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Elements of entrainment
Robert E. Breidenthal
University of Washington
Seattle, WA 98195-2400 USA
Abstract
A unified model is proposed for the fundamental behavior of turbulent entrainment over a
broad class of flows. The entrainment velocity is expressed as the ratio of a relevant length
scale to a time scale for all flows, a generalization of the original entrainment hypothesis of
Morton, Taylor, & Turner. This generalization appears to bring the theoretical effects of
acceleration, compressibility, confinement, rotation, stationarity, and stratification in accord
with observation.
Introduction
Turbulence has been called the most important unsolved problem in all of classical physics.
From astrophysics to oceanography, aeronautics to combustion, turbulence is ubiquitous.
Yet in spite of its central role in science and engineering, turbulence has defied solution for
over a century.
The most important property of turbulence is entrainment. Both transport and mixing in
turbulent flows are controlled by entrainment. Boundary layer heat transfer and skin friction
are the transport of energy and momentum at a wall. The vertical transport of water and
energy in the atmosphere and ocean are determined by stratified entrainment. In high
Reynolds number flow, the mixing is entrainment-limited, so much so that the molecular
diffusivity can change by three orders of magnitude while the molecular mixing rate changes
by only a factor of two. Entrainment determines most of what we really want to know about
a turbulent flow.
Entrainment hypothesis
Half a century ago, Morton, Taylor, & Turner (1956) proposed the most successful
hypothesis for entrainment. In order to model a thermal rising from the sudden release of
buoyant fluid, they argued on dimensional grounds that the local entrainment velocity v
e
into
the thermal at any station must be proportional to the rise speed W of the thermal at that
station. There is simply no other speed available on which to base the entrainment velocity
(see figure 1). In this way, the thermal grows linearly with height, in accord with
observation. Furthermore, their hypothesis is equally valid for a wide variety of other
classical flows that might be termed “ordinary turbulence,” correctly accounting for the
entrainment rate in the plume, shear layer, jet, wake, etc.
However, the entrainment hypothesis sometimes fails. For example, when the speed of
sound becomes comparable to the velocity jump across a shear layer, the entrainment rate
precipitously declines by a factor of five (Papamoschou & Roshko 1989). This cannot be
explained by the original entrainment hypothesis. The entrainment rate is also strongly
affected when acceleration, confinement, rotation, or stratification become appreciable. This
paper is an attempt to extend the entrainment hypothesis into a more general theory.
Entrainment process
Entrainment was thought to be a small-scale nibbling process at the edge of a turbulent
region. Corrsin & Kistler (1955) proposed a “superlayer” there, across which fluid was
thought to be entrained by small-scale nibbling. Shadowgraph images of the supersonic
round wake of a projectile seemed to support this notion. However, shadowgraph images of
the plane shear layer revealed the engulfment of large tongues of fluid by the largest vortices
in the flow (Brown & Roshko 1974, Roshko 1976). The two-dimensional geometry of their
shear layer allowed a more clear view of the entrainment process. Instead of polite little
nibbles, their images revealed that the turbulence really entrains like a hungry teenager taking
big gulps of fluid. These large engulfed tongues of pure, unmixed fluid are transported by
the large-scale vortices entirely across the layer (Konrad 1976).
Entrainment rate
The entrainment rate v
e
is a velocity. From dimensional considerations, it must therefore
always be expressible as the ratio of a relevant length scale to the rotational period!
"
of the
eddy responsible for entrainment. If there is engulfment, then the relevant length scale must
be the size !
of the entraining eddy.
v
e
= const.
!
Of course, the dimensional argument cannot establish the value of the constant of
proportionality. If there is no engulfment, such as at a solid wall or at a strongly stratified
interface, the length scale must be a diffusive one, the square root of the product of the
diffusivity and an eddy time.
For ordinary, incompressible, free shear flows, the entrainment rate must be proportional to
the ratio of the size of the largest eddies to their rotation period. This is a direct consequence
of Roshko’s engulfment, whereby the first step of engulfment by the largest eddies is rate-
limiting. The subsequent processing of the engulfed fluid by all smaller eddies is both
proportional to and sufficiently fast compared to the largest eddies that only the largest
eddies matter. Since the largest eddies control the rate, we do not need to know much about
anything else. This happy circumstance vastly simplifies matters, such as modeling the
mixing (Broadwell & Breidenthal 1982).
