New Perspectives in Turbulence.
Intermittency, a basic property of fully developed turbulent flow, decreases with growing viscosity; therefore classical relationships obtained in the limit of vanishing viscosity must be corrected when the Reynolds number is finite but large. These corrections are the main subject of the present paper. They lead to a new scaling law for wall-bounded turbulence, which is of key importance in engineering, and to a reinterpretation of the Kolmogorov--Obukhov scaling for the local structure of turbulence, which has been of paramount interest in both theory and applications. The background of these results is reviewed, in similarity methods, in the statistical theory of vortex motion, and in intermediate asymptotics, and relevant experimental data are summarized.
Introduction
In February 1996 I had the privilege of meeting Prof. G.I. Barenblatt, who had just
arrived in Berkeley. In our first extended conversation we discovered that we had been
working on similar problems with different but complementary tools, which, when wielded
in unison, led to unexpected results. We have been working together ever since, and it is
a pleasure to be able to present some of the results of our joint work at this distinguished
occasion.
The present talk will consist of three parts: (i) An application of advanced similarity
methods and vanishing-viscosity asymptotics to the analysis of wall-bounded turbulence,
(ii) a discussion of the local structure of turbulence with particular attention to the higher-order
structure functions, and (iii) a discussion of a near-equilibrium statistical theory of
1 Supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research,
U.S. Department of Energy, under contract DE-AC03-76-SF00098, and in part by the National Science
Foundation under grants DMS94-14631 and DMS89-19074.
turbulence, which motivates and complements our reading of the numerical and experimental
data. The basic premise is that, as the viscosity tends to zero and the solutions of
the Navier-Stokes equations acquire poorly understood temporal and spatial fluctuations,
certain mean properties of the of the flow can be seen to take on well-defined limits, which
can be found by expansion in a small parameter that tends to zero, albeit slowly, as the
viscosity tends to zero.
In the case of wall-bounded turbulence, our argument and the data show that the
classical von K'arm'an-Prandtl law should be replaced, when the viscosity is small but
finite, by a Reynolds-number-dependent power law. In the case of local structure, an
analogous argument shows that the Kolmogorov scaling of the second and third order
structure functions is exact in the limit of vanishing viscosity, when the turbulence is most
intermittent and least organized. When the viscosity is non-zero (Reynolds number large
but finite), Reynolds-number-dependent corrections to the Kolmogorov-Obukhov scaling
of the structure functions appear, due to a viscosity-induced reduction in intermittency.
For higher-order structure functions the vanishing viscosity limit ceases to exist because
of intermittency, and the Kolmogorov-Obukhov scaling fails.
The near-equilibrium statistical theory we shall present is the basis of vanishing-viscosity
asymptotics and relates the the behavior of the higher-order structure functions to the
presence of intermittency. All parts of our analysis are heterodox in the context of the
current state of turbulence research, but not in the broader context of the statistical
mechanics of irreversible phenomena.
2. The intermediate region in wall-bounded turbulence
Consider wall-bounded turbulence, in particular fully developed turbulence in the working
section, i.e., far from the inlet and outlet, of a long cylindrical pipe with circular
cross-section. It is customary to represent u, the time-averaged or ensemble-averaged
longitudinal velocity in a pipe in the dimensionless form
where u   is the "friction" velocity that defines the velocity scale:
where ae is the density of the fluid and - is the shear stress at the pipe's wall,
d
Here \Deltap is the pressure drop over the working section of the pipe, L is the length of the
working section, and d is the pipe's diameter. The dimensionless distance from the pipe
wall is
where y is the actual distance from the wall and - is the fluid's kinematic viscosity. The
length scale -=u   in (2.4) is typically very small - less than tens of microns in some of the
data discussed below. The key dimensionless parameter in the problem is the Reynolds
number
ud
where - u is the velocity averaged over the cross-section. When the Reynolds number Re
is large, one observes that the cross-section is divided into three parts (Figure 1): (1)
the viscous sublayer near the wall, where the velocity gradient is so large that the shear
stress due to molecular viscosity is comparable to the turbulent shear stress, a cylinder
(2) surrounding the pipe's axis the where the velocity gradient is small and the average
velocity is close to its maximum, and the intermediate region (3) which occupies most of
the cross-section and on which we shall focus.
The velocity gradient @ y u, (@ y j @
