Boundary Lyer Theory

202 X. Approximate rncl.hoda for steady cqrlntionu n. Applicnth~ of the morncntrlm rqr~ntion to Lho flow pnut n flnt plnLo at zcro incidcr~cc 203
the c:ont.rol s~lrfacc, consitlcrrd fixrd in spacr, is cq~lal to the skin friction on the
plate D(s) from the leading cclge (s =0) to the current section at x. The application
of thc momentum equation to this particular case has already been cliscussecl in
See. IXt It was then found, cqn. (9.26), that the drag of a plate wetted on one
side is given by
m
D(4 =be/ u(u,--u)dy, (10.1)
u-0
where the integral is to he taken at scrtion s. On tho other hantl tho tlmg mn bo
expresscd as an intrgral of the shearing stpress to nt thr wall, lnltrn nlong tl~c plntr:
X
1) (s) = b 1 r0 (x) dx .
Upon comparing eqns. (10.1) and (10.2) we obtain
0
This equation cnn bc also dcclucccl in n purrly formal way from t,hc 11onntlnr.y-layer
equntion (7.22) by first integrating the equation of motion in the x-direction with
respect to y from y ---- 0 to y = m. Equation (10.3) is, finally, obtained without difficulty
if the vclocit,y component v is eliminated with the aid of the equalion of
continuity, and if it is noticed t.hat p(au/a~),,,~ =to.
&'Iu:$
mtmI surlace
Fig. 10.1. Application of tho momcntun~ equa-
-5 u @.Y) tion to the flow pmt n ant plnto nt zero incidencc
-x
Introdncing thc morncnturn thiclir~css, a,, defined by rqn. (8.31), wc have
Tllc momcntom cqmtion in ils form (10.4) rcprosrnb n particular cnso of the gcncrnl
momentum eqnntion 01' bountlary-laycr Lhcory as given in eqn. (8.32), heing valid
for the cnse of n llat plntc nt zcro ir~ciclcncc. 1t.s phpical meaning expresses thc fact
thats Iht! shearing stmss at the wall is cqunl to thc lhss of momentum in tho bonntlary
I:lycr, because in tho cxarnple under consiclcrnlion t,hcre is no conl.ril~ution from t,llc
prcssure gmtliont.
'So far rqn. (10.4) int,roclucncl no ntlclit.iona1 :~ssnmpLions, as will be the case
wit,l~ the ajq)roximntc method, bul, 1)c:forr tliscussing this matter it might be nscful
to nolc x ~cI:LI.~oII I~CLWCCI~ to nnd S2, WII~CII is obtaincti from cqn. (10.4) by int,rotlucing
the exact value for to from eqn. (7.32). Putling tu/p Urnz =a iy/urn2 with a =0.332,
we have
E .-
'
With rcfc:rcncc to qn. (10.3) or (10.4) wc con now porfortn nn npproximnto
calcnlnt.ion of the l~ountlnry laycr nlong n Il:bt, plnlo at zcro incitlcncc. '1'11t: CRWIICO
of the npproximatc method consists in assuming a suitable exprrssion for the vclo~it~y
tlist,ribution u (y) in the boundary laycr, taking cnrc thnt it sntisfics the importnnt~
boundary conditions for u(y), and that it contains, in addition, one free parameter,
such ns a ~nitddy choscn boundary-layer thicltncss which is finnlly dctcrmincd wit.11
t,he aid of the momenlum equation (10.3).
In the particular case of n flat plate at zero incidenco now being considered
it is possible to t,ake advantage of the fact that the velocity profiles arc similar.
IIencc we put
where r] == 2/16 (s) is the dimensionless distance from the wall referred to the boundnrylayer
thicltness. The sin~ilarity of velocity profilcs is here acconnt.ed for by assu~ning
that /(?I) is a function of 7 only, and contains no additional free parameter. The
function / must vanish at the wall (7 = 0) and tend to the value 1 for large values
of 17, in view of the boundary conditions for u. When using the approximate method,
it is expedient to plnce the point. at which this transition occurs at a finite distance
from the wall, or in olher words, to assume a finitc boundary-layer thickness 6(x),
in spit.c of the fact that all cxnct solutions of t.hc houndnry-layer equations t.cntl
asympt~otically to the ptential flow associated with the particular problcnl 'l'hc
boundary-lnyer t.llislrncss has no physical significance in this conncxion, being only
a quant.ity wl~ich it is convenient to use in thc computation.
Having assrimcd tl~c vclocity profilc in cqn. (10.0), we c:~n now proceed to
rvnl~~atc tho momentum intcgml (10 3), arid we obtain
for short,, we have
ru(uW- u) dy=um2~2 = a, 8 urn2,
v-0
or d, = a, 0 .