Boundary Lyer Theory

176
IX, Nxnrt ~oi~~liot~s of ~IIC ~~cndy-state boundary-Inyer rqrmtiona
as shown in Fig. 9.9. The bonndary AIBl, parallel to the plate, is placed at such
e distancc from the body that it lies ovcrywhere in the region of undisturbed velocity,
I/,. Purthorrnorc, t,hc pressnrc is constant over the whole of t,he control surface,
so t01at j~rcssurc forces (lo not contribute to the mornenturn. When calc~lat~ing
the flux of momontunl across the contml surfacc it is necessary to remcmber that,
owing to ront,innity, fluid nu st loxvc t,l~rongh tho hountlary AIBl; tho q~lantit~y of
fluid leaving Ll~rongl~ A1lll is cqu;rl t.o tho tliffcrent:c I)clwccn tht ontcring Lhro~rglt
AIA and loaving through BIR. 'rho boundary AT3 contribntcs no term to t.hc
nom men tam in the x-diraction becanso, owing to symmetry, the transverse velocity
vanishes along it,. The momentnm balancc is given in tabular form on the next page,
and in it the convc~ltion is followed t.11:bt inflowing masscs are considcrcd positive,
and ontgoing masscs arc taken t,o bc negative. The width of the plate is denoted
by b. 'l'hc tot,al flux or morncntnm is cqnal to the drag D on a flat plate wetted on
orlo sitlc. 'l'hus we have
03
D =be/u(~,-u)dy.
v-0
Intrgration may bo prrformctl from y = 0 to y = oo instcad of to 2/ = It, because
for ?/ > h thc intcgrantl in eqn. (9.26) vanishes Ilrnce thc drag on a plate wetted
on both sides bcromrs
+
2 D = b e / u(u,-u) dy . (9 27)
- m
This cqnat,ion applies to any symrnet,rical cylintlrical body ant1 not only to a flat
plat,o. Tt is t,o bo rcrncmbcrctl that in the more general case thc intcgral over the
profile in t,he wake must be t,aken at a sufficiently distant sect.ion, and one across
whirh t,ho st.at.ic pressure has it.s undisturbed value. Since near a plate there are
no pressure tlill'crrnccs cit,hcr in t,l~e longit~ldinal or in the transverse direction,
ccln. (9.27) npplins t,o any tlist.ancc brhintl the platc. Furthermore, eqn. (9.27) may
11c: nppltc(i t.n any section x of tlhc I)o~lntlary layer, when it gives the drag on the
portion of t-l~c plate between the leading ctlgc and tlltat sect,ion. The physical meaning
of tho ir~t~cgml in eqn. (9.20) or (9.27) is that it rcprcscnts tho loss of momentum
due to frict,ion. It is itlcntical with the intcgral in eqn. (8.31) which dcfir~ed the
mome?ltum thickness a, so that eqn. (9.26) can he givcn tllc alternative fbrm
Wc shall now proccrd to calculate tthc velocity profile in the waltc, in particular,
9.1, a. large dist.ance x t)ehintl the trailing edge of the flat plate. The calculation must
bn p(:rformcd in t,wo sLcps: 1. Through an expansion in thc downstream direction
from I.he Irntling t.o t,hr tmiling ctlgr, i. c. I)y n ~:~lculation which inv?lvc:s thc cont.inu:~.t.ion
of tJ~o Illilsius profile on thc plalo near d.hc tmiling cclgo, antl 2. Through an
expansion in t,hc nl)st,rrarn direction. 'fhe lattw'is a kind of asymptot,ic'int,egration
for x Inrgc tlistancc behind thr plate and is valid irrespective of the shpe of the
1)orlp. It. will 1)c nrrrssnry hrrc 1.0 n~nkc lhc nssrrmpt,ion t,llat t.he vc1orit.y difference
in t.11~ wn kc
711 (", !/) ' U, - - u(z, y)
(0.29)
e. Flow ill ll~e wake of flnt plntc at mro incitlrncc 177
Croswxxtion I Rnte of flow I Dlonient~~ni in dircclion r
C -- Control srlrfnrc 2 Rnte of flow = 0 ::Mornctit~~m flus -= Drng
is stnall rotnparctl wit.11 Urn, so thnt q~~n.tlrn,t ic nntl highrr t~crtns in 711 IIIIIY hr 11t~gltv~t.rt1.
, ,
I l~c ~~occ~luw mnltrs 11xc of n nict,l~otl ol' c:o~~l,inuir~g n. Iznown solul.ioii. 'l'ltc~ (:ILI(:u-
Int,ion st,arts with t.11~ p~viile at the t.miling ctlge, calculnt.ct1 with 1.11~ aid ol' Jllnsius's
~ncthotl, and we sha.11 refrain from furthrr disrussing it hrre. 'I'hc asympt,ot,ic cxpmsion
in t.he upst,rraln direction was calcnlatcd by W. Tollrnicm 1091. Sinrt: it, is
t,ypical for problems oF flow in t,hc wake, antl since we shall mdte nse of it in t,hc more
ilnport,nnt, tmbulcnt case, we propose to devot,c some t,itnc t,o an account, of it.
As thr prrssnre trrm is rqr~al to zero, the bonntlary-layrr cynntiot~ (9 2)rombinetl
wit11 rqn (9 29) gives
'I'he partial tlilli:rr~~t.inl cqunt,ion call, here 1.00, be tmnsformctl into an or(li11iir.y
tliffcrcnlinl ecpat,ion by n snit,a,blc? t,mnsrormnt,ion. Sirl~ilnrly to 1.11~ assuml)tion (7.24)
in 13lasirrs's mct.l~od for t,hc 11x1 plate wr put.
antl, in adtlit.iot~, wr assnme t.hxt( u, is of' the forin
tl1 = U-c (-;)-kg(,]),
whew 1 is the lrngt,ll of thr platc, Fig. 9.9.
Tho power -- .j for 1: in eqn. (9.31) is just.ifict1 on the ground that the ~no~nent.urn
int,cgrnl whicll givrs t,hc drag on tho plnt,c ill oqn. (!1.27) I~IIS~, I)r intlrpondrnk of r.