@y ), in the intermediate region (3) of Figure 1 depends
on the following variables: the coordinate y, the shear stress at the wall - , the pipe diameter
d, the fluid's kinematic viscosity - and its density ae. We consider the velocity gradient
@ y u rather than u itself because the values of u depend on the flow in the viscous sublayer
where the assumptions we shall make are not valid. Thus
Dimensional analysis gives
y
ud
where \Phi is a dimensionless function, or, equivalently,
Outside the viscous sublayer j is large, of the order of several tens and more; in the kind
of turbulent flow we consider the Reynolds number Re is also large, at least 10 4 . If one
assumes that that for such large values of j and Re the function \Phi no longer varies with its
arguments and can be replaced by its limiting value \Phi(1; (this is an assumption
of "complete similarity" in both arguments, see [1]), then equation (2.8) yields
and then an integration yields the von K'arm'an- Prandtl "universal" logarithmic law
where - ("von K'arm'an's constant'') and B are assumed to be ``universal'', i.e. Re independent
constants. The assumption that B is Reynolds-number-independent is an additional
assumption. The resulting law is widely used to describe the mean velocity in the intermediate
region; the values of - in the literature range between :36 and :44 and the values
of B between 5 and 6.3- an uncomfortably wide spread if one believes in the "universality"
of (2.10).
However, there is no overwhelming reason to assume that the function \Phi has a constant
non-zero limit as its arguments tend to infinity, nor that the integration constant remains
bounded as Re tends to infinity. When either assumption fails other conclusions must
be reached. Rather than list alternative assumptions we present a model problem that
exhibits in a simple manner what goes wrong as well as the cure.
Consider the equation
dy
y
for y positive, where u is subject to the boundary condition
positive parameter. One can view ffi as a dimensionless viscosity, and thus ffi \Gamma1 is analogous
to a Reynolds number.
One could reason as follows: For
dy is approximately zero, and thus u is a
constant, which can only be the constant 1. We can derive the same result for small y
and ffi by an assumption of complete similarity: Equation (2.11) is homogeneous in the
dimensions of u and y, and thus one can view both of these variables as dimensionless. By
construction, ffi must be dimensionless. The dimensionless relation between these variables
takes the form
and if one makes an assumption of complete similarity, i.e., assumes that for ffi; y small \Phi
is constant, one finds again that u is the constant 1.
This conclusion is false. Equation (2.11) has the following solution that satisfies the
boundary condition
Note that for any positive value of ffi this solution is a power law and not a constant.
We can obtain this solution for small y and ffi by assuming that the the solution is a
power of the variable y while the form of its dependence on ffi is unknown; this leads to
. (This is an assumption of "incomplete similarity" in y and no similarity in
ffi). A substitution into equation (2.11) yields A(ffi); ff(ffi).
Further, consider the solution (2.13) and, for any non-zero value of y, its limit as
Clearly,
and thus, as e, i.e., the limit of (2.13) for y ? 0 is the constant e. As
deduced from the false assumption of complete similarity, the limit of u is a constant, but
it is not the obvious constant 1. Furthermore, for a finite value of ffi, however small, u is
not uniformly constant; it is not equal to e either for y ! ffi or for y large enough. The
approximate equality u - e holds, when ffi is small but finite, only in an intermediate range
constitutes an example of "intermediate asymptotics" [1].
Now consider subjecting \Phi in (2.8) to an analogous assumption, of incomplete similarity
in j and no similarity in Re when both arguments are large [2]:
Note a clear difference between complete and incomplete similarity. In the first case
the experimental data should cluster in the (ln j; OE) plane on the
single straight line of the logarithmic law; in the second case the experimental points would
occupy an area in this plane. Both similarity assumptions are very specific. The possibility
that \Phi has no non-zero limit yet cannot be represented asymptotically as a power of j has
not been excluded. Both assumptions must be subjected to careful scrutiny. In the absence
of reliable, high-Re numerical solutions of the Navier-Stokes equation and of an appropriate
rigorous theory, this scrutiny must be based on comparison with experimental data.
We now specify the conditions under which (2.15) may hold and narrow down the choices
of A(Re) and ff(Re) (see [4,5,6,7,9,10]). Fully developed turbulence is not a single, well-defined
state with properties independent of Re; there may be such a single state in the
limit of infinite Reynolds number, but experiment, even in the largest facilities, shows that
fully developed turbulence still exhibits a perceptible dependence on Re. Fully developed
turbulence has mean properties (for example, parameters such as A and ff in (2.15)) that
