Boundary Lyer Theory

082
XXII. Thc incomprm~il~lr L~lrbtrlolt bn~rndnry lnyrr
wherc bhr following numerical vnlncs (cf. Table 22.2 - energy method) have been
employed :
b = 0.152; v' .= 80 v; with flq(rI) = v' {[U(r1)J2 R3(xl))lCb.
(22.22 b)
This explicit formula cont,nins only t.hr extmnal free-strcam vclocity U(x) which
mny IIC known from pot,cnt,ial t.hcory or from mcasarement. The positmion .2: = xl
constit.ntes t,he st.nrt,ing point for t,he calculation.
Apart from the velocity U(xl), the constant of integrntion E3(rl) rontains also
the encrgy thickness 63(.cl). If the station xl coincides with the point of transitmion
the cnergy thickness should be ralc~llatcd over the laminar boundary layer in t.hc
rangc 0 2 x xl Here x = 0 rlenotcx thc start of t,he boundary layer; for example,
the leading edge of a plate or the stagnation point of a blunt body. It waa shown in
11141 that eqn. (22.22a) is also valid for laminar boundary layers whenb = 1, vi =
v/Q& = 0.91 7v and lC3(xl = 0) = 0 should be specificd. In this case, with a laminar
starting length, the constant of integration becomes
112
(point of transition). (22.23)
If the boundary lnyer drcntly is t,urbulent at z = XI, it is necessary to substitute
into eqn. (22.22) for E3(z1) t,he local vnlue R3(x1) = n3(x1) U(xl)/v.
In many practirnl npplicnt.ions it. is not enough to know the \)cl~aviour of t,hc
boundary-layer thickness, lwre t,hc encrgy t.hickness &(x). This is the case wit,h
ueparation-prone or separat,ctl boundary hyers. If, for example, it is necessary t,o
makc n statcmcnt about t.he possibiliby of separation, it is necessary to know the
velocity parameters along the wdl. All mct,hods discussed in Sec. XXIIbl provide
procedures for t,he calouletion of sonir xhnpe fnctor in addition to tht of a boundarylayer
t~liiclrncsx, such as the niomcnt,~~m thickness &(x) discussed t.here. The shapc
factors nre defined differently in diff'rrent methods and different differential equations
are specified for thcir ralrnlation. A revicw and intercomparison was given by J.C.
Roth [85].
l'hc diPFrrcwt,inl rq11r21~ion (22.!h, 1)) for t.hc shnpr. factors 1132(x) and Hz3(~) were
obtained by tlic coupling of thc ~~ior~~rnturn-integral and energy-integrnl equnt.ions
(22.78, b). Thc pereding dilTrrent,inl equntions det,ermine the shape factor in a unique
way proviclrcl that onc-para~net~cr velocity profiles Hlz.= f(H32) or Hlz = f(Hm)
nrc post,ulnt,ctl and npproxiniak cxprrsxions for t,he shear-stress coefficients CT and
co are s~hst.itllt.c11 from rqn. (22.Rn, b). The dct,crmining equntion (22.11b) for thc
shapr factor cnn br writ.tm in t,crms of thc naodijied shape lactor If = f(llzs) proposed
by 15. l'r~~rlrcnl~rotlt.. l'ogcl.hcr with rqn. (22.11a), this relnt,ion forms a system of
simultn.rous tliffrrrnhl equations for t,l~e Reync~lds number formed wit,h t,lte energy
t,hickncss, Ra(:v). and for thc shapt factor 11(a)! According to Table 22.1 we must
put TN -.- 2 =- const 1tr1 Ihr rnrlgy ntcthotl clisrussed herc 7'hc forms of the funrtions
cl,3(RR, 11) nnd lli3(R3. If) arr lo Irr taltrll Irom thc same talrlc Rcference[ll4]summarizci
cqns (22 1 I n) nntl (22 1 1 0) as follows :
with
and
Z
G1(x) = G(q) + j y' un-' dx,
where n is listed in Tablc 22.2 (energy method). The correction function y'(x) =
yf(R3, H) can be calculated with the aid of a(ll), b(H), u(H), /l(II), If12(x), 1fS2(H),
as well as R3(2) and N(x).
The correction y' differs by a larger or smaller amount from the value 1.0 in the
case of a turbulent boundary layer, nnd cannot bc determined with an adrqunte
degree of reliability. 13y way of approximation, we assume y'(x) = const = 1.0 and
introducc a new quantity c = const = 4.0 in order further to simplify the analytic
solution. The qunntily c has been. so dct~ermincd as to nchicve optimum ngrccmcnl
between available measurements [64] and theoretical results; see also [114]. The
modified shape factor is obtained from the equation
which is the result of somc algchraic txansformations not reproduced Itcrc. llerc thc
influence /unctions of the external velocity distribution are defined os
The initial valuw, i.0. the constants of intcgration are
We take the numerical constants as
The integral expression (22.25) for the calculation of the shape factor contains
only the external velocily distribution U(x), as was the case with the corresponding
integral expression (22.22) for the calculation of the Rcynolds number. The detm-
mination of the inllucncc function N(r) rcquircs thc pcrformnnco of n doublc inLc-
gration with respect to x. The position x = xl once again represents the starting
point of the calculation.
The constants of integration G(s1) and N(xl) contain the shape factor lI(xl)
in addition to the'velocity U(xl) and thc Reynolds number R3(xl). If the position
xl coincides with the point. of transition, it, is necessary to require that the encrgy
thickness of the laminar boundnry layer must be equal to that of the turbulent.
boundary lnyer in accordance with eqn. (22.23). On the other hand, the shapc factor
may change it.s value at the point of trnnsition. The numcricnl values of bl~c shapc
fnctor lic in t,hc mngc 1.0 2 11 2 Ils = 0.723.
2,