Boundary Lyer Theory

CHAPTER I V
General properties of the Navier-Stakes equations
Reforc pssing on to thc int,rgrat,inn of the Navitx-Stokrs cqunl.ions in the
following ch:lpt,ers, it now sncms pcrtincnt, to discnss some of their general properties.
In doing so wc shsll restrict ollrsclvcs to irrcornprcssiblc viscous fluitls.
R. J)c.rivntion nf Reynolds's principle of sindnrity from the
Nnvicr-Stokes cquntiorla
TJr~til I,Ilc prrscnt day no gcncd a.nn.lyt,ic n~rthotl.s 11:tvr I~rcotnc availnblc for the
intc-gmtion of t,hc Navirr-Sl.okcs~cl~~at~ions. I~urtl~crmorc, soluLion~ wl~inh arr vnlitl
for all values of viscosity are Irnown only for some particular cases, c. g. for Poiseuille
flow through a circular pipe, or for Couctte flow bet,ween two parallcl walls,
onc of which is at mst,, the other moving along its own plane with a constant
velocity (set: Fig. 1.1). For this reason tl~c problcm of calculatir~g the motion of
a viscous fluitl was attaclrctl by first tackling limiting cases, that is, by solving prohlcrns
for very large viscosit,ics, on the one hand, and for very small viscosities on
the other, I)cmusr in t,ltis manner thc matllcmatical problem is considerably simpli-
Actl. liowevrr, tho casr of modcratc viscosit,ics cannot be intmpolatd I~ct~ween
thsc two rxtrornes
1l:ven the limit,ing cases of vcry largc antl very small viscosities present great
mat,hemntical tlifficulties so that rescarch into viscous fluid motion proceedetl
to a largc cxtcnt. by experiment. In this conncxion t,hc Navier-Stnlrcs equations
furnish vcry uscSul hints which point to a considerable rcduct,ion in the qnantity
of cxperimcntal work required. It is ofhn possible to carry out. expcrimcnts on
models, which means that in the experimental arrangement a geometrically similar
model of tho aot11a1 body, but reduced in scale, is investigated in a wid tunnel,
or other s~ritahlc arrangement. This always raises the question of the dynamic
sim~ilnril?y of fluid mot.ions which is, evidcnt.ly, intimately connectmi with the question
of how far rcsult.~ obta.inod wit,h motlcls can Jlc ntilizcd for the prediction of
tho Id~aviour of the full-scale body.
As alrr:dy oxpl~incrl in Chap. I, two fluitl nibtions are dynamically similar if,
with gc?ornct,rit:ally sirnilnr k)oundn,ries, the velocity ficltls are geometrically similar,
i. e., .if t1tc.v have gromctricnlly similar strcnrnlincs.
This question was answcrrd in Chap. 1 for thr caw in which only inertia and
visrntts fnrt~s t:~Itc pitrl. in the process. It was found there that for the two motions
I
the RrynoItIs IIII~I~JC~S mnst be rqunl (lirynolcls's pri~~~i~)lcb of sirnil:~rit.y). 'I'his
roncllrsion was drawn by astimating thc forces in the strewn; wr now propose to
tlctlr~ce it again directly from thc Navicr-Stolrcs equations.
'rlrc Navicr-Stokes cqr1a1,ions cxpress tho condition of cqt~ilil>ri~~~~t,
II:IIIICI~
that for cnc11 pa.rticle thrc is eqrrilibriurn betwccn hly forcrs (woigI~(.), SII~~;LC~
for~cs a~ttl jncrti:~. forcrs. 'J'hc sr~rfac!c forc:c:s co11sist. of prcwurr for(*c.s (IIO~~II:~~ Ii)r(:(:s)
and frictiotl forrcs (sl1ea.r forccs). TZotljr forccs n.rc in~port,nt~t, only irl c::t.sc>s ~IIC:II
tlicro is a free s~~rfncc or whcrl l,lto tlrtlsily clisl.ril~trl.ion is it~l~orno~c:~~c:o~ts. III 111,:
(:xs~ of a hornogcnrol~s fl~titl in tltc :l.hscnc:t? oS n Srrc wtrfi~cc tltrrc is c:(l~tiIiI)t.i~tt~~
l~ctwc(~t tho wcigltb of'c:at:l~ p;~.rl.ivlc at111 it,u I~.yrlrwld,iv I )IIO.~IIII~:~ l'orc;~!, in tl~c S:I,IIIC
w:~y 3.8 at rost. Ilc:nco in 1.11~ rnot,iorl of a I~o~nogcncons Illticl, ir~ thc nt~sct~c:c of:^ I'rrc?
snrf:icc, body forces can 11r canrcllctl if prcssttrc is t,dt~n to IIIC:II~ tho (Iillcr~~ncc
I~ctwccn that in n~ot,ion a.nd at rcst. In t h following arpttnc~tt, wc sl~nll rc.st,ric:t. our
at,tc~~tion to cases for whic:h this assttn~ption is trtrc bccalisc they arc t,ltc: tnost imporhnt
oncs in n.pplicntions. Tltr~s bltc Nnvicr-Stoltcs rqnations will now c:ortt,air~
only forces clue to pressure, viscosity, and inertia.
Unclor thwc assumptions and ronvcntions ihc N~~vicr-Stolccs rqn:ttions for
:In inromprcssiblc fluid, rcstrick:tl 10 stci~dy llow nncl in vcclor fnrttt, sinl1~lil:y to
This clifl'crential equation must he indrpcnclent d the clloicc of the utli(.s for t.lrc
various physical quantities, suc:h as velocity, prcssnrc, clc., which appe:lr in it.
We now consider flows about two gcomctrically similar boclics of diKcrcnt lincar
tlimcnsions in streams of different velocitics, c. g., flows past two spt~cms in wllictl
the densitics and viscosities may also bc different. Wc shall invcstigatc the condition
for dynamic similaritfly with the aid of tho Navier-Stokcs cquat,iot~s. Evidently,
dynamic similarity will prevail if with a suitablc choice of the units of Icngf.h,
tirnc, antl force, the Navicr-Stokes cqn. (4.1) is so tmnsforn~ctl that it, I)ccomcs
identical for the two flows with geomctric:ally similar botir~tlarics. Now, it is [~ossiblc:
to free oneself from the fortuitously selechcl units if clirncnsiorllcss q~~ntltitics n.rc
introduced into cqn. (4.1). This is achievctl by snlcct.ing ccrt,:~in suitnhlc c:har:rt:taristic
mxgnitudcs in thc flow as our ~rrtil,s, antl by refixring all otlwrs t,o t11c:nr.
., .Ll~us c. g., thc frcc-slrcarn vcloci1,y anrltl tllc tlianlcl.cr of Ll~r sphrrc: cnlt IJC srl(:c:t.t:tl
as the rcspcctivc 11ni1.s of vcloc:it.y and Icngth.
1~r.t V, 1, and pl clcnotc tl~csc characteristic rcfcrcnco magt:itrltlrs. II' we now
introtltlcc into thc Navicr-Sl.okcs oqn. (4.1) thc tlirrrrnsionlrss ri~tios