So for such flows equation (1) becomes
The vortices are temporarily self-similar if their next rotation period is proportional to their
last one. Otherwise there would be a special, distinguished time scale, a contradiction of
self-similarity. Figure 2 illustrates the evolution of the rotation period of temporally self-
similar turbulence. The line must be straight and !
must be a constant. For all ordinary,
unforced turbulence,! < 0 and slopes upward.
exponential jet
The line is horizontal if the next rotation period is the same as the last ( !=0). This can
achieved in an exponential jet, where fluid is ejected from a nozzle with a speed V
J
( t )that
increases exponentially in time,
V
J
( t ) = V
J 0
e
t
!
e
where V
J 0
is the nozzle speed at t = 0. Because of this forcing, every large-scale vortex in
the exponential jet rotates with the same period, equal to the e-folding time!
e
imposed on
the flow, no matter how old or how far from the nozzle. The vortices never age. It is a kind
of perpetual youth.
Remarkably, acceleration reduces the normalized entrainment rate. A convenient way to
measure entrainment at large Reynolds number is with a fast chemical reaction that destroys
a visible chemical in the nozzle fluid when mixed with the ambient fluid. If the mixing is
entrainment-limited, changes in the visible “flame length” reflect changes in the normalized
entrainment rate. Compared to the ordinary jet, the exponential jet has about a 20% greater
flame length (Kato et al. 1987). In fact, such acceleration is the only known method for
affecting the far-field entrainment rate of the incompressible jet, as noted by Zhang & Johari
(1996). Their detailed images of jets with modulated nozzle speed demonstrate that
acceleration only influences the entrainment rate when the imposed change in velocity during
one vortex rotation is comparable to the initial velocity. In other words, the logarithmic
derivative must be appreciable.
super-exponential forcing
The third category is the line sloping downward in figure 2 ( !>0). In spite of getting older,
the vortices spin ever faster. After a finite time, the spin rate becomes infinite and the
rotation period vanishes.
One might anticipate that the entrainment rate would be further reduced as !
increases.
Using dimensional and heuristic arguments, one theory has been proposed (Breidenthal 2003
with different notation). The dimensions of the dissipation rate per unit mass are
(length)
2
(time)
- 3 . Every canonical turbulent flow has a conserved quantity Q. For example,
in the shear layer, it is the velocity difference! U. If the dimensions of Q are in general
(length)
m
(time)
, the dissipation rate is proportional to Q
2
m
v
" 3 "
2 n
m
$
%
&
'
(
, where the vortex period
is!
v
For super-exponential forcing,
Q = Q
0
e
t
!
0
" # t
where Q
0
is the value of Q at t = 0.
Define D to the dissipation rate normalized by that of the unforced flow. From heuristic
grounds, we conjecture that the quantity is the natural scaling of effect of! on D. If so,
then
2 n
m
dD
D
d!
2 n
m
D = e
!
"! "
$
where!
"
is the value of! for the unforced flow.
Compressibility
It has long been known that a compressible flow grows more slowly than an incompressible
one. Papamoschou & Roshko (1989) found that the spreading angle of a turbulent shear
layer dropped by a factor of about five as the Mach number increased. Linear stability theory
may provide an indication of the entrainment behavior, since the underlying instabilities
drive the basic flow. However, the indication can only be qualitative, in as much as the finite
amplitude eddies are fully nonlinear.
Bogdanoff (1983) recognized that the important parameter for the instability is a
“convective” Mach number, the Mach number of the outer flow with respect to the speed of
the instability waves. A hint that this is the correct approach comes from the flow models of
Brown (1974), Coles (1981), and Dimotakis (1986), discussed below.
One heuristic model that addresses the fully nonlinear flow supposes that nonsteadiness is
essential to entrainment. This is a hint of this in the results of the Oster-Wygnanski (1982)
experiment, where the vortices in a shear layer are forced to be equally spaced. For a certain
time, these vortices are steady, resembling Kelvin’s cat’s eye pattern (Kelvin 1880), with no
term entrainment here to include the entire physics of transport and molecular mixing in a
confined vessel.
Consider the probability density function (pdf) for the concentration of an inert scalar mixing
with a second fluid in some general flow sketched in figure 3. Initially the pdf consists of
two delta functions at the extrema, corresponding to the two pure fluids. As the turbulence
mixes some of the two pure fluids together at intermediate concentrations, forming a central
Gaussian in the pdf. For a self-similar free shear layer with two infinite supplies of pure
fluid, the pdf would reach a steady state (Konrad 1976, Broadwell & Breidenthal 1982).
