Nonlocal interactions versus viscosity in turbulence
Nonlocal interactions versus viscosity in turbulence
A. Bershadskii
ICAR, P.O. Box 31155, Jerusalem 91000, Israel
It is shown that nonlocal interactions determine energy spectrum in isotropic turbulence at small
Reynolds numbers. It is also shown that for moderate Reynolds numbers the bottleneck effect is
determined by the same nonlocal interactions. Role of the large and small scales covariance at the
nonlocal interactions and in energy balance has been investigated. A possible hydrodynamic mech-
anism of the nonlocal solution instability at large scales has been briefly discussed. A quantitative
relationship between effective strain of the nonlocal interactions and viscosity has been found. All
results are supported by comparison with the data of experiments and numerical simulations.
PACS numbers: 47.27.-i, 47.27.Gs
I. INTRODUCTION
Classic wind-tunnel experiments [1] showed that there is no scaling behavior in a presumably isotropic turbulence
at low Reynolds numbers. Moreover, recent high-resolution numerical simulations [2] performed for low-Reynolds-
number (Rλ ≈ 10− 60) show no hint of scaling-like behavior of the velocity increments even when ESS [3] is applied.
For moderate Reynolds numbers numerical simulations show ’excess’ power just before the dissipation (Kolmogorov’s)
wavenumber kd (a hump in the compensated energy spectra), see for instance [4]. This non-scaling effect (bottleneck
effect [5]) is usually related with reducing efficiency of the energy cascade toward kd [5],[6],[7]. For the small Reynolds
numbers applicability of the energy cascade idea is problematic for entire range of scales. It is shown in recent paper
[8] that nonlocal interactions become dominating in comparison with the local ones just in the near-dissipation range
of scales (cf. also [9]). In this range the viscous effects cannot be neglected and scaling asymptote corresponding to
the nonlocal regime cannot be observed [8] (in the inertial range the local interactions are presumably dominating
ones and the nonlocal scaling asymptote also cannot be observed). However, the nonlocal scaling asymptote can be
used as a zeroth term in a perturbation approach taking into account the viscosity effects. There are many ways
to develop such perturbation approach. For instance, in the paper [8] a perturbation approach giving logarithmic
corrections to the scaling was developed, provided by significant role of the kink instabilities of the vortex filaments at
moderate and large values of Reynolds number [10]. This ’logarithmic’-perturbation approach is shown to be effective
in a vicinity of the crossover scale rc, where exchange of stability between local and nonlocal regimes takes place
[8]. This vicinity is rather wide at moderate values of Reynolds number, when an overlap between these regimes is
a strong phenomenon [8]. For small Reynolds numbers, however, the kink instabilities of the small vortex tubes is
suppressed by the strong viscosity (see, for instance [11] and references therein). Therefore, for the small Reynolds
numbers another, adequate just for this case, perturbation approach should be developed. An approach of such type
is suggested in present paper. Starting from this approach and using comparison with results of numerical simulations
[12] the laboratory experiments [1] it is shown that energy spectrum for small Reynolds numbers is determined
by the nonlocal interactions even in isotropic turbulence. The same nonlocal interactions provide a hydrodynamic
mechanism for the so-called bottleneck effect for moderate Reynolds numbers. It is also shown that large and small
scales covariance at the nonlocal interactions plays a significant role in these phenomena. A quantitative relationship
between effective strain of the nonlocal interactions and viscosity has been found using dynamical equations.
