Page 107 - Bird R.B. Transport phenomena

92 Chapter 3 The Equations of Change for Isothermal Systems

30 Fig. 3.6-2. Critical Reynolds number for the tangen-
tial flow in an annulus, with the outer cylinder rotat-
20 ing and the inner cylinder stationary [H. Schlichting,
Boundary Layer Theory, McGraw-Hill, New York
о
X \ (1955), p. 357].
10
\ /
7
\ /
cf

5 10 20 50
( 1 - K ) X 10 2

Equation 3.6-32 describes the flow accurately for small values of ft,. However, when
3/2
2
ft, reaches a critical value (ft/ /Crit ~ 41.3(/г/Я (1 - к) р) for к ~ 1) the fluid develops a
secondary flow, which is superimposed on the primary (tangential) flow and which is
periodic in the axial direction. A very neat system of toroidal vortices, called Taylor vor-
tices, is formed, as depicted in Figs. 3.6-3 and 3.6-4(b). The loci of the centers of these vor-
tices are circles, whose centers are located on the common axis of the cylinders. This is
still laminar motion—but certainly inconsistent with the postulates made at the begin-
ning of the problem. When the angular velocity ft, is increased further, the loci of the
centers of the vortices become traveling waves; that is, the flow becomes, in addition, pe-
riodic in the tangential direction [see Fig. 3.6-4(c)]. Furthermore, the angular velocity of
the traveling waves is approximately |ft/. When the angular velocity ft, is further in-
creased, the flow becomes turbulent. Figure 3.6-5 shows the various flow regimes, with
the inner and outer cylinders both rotating, determined for a specific apparatus and a

Inner cylinder
"" rotating *"*"•

(6)

(Q) (6)
(6)

Outer cylinder fixed ~ (a)

Fig. 3.6-3. Counter-rotating toroidal vor- Fig. 3.6-4. Sketches showing the phe-
tices, called Taylor vortices, observed in the nomena observed in the annular space
annular space between two cylinders. The between two cylinders: (a) purely tan-
streamlines have the form of helices, with gential flow; (b) singly periodic flow
the axes wrapped around the common (Taylor vortices); and (c) doubly periodic
axis of the cylinders. This corresponds to flow in which an undulatory motion is
Fig. 3.5-4(W. superposed on the Taylor vortices.