Boundary Lyer Theory

074 XXIJ. Tho incomprc~~ihlo t,t~rbnlcnt boundnry lnycr
uniqwly, as evidcnced by the graph of Fig 22.5 This fact is expressed by a relation
Iflz = /(If32), if a dight, residual rlcpendence on the Reynolds number is neglected.
Guided hy the prenrding ohservaf ion, IS.'rruckenbrodt [I 141 introducetlt the nrodt/i~d
shape factor
The reference valuc (IIz3)m = (l/Naz)m hns been chosrn as t.he loner limit, of integmtion
because it. reprcsent.~ an avcrage value for flows wit'hout a pressure gradient,. 111
the casc of t~urbulcnt boundary layers wn choose = 1.3. The nunierical evaluation
of the relation in eqn. (22.4) can tie undert,alren on the basis of a relation
intlicntcd 1)y 11. Fernholz [33]. The result is seen plotketl in Fig. 22.6.
In the case of flow with zero pressure gradient, we find that I/ = If, = 1 by
definition (mean value in the casc of tnrbulrnt flow). Flows wit11 atlversr pressure
gradicnta (pressure rising in the downstream direction) are characterized by If <
H < 1, wlierc.ns for accelerated flows (pressure decreasing) we find that 1 < H <
Hop where Ifs denotes tlie shape factor for the velocity profile with incipient separ-
ation, and Ifo denotes the shape factor of a two-dimen~ional stagnation-flow profile.
According to I
(Hlz)s > 4.0 or 11s w 0.723, wl~creas A. \\'nlz [I 161 proposcs the values 1.50 < (1132)s <
1.57, or 0.736 < If,. < WXil. According 1.0 A. I\. 'I'ownseritl (I Ion] (cf. St.vatfortl (1041)
a vanishing xhcar st.rcss occurs for (I/12)s - 2.274 or IIs = 0.784 in tlie case of
profiles crcat,ed by an ext,crnal flow with U(x) - sP with p = -0.234. The various
shapc fact,ors for incipient separation haw been indicat,ed. in Fig. 22.6. The values
of the modified factor Ils fluctuate n~uch less than those of (Ill& and (H23)S.
Refrrcnce 11141 indicates t.l~at acparalion can occur for
The rangc Ifs - 0.723 < II < 0.761 = If; I
describes vrlority prolilcs thnt nre prow to separiatc.
: l'hr ri~rnirric*nl
nv:~il:~blc nL llw I~nre.
roncltnntn in rqn. (22.51,) hnve heen ndjusted to represent the experiments
b. Thc cnlculntion of two-dimcnsionnl tr~rbulcnt honndnry Injqyv 675
Fig.22.5. l'hr ratio of houndnry
lnyrr thirknrclncn Ifaz = &/rh plottcd
against If12 = fill&, aftcr J.C. ltottrr
1831 and I