(C)
The Kolmogorov 4/5 - law
We have focussed so far on the absolute structure functions, that were used to obtain bounds
on the energy flux ⇧` . However, there are other types of structure functions of interest, some
of them more directly related to energy flux and which, in fact, provide alternate definitions of
it. Rather than define “large-scale energy” by
ē` (x, t) = 12 |ū` (x, t)|2
one can instead make an alternate definition by filtering just one factor
1
u(x, t) · ū` (x, t)
Z2
1
=
dd r G` (r) u(x, t) · u(x + r, t)
2
e` (x, t) =
where
er (x, t) = 12 u(x, t) · u(x + r, t)
is the so-called point-split kinetic energy density. The quantities e` (x, t) and er (x, t) are welldefined whenever u has finite mean energy:
Z
Z
1 T
1
dt
dd x u2 (x, t) < +1.
T 0
2
V
We now derive a balance equation for the point-split kinetic energy density of a Navier-Stokes
solution, as follows

1
1
0
@t ( u · u ) + r · ( u · u0 )u +
2
2
1
=
rr · [ u | u|2 ]
4
1
1
1
(pu0 + p0 u) + |u0 |2 u ⌫r( u · u0 )
2
4
2
1
0
0
0
⌫ru : ru + (f · u + f · u)
2
with the notations
u = u(x, t), p = p(x, t)
u0 = u(x + r, t), p0 = p(x + r, t)
u = u0
u = u(x + r, t)
16
u(x, t)
Remarks:
#1. This identity was first derived by L. Onsager in the 1940’s in a space-integrated form.
The result was communicated to C. C. Lin in a letter in 1945, but never formally published
by Onsager. The space-local form was derived by J. Duchon & R. Robert, Nonlinearity 13
249-255(2000) in a smoothed version, discussed a bit later.
#2. The relation is analogous to the energy balance equation that we derived in the filtering
approach:
@t ( 12 |ū` |2 )
+r·

1
2
2 |ū` | ū`
⌧ ` · ū`
+ p̄` ū`
⌫r( 12 |ū` |2 ) = rū` : ⌧ `
⌫|rū` |2 + f̄` · ū`
Proof of the identity: Take
@t u + r · (uu)
@t u0 + r · (u0 u0 )
⌫4u + rp = f , r · u = 0
⌫4u0 + rp0 = f 0 , r · u0 = 0
Dot the first by u0 and the second by u and add together, to obtain
@t (u · u0 ) + u · [r · (u0 u0 )
⌫4u0 ] + u0 · [r · (uu)
⌫4u]
+ r · (p0 u + pu0 ) = f 0 · u + f · u0
Now the viscous term is reorganized as
u · 4u0 + u0 · 4u = r · [ui ru0i + u0i rui ]
= r · [r(u · u0 )]
2rui · ru0i
2ru : ru0
Lastly we discuss the crucial nonlinear term. Note first that
u · [r · (u0 u0 )] + u0 · [r · (uu)] = ui @j (u0i u0j ) + u0i @j (ui uj )
= ui @j (u0i u0j ) + @j (u0i ui uj )
= 4 + r · [(u · u0 )u]
with
4 ⌘ ui @j (u0i u0j )
17
ui uj (@j u0i ))
ui uj (@j u0i )
.
By incompressibility,
4 = ui u0j (@j u0i )
= ui (u0j
ui uj (@j u0i )
uj )(@j u0i )
= ui uj (@j u0i )
Also by incompressibility rr · ( u) = rr · u0 = 0, so that
rr · [ u | u|2 ] = ( u · r0 )| u|2
= 2( uj @j )u0i · (u0i
= 2 uj · u0i @j u0i
ui )
2ui uj (@j u0i )
u · r(|u0 |2 )
24
= r · [|u0 |2 u]
24
=
Thus,

1
@t (u · u ) + r · (u · u0 )u + (pu0 + p0 u) + |u0 |2 u ⌫r(u · u0 )
2
1
rr · [ u | u|2 ] + 2⌫ru : ru0 = (f · u0 + f 0 · u)
2
0
QED!
Multiplying the point-split identity through by G` (r) and integrating over r gives a corresponding balance equation for the regularized energy density 12 u · ū` :

1
1
1
1
1
1
@t ( u · ū` ) + r · ( u · ū` )u + (pū` + p̄` u) + (|u|2 u)`
(|u|2 )` u ⌫r( u · ū` )
2
2
4
4
2
Z2
1
1
d
2
=
d r (rG)` (r) · u(r) | u(r)|
⌫ru : rū` + (f · ū` + f̄` · u)
4`
2
The above equation can be shown to be valid even for singular Leray solutions of INS, if the
space-time derivatives are interpreted in the sense of distributions. (See Appendix.)