II. PERTURBATIONS TO SCALING
Let us following to the paper [8] consider a dimensional function E(k) of a dimensional argument k. And let us
construct a dimensionless function of the same argument
α(k) =
E−1dE
k−1dk
. (1)
If for kd ≫ k we have no relevant fixed scale (scaling situation), then for these values of k the function α(k) must
be independent on k, i.e. α(k) ≃ const for kd ≫ k. For turbulence k could be a wavenumber and kd could be a
dissipation wavenumber (kd = 1/η, where η = (ν
3/〈ε〉)1/4 is so-called viscous scale [24]). Solution of equation (1)
with constant α can be readily found as
E(k) ∼= ck
α (2)
http://arxiv.org/abs/0704.1904v4
0.01 0.1 1
0.01 0.1 1
0.01 0.1 1
0.01 0.1 1
FIG. 1: One-dimensional spectra measured in nearly isotropic turbulence downstream of a grid at small Reynolds numbers
(the data are reported in [1]). The solid curves are drawn in the figure to indicate correspondence of the data to the equation
(10) (non-local regime).
where c is a dimension constant. This is the well-known power law corresponding to the scaling situations.
Let us now consider an analytic approach, which allows us to find corrections of all orders to the approximate power
law, related to the fixed scale kd. In the non-scaling situation let us denote
f ≡ ln(E/A), x ≡ ln(k/kd) (3)
where A and kd are dimensional constants used for normalization.
In these variables, equation (1) can be rewritten as
= α(x) (4).
In the non-scaling situation x is a dimensionless variable, hence the dimensionless function α(x) can be non-constant.
Since the ’pure’ scaling corresponds to k/kd ≪ 1 we will use an analytic expansion in power series
α(x) = α0 + α1(k/kd) + ...+
αn(k/kd)
n + ... (5)
where αn are dimensionless constants. Choice of the small parameter for analytic perturbation approach is determined
by physical situation, which one intends to consider. For instance, formoderate Reynolds numbers it would be generally
preferable to consider the x−1 as a small parameter [8]. In this paper, however, we intend to consider energy spectra
in isotropic turbulence at small Reynolds numbers and corresponding phenomena in a relatively close vicinity of the
dissipation scale at moderate Reynolds numbers. The kink instabilities of the vortex filaments, which are a significant
factor near rc (see above and [8]) are presumably not significant in situations with strong viscous effects [11]. Therefore
0.01 0.1 1
0.01 0.1 1
0.01 0.1 1
0.01 0.1 1
rc≅ηs
FIG. 2: Three-dimensional spectra from a DNS performed in [12] for different Reynolds numbers up to Rλ = 125. The solid
curves in this figure corresponds to the best fit by equation (10) (nonlocal interactions). The value of α1 = 6.0 ± 0.1 is the
same in all cases.
the choice of x−1 does not seem to be relevant here. On the other hand, parameter k/kd seems to be less related
to the specific hydrodynamic structures dominating processes in turbulence and more relevant to a sheer taking into
account corrections to the scaling providing by viscosity.
After substitution of the analytic expansion (5) into Eq. (4) the zeroth order approximation gives the power law (2)
with α ∼= α0. First order analytic approximation, when one takes only the two first terms in the analytic expansion
(5), gives
E(k) ∼= ck
α0eα1k/kd . (6)
(cf , for instance, [13]-[16]). Corrections of the higher orders can be readily found in this perturbation approach.
III. NONLOCAL INTERACTIONS
Let us recall that in isotropic turbulence a complete separation of local and non-local interactions is possible in
principle. It was shown by Kadomtsev [17] that this separation plays a crucial role for the local Kolmogorov’s cascade
regime with scaling energy spectrum
E(k) ≃ K 〈ε〉2/3k−5/3 (7)
where 〈ε〉 is the average of the energy dissipation rate, ε, k = 1/r is the wave-number, and K is the so-called
Kolmogorov constant. This separation should be effective for the both ends. That is, if there exists a solution with
the local scaling (7) as an asymptote, then there should also exist a solution with the non-local scaling asymptote.
0.001 0.01 0.1 1
0.001 0.01 0.1 1
rc ηs
FIG. 3: As in Fig. 2 but for Rλ = 460. The upper part of the figure shows the same energy spectrum as the lower one but in
the compensate (according to the Kolmogorov’s scaling) form. The solid curves are the best fit corresponding to the nonlocal
approximation (10) with the same α1 = 6.0± 0.1 as in Figs 2.