However, if only one fluid supply is infinite, such a finite jet injected into an infinite
reservoir, then eventually there is only one delta function in the pdf. If both fluid supplies
are finite, then the two delta functions both disappear, and the pdf consists of a central
Gaussian, the width of which is the rms concentration fluctuation. As the turbulence further
mixes the fluid, the Gaussian progressively narrows and the fluctuations decline.
Here the simplest two assumptions are that both the flow and the mixing are self-similar
(Breidenthal et al. 1990). The former requires that the vortex rotation period is proportional
to its age, as we have seen above. The latter implies that the concentration fluctuations
decline by a factor of e at each rotation. The simple result is that the concentration
fluctuations should be proportional to a characteristic time scale! divided by time.
c
c
= const.
t
The characteristic time scale is determined by dimensional considerations of the problem.
For example, if one fluid is initially in a spherical chamber and a second fluid is momentarily
injected into the chamber,! depends on the jet impulse and the chamber diameter. The
characteristic time! must also equal the vortex rotation period at the moment t =! when all
pure fluid has disappeared and the large-scale vortices have filled the chamber.
Measurements of concentration fluctuations are consistent with (6), in spite of the fact that
the actual vorticity field appears to decay exponentially instead of as inverse time (Aarnio
Density ratio
The coherence of large-scale structure in turbulence was discovered by accident. Brown &
Roshko (1974) were attempting to find out about the compressibility effects on entrainment.
It was known that supersonic jets exhibited an anomalously low spreading angle. It was not
clear if this was due to Mach number or to the density ratio of the supersonic experiments.
Since density ratio was easier to control, they elected to measure its effect on spreading angle
in incompressible flow by taking shadowgraph pictures. While the most important result of
their experiment was the coherent structure revealed by their pictures, they also determined
that density ratio has a remarkably weak effect on entrainment rate. The density ratio must
vary by a factor of 49 to achieve a factor of two change in spreading angle. This proved that
the main influence on jet spreading angle was Mach number.
A simple picture readily accounts for the effect of density ratio on entrainment into a shear
layer. Coles (1981) drew the shear layer in the Lagrangian frame of the vortices (see figure
4). Fluid enters a vortex from each stream due to the relative speed of the stream with
respect to the vortex. Brown (1974) showed that the relative speed ratio comes from
consideration of the stagnation streamlines. Assuming quasi-steady inviscid flow, the total
pressure on both streamlines must be constant and equal. Furthermore, the streamlines far
from the stagnation point are quasi-parallel, so that their static pressures must be equal. The
result is the dynamic pressures of the relative flows far from the stagnation point are equal.
So the speed ratio in this frame is just the inverse square root of the density ratio. Dimotakis
(1986) neatly summarizes the effects of both density and velocity ratio on both the spreading
angle and the entrainment ratio from the two sides of the layer.
Rotation
Bradshaw (1969) noted that when a fluid rotates, the higher speed fluid tends to want to
move to the outside of the turn. This corresponds to a state of lower kinetic energy for the
same angular momentum. The difference in kinetic energy between the two states can
dissipated into thermal energy in accord with the second law. On the other hand, if the
higher speed fluid is already on the outside of the turn, a rotating flow acts as if it is
stratified. This occurs even when the fluid has uniform density. This effective stratification
inhibits entrainment.
Cotel (2002) used Bradshaw’s analogy to explain the remarkable behavior of aircraft trailing
vortices. Even many kilometers behind a large aircraft, the wingtip vortices are compact,
laminar cores of only about a meter in diameter, in spite of the large Reynolds number. The
radial transport of momentum is strongly inhibited by the effective stratification due to the
rotation.
Stationarity
When a vortex is near a surface, the motion of the vortex with respect to the surface becomes
important. The entrainment rate across the surface depends on the amount of stationarity of
the vortex. Even a small amount of vortex movement completely changes the physics.
Cotel & Breidenthal (1997 & 1999) first identified this effect at a stratified interface. The
entrainment rate across a stratified interface was much different for an impinging vertical jet
compared to other turbulent flows, such as from an oscillating grid. The impinging vertical
jet entrained fluid across the interface with stationary, lateral vortices, in contrast to the
moving vortices from an oscillating grid or horizontal jet.
