Boundary Lyer Theory

128 VII. Ihr~nclary-layer rq~lnt,ions for t\vo.ditnrnuiotd flow; ho~~ndary lnyer on n plate
layer, t.hc so-cnllctl bounclary layer. In this manncr there are two regions to consitlcr,
rvrn if t,l~o tlivision t~et,wrcn t,hcm is not very sharp:
1. A vcry thin lnyrr in the immct1i:~tc ncighbourl~ood of the body in which thc
vrlocit,y grn.tlicnt normal to t.hc wall, a?~/r?y, is very largc (hnimrlrr~y hjer).
111 t.l~is rrgiolr t,ho vrry small viscosiLy ,u of Ihc flr~itl cxcr1,s :rn rssc:nt,i:~l infl~rnnc.~
ill so f:~.r :LS 1.11~ sl~caring sI.rcss t -7 l~(i)~/i)y) tn:~y :ISSIIIIIO Inrgt: V:L~IICH.
In gcncml it, is possible to st,at.c l.hat the Ll~icltncss of thc bonndary layer intmvtscs
wit.11 viseosit,y, or, more generally, Lhat, it decreases as the Reynolds numbcr
incrensrs. It. W:LS SCC~ from scvcrnl cx:lct solutions of t,he Navicr-Stolzcs cqrtntiorls
I)rrser~tcd in C11n.p. V that t,ltc bonntlary-lnycr thickncss is prop~rt~ional to the sqliarc
rool, ol' Izinc:m:tl,ic visoosit,y :
6-fi.
\\'llcrr malci~~g t.lrc si~nplific:~Lions t,o I)c inl.roducrd into I.he Nnvicr-St.ol;ns eqnnt.ior~s
it, is nssr~lnrcl t.l~at, this thickncss is vvry small cornpnretl with a still ~~nspeciliotl
linrnr tlimc?nsiot~, T,, of thc hly:
0 < 1, .
\Va shall now proccctl to discuss the simplification of the Navicr-Stokes
rq~tnt,ions, awl in ortlcr t.o achieve it, we shall make an estimate of the order of
rn:~gnit,ntlc of c:~,cl~ tnrn~. 111 the two-dimensional problem shown in Fig. 7.1 we
sl~:l.li Itcgin I)y RSSI~III~~I~ Lhc wall tio be flat :~nd coinciding with tho x-dircctrion, t h
1,-:txis being I~crl)cndic~llar to it. Wo now rcwritc the Navier-Stokcs equations ill
tli~r~r~lsionlrss I;)r.m by rehrring dl .;clonit,ies to tl~c free-st.rcnm vclocit,y, V, and
I)jr rcfrrring all lilwtr tlimnnsions 1.0 a rhar:~etcrist.ic Icrrgt,h, L, of t h I)otly, wltioh
is so srlccl,cd ns t.o rnsnrc that the tlimcnsionlcss ricri~at~ive, au/i)z, clocs not, cxcocd
rrt~it,~~ in tlrr rrgiot~ r~ntlcr consitlcmt,ion. 'rl~c pressrlre is made dirncnsinnlnss with
p 1f2, nntl I irnr is rrfcrrctl to r,/ If. lPurt,I~cr, the cxprcssiorl
With the :~ssnmpI~iotrs mndo ~~rwiously t.llc: tlirnrtrsionlrss l)o~~t~~l:~ry-l:~~
t,hiclzncss, dll,, for which \vc slra.ll retain t,hc syml)ol (\, is very small wit,ll rrsprct.
1,o ~tnity, (0 < I).
\\.'c sl~all, further, assume that the nolr-stcndy :~ccclcr:~t,ion i)~/al is of the! sn.tnct
ordcr n.s (.he convective term 11. aupr which mcnns t.llat vory s~ttltlcn nccolcrnt.iotrs,
s~lcl~ as occur in vcry lnrgc prcssnrc waves, arc cxclr~ctctl. Zn accort1:~rlcc wit,ll orlr
previous n.rgtlrncnt sornc of the viscous terms must be of thr samc ordcr of rnngnit,ntlr
:IS t.hc incr1,i:t t,crms, at lcnst in t,lrc immctlintc ncighl)ourl~ood of the wall, nntl ill
spi1.c of t,lw srn:~llt~css of (,he fact,or 1/R. JIcncc sornc of t,hc second deriv:it.ivcs of
vrloci1,y must t~ccornr vcry Ixrgo nnar tJtc wall. In nt:cortl:tncc with what w:~s wit1
Iwforc: this can only :~pply to a27r./ay2 ant1 i)2~~/ay2. Since t.ho component of vc:loci~,~
p:~,rallrl t80 tl~c w:dl increases from zero :tt. t,ltc wall 10 t,ltt: vnluc: I in t211(. frwsl.rc::~rn
arross 1.l1r 1:Lycr of t,hic:lrncsn (?, we 11;tvc:
wl~c~rens i)v/if?j ,- 016 - I ant1 iPv/if?~~ - l/d. If tltcsc v:drtes :~ro inscrLctl ittt.o c:qirs.
(7.2) at~d (7.3), it. follows from the first cq~~nt,ion of molion Il~at the viscous f'orrrs ill
t,lrc honlltl:wy layer can bccomc of t.hc samc ortlcr of rnngl~ihtlc :ts thr inrrt,i:t ii)rc.c:s
only if tho Itrynoltls nurnl)cr is of the ortlcr 1/02:
.
ll~c . first, cquat,ion, tlt:~t. of cont,inuit,y, rcrnnins unnltcretl for vcry lnrgc Ilcynoltls
ntlmbcrs. The srcontl cquathm can now be sirnj)lifirtl Ily ttcgl(:ct.it~g iPi~/r?.c.2
wit,h respect to a2i~/ay2. From t,he t,hirtl eqnation wc may infrr t,hnt i)p/al/ iu of t,Ire
ordcr d. The pressure incrcnse ncross the bonndnry Inycr wl~icl~ woultl he ot~t,:r,ined
I)y int.cgrn,t,iny!
'l'lt~ls 1.11~ I)I.~.SSIII.~\
l.lw third tqnirt,ion, is of 1.11c otdrr 02, i. V. very st~~n.ll.