Boundary Lyer Theory

672
XXll. 'l'llc incon~prcssiblc turbulent bol~ndary lnycr 1). 7'110 cnlci~lnLion of two-rlimcnsional lrtrbulont boundary lnynrs 673
involved when differential equation methods are used is substantially larger than in
tho case of integral methods. The former require the use of a very large digital
computer equipped with a large memory, whereas the latter can be done on a small
calcrllator or, even, with the aid of a slide rule.
In the following paragraphs we shall confine ourselves to the des~ript~ion of
methods which rcsult merely in the calculation of time-averaged values of such
variables of the turbulent flow as t,he velocity, the local shearing stress and the region
of separation, because we subscribe to the view that only such mean values are of real
interost to the engineer. Thus we rcfrain from calculating all those quantities that
result from fluctuations, for example the correlation coefficients, the intensity of
turbulence and its scale. Readers interested in these aspects are referred to more
specialized publications, e. g. [lo, 813.
Rcsearch into turbulent boundary layers was considerably advanced by the
Stanford Univcrsity Conference organized by S. J. Kline in 1968. The results achieved
at the time have been published in two large volumes edited by S. J. Kline, M.V.
Morkovin, G. Sovran, D. J. Cockrell, D. E. Coles and E.A. Hirst [64]. In the appended
[79] "morphology" prepared by W. C. Reynolds, the reader will find a de~cription
of 20 integral and 8 differential methods and characterized according to their respective
physical basis (status = of 1967). They differ, principally, in the empirical closure
functions which are introduced in ordcr to malre the system of equations solvable. In
addition, the conference had at its disposal 33 sets of experimental data which served
as testing material for the computational algorithms. About ten years later, W.C.
Reynolds [81] provided once again a summary review of the very large number of
computational schemes; this appeared in his contribution to the Annual Reviews of
Fluid Mechanics of 1976 (cf. the same author's 1974 contribution in Chemical Engineering
[80]). In 1974 there appeared the book by F.M. White [I191 which describes
20 integral and 11 differential procedures. It is difficult, and we shall not attempt, to
select a "best method" from among the very large number proposed so far.
A summary of many of these methods, principally integral ones, was prepared
earlier by A. Walz [116] and J.C. Rotta [86, 871. A review of differedial methods is
conlaincd in P. Bmdshaw's contributions [9, 12, 13, 141. Further, the book by T.
Cebrci and A.M.O. Srnit,l~ [20] and two earlier papers by the same authors [18, 191,
contain good reviews of many calculational procedures. The two earlier reviews by
L. S. C. I$Y .
2. Truckenbrodt's integral method. Before we proceed with the description of the
details of E. Truckenbrodt's [I141 method, we find it helpful for its understanding to
preface it with a few historical remarks. As already mentioned earlier, all computa-
tional algorithms for turbulent boundnry layers rcly on ecrtnin empirical relations.
As time progressed, and, in particular, since thc middlc of t,hc tl~irtics, tho empirical
basis, and hencc also the semi-empirical and theoretical computational proccclurcs,
underwent a process of continuous improvement.
The first method for tho calculntion of t,urbulcnt boundnry Inycrs wit.11 prcssure
gratlicnts was formulated by E. Gruschwitz [40] in 1931. The cxpcrimcntd data on
which this mrthod wns basod wcro Intcr irnprovcd by A. 1