Boundary Lyer Theory

ccnt,rred nt, (m -1- 1, n). 1'110 two t,xptwsions are t.hertwpon cornt)ined in such a way
thnt ternls of ortlcr Aq2 are elitnina1.rtl. The corrcsponclir~g difference qnot,icnks can
I,c given the, form (index we 1 I omitted) :
a',, 1
{ I I I 2 T I I 1 7 I] I 0 ( A . 1 ) (9 69)
il,l 2 /I 11,'
where
p1 - 1
l2 (I -+- K), rPz - L ~,
.-- 2 r1 J'~, I>,, -= I .
Eq~tn.tions (9.69) nncl (9.70) rcrlr~cc t.o t,hc st.nntlartl form for cent,ral difl'crenccs when
K = I.
For the (-tlcrivnt,ivrs in cqnnt.inn (9.64) a simple bnclrwnrtl tlini.rrner formule is
used
y - Fsr 11. n -- Fnr, n
E -
-I om. (9.7 I )
At --
The 1nrgc.r I,rtlnc.at,ion error which appcn.t,s here is balancctl by t,hc it.crat.ivc scltclne
proposc(1 for solving thr tlilkrcncc c-qt~at,ion. 'l'llc non-lincn.r t,ernis in r(lnnt,ion (9.04)
Imvc to be rcplaced by lincarizccl diflkrencc quoticnt.s. Tho tcrlns fFIl and FFg may
serve as exntnples and thcy are writ,t,cn as
l'hc lincnrizrtl Iini1,r-clifl'ercnw qnotionts given nhove are su11st~it.ut~erl into the
tlifl'crentinl ty~tnt.ion (9.G4) nnd Lhe result is multiplied througlt by A E to give n
tlilYcrcnce equa.t,iotl. 'J'his is writden ns follows
i. The n~ctl~otl of finite dilTwcnces 191
111 cqun1,ions (9.75), 6 and 0 nre cvnlunlctl nt (111. 1- I), mtl ot~ly the vn.rir~l,lcx wit.11
sttp(wcripll i ntt 11~1cInlc~1 through st~cccssivt: it,crntions. To s~~xxl-III) t,Itv il~r~~:t,l,ion
proces.s t,he tcrrns (/S)t can be licJ)t constnrlt. (equnl to t . 1 ~ &luc nt t.hc prrvious
shtion) unt,il initial convergence is nchicved.
Method of nolution: Equations (9.74) rcprc~ont~ n ~ oof t N-1 si~ntlll.nncwus r~.lgc:-
I~rnic equntionrr for the unlrnown k;ntl,n (n = 2, 3, . . ., N). At, cnch levcl IL t.l~rcc
unknown quantities nppenr, namely Fnajl, .-I, Fm.kl,n and Fniit, ,,+I, but sincc
F,+I,~ and Fm+l,~ nre known from the bonndnry conilitions, 1.11~ totnl nurnbcr of
cquntions equals the nunlber of unlrnowns. The set of nlgclmic cqunt,ions rnn be
writt.cn in so-callod tllrcc-tlingonnl matrix form. MnLriccs of Illis ttypcwhcro oK-tlir~go~tnl
elements vnnish outdc the three-tlingonal band can bc inverted bg n sirnplc: and
direct nlothod well suit.cd for digital con~putcrs.l'o end tlriseqna.tion (0.74) is rcwrit,toll
in "stantlard form" (subscripts (m -1 I) ornittcd)
Thc botlnclary conditions arc
F1 = 0 nntl PN = 1, (9.70)
wllere IL = I tlenotcs tbc wall and ?t = N thc edge of the bountlary Inycr. J1, is asst~tn-
etl now t,llat a solution existst in tllc form
The boundary condit,ion F1 = 0 nnd t,he rcyuirernent t,llnt rquation (9.77) sl~ould
rcrnnin valid indcpcntlcnt,ly of the sl,cp size /Iq leads to
A direct, colisrquencc of rqnntion (9.77) is that
When the preceding expressions are substitutml int20 oqn. (!).741,), 1.11~ following
relohion is obt.aincd
By tncnns of equalion (9.81) and the condition (9.78), it becomes possil~lc t.o cotnpot,~