We now consider the limit of vanishing viscosity ⌫ ! 0. If the Navier-Stokes solution u⌫ ! u
as ⌫ ! 0 in the space-time L2 -sense, i.e.
18
R
dt
R
dd x |u⌫ (x, t)
u(x, t)|2 ! 0, as ⌫ ! 0
then it is not hard to show that the limiting velocity u is a solution of the incompressible Euler
equations
@t u + r · (uu) =
rp + f , r · u = 0
in the sense of space-time distributions. If it is furthermore true that
R
dt
R
dd x |u(x, t)|3 < +1,
then the point-split balance holds also for the Euler solution in the distribution sense:

1
1
1
1
@t ( u · ū` ) + r · ( u · ū` )u + (pū` + p̄` u) + (|u|2 u)`
2
2Z
2
4
1
d
=
d r (rG)` (r) · u(r) | u(r)|2
4`
1
(|u|2 )` u
4
(?)
For simplicity, we consider the case with no external force, f = 0. We now consider the limit
R R
` ! 0. Under the same basic assumption, that dt dd x|u(x, t)|3 < +1, it is not hard to
show that the LHS of equation (?)
1
@t ( u · ū` )
2

1
1
r · ( u · ū` )u + (pū` + p̄` u) +
2
 2
1 2
1
! @t ( |u| ) + r · ( |u|2 + p)u ⌘
2
2
+
1
(|u|2 u)`
4
D(u),
1
(|u|2 )` u
4
as ` ! 0
in the sense of distributions. Just to consider one typical term,
Z
Z
1
dt d x r'(x, t) · ( u(x, t) · ū` (x, t))u(x, t)
2
1
1
 sup |r'| · ||( u · ū` )u ( |u|2 )u||L1
spacetime
2
2
d
Using 12 (u · ū` )u
1
2
2 |u| u
= 12 [u · (ū`
||[u · (ū`
Z
dt
Z
1
dd x r'(x, t) · u(x, t) |u(x, t)|2
2
u)]u and the Hölder inequality
u)]u||L1  ||u||2L3 ||ū`
u||L3
One can then show that the upper bound ! 0 as ` ! 0. This implies that
⇥
⇤
⇥
⇤
r · ( 12 u · ū` )u ! r · 12 |u|2 u
in the sense of distributions. The other terms are treated in a similar fashion. But since the
LHS of equation (?) converges to
D(u) in the sense of the distributions, so does the RHS!
That is,
19
1
D(u) = lim
`!0 4`
Z
dd r (rG)` (r) · u(r)| u(r)|2
(1)
in the sense of distributions. To summarize, we obtain the above formula for the anomalous
dissipation D(u) that appears in the energy balance relation

1 2
1
@t ( |u| ) + r · ( |u|2 + p)u =
2
2
D(u).
(2)
for the singular Euler solution u(x, t). This result is quite interesting in its own right and not
just as a step in the proof of the Kolmogorov 4/5-law. It is a precise mathematical formulation
of Onsager’s idea that Euler solutions which arises in the zero-viscosity limit of turbulent flow
may not conserve energy. We could derive the same balance equation (2) by starting from
the balance equation for 12 |ū` |2 and taking the limit ` ! 0. This would give us another valid
expression
D(u) = lim ⇧` (in the distribution sense)
`!0
for the anomalous dissipation D(u). In fact, the RHS of equation (?)
Z
1
D` (u) =
dd r (rG)` (r) · u(r)| u(r)|2
4`
(3)
is another way of measuring energy flux to small scales, alternative to ⇧` . We can get from
equation (1) for D(u) Onsager’s bound on energy flux to small scales. For example, if u has
Hölder exponent h, then it follows from equation (3) that
D` (u) = O(`3h
1)
the same bound derived earlier for ⇧` . These bounds imply the assertion of Onsager in his 1949
paper that Euler solutions must conserve energy if the velocity has Hölder exponent h > 1/3.
Using Lp norms, one can easily show also that energy is conserved if
⇣p > p/3) for p
p
3. Under any of these regularity assumptions, D(u) ⌘ 0!
20
>
1
3
(equivalently,
Let us now return to our derivation of the 4/5-law, by obtaining a simplified expression for D(u)
for the case of a spherically symmetric filter kernel G that depends upon only the magnitude
r = |r|:
G(r) = G(r)
so that
rG(r) = r̂G0 (r).
In that case, one can go to spherical coordinates in d-dimensions
R1 d 1 R
1
D` (u) = 4`
dr S d 1 d!(r̂)r̂ · u(r)| u(r)|2 (G0 )` (r)
0 r
where S d
1
is the unit sphere in d-dimensions and d! is the measure on solid angles. Now
introduce
uL (r) = r̂ · u(r) = longitudinal velocity increment
and
Z
1
h uL (r)| u(r)|2 iang =
d!(r̂) uL (r)| u(r)|2
⌦d 1 S d 1
= angular average of uL | u|2
where ⌦d
1
is the (d
1) -dimension volume of S d
D` (u) =
1
⌦d 1
4` Z
= ⌦d
1
Z
1
rd
0
1
1
1.
