Boundary Lyer Theory

206 X. Approxitnnte rnct.l~otls for steady equations
Table 10.1. Rrsultn of the calcrllation of the bolnltlary layer for a flat plate at zero incidence
baaed on approximab thcory
Vclonity
dint,ribtrtion
ll/U = f (11) i
It is seen that t.hc npprosinmtc mct.liocl Icacls to sa.t,isfnctory rcsult,~ in the case
of a flat plate at zero inciclcncc, and the extraordinary simp1icit.y of the calcnl:~tior~
is cluite remarkable, compared with the complcxit,y of thc exact solution.
We now propose to clcvelop thc approximate method of thc preceding section
so t,llnt it can l)c applied to t.hc general problrm of a two-tlirncr~sional hotrntlnry
layer with prcssurc gradient. The tnct,l~od in its original for~n was first intlicatctl I)y
1C. l'ohlhar~scn [Is]. The succeccling tlcscriptiotl of thc method is based on iLs mnrc
motlcrn form as developcrl by tT. TIolstcin and T. 13011len [GI. \Vc now choose, as
before, a system of coortlinat.cs in which x c1enot.c~ t,hc n.ro mcasured along the wcttetl
wall and whcrc y tlcnotos the tlisLancc fronl t,hc wall. 'rhc hsic crlnat,ion of thc monrent,nm
theory is ol)t,ninctl by intcgrnthg the eqr~:~l,ior~ of motion wit#h rcspcct tm y
from t.hn wall atf ?/ -=- 0 t.0 a ccrt8i~.in tlistanca h(x) which is a~sntn(:d to be outside
t.11~ I)o~tndary layer for all val~ros of x. With this r~otat~ior~ the momentum cqttat.ior~
'
11a.s the form nlrcntly givcri in (8.32), namely
This eq~~ationgives an ordinary diffrrrni ial rqnation for the ho~~t~tlnry-la~pr thiclrncss,
as was tire rase with thc flat plntc in lJic prccctling scc.lion, provitlccl that a
I
b. The approxirnnto nirthod duo to TIN. von Jchrnlhn and K. Polllha~~ncn
form is assumed for the vclocitty profile. This allows us to calculate the momentum
tl~ickncss, the displacement thickness, and the shearing stress at thc wall. In choosing
a suitablc velocity fi~riction it is necessary to talrc into account the same considerat,ions
ns beforc, nnmnly thosc regarding the no-slip condition at t,hc wall, as wcll as thc
reqi~ircment,~ of cont.inclit,y at, the point whcrc this sol~tt,ion is joinctl to tho poLcnti:d
soIut,ion. I~t~rI.l~t~rn~orr,
i11 I,hc prcsonco of IL pr(-ssuro grmlic~tt~ tho f~~nc:I,ion n~~tst
atln~it the cxisbc~icc of profilcs with and without a point of inllcxion corrcspontling
t.o t,hcir occnrrencc in regions of nsg,zt,ive or positive pressure gradients. In ortlcr
to kc in a posit,ion to cn.lcr~latc tho point of scpamtion with tho aid of thc npproxin~at~c
n~etl~otl thc existence of a profile with zero gratlicnt at thc ~ ~ (au/ay),-,=0 1 1 must
also be possible. On t,hc ot.hcr l~ancl functions postulating similarity of vclocity
profiles for various valws of x may no lorigcr be prcscribctl. Following I