Boundary Lyer Theory

72 I\'. C:rncr:tl prol,e~lirs of thc Navier-Stokrs rquat.ions c. The Navicr-Stoltes equations intcrprctd as vortirit,~ t.rnnsport eqr~ntio~~s 73
'J'his princildo was tlisc:ovrrctl I)y Osbornc Iteynoltls when he invrst,igxt.ctl fluitl
~nolio~l thro~~gh ~'ipcs ""(1 is, t IlcrcSorc, ltnown ns the Reynolds priucipla o/ similnriby.
'I'hc rli~nrt~sio~~lrss ratio
e." = v z _ R
Cc
v
(4.3)
is cnllctl the Itcynoltls nrlni\)cr. JTere tho ratio of the dynamic viscosit,y 11, tm the
clcr~si(,y e, tlcr~otctl by v = ,I./@, is the Itincmatic viscosity of the fl~~itl, int.rotl~lccd
cn,rlicr. S~~niming np we can state that, flows nhout geon~ctrirally sirrlilnr bodies
are tly~~n.miaally similar whcn the Rcynoltls numbers for the flows arc equal.
Thus Itcynoltls's similarity principle has been deducctl once nlorc, t,his t,imc
from t,he Navicr-Strokes cq~mtions, having I~ccn previously derived first from an
c:st.irnnt,ion or Sorccs :in(] sccontlly from dimensional analysis.
b. I.'ricliordenn flow as LL801u1io~~n" of the Navicr-Stokes equations
It nay bo worth not,ing, prent.hrt.ically, that, the .solutions for incomprcssil~lc /riclionless
flown may also bc regarded as exact solutionn of tho Nnvier-Stokes cquat,ionn, bcca~~sc in such
rases tho frictional tcrnls vanish itlont.irnlly. In the case of incomprcssiblc, fricl.ion~csn flows tho
vr1oc.il.y vector can he rrprrscntn?tl an tho grntlicr~t of a potcnt.ial:
w = grad di ,
whrrr t.he potential @ RR~~S~IOS t,hc L:lplacc cquat.ion ,'
V2@=0.
We thn nl~o have grad (V2 @) - V2 (grad @) = 0, that is, V2 w = 0 .
t See foot.nota on 1). 48.
Tl~us the frictional terms in eqn. (4.1) vanish identically for potential flows, but generally
speaking both boundary conditione (3.36) for the velocity cannot thcn be satisfied sin~ulta~~cot~sly.
If the normal con~ponent must ccsmtmu prencribed vnlucs along n bouncinry, thn, in potential
flow, l.l~o t,iw ont.inl oon~ponont i~ tl~oroby tlolorn~i~rnd no 1,Ilnt I,lm 110 dip oo11c1iI.io11 IVII~IIOI~ IN)
sdislicd nt Lf~o mtnm l,i~no. Jd'or Lhis reason ow cnnnok regnril pohntinl ilowe a" pl~ysidl~
moaningfill nolutiona of 1.110 Nnvicr-Stokon cquntionn, bocnuno tlmy do not nnt.inry thc ~w~:scril,rd
boundary conditions. l'hcro exist^, howcver, an important cxccption to tho prccccling ~tx~cmcnt
which occurn whon tho solid wall is in motion and when this condition docs not apply.
The shylest parlicular case is that of flow pant a rotating cylinder wl~cn the pofential ROIIItion
does constit,utc a meaningfnl solution to the Navicr-Stokcs cquntions, as explainctf ill
grcatcr detail on p. 80. The rcadcr may rcfor t,o two papers, one by G. 1InnieI [4] and onc by.
J. Aclteret [I], for fnrt.ller details.
The following sect,ions will be rest,ricted to the consideration of plane (two-din\rnsional)
flows because for such caocs only is it possible t,o inclicato son~e gcncral properties of Lhc Navicr-
Stokes equations, and, on Mia oClrcr hand, plane flows ronstituh by fir tho lnrgrst clans of
prohlcrns of prartirnl i~nportance.
c. The Navier-Stokes equations interpreted as vorticity transport equatinns
In t,he case of two-dimensional nori-stcatly flow in the x, y-pla~lc t,l~o vcloc:it.y
vector bcco~ncs
and the system of rquat,ions (3.32) and (3.33) trnnsforms into
whicli furnishes three equations for u, v, and p.
We now introduce the vector of vorticity, curl W, wl~ich rctluccs to t.hc one
component about the z-axis for two-dimensional flow:
I~rict,ionless motions are irrotat.ionn1 so that curl cct = 0 in s~lcll cascs. Eli-
minating pressure from eqns. (4.4a, b) we obtain
or, in short,hand form
This equation is referred to as the vorlicity transport, or transfer, equatzor~ It stalvs
that the subskmtive variation of vorticity, which consists of tlw lord ant1 rot~vrcl,~ct~