Boundary Lyer Theory

244 XI. Axially symmrtrical nnd tl~rcc-dirnertsionnl boundary lnyers
Fig. 11.10. Ihg coefficirntr,
oi n rotnt.ing spl~crr in axinl
flow in trww of tl~c ltcJrnolds
number R and rotntion
Imramrtrr .Q - ioR/lI,
h t.l~corct.iwl cxpl;~nntiou of t,hc vrry cwnplcx thrrc-dimrnsionnl cll'ccb in the boundary
Iayrrof rotating I~odics of rcvolut.ion in axid flow is contained in the papers by H. Schlichting [00],
IC. 'I'rnckrnhrotll~ [I181 antl 0. I'arr 1841; thcse authors onployed the approximato method
!:xplainod earlior. It is t.rne that the boundary layer of a rotating body of revolution
In axial flow still rctains it^ axial syn~mctry, hnt owing to the rotation there appears a peripheral
vc1ocit.y cotnponent in addition to that in the mcridional direction. For this reason, the calculation
for such a I)o~lntlsr,y layor must int,roduce a ~norncntom eqnat.ion in the circumferential direction
(11-direr:t,ion) in atlclit.ion Lo that in tho n~crirlional direction (x-direction). Assuming that the
a~~gulnr vclocit,y of t.11~ I~ody is io, antl ilcnoting t,he coordinate at right angles to the wall by y,
wr ran writ.(: 1.11~ 1.~0 erluat.ions of n~otn~nl~un~ in tho form
r.
I hr component,^ or the shearing stress at thc wall are then given by
~ig. 11.11. Position of line of laminar
separation on a spl~ero rotating in axinl
stream, after N. IF. Hosltin [SO]
c. Iblation hel:ween axially ~y~ntnetrical and t\\o-cli~~~c~~aio~~;rl I,onntl;rry I;ryr~s 245
and tho displacement and momentum thicknesses arc defined as
m m
In the procctling equations, the local pcriphernl velocity w, - r u) hm been cl~onen ,w n rofcmnrx:
veloc~ity for the a7.imutal con~poncnt, w,(x, 2). 'I'ho preceding equations ~nnke it possi1)lo Lo pcrforrn
cnlculntions for Inminar as well as for turbulent flows, it being necessary to introduce difircnt
expression8 for the shearing stress at the wall in the latter cme (see ref. [R4] and Sec. XXllc).
In some of the cases, it proved possible to evaluate the drag coefficient in addiLion to t,he t,nrning
tnonwnt,, the former decreasing as the parameter mR/Um is increased. In this connexion, the papers
I)y C. It. Illir~gwortl~ [54] and S. T. Clru and A. N. TilTbrcl [13] may nlso hc stdied. The approxilnate
procedure conceived by H. Schlichting [98] was extenrlcd to compressible flows by .I. Y;rnlnga
[125]. The preceding investigntions have bcen extcnded for laminar as well ns for t.nrl)ulent.
tlows by theoretical and experin~ental investigation^ described in ucveral papers by ,Japanese
authors [29n, 10, 01, 79, 801.
l'rohlcn~s connected with laminar flow nbout a uphere rotating in a flnid at, resL IIGVO IICCII
discussed by I.. Ilowarth [51] and S. I). Nigam [All. An extension to the case involving ellipsoids
of revolut.ion wns provided by B. S. Fadnis p6]. Near tho poles, the flow is the same as
on a rotating disk and near the equator it is like the one on a rotatin cylinder. The aecornpanyi~~g
secondnry st re an^ causes fluid particles to flow into tho boundary yaycr near the poles, nntl out.
of it at the equator. The rate of this secondary flow increases with increasing slenderness, the
cquabrial area and peed of rotation remaining constant. However, the phenomena in the
plnne of tho equator where the two boundary layers impinge on each other and are thrown
outwards can no longer be analyzed with the aid of boundary-lnyer theory, el. W. 11. If. Banks [5a].
Further theoretical and experimental investigations of t.his problem have been later under-
taken by 0. Sawatzki [94] and by P. Dumsrgue et al. [21a]. Reference [94] describes n~edsure-
rnenls d the torque exerted on a rotating sphere in the rango of Ibynolds number 2.105 < R <
1.5 x 106 which goes far beyond the laminar regime. Tho invwtigntion of Ref. [21 a] included
the vi~unlizntion of the spiral strenmlines near the wnll on n sphere nnd on cones of various in-
cluded angles as they occur'in laminar flow.
It has been observed that in axial turbomachines there may, under certain circumstances,
appear an extended zone of dead fluid in tho whirl behind the row of stationary blades antl
ncnr the hub. This phenomenon was described in great detnil by K. J3antmert and H. Klaeukens
[5]. The origin of this dead-water area is conneckd wiLh the radial increase in prcssurr in Iho
ontwnrtl direction which i~ due to the whirl. Owing to tho whirl the axinl pressure inerrme nrnr
lhe huh in the bladelorn annulus behind the guides is much greater than at the outer wall. The
influence of tho houndory layer is here only ciecontlnry. ALLonLion rn!ly, further, ho drawn Lo
an invesbigotion duo Lo I