We thus find that
dr (G0 )` (r)h uL (r)| u(r)|2 iang
⇢d d⇢ G0 (⇢)
0
h uL (r)| u(r)|2 iang
4r
r=`⇢
where ⇢ = r/`. We know that the limit of the LHS exists as ` ! 0 in the sense of distributions
and gives D(u). Taking the limit on the RHS, we see that
h uL (r)| u(r)|2 iang
4r
! D⇤ (u) , as r ! 0
with
⇤
Z
1
⇢d d⇢G0 (⇢)
0
Z 1
⇤
= D (u) · ( d · ⌦d 1
⇢d 1 d⇢G(⇢)) by integration by parts
0
Z 1
⇤
=
d · D (u) since ⌦d 1
⇢d 1 d⇢G(⇢) = 1
D(u) = D (u) · ⌦d
1
0
21
We conclude finally that
h uL (r)| u(r)|2 iang
=
r!0
r
lim
4
D(u)
d
The results in the above form were given in the paper
J. Duchon & R. Robert, “Inertial energy dissipation for weak solution of incompressible Euler and Navier-Stokes equations,” Nonlinearity, 13 249-255(2000).
It is possible, by an elaboration of these arguments, to derive expressions for D(u) that involve
only uL (r), or mixed expressions that involve uL (r) and the transverse velocity increment
uT (r) = u(r)
uL (r)r̂
which satisfies r̂ · uT (r) = 0, or
u2T (r) = | uT (r)|2 /(d
1),
the magnitude of the transverse velocity increment per component. These are, in d-dimensions,
h u3L (r)iang
r!0
r
h uL (r) u2T (r)iang
lim
r!0
r
lim
=
=
12
D(u)
d(d + 2)
4
D(u)
d(d + 2)
For the derivation, see G. L. Eyink, Nonlinearity 16, 137-145(2003). The idea is to look at
separate equations for point-split energy densities u(x, t) · uL (x + r, t), u(x, t) · uT (x + r, t).
Example: Burgers Equation
The above discussion has been a bit abstract, so that it is useful to consider a concrete example.
All the previous results have exact analogies for singular/distributional solutions of the inviscid
Burgers equation, which can be shown to satisfy the energy balance equation
22
@t ( 12 u2 ) + @x ( 13 u3 ) =
with
1
D(u) = lim
`!0 12`
Z
+1
1
D(u)
dr(G0 )` (r) u3L (r)
in the sense of distributions. Alternately,
limr!0
where h u3L (r)iang = 12 [ u3 (+|r|)
h u3L (r)iang
|r|
=
12D(u)
u3 ( |r|)]. For the Khokhlov sawtooth solution in the limit
⌫ ! 0 it is straightforward to calculate explicitly that, with r > 0
h u3L (r)iang =
1
[(r/t 4u)3 ( r/t)3 ] [
2
1
+ [(r/t)3 ( r/t + 4u)3 ]
2
r,0] (x)
[0,r] (x)
so that
h u3L (r)iang
r
!
[ 12 (4u)3 + 12 (4u)3 ] (x) =
(4u)3 (x) , as r ! 0.
12"(x), where "(x) = limr!0 ⌫(@x u⌫ )2 =
Notice that this is equal to
1
12 (
u)3 (x) is the
distributional limit of the viscous dissipation in u⌫ (x, t) as ⌫ ! 0.
A similar result can be obtained for the ⌫ ! 0 limit of Leray solutions of the Navier-Stokes
equation. These satisfy a local energy balance of the form

1 ⌫2
1
@t ( |u | ) + r · ( |u⌫ |2 + p⌫ )u⌫
2
2
1
⌫r( |u⌫ |2 ) =
2
⌫|ru⌫ |2
(or, 
⌫|ru⌫ |2 ),
(?)
For simplicity, we shall only consider the case where “=” holds above rather than “”. (For the
general case, see Appendix.) Let us assume that u⌫ ! u as ⌫ ! 0 in the L3 -sense in spacetime,
i.e.
R
dt
R
dd x |u⌫ (x, t)
u(x, t)|3 ! 0.
This is stronger than the L2 -convergence assumed earlier, so that, again, the limiting velocity
u is an Euler solution in distribution sense. Furthermore, it is now possible to check that the
LHS of equation (?) above has the limit
⇥
lim⌫!0 @t ( 12 |u⌫ |2 ) + r · ( 12 |u⌫ |2 + p⌫ )u⌫
⇤
⇥
⇤
⌫r( 12 |u⌫ |2 ) = @t ( 21 |u|2 ) + r · ( 12 |u|2 + p)u
23
in distribution sense. The argument is very similar to that which we gave earlier for the limit
` ! 0. Furthermore, the limit is exactly the same, i.e.
