Page 292 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Mixing 247

on infinitesimal particles, as described by the Navier–
Stokes equation. The continuous advection of vortex
tubes past any given point with changing orientation is
a plausible explanation for the Eulerian (i.e., fixed
coordinates) observation of turbulence as illustrated
in Figure 10.5. Figure 10.6 illustrates the alternate
‘‘Lagrangian’’ view, i.e., a relative coordinate system
in which the observer moves with the motion of a given
parcel.

10.3.1.2.2 Turbulence Generation
As stated, turbulence is associated with velocity gradients
(i.e., fluid shear). To generate turbulence for mixing, bound-
FIGURE 10.6 Space-fillingvortextubes.(AdaptedfromHanson,A.T. ary discontinuities give the highest velocity gradients, i.e.,
and Cleasby, J.L., J. Am. Water Works Assoc., 82(11), 56–73, 1990.) high shear. Form separation and submerged jets are two
approaches for causing such high-shear zones (Rushton and
Oldshue, 1953b, p. 268) and are illustrated in Figure 10.7a
Meroney, 2001; Bienkiewicz, 2001). Figure 10.6 is a and b, respectively. In each case, the discontinuity causes
depiction of such an array of vortex tubes having a eddies; the shear (velocity gradient) between a given eddy
spectrum of diameters, lengths, orientations, and and the surrounding fluid results in an ‘‘eddy cascade.’’
energy levels. The tubes also interfere with one The turbulence generated by the separation seen in (a) is
another, causing a given tube to combine with another termed ‘‘wake turbulence’’ and causes ‘‘wake mixing’’
and perhaps breaking (Bienkiewicz, 2001). In addition, (Stenquist and Kaufman, 1972, p. 7). In (b) there is no formal
a given vortex tube sheds smaller daughter vortices, designation but the term, ‘‘jet mixing,’’ is descriptive. These
losing some of its energy. Eventually, the vortices two themes, i.e., wake mixing and jet mixing, permeate
become small enough that viscous forces predominate virtually all mixing technologies.
and the kinetic energy of the vortex dissipates as heat
that occurs in the ‘‘viscous sub-range’’ of the ‘‘energy . Wake turbulence: Wake turbulence is created by
spectrum’’ (Section 10.3.1.2.3). The vortex tangle is devices such as a flat-blade radial-flow impeller,
thus a part of a dynamic system and has a deterministic an orifice plate in a pipeline, a sudden pipe expan-
character in that it is due to the forces and acceleration sion, etc. Figure 10.8 illustrates the process of eddy

(a)

Core of jet

Nozzle

(b)

FIGURE 10.7 Types of boundary discontinuities. (a) Flat plate causing wake turbulence. (b) Submerged jet generating eddies.