Of course, the two solutions with these asymptotes should be alternatively stable (unstable) in different regions of
scales. It is expected, that the local (Kolmogorov’s) solution is stable (i.e. statistically dominating) in inertial range
(that means instability of the non-local solution in this range of scales).
Roughly speaking, in non-local solution for small scales r only non-local interactions with large scales L (1 ≫ r/L)
are dynamically significant (the interaction among the small scales is negligible compared with interaction via large
scales) and the non-local interactions is determined by large scale strain/shear. This means that one should add to
the energy flux 〈ε〉-parameter (which is a governing parameter for the both solutions) an additional parameter such
as the strain s for the non-local solution. As far as we know it was noted for the first time by Nazarenko and Laval
[18] that dimensional considerations applied to the non-local asymptote result in the power-law energy spectrum
E(k) ≃ c
k−1 (8)
both for two- and three-dimensional cases. Linear dependence of the spectrum (8) on 〈ε〉 is determined by the linear
nature of equations corresponding to the non-local asymptote that together with the dimensional considerations
results in (8) [18]. Interesting numerical simulations were performed in [19]. In these simulations local and non-local
interactions have been alternatively removed. For the first case a tendency toward a spectrum flatter than ’-5/3’ is
observed near and beyond the separating scale (beyond which local interactions are ignored), that supports Eq. (8).
Following to the perturbation approach suggested above both local and non-local regimes can be corrected. The
first order correction is
E(k) ≃ K〈ε〉2/3k−5/3e−β(k/kd) (9)
E(k) ≃ c
k−1e−α1(k/kd) (10)
for the local and non-local regimes respectively (K, c, α1 and β are dimensionless constants).
Figure 1 shows one-dimensional spectra measured in nearly isotropic turbulence downstream of a grid at small
Reynolds numbers (the data are reported in [1]). The solid curves are drawn in the figure to indicate correspondence
of the data to the equation (10) (non-local regime). The experimental data, however, is not controlled enough to
study fine properties of the isotropic turbulence. Therefore below we will mainly use the data obtained in numerical
simulations.
It is shown in [8] that there is an ’exchange of stability’ phenomenon at certain kc = 1/rc. That is, for k < kc the
Kolmogorov’s regime is stable and the non-local regime is unstable, whereas for k > kc the Kolmogorov’s regime is
unstable and the non-local regime is stable. For this scenario, at k = kc the Kolmogorov’s regime is still asymptotically
scale-invariant (i.e. Eq. (7) gives an adequate approximation for this regime), while for the non-local regime the first
order correction is substantial (i.e. Eq. (10) should be used at k > kc for the non-local regime). In this scenario the
Kolmogorov’s regime plays significant role in the viscous stabilization of the non-local regime for k > kc (see [19]), but
for these k the non-local regime becomes statistically dominating instead of the Kolmogorov’s one (it is well known
that only for energy spectra steeper than k−3 it can be rigorously proved that the dominant interactions are nonlocal
(cf Eq. (10) that provides such steepness).
It should be noted that in [20] a second scaling solution E(k) ∝ Pk−1 was obtained in addition to the Kolmogorov
scaling using the Clebsch formulation of hydrodynamics. The prefactor P = (〈ε〉ν)1/2 denotes a certain flux in the
wavenumber space [20]. It will be shown below that the prefactor in Eqs. (8),(10) can be transformed into the one
formally equal to P (see Eq. (15) and Fig. 4). This could mean that the studied nonlocal regime (more precisely:
its scaling asymptote) is closely related to the Yakhot-Zakharov scaling solution introduced in the Ref. [20]. Such
identification (if valid) allows us understand instability of the nonlocal regime for large scales (small k). Indeed, it is
shown in [21] that reconnection process breaks conservation of the integral determining the flux P for the large scales
(small k) and , therefore, there is no possibility for realization of the scaling asymptote k−1 of this solution. Following
to Newell [21][22] the nonconservation of the integral in large scales generally follows from nonlocality of the viscosity
term in the equations formulated for the Clebsch variables. Since for the large scales the nonlocal regime should be
represented by its asymptote E ∝ k−1 (as an intermediate asymptote [23] the nonconservation of the integral makes
this regime unstable for the large scales (small k).