D(u)! Since the limits of the LHS and
the RHS of equation (?) must be the same, we obtain
D(u) = lim⌫!0 ⌫|ru⌫ |2 = lim⌫!0 "⌫ (Duchon & Robert, 2000)
in the sense of distributions. Notice the RHS of the above expression is non-negative, so that
its limit also must be:
D(u)
0
More precisely, D(u) is a nonnegative distribution, which satisfies
R
dd x dt '(x, t)D(u)(x, t)
0
for every nonnegative test function ' ( C 1 with compact support). It is known that every
nonnegative distribution is given by a measure, i.e.
R d R
RR
d x dt '(x, t)D(u)(x, t) =
µ(dx, dt) '(x, t)
This “dissipation measure” has been much studied experimentally and observed to have multifractal scaling properties, as we discuss a bit later! If ' is nonnegative and also normalized
R R d
dt d x '(x, t) = 1,
then we can interpret
R
dt
R
dd x '(x, t) ⌫|ru⌫ (x, t)|2 ⌘ h⌫|ru⌫ |2 i'
as an average in spacetime over the compact support of ', weighted by '. The above result
then says that
lim⌫!0 h"⌫ i' = hD(u)i'
Our earlier results can be stated in a similar fashion, e.g.
limr!0 lim⌫!0
h[ u⌫L (r)]3 i',ang
r
=
12
d(d+2) hD(u)i' .
We thus see that, taking first ⌫ ! 0,
h u3L (r)i',ang ⇠
24
12
d(d+2) h"i' r
This is the famous Kolmogorov 4/5-law (since the coefficient
12
d(d+2)
=
4
5
for d = 3), derived by
Kolmogorv in the third of his celebrated 1941 papers on turbulence. The related results
h uL (r) u2T (r)i',ang ⇠
h uL (r)| u(r)|2 i',ang ⇠
4
h"i' r
d(d + 2)
4
h✏i' r
d
are called the Kolmogorov 4/15- and 4/3-laws, respectively. These were derived by Kolmogorov
in the statistical sense, averaging over an ensemble of solutions assuming statistical homogeneity
and isotropy. He employed in his derivation an equation derived earlier for the 2-point velocity
correlation hui (x, t)uj (x + r, t)i by von Kármán and Howarth (1938), so that this is sometimes
called the Kolmogorov-Kármán-Howarth relation. The result presented here (following largely
the derivation of Duchon & Robert, 2000) is much stronger, because there is no average over
ensembles and no assumption of homogeneity and/or isotropy. It seems to have been Onsager
in the 1940’s who realized that such relations should hold for individual realizations, without
averaging. He derived the formula
D` (u) =
1
4`
R
dd r (rG)` (r) · u(r)| u(r)|2
and discussed its limit for ` ! 0. In the statistical framework, the corresponding result
rr · h u(r)| u(r)|2 i ⇠
4h"i , as
r!0
was derived by A. S. Monin (1959) and is sometimes called the Kolmogorov-Monin relation.
It does not assume isotropy. There is another derivation of the 4/5-law by Nie & Tanveer
(1999) without statistical averaging. It uses also spacetime averaging and angle-averaging. It is
stronger than the result presented here in that it includes viscous corrections, but it is weaker
than the presented local results, since it requires a global spacetime average.
25
on with compact spectral support
consequent anisotropies can change the extent of the
The microscale Reynolds number
scaling so drastically. This matter will be discussed
othesis was not necessary.
elsewhere in more detail. We have examined many
olds number is only moderately
structure function plots and consistently used least-square
t and simulations, and a critical
fits to the R range of Fig. 2 to obtain the numbers to
aling range. A traditional way—
be quoted below, and verified that the relative trends are
—is to obtain the scaling region
robust even for the K range as well as for the ESS method.
⌅r versus r [3]. Unfortunately,
One noteworthy feature of the plus/minus structure
functions is shown in Fig. 3, which plots the logarithm
procedure is valid exactly in the
of the ratio Sq2 ⌅Sq1 against log10 r for various values
Experiments
ropies such as occur
[10] in pipeand Simulations
of q. It can be seen readily that the ratio Sq2 ⌅Sq1 is
al correction is needed [11]. We
? K. Sreenivasan
et
al., PRL,
greater
thanvol.77,
unity 1488-1491
for all r (1996)
, L whenever q . 1 and
t of scaling in the energy
spectral
smaller
than
unity
whenever
q , 1. Here L is the soo-called extended scale similarity
called
integral
scale
of
turbulence
characteristic of the
on of relative scaling [13] and,
These
authors
present
data
for
the
4/5-law
obtained
from
experimental
large
scale
turbulence.
