Boundary Lyer Theory

214 X. Approximato mothotls for shady equations
or, u~ing the ~lumerical values for a and b given earlier:
I'hus the solution of eqn. (10.36) is secn to reduce lo a simple quadrature. An analogous
quadrature will br uscd in C!hnp. XXIl for thr ~olution of thc cquntions of
turbulent. flow.
-0 R -am -OM -am m 602 o am onc m om
K
Fig. 10.6. Tho auxilinry funchn F(K) for the col-
culstion of lnminnr boundnry lnyer by thc method
of JIolntcin and Bohlen [5]
(1) train* eqn. (10.35):
(2) lincar nlrproximntioll F(R) -- 0-470 - 6 K;
S = ahgnntion point:
lf -- vclori1.y milximum
c. Con~pnri~on between the npproximntc nnd cxnct solutions
I. Flat plnte at zero incidence. It is easy to see from cqn. (10.22) that the
Pohl11a11ncn i~pproximation becomes cquivalcnt tjo 1Sxn.mplc 3 in 'Ihblc 10.1 for the
case of a fI:~l. piat,c at zero incidence. l'his ease can also t;c obtainctl directly from
eqn. (10.36), whcre U(z) = U, U' = 0 and hence R = A = 0, so tlmt ejn. (10.36)
givcs dZ/tlz = F(O)/U, = 0.4698/U,. Taking into account t,hat Z = 0 at s = 0
it follows that R =0.4098 z/fJ,, or a, - 04XH3 I/l.&/ii, in ngrccmcnt wi0h Tal)lc
10.1. Table 10.1 contains cx.zct and approxi~nat~c values of the boundary-laycr pnrarnct.crs
for tho purpose of comparison. Tt is sccn tb'nt agreement is very satrisfactory.
2. Two-clin~cn~io~~al stngnntion flow. The cxart solntion of 1,he problem of
I wo~dimrnsional stagnation flow for wllirll U (r) = U' . r, was given irl Sec. V 9.
r 7
Ihe exart vnlnrs of displacrmcnt ll~iclzness, momentum thickness and shcnring
strrss at tho wall, calrulalcd with the aid of that theory, arc given ill 'I'aljlc 10 3.
c. Compnrison lwtwccn tho upproximutc and oxnct uolrlth~ 215
Tablo 10 3. Compnri~on of exnrt nnd npproximnto vnlr~rs of Lhr honndury-lnyer pnrurnctors for
tho caso of tim-dimen~ionnl stngnntion /lout
oxnct, solul.iori
In tho approxi~nnt~c mcthotl wc havc Z0 = R,/U1, and from cqn. (103) it follows
- -. -
that t11r momentum tl~ickness is given by (T2 iVfi = dif; = i0.0770 = 0.278 .
It is seen from cqn. (10 81) that the displacement thickness is approximatctl by
6, JiF/v = /,(~i,,) 1/ii, = 0.641 anti cqn. (10.32)gives t0/p u - 47~' = /z(Ko)liR, -
0 332/0.278 = 1.1!) for tllc sllcnring slrcss at the wall. 'L'h ngrccnlcr~l bclwccn
the approxirnnte ant1 exact values is here also complet,cly satisfactory.
3. Flow past n circulnr cylinder. A comparison of the result of the approximate
calculation for a circular cylindcr with the mlution due to Ilicmcnz (See. IXc)
was given by I