Boundary Lyer Theory

5l)O XIX. Tl~rorrt,icnl nmumptions for t,hc mlculntion of td~i~lrnt Ilows
'I'll(: nnivc?rsnl vc~loc:it~y-elisl.ril~~~t,iot~
I:Lw, nqn. (I9.:1:,), wl~ic:l~ has now I)ecn dcrivctl
for th: (:tist? of :L fht, wnll (rcc:t.:ingul:w c:l~:~.nnol) rct.airw its fu~nlnmcntnl irn1)orLancc
for flows t,l~rough circular pipes, :w will be wcn in t,l~c next chnpt,c:r. We may now
st.ntc, in anticipation, that it loatls to pot1 agrccmcnt wit11 cxperimcnt.
Tn nonclutling this c:hnptrr it may 1)e worth stmssing once again tht thc two
nnivcrs;il vclocit,y-clis(.ril)~~ttio~i Inws in cqns. (19.21) ant1 (10.27) were obtained for
t,~irl)nlcnt~ flow, and took i11t.o nc:nount,, ap:lrt from t,hc small sub-layer mar the
wnll, only tmrhnlont shmring strosscs, ant1 it sho111tI 1)e realized trIrnt3 snch an assumpt,ion
is s:ttlisfi(:(l x,l:~rg!r l~cyn?ltls nurnl)crs~only, (h~scq~~ent~ly tile vclonit~y-tlist,ri-
1)ul.ion law, p:~rt.ic:~~ln,rly t.l1:11, in cqn. (I!).33), must I n rcjinrtlctl as nn asymptlotfie
law a.pplic:xi)lc to very 1:~rge ltcynoltls numbers. For smallgr-Rcyr~oltls- ritlnll~c~~
w11c-1~ .I?n~it!:~r. fi.ict,i(?l~ I:xo~(,s,. somc ~II~~IICII~C outside th~very~-tl~it~ ,q~b-l:~ycr,
- .- - . . -. t:xp .
riihcnt Icads to a power law of the form ,..
. ..
.
wlwrr the exponent 21, is approximately rqunl to :, but varies somewhat wit.11 the
Itrynolris nnml)cr. 'l'his point will also be t,akcn up agnin in the succeeding cl~aptcr.
'rhc c:~e of so-cxllod Co~wtt,c flow I)ct.wcen two parallel flat plates which nre
tlisplnccxl rcl:it,ivc t,o circl~ other (Wig. 1 .I) const.it,ut,rs a very simple cxaniplc of a
flow it1 whicl~ the sl~c::~ring stress rcrnnins c:or~st~ant,. 'l'he sl~c~wing st.rrss T rrrnains
Fig. 19.3. Vdority profilcs
in prallel Coucttm flow
betwren two parallel plates
moving in opposite dirrctions,
after H. Rcichardt
[25, 261
At R - I200 Lhc flow Is Inmimr;
nt R .- 2000 nnd 34.000 ll~c flow
Is L~~rbulcnt
rigorously constant in trrrbnlcnt ns wcll ns in laminar flow, xncl is cq~~al l,o t,11:1t, :it,
thc wall, to. 11. Ltcicl~a.rtlt. 126, 261 carrictl out an oxtcnsive invcst.igat,ion of this cSasc;
somc of his rcsulh can bc inferred from Fig. 19.3 which sl~ows several vclorit,y pwfilcs
observed in Couctt,c flow. 'l'l~c flow rcmzins laminar ns long ns the I~C~IIOIIIS
number R < 1600 sncl thc velocity distribution is t,lwn linear to a good tlcgrrc of
approximation. When the Reynolds r~umbcr R excccds tl~c value 1500 the flow is
turbulcnt. Tllc turt)~~lcnt velocity profilcs arc very flat near the centre ant1 bccornc
very steep near the walls. A profilc of this kind is to be cxpcctccl in tnrl)ulont flow
if it, is rcmcnibcrcd thxL t.11~ shearing stress nonsists of n I:~.tninnr c:ont.rit)~~t.ion ,
, anfl :i t,~~rbulcnt~ conI.rihl,ion
clue to turbulent mixinp. Ilcncc
wl~crc A, donotes the mixing cocfficicnt tlcfinctl in rqn. (19.1). 111 t,llis matinor the
velocity gradient turns out to be proportional to I/(p -t A). Since A varies from
zero at the wnll to its maximum in the centre of thc cl~anncl, the velocity profilc
must bccomc stmp at tltc wall and flat at t,Im centre, as confirmctl 1p.y thc plots in
Fig. 19.3. The turbulcnt mixing cocfficicnt increases with an increasing Reynolds
number and the curvature of the vclocitpy profile bccomcs, correspondingly, more
pronouncrtl; compare the paper I)y A. A. Szcri [nlnl.
f. Further dcvelopmcnt of theoreticnl hypotheses
The cnlculation of Lurbulcnt flows on 1.111: bn4k of t h difl'crcnt s~n~i-c~npirirnI l~yl)ot.l~(:scs
discusset1 previously, and mrricd out in rict,nil in thc succoctiing cl~npkrs, is not sntisfact.ory
in so far as it is still itnpossiblc to analyze t1ifli:rcnt kinds of turbulont llow \vitl;'iho &I of t.hc swnc
hypothesis concerning trrrbulcnt friction. Ipor cxatnple, Prnntltl'a hyp~tl~c~is 011 1.11~ mixing length,
cqn. (19,7), fnih cotnplctely in-tl!? casc of RO-cnlld isotropic. turbulence ris it, c?xint,s bcl~i~~cl n
scrccn of $tic ~cs11, bccnuw in tliia cost the: vciociLy giatlicnL of the biuic flow ik ct111nI 1.0 zc:r~
cvery.wJ~crc, Tho liyPo~llcscs lor bhc cilct;lniio~~ or clovclopccl LurlmIcnt flow, (~~HoIIRR('II in Sees.
XIXb and c, have been considerably cxtentlcd by I,. I'ranclt,l 1221 in an attempt to rlcrivo n universally
valid system of equations (turbulnnt flow near wall, frco turbulcnt flow, isotropic tmrhu-
Icncc).
Energy eqemtioa: L. Prnntlll bnsctl his
-
ncw dcvelopmcnt - on t.lw consitlcrntion of t.hc kinotic
encrgy of turbulcnt fluctuation, R = o(r'2 + 11'2 + z), nnrl cnlc~rlatctl thc rltange of t.1~
energy of the suhsirliary motion with tinrc, UR/J)l, for n particle which nlovcs with Lhc basic
stream. This is con~poncd of t,l~rec Lcrnw: of the decrcnsc. in cncrgy t111c to internnl fric.t.ion in the
motion of tho lumps of fluid, of thc tmnsfcr of cncrgy from tho hnsic motion Lo tl~o sut)sidiilry
n~ot,ion - this term heing proportionnl to (dlJ/dy)z - and, linnlly, of thc trnnsfcr of kinetic
energy from the more turbulent to the lem tmbulcnt zones. The encrgy balnnce I)ct,wren t,llc.~e
tl~rcc terms leads t,o a differential equation for the cncrgy of the t,urbulcnt'sr~h~itlinry motion
which must be added to the systen~ of differential equations for the Incan niotion; it has the forni
cliasipntion production diK~~nion