By
definition,
the
ratio
should be observations
ty of the results to the scaling
exactly unity for q
1. For one-dimensional data such
the centerline
of flow
throughhere,
a pipe
at bulkfrom
Reynolds
number Re
as those
considered
it follows
the definition
of = 230, 000,
ot of the compensated of
spectral
generalized
dimensions Dq that the ratio of the minus to
. p
data from the experiment; in
and from a plus
5123 structure
DNS of homogeneous
forced
turbulence
at
Re
=
220(
=
Re).
functions scales as
hypothesis, spectral frequency is
2
1
Scaling exists over a decade
or for velocity increments
q 2Dq ⇤
.
⇥r⌅L⇤
The data
were⇥q21⇤⇥D
not angle-averaged,
so this is also a test of
s as the K range. Figure 2 plots
For consistency with the observation that Sq2 ⌅Sq1 is
t r for both the experiment
andto isotropy
return
at small scales.
greater than unity for q . 1 and less for q , 1, one
ows are at comparable Reynolds
should have
ng region (to be called the R
maller for the experiment than for
Dq2 , Dq1
y smaller than the K range. The
end is roughly the same in all
in the homogeneous simulation
e in the experiment extend to
lower frequencies) than does
ge. That the scaling in one-
ity of u multiplied by f 5⌅3 , where f
show the flat region. Scaling occurs
. There is no perceptible difference
ponent slightly different from 5⌅3 is
frequency rolloff.
FIG. 2. The quantity ⇧Du3r ⌅r as a function of r. Squares,
experiment; circles, simulations; dots indicate Kolmogorov’s
4
It is believed that the slight bump in the left
5 th law.
part of the experimental data is the bottleneck effect [see G.
Falkovich, Phys. Fluids 6, 1411 (1994); D. Lohse and A.
Mueller-Groeling, Phys. Rev. Lett. 74, 1747 (1995)]. While
the bottleneck effects discussed in these two papers refer
especially to second-order structure functions (or to energy
spectrum), a similar effect is likely to exist for the third-order as
well. This is typical of most measurements [see, for example,
Y. Gagne, Docteur ès-Sciences Physiques Thèse, Université de
Grenoble, France (1987)].
1489
26
? T. Gotoh et al. Phys. Fluids 14, 1065-1081 (2002)
This paper presents data from numerical simulations up to 10243 resolution of homogenous forced turbulence, and Re in the range 38
460. Tests were made of
both the 4/5-law and the 4/3-law, again without angle-averaging.
02
Velocity field statistics in homogeneous
–Kolmogorov equation when R -
1071
FIG. 12. Kolmogorov’s 4/5 law. L/ . and -/. are shown for R - !460. The
maximum values of the curves are 0.665, 0.771, 0.781, and 0.757 for R !125, 284, 381, and 460, respectively.
1072
Phys. Fluids, Vol. 14, No. 3, March 2002
FIG. 14. Kolmogorov’s 4/3 law. L/ " and #/" are shown for R # #460. The
maximum values of the curves for the 4/3 law are 0.564, 1.313, 1.297, and
1.259 for R # #125, 284, 381, and 460, respectively.
sting that the second order
ensemble, which indicates the persistent anisotropy of the
as r 2/3. The scaling expo- Equation !24" is recovered by substituting Eqs. !22" and !23"
into
Eq.
!27".
larger scales. The above findings are consistent with the curis paper. The isotropic reFigure 11 shows the results obtained when each term of
rent knowledge of turbulence developed since Kolmogorov,
d D 2222!3D 2233!D 3333 ,
54
?
M.
Taylor
et
al.
Phys.
Rev.
E
68,
026310
(2003)
Eq.
!24"
is
divided
by
#̄
r
for
R
!460.
Curves
in
which
r/
.
although confirmation of some aspects of turbulence using
were not computed.
is larger than r/ . !1200 are not shown, because the sign of
actual data is new from both a numerical and experimental
D
changes.
A
thin
horizontal
line
indicates
the
KolmogLLL
3
of view.56,59–65forced turbulence, with
MOGOROV EQUATION
Another numerical study in a 512 DNS of point
homogenous
orov value 4/5. When the separation distance decreases, the
It is interesting and important to observe when the Kolscales is described by the effect of the large
scale forcing
in thepaper
present DNS
mogorovof4/5angle-averaging
law is satisfied as and
the Reynolds
Re ⇠
263.usedThis
studies the e↵ects
obtainsnumber
= 249
KHK" equation,
decreases quickly, while the viscous term grows gradually.
increases.6,66–69 Figure 12 shows curves of !D LLL (r)/(¯! r)
quickly
The third
order longitudinal
structure
function D LLL
results
with such
averaging
comparable
to those
of Gotoh
al.(2002)
nearly
for various
Reynolds et
numbers.
