(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 e¯` (x, t) = 12 |¯ u` (x, t)|2 one can instead make an alternate definition by filtering just one factor 1 ¯ ` (x, t) u(x, t) · u 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 |¯ u` |2 )
+r·
1 ¯` u` |2 u 2 |¯
¯` ⌧` · u
¯` + p¯` u
¯` ⌫|r¯ u` |2 + ¯f` · u
⌫r( 12 |¯ u` |2 ) = r¯ u` : ⌧ `
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 correspond¯ `: ing balance equation for the regularized energy density 12 u · u
1 1 1 1 1 1 ¯ `) + r · ( u · u ¯ ` )u + (p¯ ¯ `) @t ( u · u u` + p¯` u) + (|u|2 u)` (|u|2 )` u ⌫r( u · u 2 2 4 4 2 Z2 1 1 ¯ ` + ¯f` · u) = dd r (rG)` (r) · u(r) | u(r)|2 ⌫ru : r¯ u` + (f · 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 ¯ `) + r · ( u · u ¯ ` )u + (p¯ @t ( u · u u` + p¯` 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 · u 2
1 1 ¯ ` )u + (p¯ r· ( u·u u` + p¯` u) + 2 2 1 1 ! @t ( |u|2 ) + 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 ¯ ` (x, t))u(x, t) dt d x r'(x, t) · ( u(x, t) · u 2 1 1 ¯ ` )u ( |u|2 )u||L1 sup |r'| · ||( u · u spacetime 2 2 d
¯ ` )u Using 12 (u · u
1 2 2 |u| u
= 12 [u · (¯ u` ||[u · (¯ u`
Z
dt
Z
1 dd x r'(x, t) · u(x, t) |u(x, t)|2 2
u)]u and the H¨older inequality u)]u||L1 ||u||2L3 ||¯ u`
u||L3
One can then show that the upper bound ! 0 as ` ! 0. This implies that ⇥ ⇤ ⇥ ⇤ ¯ ` )u ! r · 12 |u|2 u r · ( 12 u · 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 |¯ u` |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 H¨older 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 H¨older 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) = ˆrG0 (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!(ˆr)ˆr · 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) = ˆr · u(r) = longitudinal velocity increment and Z
1
h uL (r)| u(r)|2 iang =
d!(ˆr) 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)ˆr
which satisfies ˆr · 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 K´arm´an and Howarth (1938), so that this is sometimes called the Kolmogorov-K´ arm´ an-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 arm´ n& taken theK´ flow hasahad 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⌫ ⌫ ˆ ) + p⌫ (x, s)ˆ = ds u (x, s)(u⌫ (x, s) · n n ⌫ (x, s) dA, @n @⌦ Z t0 Z ˆ dA = ˆ dA = 0 u⌫ (x, t) · n u⌫ (x, t0 ) · n @⌦
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
¯ ⌫` + r · (u⌫ u⌫ )` = @t u
¯ ⌫` = 0 r¯ p⌫` + ⌫4¯ u⌫` , r · u
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 tT 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 + (¯ u` · r)u =
rp + ⌫4u
¯ ` = G` ⇤ u appearing in the advection term has been smoothed. Leray where only the velocity u 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 u 1 1 1 ¯ ` ) + r · [( u · u ¯ ` )u + (p¯ @t ( u · u u` + p¯` u) 2 2 2 1 1 1 ¯ `] + ((|u|2 u)` (|u|2 )u ⌫r u · u 4 Z 4 2 1 = dd r rG` (r) · u(r)| u(r)|2 ⌫ru : r¯ u` 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.
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