vary with Reynolds number like K 0 are constants and " is a small
parameter that tends to zero as Re tends to infinity, small enough so that its higher powers
are negligible, yet not so small that its effects are imperceptible in situations of practical
interest; the latter condition rules out choices such as We expect A(Re) and
ff(Re) in (2.15) to have the form
are universal constants; we have implicitly used a principle derived
from the statistical theory of section 4, according to which the average gradient of the
velocity profile has a well-defined limit as the viscosity - tends to zero [5,6,10]. This is the
the vanishing-viscosity principle. We expand A(Re); ff(Re) in powers of " and keep
the first two terms; the result is:
Substitution of (2.17) into (2.15) yields
For this expression to have a finite limit as - tends to zero one needs ff must
tend to zero as Re tends to infinity like ( 1
Re ) or faster. The assumption of incomplete
similarity, experiment, and the vanishing-viscosity principle show that the threshold value
Re is the proper choice. Use of this choice in equation (2.18) and an integration
yield
Re
where the additional condition motivated by the experimental data, has been
used.
According to this derivation, the coefficients C are universal constants, the same
in all experiments of sufficiently high quality performed in pipe flows at large Reynolds
numbers [2,3]. In [12] the proposed law for smooth walls (2.19) was compared with the
data of Nikuradze [26] from Prandtl's institute in G-ottingen. The comparison yielded the
coefficients ff
with an error of less than 1%. The
result is
We now wish to use the law (2.20) to understand what happens at larger Reynolds
numbers and for a broader range of values of j than were present in the experiments
reported by Nikuradze. If this extrapolation agrees with experiment, we can conclude that
the law has predictive powers and provides a faithful representation of the intermediate
region. We have already stated that the limit that must exist for descriptions of the mean
gradient in turbulent flow is the vanishing-viscosity limit, and thus one should be able
to extrapolate the law (2.20) to ever smaller viscosities -. This is different from simply
increasing the Reynolds number, as - affects j and - u as well as Re. The decrease in the
viscosity corresponds also to what is done in the experiments: If one stands at a fixed
distance from the wall, in a specific pipe with a given pressure gradient, one is not free to
vary
ud=- and independently because the viscosity - appears in both, and
if - is decreased by the experimenter, the two quantities will increase in a self-consistent
way and - u will vary as well. As one decreases the viscosity, one considers flows at ever
larger j at ever larger Re; the ratio 3
Re tends to 3=2 because - appears in the same way
in both numerator and denominator. Consider the combination 3 ln j=2 ln Re in the form
=\Theta
d
According to [3], at small -, i.e. large Re, -
u=u   - ln Re, so that the term ln(-u=u   ) in
the denominator of the right-hand side of (2.21) is asymptotically small, of the order of
Re, and can be neglected at large Re; because the viscosity - is small, the first term
in both the numerator and denominator of (2.21) is dominant, as long as the
ratio y=d remains bounded from below, for example by a predetermined fraction. Thus,
away from a neighborhood of the wall, the ratio 3 ln j=2 ln Re is close to 3/2 (y is obviously
bounded by d=2). Therefore
is a small parameter as long as y ? \Delta, where \Delta is an appropriate fraction of d. The
quantity exp(3 ln j=2 ln Re) is approximately equal to
exp
Re
'-
Re
'-
Re
\Gamma2
According to (2.20) we have also
exp
and the approximation (2.22) can also be used in (2.23). Thus in the range of y where
y ? \Delta, but y ! d=2, we find, up to terms that vanish as the viscosity tends to zero,
and
Equation (2.25) is the asymptotic slope condition: As - ! 0 the slope of the power
law tends to a finite limit, the limiting slope, which is the right hand side of (2.25). The
von K'arm'an-Prandtl law also subsumes an asymptotic slope condition, with a limiting
slope 1=-; the limiting slope in equation (2.25),
approximately
larger than a generally accepted value for - \Gamma1 . One can view equation (2.24) as an
asymptotic version of the classical logarithmic law, but with an additive term that diverges
as the Reynolds number tends to infinity, and of course a different slope.
The family of curves parametrized by Re has an envelope whose equation
tends to
with
close to the values of - found in the literature. The corresponding
value of 1
- is exactly
e times smaller than the value on the right-hand side of (2.25). It is
clear that the logarithmic law usually found in the literature corresponds to this envelope;
indeed, if one plots points that correspond to many values of Re on a single graph (as is
natural if one happens to believe the von K'arm'an-Prandtl law (2.10)), then one becomes
aware of the envelope. The visual impact of the envelope is magnified by the fact that the
small y part of the graph, where the envelope touches the individual curves, is stretched