In this at
figure,
the 4/5 law
!24" rises to the Kolmogorov value, remains there over the inerapplies when the curves are horizontal. The portion of the
twice
ther/ Reynolds
. ,50 and 300",number.
and then decreases. In
tial range
!between
curves in which r/ " "1200 is not shown. Although there is a
Z(r) denotes contributions the inertial range, the force term decreases as r 3 according to
small but finite horizontal range when R # "284, the level of
Eq. !26", while the viscous term increases as r / 2 "1 ( / 2 %1)
the plateau is still less than the Kolmogorov value. The
when r decreases. 0Since each term in the figure is divided by
maximum values of the curves are 0.665, 0.771, 0.781, and
( #̄ r), the slope of each curve is 2 and / 2 "2, respectively.1
0.757 for R # #125, 284, 381, and 460, respectively. The
The sum of the three terms in the right hand side of Eq. !24"
value 0.781 for R # #381 is 2.5% less than 0.8. An
divided by #̄ r is close to 4/5, the Kolmogorov value. The
asymptotic state is approached slowly, which is consistent
os kr sin kr
deviation of the sum from the 4/5 law at the smallest scales
4 "3
5 F ! k " dk.
with recent studies. However, the asymptote is approached
27
kr "
! kr "
is due to the slightly lower resolution of the data at these
faster than predicted by the theoretical estimate.66,69 The
!25" scales !k max . is close to one". At larger scales greater than
r/ . !700, the deviation is caused by the finiteness of the
m F(k) is localized in a
#
FIG. 16. Variation of P
curves is the same as in
slow approach is d
order structure fun
canceled by negati
of the $ u r PDF con
of the R # #460 cur
value would be exp
the R # #460 run w
The generalize
tion Eq. %27& is also
shows each term o
line indicates the 4/
data and theory i
slowly approaches
Fig. 14. The maxim
are 0.564, 1.313, 1
and 460, respective
VII. STRUCTURE
EXPONENTS
The velocity st
S Lp % r & # ' ! $ u r ! p
PHYSICAL REVIEW E 68, 026310 "2003#
TAYLOR, KURIEN, AND EYINK
FIG. 6. The nondimensional third-order longitudinal structure
function, computed from a single snapshot of the stochastic dataset,
vs the nondimensional scale r/ & . The dots indicate the values of the
structure function computed at various !r j . The thick curve is the
angle average. The horizontal line indicates the 4/5 mark.
FIG. 7. The nondimensional third-order longitudinal structure
function computed from a single snapshot of the deterministic
dataset vs the nondimensional scale r/ & . The various symbols and
lines mean the same as in Fig. 6.
are quite different, while the angle-averaged results are quite
reasonable and similar to each other as well as similar to the
results obtained from long-time averaging of the coordinate
directions presented in Ref. $5% and shown for our data in
Sec. III D. Thus, we conclude that angle-averaging the data
from a single snapshot yields a very reasonable result. Similar results "not plotted# are obtained for the 4/3 and 4/15
given applies even if D(u) ⌘ 0, i.e.
laws.
aging in time. Excellent agreement is obtained in the inertial
range, with some departure at larger scales.
In Fig. 4, we show the second-order isotropy relation for
our stochastic dataset, and in Fig. 5 we show the third-order
relation for the deterministic dataset. This data is computed
angle-averaging
over derivation
a single snapshotthat
of the flow.
One lastby remark:
The
we The
have
vanishes
agreement is excellent, both in the inertial range and at the
largest scales.
comparison, this
the figures
also in
show
everywhere.
For For
example,
holds
a the
smooth solution C.ofTemporal
the Euler
variance equations, for which
same relations from the same snapshot but using only a
single
coordinate
direction
instead
of
angle-averaging.
In
illustrate the variance
time of the third-order longi3 ), so that D To
2 ) ! in
h u3L (r)ithat
⇠there
h uareL (r)
u2T (r)i
⇠ O(r
= O(`
0 and
as without
` ! 0.
Another example
ang
case,
significant
differences
for scales
well into
` (u)
tudinal
structure
function,
with
angle-averaging,
the inertial range. Thus, the angle-averaging technique apwe plot the peak value as a function of time for each dataset
pears tosolutions
be extremely where,
effective inunder
extractingvery
the isotropic
is 2D Euler
general assumptions, D(u) ⌘ 0 and there is no energy
component of anisotropic data even at large scales, where
anisotropy remains after time averaging over many snapcascadeshots.
to small
E.g.
seefor
Proposition
Similar scales.
results were
obtained
the second-order6 in Duchon & Robert (2000). There is a nontrivial
isotropy relation from the deterministic dataset and for the
4
extension
of the
to 2D
but with h u3L (r)iang positive, corresponding to inverse
third-order
isotropy
relation
from turbulence,
the stochastic dataset.
