Boundary Lyer Theory

CHAPTER XXII
The incompreesible turbulent boundary layer with
preseure gradient J-
In tho present chaptcr we sliall discnss the bchaviour of a turbulcnt boundary
layer in the prwrnce of a positive or nrgativc prcssr~rc gradient along thc wall,
thus providing an extension of thc sobjcrt matter of the preceding chapter in which
the boundary layer on a flat plate with no pressure grhdicnt was considered. The
present case is pzrticnlarly important for thc calculation of the drag of an aeroplane
wing or a tutbinc blade as well as for thc untlcrstanding of the processes
which takc plarc in a tliffuscr. Apart from skin friction we arc intcrcstctl in knowing
whether the boundary layer will scparal.c under given rircumstanccs and if SO,
wc shall wish to detcrmine tl~c point of separation. The existmcc of a ncgstive
and, in particular, of a positive prcssnrc gratlicnt exerts a strong influcncc on the
formation of the laycr just as was thc case with laminar layers. At the present timc
these very complicated phcnomcna arc far from being understood complctcly but
there are in cxistcncc several scmi-empirical mctbods of calculation which lead to
comparatively satisfactory results.
In the year 1962, J.C. Rotta [86] prepared a comprehensive and careful review
of this vast ficld of knowledge. In order to develop methods of calculating incom-
pressible, turbulent boundary layers with pressure gradients it is necessary to derive
from experiment relations which go beyond thosc employed for pipes and flat plates
at zero incidence. For this reason we shall begin by giving a short account of some
experimental results.
a. Some cx~mrirncntal results
ILrly systc~natic cxpcrimcnts on two-dimensional flow^ with pressure drop and
prcssuro rise in convcrgcnt, and divcrgcnt clianncls with flat walls have been carried
out by F. Doench [28], J. Nikuradse [71], II. Hochschild [45], R. Kroener 1571 and
J. Polzin [76]. Measurements on circular diffusers, and particularly on the efficiency
of the process of energy transformation, are described in papers by F. A. L. Winternitz
and W. J.Ramsay [123]. These experiments demonstrate that the shape of the velocity
profile dcpcnds very strongly on the pressure gradient. Figure 22.1 shows the
velocity profiles which were mcasurcd by J. Nikuradse during his g~~erirnent.3 with
t Tho new veruion of tliiu chnplcr wo.8 propnrcd by Profemor E. Truckenbrodt whose nssistance
I I~ercl~y grnbhlly ac:knowletlgo.
a. Some experimontnl rusulta
Fig. 22.1. Vclocit,y diutri1~11Lion in coni~ergcnl
and. divergent cl~nnnols with flat wall^, as
n~cmrirccl by J. Nikurnduo [71]
- ImIr Included nnglr; It - wicllll of ctlnnrlcl
-1.0 -0.6 -0.2 0 0.2 0.6 LO
L6
Fig. 22.2. Velocity distribulion in a divergo11
chnnnel of ldf includotl angle n = 6" and
a = Go, as measured by .J. Nilruradse [71].
The lnck of Qmmetry in the velocity distribution
signifies incipient separation
Fig. 22.3. Volocity distribution in n diaergent
cl\aiincl of hnll inclrtdect analc n =: X",
rnctwr~red by .I. Nikr~ratlnc [71]. Itcvcrsc flow
is coniplebly dcvclopcd. Tlicr flow oncillntcs
nt hgt?r iiikrvah het.wm\ pnthrll~ (a) twd
(b)