out by the effect of - on the values of ln j. Also, measurements at very small values of
y where the difference between the power law and the envelope could also be noticeable
are missing because of experimental difficulties very near the wall. Thus, if our proposed
power law is valid, the conventional logarithmic law is an illusion which substitutes the
envelope of the family of curves for the curves themselves. The discrepancy of
e between
the slope of the curves and slope of the envelope is the signature of the power law, and
helps to decide whether the power law is valid. The situation is summarized in Figure 2,
which shows schematically the individual curves of the power law, their envelope, and the
asymptotic slope.
Historically, the understanding of the flow in the intermediate region of wall-bounded
turbulence has been influenced by the "overlap" argument of Izakson, Millikan and von
Mises (IMM) (see e.g [25]). This argument in its original form contains an implicit assumption
of complete similarity, and once freed from it yields yet again the asymptotic
slope condition (2.25). For details, see [5,7,10].
Detailed comparisons of the the power law and the von K'arm'an-Prandtl laws with
experimental data are available in refs. [9,10]. For completeness, we exhibit in Figure
3 a set of experimental curves from the Princeton superpipe experiment [34]. Note its
qualitative similarity to Figure 2. In particular, despite a flaw discussed in detail in [10],
these experiments do indeed exhibit a separate curve for each Reynolds number, and a
well-defined angle between the curves and their envelope.
The applicability of the our analysis of the intermediate region of pipe flow to the
velocity profile in a zero-pressure-gradient boundary layer is discussed in [11].
3. Local structure in turbulence
The analogy between the inertial range in the local structure of developed turbulence
and the intermediate range in turbulent shear flow near a wall has been noted long ago (see
e.g. [14,32], and it motivates the extension of the scaling analysis above to the case of local
structure, where the experimental data are much poorer. In the problem of local structure
the quantities of interest are the moments of the relative velocity field, in particular the
second order tensor with components
difference between x and x + r. In locally
isotropic incompressible flow all the components of this tensor are determined if one knows
is the velocity component along the vector r.
To derive an expression for DLL assume, following Kolmogorov, that for
it depends on h"i, the mean rate of energy dissipation per unit volume, r, the distance
between the points at which the velocity is measured, a length scale  , for example the
Taylor macroscale   T , , and the kinematic viscosity -:
where the function f should be the same for all developed turbulent flows. If r is large,
other variables may appear, as a consequence of external forces or of boundary conditions.
The most interesting and the most important argument in this list is the rate of energy
dissipation ".
Introduce the Kolmogorov scale   K , which marks the lower bound of the "inertial"
range of scales in which energy dissipation is negligible:
Clearly, the velocity scale appropriate to the inertial range is
and this yields a Reynolds number
The inertial range of scales is intermediate between the scales on which the fluid is stirred
and the scales where viscosity dissipates energy, and is the analog of the intermediate region
in wall-bounded flow. In this range the scaling law that corresponds to (2.15) is:
r
where as before, the function \Phi is a dimensionless function of its arguments, which have
been chosen so that under the circumstances of interest here they are both large.
If one now subjects (3.6) to an assumption of complete similarity in both its arguments,
one obtains the classical Kolmogorov 2=3 law [21]
from which the Kolmogorov-Obukhov "5/3" spectrum [27] can be obtained via Fourier
transform. If one makes the assumption of incomplete similarity in r=  K and no similarity
in Re, as in the case of wall-bounded flow, the result is
r
where C; ff are functions of Re only. As before, expand C and ff in powers of 1
Re
and
keep the two leading terms; this yields
Re
r
(ff 0 has been set equal to zero so that DLL have a finite limit as - ! 0).
In real measurements for finite but accessibly large Re, ff small in comparison
with 2/3, and the deviation in the power of r in (3.9) could be unnoticeable. On the other
hand, the variations in the "Kolmogorov constant" have been repeatedly noticed (see
[25,29,31]). Complete similarity is possible only if A 0 6= 0, when one has a well-defined
turbulent state with a 2/3 law in the limit of vanishing viscosity, and finite Re effects can
be obtained by expansion about that limiting state. In the limit of vanishing viscosity
there are no corrections to the "K-41" scaling if equation (3.9) holds; this conclusion was
reached in [15] by the statistical mechanics argument summarized in section 4 below.
Kolmogorov [21] proposed similarity relations also for the higher order structure functions