Figure 2 shows three-dimensional energy spectra calculated using data from a high-resolution direct numerical
simulation of homogeneous steady three-dimensional turbulence [12] for different Reynold numbers up to Rλ = 125.
The solid curves in this figure corresponds to the best fit by equation (10) (nonlocal interactions). The exponent
α1 ≃ 6.0±0.1 for all considered values of Rλ. The arrows indicate the rc = 1/kc scales calculated using corresponding
DLLL(r) (see [8]).
In figure 3 we show the data obtained in the same DNS as those shown in Fig. 2 but for Rλ = 460. In the upper
part of this figure we show the energy spectrum in the compensate (according to Kolmogorov’s scaling Eq. (7)) form.
One can clear see the hump corresponding to the bottleneck effect. The solid curves are the best fit to the the same
nonlocal spectrum (10) with the same (universal) value of α1 ≃ 6.0 ± 0.1 (see also below). One can see that the
bottleneck effect is determined by just the same nonlocal interactions as the above considered energy spectra for small
Reynolds numbers.
IV. STRAIN AND VISCOSITY
It is shown in Ref. [8] that for sufficiently large Reynolds numbers, providing a visible inertial interval, there is
an overlapping between the two regimes: non-local and local (Kolmogorov). This overlapping is based on the very
nature of the stability exchange between the two statistical regimes. However, in analogy with the viscous scale
η = (ν3/〈ε〉)1/4 [24] there should be a scale ηs such that for scales r < ηs contribution of the local interactions will
be drastically decreased in comparison with the nonlocal ones. In the analogy with η one can calculate ηs using the
dimensional considerations as
(cf for shear flows [25]).
The dynamical equations provide us with a relationship [24]
〈ε〉 = 2ν
k2E(k)dk (12)
For E(k) given by Eq. (10) the dissipation function k2E(k) has its maximum at k = kd/6. If kd ≫ ks = 1/ηs (see
below) the maximum of the dissipation function k2E(k) is located just between ks and kd. Therefore, we can estimate
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
(<ε>ν)1/2
38 - 460
FIG. 4: The prefactor in the nonlocal approximation to the energy spectra (10): c〈ε〉/s - circles, versus (〈ε〉ν)1/2. The straight
line with the slope equals to 18 indicates agreement with Eq. (15).
(12) as
〈ε〉 ≃ 2ν
k2E(k)dk (13)
Substituting (10) (with α1 = 6) into (13) we obtain relationship
〈ε〉1/2ν−1/2 (14)
Using the relationship (14) one can estimate the prefactor in the nonlocal approximation to the energy spectra (10)
≃ 18(〈ε〉ν)1/2 (15)
Now using the data of the DNS [12] (cf Figs. 2,3) let us calculate the prefactor c〈ε〉/s. Results of these calculations
are shown in figure 4 as circles (Reλ = 38, 54, 70, 125, 284, 380, 460). This figure shows the prefactor c 〈ε〉/s against
(〈ε〉ν)1/2. The straight line with the slope equals to 18 indicates agreement with Eq. (15).
It should be noted that in the DNS [12] 〈ε〉 ≃ const for Rλ ≥ 70, in agreement with the well known Kolmogov’s
hypothesis [24]. Therefore, for Rλ ≥ 70 (when 〈ε〉 ≃ const [12]) it follows from Eq. (14) that s ∝ ν
−1/2. This
relationship provides us also with dependence of the strain s on Rλ.