5 -law
energy cascade.B. E.g.
see D.a single
Bernard
Angle-averaging
snapshot (1999).
We now present results using angle-averaging to compute
the third-order
longitudinal structure function in the 4/5 law.
Additional
References:
Figures 6 and 7 show the result of the angle-averaging procedure described above for single snapshots of the stochastic
and deterministic datasets, respectively. The snapshots are
T.after
von
taken
theKármán
flow has had &
timeL.
to Howarth,
equilibrate. The“On
value ofthe statistical theory of isotropic turbulence,”
the mean energy dissipation rate ! was calculated from the
snapshot.
This isSoc.
to be Lond.
contrasted A
with164,
previous
works in (1938).
Proc.
Roy.
192-215
which ! is a long-time or ensemble average. We have therefore computed a version of the 4/5 relation which is local in
FIG. 8. The angle-averaged "solid line# and single-direction
time. The dots represent the data from all 73 directions at all
"dotted line# values of the peak of the nondimensionalized thirdA. ofN.r that
Kolmogorov,
“Dissipation
of energy
in locally
isotropic
values
were computed. The
final weighted angleorder longitudinal
structure
function for turbulence,”
deterministic dataset, asDokl.
a
average of Eq. "14# is given by the thick curves in both Figs.
function of nondimensional time t/T, where T!2E/ ' is the eddy6 and 7. Nauk.
One can seeSSR
that the32,
results16-18
from different
directions
turnover time.
Akad.
(1941).
026310-6
28
A. S. Monin, “Theory of locally isotropic turbulence,” Dokl. Akad. Nauk.
SSSR 125 515-518(1959); see also, A.S. Monin & A. M. Yaglom, Statistical Fluid
Mechanics, vol.2 (MIT, 1975), p.403.
Papers on Deterministic Versions of 45 -law
Q. Nie & S. Tanveer, “A note on the third-order struncture functions in turbulence,” Proc. R. Soc. A 455, 1615-1635(1999).
J. Duchon & Robert, “Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes,” Nonlinearity, 13 249-255(2000)
G. L. Eyink, “Local 45 -law and energy dissipation anomaly in turbulence,” Nonlinearity, 16 137-145(2003).
G. L. Eyink, “Onsager and the theory of hydrodynamic turbulence,” Rev. Mod.
Phys. 78 87-135(2006), Section IV, B.
2D Analogues of the 45 -law
D. Bernard, “Three-point velocity correlation functions in two-dimensional forced
turbulence,” Phys. Rev. E 60 6184-6187(1993).
A. M. Polyakov, “The theory of turbulence in two dimensions,” Nucl. Phys. B
396, 367-385(1993). This paper, in particular, discusses the analogy of the 45 -law
with conservation-law anomalies in quantum field theories.
29
APPENDIX
The purpose of this appendix is to discuss more carefully the issue of energy balance/energy
dissipation for Leray Solutions of the incompressible Navier-Stokes (INS) equations, which are
possibly singular. It was proved by
J. Leray, “Essai sur Le Mouvement d’un fluide visqueux emplissant l’espace,”
Acta. Math. 63 193-248 (1934)
that solutions of the INS
@t u⌫ + r · (u⌫ u⌫ ) =
rp⌫ + ⌫4u⌫ , r · u⌫ = 0
always exist in the sense of spacetime distributions, i.e.
Z T Z
dt dd x[@t '(x, t) · u⌫ (x, t) + r'(x, t) : u⌫ (x, t) ⌦ u⌫ (x, t)
t0
Z
⌫
⌫
+ (r · '(x, t))p (x, t) ⌫4'(x, t) · u (x, t)] = dd x '(x, t0 ) · u⌫0 (x),
Z
T
dt
t0
Z
dd xr (x, t) · u⌫ (x, t) = 0
for any initial condition u⌫0 with finite energy: ||u⌫0 ||L2 (V ) < +1. Here ' = ('1 , · · · , 'd ),
are
C 1 functions on spacetime with compact support. Note that the above rather abstract-looking
formulation is actually very physical! It is equivalent to the following balance equation for the
momentum in a bounded region ⌦ ✓ V with smooth boundary @⌦ and times t > t0
Z
Z
u⌫ (x, t)dd x
u⌫ (x, t0 )dd x
⌦Z
⌦
Z 
t
@u⌫
⌫
=
ds
u (x, s)(u⌫ (x, s) · n̂) + p⌫ (x, s)n̂ ⌫
(x, s) dA,
@n
@⌦
Z t0
Z
u⌫ (x, t) · n̂ dA =
u⌫ (x, t0 ) · n̂ dA = 0
@⌦
t0
@⌦
for all possible choices of ⌦, t, t0 . These equations just state that the momentum change in a
bounded region ⌦ comes from flux of momentum across the surface @⌦ and that there is no
net flux of mass across the surface. Another equivalent formulation of the notion of notion of
“weak” or distributional solution is that the “coarse-grained INS equations”
30
@t ū⌫` + r · (u⌫ u⌫ )` =
rp̄⌫` + ⌫4ū⌫` , r · ū⌫` = 0
should hold for all ` > 0.