repeated p times; the scaling gives D Experiments
in a small wind-tunnel by Benzi et al. [13], show some self-similarity in these
higher-order functions, obviously incomplete, so that D LL:::L is proportional to r ip , with
exponents i p always smaller then p=3 for p - 3, so that i
instead of 1:67, i instead of 2:33, and i
of 2:67. It is tempting to try for an explanation of the same kind as for 2:
Re
in other words, to assume that at Re = 1 the classic "K41" theory is valid, but the experiments
are performed at Reynolds numbers too small to reveal the approach to complete
similarity. If this explanation were correct, the coefficients ff p would be negative starting
with 4, where there would be a reversal in the effect of the Kolmogorov scale (or
whatever scale is used to scale the first argument in \Phi).
As is well-known, for the Kolmogorov scaling is valid with no corrections. For
must proceed with caution. We would like to present a simple argument that
casts doubt on the good behavior of the structure functions for in the vanishing-
viscosity limit. As Re ! 1 the intense vorticity and the large velocities in the fluid
become concentrated in an ever smaller volume [15]. This is what we call "intermittency".
is the fraction of the volume of a unit mass of fluid where the kinetic energy - u 2
is large, then
; one can see that fourth moments such as hu 4 i diverge as
This casts a strong doubt on the good behavior of the fourth-order structure functions as
the viscosity tends to zero; in the absence of such good behavior our expansions in powers
of 1
Re
cannot be justified and the explanation of the experimental data must proceed
along different lines. Note that is the power where the sign of the power of r
in
an expansion in powers of 1
Re
would change.
The analysis just given of the second-order structure function contradicts the conclusions
of the Benzi et al. [13], according to whom the asymptotic exponent in (3.9) is independent
of Re and different from 2=3. We wish to point out however that, as we understand the
discussion in [13], the exponent was found to be different from 2=3 only once it was assumed
that it was not dependent on Re; to the contrary, even a cursory view of Fig. 3 in ref.
[13] shows a marked dependence on Re. We are looking forward to an opportunity to
reexamine these data in the light of our hypotheses.
There is a key difference between a derivation of the Kolmogorov-Obukhov exponent
from an assumption of complete similarity and its derivation as the vanishing-viscosity limit
of an expression derived from an assumption of incomplete similarity. Complete similarity
typically holds in statistical problems that are well-described by mean-field theories, while
incomplete similarity typically applies to problems where fluctuations are significant. This
remark is consistent with our conclusion, presented in [7,10], that the Kolmogorov scaling
already allows for intermittency, and that its application to higher-order structure functions
is limited by this very intermittency.
4. A near-equilibrium theory of turbulence
At large Reynolds numbers Re the solutions of the Navier-Stokes equations are chaotic,
and the slightest perturbation alters them greatly. The proper object of a theory of turbulence
is the study of ensembles of solutions, i.e.,of collections of solutions with probability
distributions that describe the frequency of their occurrence. We now outline a near-equilibrium
theory of ensembles of flows on those small scales where the scaling theory of
the previous section applies. This theory justifies the use of vanishing-viscosity asymptotics
for appropriate moments of the velocity field and of its derivatives and supports the
conclusions of the previous section regarding the behavior of the higher-order moments
and structure functions. It is equivalent to earlier near-equilibrium theories [15], but the
specific approach and the presentation are new; fuller detail can be found in [8].
We describe turbulence in terms of a suitable statistical equilibrium. In statistical
mechanics, statistical equilibrium is what one finds in an isolated system if one waits long
enough. One way of characterizing this equilibrium is by assuming that all states of the
system compatible with the system's given energy can occur with equal probabilities; this
is the "microcanonical ensemble". In turbulence the appropriate energy is the kinetic
energy of the flow. An equivalent characterization is in terms of the "canonical ensemble",
in which the probability of a state is proportional to exp(\GammafiH), where H is the energy of
the state and fi is a parameter. In the canonical ensemble the energy is not fixed, and one
can view the ensemble as describing a portion of an isolated system at equilibrium as it
interacts with the rest of the system. The two ensembles are equivalent in the sense that,
with a proper choice of the parameter fi and in a system with enough degrees of freedom,
averages calculated in either ensemble are close to each other.
The parameter fi is generally called the "inverse temperature" of the system. In many
physical systems indeed proportional to what one intuitively perceives to be
the temperature, as it can be gauged by touching a system with one's finger. However, the
parameter fi can be viewed more abstractly, as the parameter that makes the two ensembles
equivalent; in incompressible turbulence, in which there is no interaction between the
macroscopic flow and the microscopic motion of the molecules of the fluid, the fi that one
