Boundary Lyer Theory

204 X. ApproximnLc rnetl~ods lor steady equations
'I'hc value of the displacement thickness O1 from cqn. (8.30) will now also be calcn-
laktl as it will be required later. Putting
wc: oltt,n.in
1
a2 = J (1 - l) dtl,
0
(10.10)
0, --a, d . (10.1 1)
I?'l~rt,hrrniorr, thc visrous shearing stress at the wall is given by
Introtlllcing thcsc valnrs into the niomcntum equation (10.4), wc obt,ain
Int.cgrat,ion from 0 0 at z -= 0 givrs t.hc first, result. for tho approximato thcory
in t.11~ form
ITrnc~ tllo shearing strrss at the wall from cqn. (10.12) beronics
Finally, the l.otd drag on a plato wrttetl on both sides mri be written as
I
2 I1 -- 2 h J to
0
tlx, i. c.
:1n(1 fro111 r(lns. (10.1 1) and (10.14) we obtain the tlisplaccmerit t,hicknrss
A comparison of t.11~ approximaf,c oxpressions for the Itonndary-laycr thickrwss,
li)r the shraring st.ross at tho wall, ant1 for drag with thc respcctivc formulae of
t.11~ :lrc:ur:~t,c throry, rqns. (7.37), (7.31) ntd (7.:13), shows that Lhc use of tho iritcgr:d
rnorner~l~um cqui~tion lcatls in all cn.ses to a peufcctly correct fornlulation of the
cqmtions. In other words, the dcpcnrlcnccof tliese'quantitics on the current length, x,
the frcc-stmxm vclocit,y, Urn, antl the coeffioiont of kinematic viscosity, v, is correctly
tlctll~twl. li'urt,I~crniore, the rclation 0ct.wecn momentum thickness and shearing
strrss nt, ttw wall givrn by rqn. (10 5) ran also be dcducrd from the approximate
rnlrulation, as is rnsily vcrifird. The still-unknown coefficients a,, a, and P, can only
o. Application of tlir mo~nrnt~~rn rqr~ation to Lllr flow past a flat plr~lr at mro incidrnrr 205
hc calculated if a specific assumption regarding the vclocit,y profilc is matlr, i. r.
if t,lte function I(??) from eqn. (10.6) i~ given explicitly.
Whcn writ.ing down an expression for f (q), it is ncrrssary 1.0 sat,isljr cc.14nin
boundary condit,ions for z~(y), i. c. for /(?I). At lcast the no-slip t:onclit.io~~ IL -- 0
:~t y -- 0 antl tho condit.ion of continuity whorl passing frorn t,ltc hottnd:tr~y-laycr
l)rolilo l,o I.hc ~ml,(:~)l.i~rl vl:loc*il.y, TI . (1 III, - , 0, tnr~sl~ lw s~~lisli(vl. I~III~IIVI~ VOII.
(litlions might inclutlc t.hc continuity of thc tangent ant1 curvalurc :IL tl~o 11oirlL,
wlicrc t,lic twr~ solutions aro joined. Tn othrr words, wc may scrlr to satisfy tha con-
ditions a~l./i)?l =: 0 and 321~/a?/~
= 0 at y = 8. In tho case of :L plate tho cot~tlit~io~t ,
that a2u/tJy2 = 0 at y = 0 is also of importancr, and it ran I)o scon frorn rqn. (7.15)
tl1a.t it is satisfied by tho exact solution.
Numerical cxamplcs :
Wc now propose to test the usefulness of the prccctling approxirnak mct.hotl
wiLh the nit1 ofscvrrnl rxnmplcs. Tho q~~alit:y of thc rcsnlt tlcpcntls to a grcat cxI.cn1,
OII t,hc assurnpI,ion which is matlc for thc volocity f~lndion (10.6). 111 ally c::~sv,
as already mcnt,ionecl, the funct,ion /(q) must vanish at 17 = 0 in view of the noslip
condit,ion at the wall. Moroovcr, for large values of 17 we tilust havc /('/) = 1.
Tf only a rough approximation is tlcsirccl, the transition to the valuc /(q) = 1 may
occur with a discontinuous first, tlcrivativc. For a bettcr approximation, corrLinnit,y
in dj/dfl may bc postjulatcd. lndcpcntlcnt~ly of the pnrticular assumyt,ion for l(q)
the cruant,itks
must Itc pnrc numI)crs. Thry can bc easily calculated from cqns. (10.8) to (10.17)
Fig. 10.2. Vclocity tlislrilmhn in t.hc boundary
layer on n flat plntr nt xrro i~icitlanrc!
(1) Lincrr aplrroxirnntion
(2) Cubic npl~rrrxirnntiou Irom Tablr 10.1
Tablo 10.1 contains results of scvcral calcrtlations wil.11 a.lt,crnativc veloc:it,ydistribution
functions. Tho first two fun~t~ions nrc illuslr:l.tod with tlrc aid of I'ig. 10.2.
'I'hc linear funct,ion sat,isfics only the conditions f(0) -- 0 antl /(I) -= I, wllcrcas tho
cubic function satisfies in addition tho conditions /'(I) - 0 :~nd /"(0) :x 0; finally,
a fonrth tlcgrcc polj~nornial can be made to satisfy the atldjtional contlition /" (1) =-- 0.
Thc sinc function satisfies the same I~oundary conditions as the polynomial of
folirtli dtgrcc, except for /"(I) = 0. The polynoniials of third antl fourth tlrgree
and the sine-function lead to values of shearing slrcss at Iho wall which arc in
error by loss than 3 per cent and may bc considrrcd ent,ircly atlcquatc. 'The valnrs
of the djsplaccmerit thickncss 6, show acccptablc agrecmont wiLh thc corrcsponditlg
cxect values.