Boundary Lyer Theory

048 XXI. Turbulent boundary layers at zero preaeure gradient
boundary-layer thickness 6 is proportional to 1/72;, and hence, independent of
the radius, and that the torque, M, which is proportional to p R3 U/S, must be
U2 n3(U R/v)-11" The exact solution for
the laminar casc showed, further, that t,he dikensionless torque coefficient, dcfined
as
given by an expression of the form M - Q
for a disk wetted on both sidcs, is given by cqn. (5.56), and is equal to
C, = 3.87 R-"' (laminar) , (21.28)
where R = R2rn)/v is thc Roynolds numhcr, Fig. 5.14.
It is now proposcd to make the same estimation for the turbulent casc basing it
on the same resistance formula for turbulent flow as was used in the case of the flat
plate, i. e., in the simplest case, on the +-th-power law for the velocity distribution.
A fluid particle which rotates in tho boundary layer at a distance r from the axis is
acted on by a centrifngal forco per unit volume of magnitude e r w2. The centrifugal
force on a volume of area dr x ds and height S becomes e r w2 dr x ds. The shearing
stress to forms an anglc 0 with the tangential direction and ita radial component
must balance the centrifugal force. IIcnce we have to sin 0 dr xds =erwV~Jr xds or
On the other hand, the tangential component of shearing stress ran be expressed
with the aid of eqn. (21.5) which was used in the case of a flat plate, replacing U,
by the Lmgential velocity r o. Thus
to cos 0 - e (or)"' (v/S)ll' .
Equating T, in thcsc two cxprcssions, we find that
8 - ral' (v/Up6 .
It is sccn that in the turhnlcnt casc thc bonntlary-lnyer thickness increases outwards
in proportion to r3/bnd docs not rcmain constant as in the laminar case. Further,
the torque becomes M - to R3 N e R W~(V/CO)~/~ RRI6 B3 so that
Th. von ICiirrnhn [30] investigated the tnrbulcnt boundary layer on a rotating disk
with the aid of an approximate method based on the momentum equation and
similar to the one applied in the preceding section ijb the study of the flat plate. The
variation of the tangential velocity component through the boundary layer was as-
surnctl t40 obcy the 4-th-power law. The viscous torque for a disk wetted on both
sidcs'wa.. shown to be equal to
b. Tl~c rot~ting disk G49
and tthc torquc coefficient dcfined in cqn. (21.27) bccomcs
C, = 0.146 R- turbulent) . (21.00)
This equation has been plotted in Fig. 6.14 as curvc (2). It shows very good agrccmcnt
with the experimental rcsults cluc to W. Schmidt and G. ICctnpft for
R > 0 x 10" Tho numerical factor in the cquntion for ((he 1)ound:~ry-Iayrr t hi&n~ss
which was left unclctcrrninctl bccomcs
3 = 0.520 r (v/r2 o)' 15, (2 I .3 I )
and the volume of flow in the axial direction is given by
as comparcd with cqn. (5.57) for laminar flow.
An approximate calculation based on the logarithmic vclocity-dist~ribr~l,iot~ Inw
u/v* = A, In(?/ v,/v) -1- Dl was performed by S. Coldsteiri [21], who found i.hn following
formula for the torque :
1
= 1.97 log (R 1/c) + 0.03 (turl)ulcnt)
7"M
It is n~tewort~hy thnt this equation has tho same form as the ~~niversnl pipe-rraistancc
forrn~rln, cqn. (20.00). Tho nnmcricnl fac1,or~ have bccn 'atlj~rstctl lo obLl~i11 I,II(- IwsI~
possible agreement with experimental rcsults. This equation is sccn plotted as cnrvc
(3) in Pig. 5.14. On this topic see also P. S. Granville [22].
2. The disk in a housing. The dislc in turbines or rotwy compressors 111os1Iy
revolve in very tight housings in which the width of thc gap, a, is small compared
with the radius, R, oC the disk, Pig. 21.7. Consequcntly, it was found necessary to
investigate the case of a disk rotating in a housing.
Laminar flow. The relations become particularly simple when thc flow is laminar,
R < lo5, and when the gap is very small. If the gap, s, is smaller than the boundarylayer
thickness the variation of the tangential velocity across the gap becomes
linear in thc mmncr of Co~~ctto-flow. lFcncc, tho shcaring strcss at a distancc r from
the axis is equal to T = r(up/s and thc torquc of the viscous Corccs on onc sitlc of
a disk is given by
n
Consequently for both sides we have
J
2M
=n w R4,+ ,
and the torque coefficient from eqn. (21.27) becomes
t Soc refe. [10] and [31] in Chap. V.