obtains cannot be gauged by the sense of touch. In a given system, fi is a function of the
energy E and of whatever other variables are needed to describe the system. Note than
in other realms of physics, for example in the kinetic theory of gases, one is well used to
relating temperature to a suitable kinetic energy.
Turbulence as a whole is generally not an equilibrium phenomenon: For example, if
one stirs a box full of fluid and then isolates the resulting flow, the outcome after a long
time is not turbulence in a statistical equilibrium but a state of rest; an isolated turbulent
flow is one without outside forces or an imposed shear to keep it flowing. However, on the
small scales in which we are interested, the relevant question is whether the motion has
enough time to settle to an approximation of a statistical equilibrium in which one can
assume that all the states with a given kinetic energy are equally likely to appear. The
small scales have enough time when their characteristic time (length/velocity) is short
enough compared to the characteristic time of the large-scale motion. An inspection of
the Kolmogorov scaling given in the previous section shows that the characteristic time of
an eddy of size r is proportional to r 2=3 , and small enough eddies (i.e., vortices) do have
enough time to settle down to an equilibrium distribution. The task at hand is to construct
the statistical equilibrium appropriate for turbulence, in particular specify its states. The
question of how then to perturb it so as to take into account the irreversible aspects of
turbulence has been treated elsewhere [16,17] and will not concern us here. Note that in
most of the turbulence literature one speaks of the small scales reaching "equilibrium"
when the energy distribution among them approximates the Kolmogorov-Obukhov form;
here, for the moment, we mean by "equilibrium" a statistical equilibrium, in which all
states have equal probability; we shall shortly claim that these two meanings are in fact
identical. This is of course possible only if at the statistical equilibrium there are more
states with much of the energy in the larger scales than states with much of the energy in
the smaller scales.
To agree with observation, a a hydrodynamical statistical equilibrium must have a finite
energy density in physical space. To construct an equilibrium with this property, we start
as in the construction employed in ref. [24] for vorticity fields; for simplicity we describe
it in two space dimensions. Consider a unit box
boundary conditions. Let be the velocity field and / a stream function; divide
the box into N 2 squares of side h, in each square define a value of a discrete stream
describe the location of the square; then define a discrete velocity
field u ij by one-sided difference quotients of /, so that that the velocity is divergence-free.
(In three dimensions there is one more index and the stream-function is replaced by a
vector potential). The parameter h is an artificial cut-off, and we now present a procedure
for letting this cut-off tend to zero while producing sensible fluid mechanics in the limit.
Replace the energy
R
square
dxdy by its discrete counterpart
. For
a fixed value of h, pick a value of E h , and, as a first step, assume that the values of u are
equidistributed among all states with use a microcanonical ensemble.
One can check that on the average, each one of the boxes has the same energy ju ij
One may think that if one lets h ! 0 while keeping E h constant, the limit is an ensemble
with a finite energy per unit volume; this is not so. The sequence of ensembles one obtains
as reasonable limit: As h ! 0 the number of degrees of freedom tends to
infinity, and there is no sensible way to divide a finite energy equally among an infinite
number of degrees of freedom; indeed, if the energy per degree of freedom is zero the
limiting ensemble has zero energy and no motion, and if the energy per degree of freedom
is positive the limiting ensemble has an infinite energy (for a more thorough mathematical
discussion, see [6,8]). One can also see that the limit of these microcanonical ensembles is
meaningless by considering the corresponding canonical ensembles: One can check that as
the parameter fi in the sequence of canonical ensembles tends to infinity; one can
show that the only ensembles with infinite fi have either no energy or an infinite energy.
To find a way out of this dilemma one must modify these ensembles as h ! 0 so as to
ensure that the limit exists. We do so by looking at what happens to the parameter fi and
keeping it bounded; furthermore, we do so on the computer. This is the key point: To
obtain a sensible continuum limit, we keep fi bounded by keeping the energy from becoming
equally distributed among the degrees of freedom, and this produces an average energy
distribution among scales that agrees with the Kolmogorov-Obukhov law and produces the
Kolmogorov scaling of the low-order structure functions. One can also show (see [8]) that
this very same procedure is needed to produce ensembles whose members, the individual
velocity fields, do not violate what is known about the solutions of the Navier-Stokes or
Euler equations.
To proceed, we have to be able to calculate fi given h and E h . Averages with respect to
microcanonical ensembles can be calculated numerically by an algorithm known as "mi-
crocanonical sampling" [18]: Introduce an additional variable, a "demon", which interacts
with all the degrees of freedom in some random order. In each interaction, the demon