Importantly, Leray showed that his distributional solutions satisfy the following fundamental
energy inequality for all t
t0
1
⌫
2
2 ||u (t)||L2 (V )
+⌫
Rt
t0
ds||ru⌫ (s)||2L2 (V )  12 ||u⌫0 ||2L2 (V )
This inequality states that the total energy at time t plus the integrated energy dissipation
up to time t must be less than or equal to the initial energy 12 ||u⌫0 ||2L2 (V ) . If the solutions are
everywhere smooth (“strong”), then the inequality “” becomes equality “=”. However, if
the solutions are singular (“weak”), then there may be strict inequality, which corresponds to
“extra dissipation” due to the singularities, in addition to the viscous dissipation. Note that it
is a consequence of this fundamental energy inequality that
sup ||u⌫ (t)||L2 (V )  ||u0 ||L2 (V ) < +1
t0 tT
Z T
dt||ru
t0
⌫
(t)||2L2 (V )

||u0 ||2L2 (V )
2⌫
< +1
for a fixed initial condition u0 with finite energy. From these estimates, some other basic bounds
follow, such as
RT
t0
2
dt||u⌫ (t)||3L3 (V )  (const.) L⌫ ||u0 ||3L2 (V ) .
This inequality means that Leray solutions are sufficiently regular that one can consider a local
energy balance in spacetime. To construct his solutions, Leray considered the limit as ` ! 0 of
the modified equation
@t u + (ū` · r)u =
rp + ⌫4u
where only the velocity ū` = G` ⇤ u appearing in the advection term has been smoothed. Leray
showed that the above equation has regular solutions (u?` , p?` ), which lie in a bounded (and thus
weakly compact) subset of L2 ([0, T ], H 1 (V )). Hence, weak limits u?` ! u exist in this space,
along subsequences of `. Such u can be seen to be distributional solutions of the incompressible
Navier-Stokes equation
@t u + r · (uu) =
31
rp + ⌫4u
and, since (by weak lower semi-continuity)
R
R
lim inf `!0 dd x dt|ru?` (x, t)|2 (x, t)
for non-negative test functions
with D(u)
R
dd x
R
dt|ru(x, t)|2 (x, t)
, also obeys the energy balance
⇥
⇤
@t ( 12 |u|2 ) + r · ( 12 |u|2 + p)u ⌫r( 12 |u|2 ) = D(u)
⌫|ru|2
0. By smoothing the solution u one can write an energy balance also for ū` :
1
1
1
@t ( u · ū` ) + r · [( u · ū` )u + (pū` + p̄` u)
2
2
2
1
1
1
2
+
((|u| u)`
(|u|2 )u ⌫r u · ū` ]
4 Z
4
2
1
d
=
d r rG` (r) · u(r)| u(r)|2 ⌫ru : rū`
4`
Taking the limit ` ! 0, we recover the previous energy balance with an explicit expression for
D(u):
D(u) = lim`!0
1
4`
R
dd r(rG)` (r) · u(r)| u(r)|2
This formula makes it clear that D(u) in the Navier-Stokes solution is connected with velocity
increments. Note that, in general, in the presence of such singularities, the total Navier-Stokes
dissipation is
D(u⌫ ) + ⌫|ru⌫ |2
(where we have now added the supperscript ⌫ to indicate the viscosity value) and in the ⌫ ! 0
limit it is this total dissipation which converges to the anomalous dissipation in the Euler
solution
D(u⌫ ) + ⌫|ru⌫ |2 ! D(u).
For more details, see Duchon & Robert (2000)
32
General References on Leray Solutions of INS
An English translation of Leray’s paper of 1934 is available at the Cornell website of mathematician Bob Terrell:
http://www.math.cornell.edu/⇠bterrell/leray.pdf
A key paper on “partial regularity” of Leray solutions, improving the earlier work of V. Sche↵er
(1977) is
L. Cafarelli, R. Kohn & L. Nirenberg, “Partial regularity of suitable weak solutions
of the Navier-Stokes equations,” Commun. Pure Appl. Math. 35, 771-831 (1982)
Many good textbooks presentation of the Leray theory are available. I’d recommend
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis (AMS, 2001)
and
R. Temam, Navier-Stokes Equation & Nonlinear Functional Analysis (SIAM, 1995)
which make nice connections with numerical analysis, and
G. Gallavotti, Foundations of fluid dynamics (Springer, 2013)
which is a good discussion for (mathematically inclined) physicists.
33