In order to quantify the stationarity, Cotel defined a new parameter. The persistence
parameter T is essentially the ratio of the rotational to the translational speed of the vortex
with respect to the surface (figure 5). When T is much less than one, the flow is in the
nonpersistent limit. When T is much greater than one, the flow is said to be persistent. For a
vortex near a surface, there is no more important parameter than this.
Richardson, Reynolds, Schmidt, Prandtl, and persistence parameters. For simplicity, we will
only consider the limit of a thin stratified interface.
The Richardson number Ri (of the largest eddies) is defined as the ratio of the potential to the
kinetic energy of the largest eddies at the stratified interface. One can also define the eddy
Richardson number Ri
!
of a smaller eddy of size !. For a Kolmogorov spectrum, the eddy
Richardson number increases with eddy size.
If Ri << 1, the potential energy is dominated by the kinetic energy and stratification is not
important for any eddy. If Ri > 1, there are at least two possibilities. Depending on the
Reynolds number, the smallest eddies at the Kolmogorov microscale!
0
may have an eddy
Richardson number Ri
! 0
greater than one. If so, then they and therefore all eddies have
insufficient kinetic energy to engulf a tongue of fluid across the interface. Consequently, in
this limit of strong stratification the interface must be essentially flat. All fluxes are purely
diffusive. From dimensional considerations, we can define a corresponding effective
entrainment velocity to be the square root of the ratio of the diffusivity divided by some eddy
rotation period. The diffusivity corresponds to the flux in question, i.e. mass, momentum, or
energy.
There are many choices for the eddy rotation period, ranging from that of the largest to the
smallest eddy. Clearly, eddies in the middle cannot be rate limiting, since there is no basis to
select one over another. So only the largest or the smallest eddy could be correct. Cotel
proposed that in the persistent limit, the correct choice is that of the largest eddy.
Remarkably, the fluxes would then be completely independent of any fine-scale turbulence.
While this prediction may not yet have been tested in stratified flow, it does seem to work in
the corresponding wall flow discussed above. The heat flux is laminar because the persistent
vortices make the flow laminar.
In the non-persistent limit, the fluxes would be controlled by the smallest-scale eddies,
corresponding to ordinary turbulent flow. This is in accord with many observations at
stratified interfaces and the boundary layer.
If the smallest scale vortices have an eddy Richardson number less than unity, then the
interface is not flat. The eddy whose Richardson number is equal to about unity can engulf
fluid across the interface. It determines the entrainment rate.
Dramatic evidence of the importance of persistence on stratified entrainment was measured
by Cotel et al. (1997). Following a suggestion by L. Redekopp (private communication
1995), they tilted an impinging jet and precessed it. The entrainment rate was reduced by
orders of magnitude compared to that of the vertical jet. The effect is not only large, but
counter-intuitive.
Conclusions
The entrainment rate of a turbulent flow can always be expressed as the ratio of a length to a
time scale corresponding to the entraining eddy. This is a generalization of the entrainment
hypothesis of Morton, Taylor, & Turner that accounts for a variety of effects, such as
acceleration, compressibility, confinement, stratification, and stationarity.
Acknowledgements
The author would like to acknowledge the contributions of his instructors, colleagues, and
students.
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Figure captions
Figure 1. Entrainment velocity v
e
is proportional to the thermal rise speed W according to
the entrainment hypothesis (Morton et al .)
Figure 2. Temporal evolution of the vortex rotation period for self-similar flow
Figure 3. Probability density function of the concentration field of a passive scalar is
composed of contributions from the pure fluid, Taylor layers, and the vortex
cores (Broadwell)
Figure 4. Sketch of the flow in the shear layer for an observer moving with the vortices
(Brown, Coles, and Dimotakis)
Figure 5. The intrinsic velocity ratio of a vortex near a surface – vortex persistence T
=U
2
/U
1
(Cotel)
Figure 6. Cat’s eye flow (Kelvin)
Figure 7. Stratified entrainment diagram in the persistent limit (Cotel)
Figure 8. Stratified entrainment diagram in the nonpersistent limit (Cotel)
Figure 3
Figure 4
c
Figure 5
Figure 6
U
U
Figure 8
Re
Ri
1 Ri
Sc
Ri
Re=Ri
1
Sc
Re
Sc
Re
1 Sc