either absorbs an energy packet of some predetermined magnitude s, s !! E h , or gives
away an energy packet of the same size. If the demon takes in an energy s, it reduces
the energy in the velocity field by modifying / ij so that the integral
R
dxdy over the
unit square is reduced by s; the effect of this reduction modifies the values of u in the
neighborhood of the point (i; j). If the demon gives out energy, it modifies / so as to
increase the energy integral. The demon is constrained so that it cannot give out energy
unless it had acquired energy in its previous history; no "loans" are allowed. The sequence
of states wrought by the demon's actions ranges over even-handedly the configurations of
the system. If one wishes to conserve an additional quantity, as we shall below, one can
do so by allowing the demon to exchange doses of the conserved quantity as it wanders
along, subject to the condition that it never give out what it does not have.
The parameter fi in the equivalent canonical ensemble can be determined in the course
of calculating averages: As the demon interacts with the ensemble it typically has some
energy stored away; the system consisting of the physical system plus the demon is isolated,
and by the equivalence of the canonical and microcanonical ensembles, the probability of
an energy E d being stored by the demon is canonical, i.e., proportional to exp(\GammafiE d );
this observation allows one to estimate fi after the demon has had a sufficient number of
interactions. In addition to its dynamical role in moving the system from state to state
for the purpose of calculating averages, the demon reveals the value of fi; if h and E h are
given, there is a well-defined numerical procedure for finding the corresponding fi.
Rather than keep fi merely bounded, we keep it constant. To do this, one needs a
variable that can be altered and whose variation controls fi. Experience and mathematics
show that one can use as control variable the integral I =
R
j-jdxdy , where - is the vorticity
calculated by finite differences and the integral is approximated by the appropriate sum.
Thus the plan is to determine an I for each h so as to keep fi at a fixed value fi goal common
to all the h. For simplicity and without loss of generality we set E
For a given h, pick a starting guess for I, say I 0 , and then produce a sequence of better
values I n by the formula
where fi without a subscript is the latest estimate of fi available from the demon and K
is a numerical parameter chosen so as to ensure that the I n converge to a limit. Before
calculating a new value I n+1 of I the demon must be allowed at least one energy exchange
with the ensemble, during which the variable I is maintained at its last value I n . Once fi
reaches the desired value fi goal the quantity I remains constant. The resulting ensemble
gives non-zero, equal probabilities to all states compatible with both the given value
and the calculated value of I; when both constraints are satisfied, the energy per degree
of freedom is no longer the same for all the degrees of freedom.
The changes in I needed to keep fi fixed as h is changing are displayed in Figure 4
for several values of . The statistical error is throughout of the order of 2%.
The values of I needed to keep fi fixed increase with As shown in [8, 16], for
small enough h the curves are independent of the value of fi, and this fact is reflected in
the confluence of the several curves in Figure 4. Note that I is calculated on the grid
by taking differences of the values of the velocity u at points separated by h, which by
the Kolmogorov-Obukhov scaling should be proportional to h 1=3 ; then one divides by h,
and takes an average; one expects I to grow with N like N 2=3 . In Figure 5 we plot the
logarithm of I vs. the logarithm of N ; the relation is well approximated by a straight
line whose slope is :65 with an error of \Sigma:05. Within the limitations of the Monte-Carlo
sampling, the Kolmogorov-Obukhov scaling is seen to be applicable in this equilibrium
model. As the Kolmogorov-Obukhov scaling applies to the low-order structure functions
in a flow with a finite but small viscosity, Figure 5 shows that low-order moments structure
function have a limit as the viscosity tends to zero. One can perform a similar analysis of
the small-scale structure of flow near a wall and conclude that the the first-order moments
of the derivatives of the velocity field near walls have well-behaved limits, a fact used above
in the discussion of the scaling of the wall-region in the pipe. Figure 4 defines the limiting
process in which h ! 0 with a limit that provides a meaningful equilibrium ensemble for
the small scales of the flow.
The fact that the construction above is numerical enhances its value rather than detracts
from it, as we expect to use similar constructions in the numerical modeling of turbulence.
An elegant argument, suggested by the work of Kailasnath et al. [19] and presented in detail
in [8], shows that the third-order structure function is calculated in the equilibrium theory
exactly; of greater interest here is what happens to moments of the velocity field of order
four and more. We have argued in the preceding section that the vanishing viscosity limit
is not well-behaved for these higher moments, and thus the good behavior of the structure
functions is unlikely and the expansion in powers of 1= ln Re is invalid. In the equilibrium
theory the fourth-order moments fail to converge to a finite limit as
In

