or1nn.l t,o tho ?/-axis. Consitlcring. for example, a mean flow given by
IZ == I:(?/), Ij --- 221 --- 0 with t14/dy > 0, tPig. 18.2, we can see that the mean product
16' 11' is tlilTcront. I'rom zero: 'l'llc pnrl,iclcs wl~ich travrl upwards in view of the trrrbulcnt,
Fig. 18.2. Transport of momentum due
to torbrdent velocity flrrctuation
-
ottly tlilli:ro~l, from 7,cro l~t, also nrg:l.l,ivc:. 'l'llc shearing stmss a,,' = - p 11,' 11' is
~)osit,ivct in tllis c.:~.sc nntl hns tJ~e snmc sign as the rrlcvnt~t Inrninnr sllcaril~g stress
T, -- 16 tl~i/tl?y. 'l'llis fact, is nlso csprcssctl by sl,a.l,ing that thcrc exists a co~wlnlion
I)c*t,wcotl 1.l1o longit.n(linal and t~mt~svcrsc Iluctuation of vc4ocit,y at a givcn point.
cl. nrrivntion of rlw ~trrss trrisor of nppnrctlt tt~rbrrlcnt friction
from thc Nnvic-r-Stokcs cqrrntior~s
ll;~\,it~g ill~~sl.r:~(.c(l t.11~ origin of t,I~r n,~l~lit,i(~~~:~l I'nrccs ca~tsc(I l>y l~~trl~nlc~~l~
Ilt~c-(.~~:~l.ion wil.lt l,11(: :lit1 of :I, pltYsic:~l :trg~trnc:nt wo sh:~,ll ttow pt.o(:(~~~I 1.0 tl(*rivc
tit(- snlllc rxl)rrssion in :I. tnoro forn1:~l ~ 1.y :~11(1 dirrd.1y from t.llr Navicr-SI,oIzcs
cvl~~:~tio~~s. 'i'lw ol,jwl. of 111~ sncxwxlirtg :I~~IIIIICQ~, is 1.0 (Icrivc thn cqun.l.ions of
~noliou \vhic41 tnr~sl, Ilr sn.l.is~ircl 1)y 1.11~ tin~c-avcrt~grs of t.llc vclocit,y com~~onc~~t.s
ii, i;, 171 :111tl of tlw pr(.ss~~rr p. '1'11~ Nnvi(~r-Sl,ok(~s cclu:~I.ions (3.32) for incompressible
llow WII I)(% r~x\vril,k~~ in tltc, rcu-111
wltcrc V2 de~~ot~cs 1,apl:wc's oprat,or. Wc IIOW in troducc t.hc 11y pol.l~cscs t.rg:~tdil~g
the decomposit.ion of velocity componcnt,~ and prcssnrc int,o tlwir tirne-avcr:~grs
and fl~~ct,~lat.ion tmrns from cqn. (18.1) antl form tirnc-averagcs in tPhc rcwtll,iltg
cqunt,ionst t,erln by tcrrn, t:lking inLo aec.onnt thc rult,s from cqn. (18.4). Sincsc
a?/az = 0 et,c. t.hc equation of continuity bccomcs
From cqns. (18.7) nntl (18.6~1) we obtain also thaL
It is seen that the time-averaged vclocity components and tl~c fluctu:lt.ing coniponents
each satisfy the incompressible eqnat.ion of cont,in~~it~y.
Tntroducing tJle assumpl,ions from cqn. (18.1) ido the rcl~t:~t,ion.s of tnot.iolt
(18.0a, b, c) we obtain expressions similar to those givcn in 1.lln ~)rccctling scv:t.ion.
Upon forming averages antl considering the rules in cqn. (1 8.4) it is not,icctl t,l~nt. the
quadmt.ic tcrn~s in thc menti values rernn.in unalt.crctl 1)ccausc l,l~ry :~rc :~Irc:t~ly
const,ant in t.imc. l'hc tcrnls which arc lincar in the Lurb~~lcnt con~poncnts sud1
as c. g. ad/at and a2u'/ax2 vanish in view of cqn. (18.3). 'rho sntnc is true of t.1~:
nlixetl t,crrns such as c. g. G . IL', bllt t.hc cl~~adr:~t.ic tcrtns in t h llirc:l,n:~l.itt:! c:o~tl-
--
ponents remain in the eqnat,ions. Upon averaging they assume t,lrc form 1.'" 71,' v' ctc.
llrncc, if the averaging process is carried out on cqns. (18.0), ant1 if silti~)lific~:~(io~ls
arising from the continuit,y equat.ion (18.7) arc int.rod~~rc*tl, the follo\~ir~g syslctn
of eqnatlions rcsults
r 7
.I 11c cl~~a(lrnl,ic terms in tnrlmlcnt vcloc:;l,y cotnpo~tr~~t.s It:~vo l)ecl~l I,r:~nsri~~wtl lo
th: rigl~t,-I~:~~n(l si(1o for :I, r ~~~son whid~ will soott I)(WIII(: :I~I)IIIWI~,. ICeps. (1S.S)
togctl~rr with t,l~e ccl~~:~I.ion nf cont,in~~if~y, (Y~II. (18.7), (I(~l.~wtlit~c 1.110 11r01~1(-111 t11111(.r
consitlcralion. 'I'hc Icftr-Irnnd sitlcs of ccpls. (18.8) arc l'~~rtn:rlly itlcnt.icd will1 (111:
steady-state Nnvier-Stokes equations (3.92), if 1,llc vc1ocil.y con~ponrnl.~ I(,, v, 111 arc