Figure

6 we display the fourth-order moment
R
juj 4 dxdy as a function of N at the
parameter value 5. Higher-order moments diverge even faster. The divergence of the
higher moments corresponds to the formation of concentrated vortical structures, like the
ones explicitly constructed in [16]. We have thus produced Kolmogorov scaling for the
low-order moments in a system that is highly intermittent in the sense that the vorticity
is concentrated on a small fraction of the available volume. The results of the equilibrium
theory are therefore consistent with the scaling analysis of the previous section, according
to which the Kolmogorov scaling of the second-order structure function is exact in the
limit of vanishing viscosity not despite intermittency but because of intermittency, while
its failure for the higher-order moments can be ascribed to the absence of a well-behaved
vanishing-viscosity limit, as a result of which the expansion in the inverse powers of ln Re
is not legitimate.
Note the small number of assumptions made in the equilibrium theory; all that was
assumed was that the fluid was near statistical equilibrium on the small scales, the fluid
was incompressible, the energy density in physical space was finite, and a probability
measure on the ensemble of flows was well-defined. The Navier-Stokes equations did not
enter the argument in the present paper (but see ref. [8]).
Finally, it is worth noting that an analysis of simplified near-equilibrium vortex models
[22,33] has provided an example where an expansion in powers of a parameter analogous
to 1 ln Re can be fully justified without recourse to experimental data.
5. Conclusions
We have reached the following conclusions:
(1) The von K'arm'an-Prandtl law of the wall must be jettisoned, and replaced by a
power law with a Reynolds-number dependent coefficient and exponent, as suggested
by an assumption of incomplete similarity.
(2) The Kolmogorov-Obukhov scaling of low-order structure functions in the local
structure of turbulence admits only viscosity-dependent corrections, which vanish
as the Reynolds number tends to infinity. There are no "intermittency corrections"
to this scaling in the limit of vanishing viscosity. The Kolmogorov scaling of the
higher-order structure functions fails because of intermittency.
(3) These conclusions are consistent with and support the near-equilibrium theory of
turbulence.

Acknowledgement

Prof. Barenblatt and I would like to thank the following persons for
helpful discussions and comments and/or for permission to use their data: Prof. N. Gold-
enfeld, Prof. O. Hald, Dr. M. Hites, Prof. F. Hussain, Dr. A. Kast, Dr. R. Kupferman,
Prof. H. Nagib, Prof. C. Wark and Dr. M. Zagarola.



--R



Cambridge University Press

Reynolds number) in the developed turbulent flow in pipes


Basic hypotheses and analysis

of local structure and for the wall region of wall-bounded turbulence





turbulence theory

asymptotics and intermittency

in preparation

Discussion of experimental data
Sciences USA



the zero-pressure-gradient turbulent boundary layer

Part 2.

Physica D 80






Applications to Physical Systems

law and odd moments of the velocity difference in turbulence



USSR









Series 4

exponents at very high Reynolds numbers




for the two-dimensional XY model

Reynolds number
Figure 1.
Figure 2: Schematic of the power law curves
The individual curves of the power law
The envelope of the family of power law curves (often mistaken for a logarithmic
The asymptotic slope of the power law curves.
Figure 3.
of the experimental data according to their Reynolds numbers and the rise of the curves
above their envelope in the (ln j
at the center of the pipe.
and are incompatible with the von K'arm'an-Pradtl universal logarithmic law
Figure 4.

Figure 5.
given by the Kolmogorov-Obukhov scaling
Figure 6.
--TR
wall-bounded turbulence;local structure;intermittency;statistical theory;turbulence;scaling