Due to the nonlinear character of the Navier-Stokes equations the so-called triadic type of interactions is dominating
mechanism of the dynamical interactions in turbulence (see, for instance, [9]). A triad corresponding to the nonlocal
interactions involves two short-wave-number modes and one long-wave-number mode (see a sketch in figure 5). Ac-
cordingly, at the nonlocal interactions two characteristic space scales are actively involved: large-scale characteristic
scale ηs (11) and small-scale (viscous or Kolmogorov) characteristic space scale η = (ν
3/〈ε〉)1/4. It is naturally that
the large scales should be normalized by ηs while the small scales should be normalized by η. This could cause
an obvious problem at the nonlocal interactions. However, the large and small scales covariance relationship at the
nonlocal interactions follows directly from the relationship (14)
≃ const (16)
(i.e. the relation ηs/η is independent on Rλ). And vice versa, the large and small scales covariance (15) results in the
relationship of the type (14). The covariance at the nonlocal interactions supports right balance between the energy
flux to the small scales and their dissipative capacity (cf, for instance, [26],[27]).
FIG. 5: Sketch of the nonlocal triadic interaction in the physical and in the Fourier space
Actually, the scale ηs = 1/ks should provide an edge of applicability of the approximation (10) to the real energy
spectra. If one compares the ηs determined by this way from the Figs. 2,3,6 one can see that indeed ηs/η ≃ const
in agreement with Eq. (16) (we have also checked this with the data for Rλ = 284, 380 [12] and the data reported
in [29]). Moreover, taking into account a continuity condition of the energy spectrum at the point k = ks (and using
Eqs. (7) and (10)):
K 〈ε〉2/3k−5/3s ≃ c
k−1s e
−α1(ks/kd) (17)
we obtain equation
K ≃ c exp[−(2c)3/2/α21] (18)
Substituting the Kolmogorov constant K ≃ 1.6 (see [12],[28]) and α1 ≃ 6 into this equation we obtain c ≃ 2. Then,
substituting this value of c into Eq. (16) we obtain ks/kd ≃ 0.037. The last value is in agreement with the available
data (see Figs. 2,3,6 and [29]).
Together with the universality of α1 ≃ 6.0 (which also is a consequence of the scale covariance) Eq. (16) determines
the well known from DNSs [12],[29] universality (independence on Rλ) of the position of the ’hump’ in the axes kη
for the bottleneck effect.
V. PASSIVE SCALAR
For passive scalar θ in the isotropic turbulence the equations (9),(10) should be replaced [8] by the equations
Eθ(k) ∝ 〈ε〉−1/3〈εθ〉 k
−5/3e−γ(k/kd) (19)
Eθ(k) ∝
k−1e−δ(k/kd) (20)
for the local and non-local regimes respectively (γ and δ are dimensionless constants, and 〈εθ〉 is the average value of
dissipation rate of scalar variance).
Figure 6 shows three-dimensional passive scalar spectrum from a DNS performed in [30] for Péclet number Pλ = 427
(the Schmidt number is unity, i.e Pλ = Rλ). The solid curves in this figure correspond to the best fit by equation (20)
(nonlocal interactions). In the upper part of this figure we show the spectrum in the compensate (according to the
Corrsin-Obukhov scaling [30]) form. One can clear see the hump corresponding to the bottleneck effect. The arrows
show position of the ηs scale. Comparing with Figs. 2,3 one can see that for the passive scalar ηs/η takes the same
universal value as for the velocity field (the large and small scales covariance (15)).
0.001 0.01 0.1 1
0.001 0.01 0.1 1
FIG. 6: The passive scalar DNS data (circles) for homogeneous isotropic turbulence described in [30], Péclet number Pλ = 427
and the Schmidt number is unity (i.e Pλ = Rλ). The solid curves in this figure correspond to the best fit by equation (20)
(nonlocal interactions).
Acknowledgments
I thank K.R. Sreenivasan for inspiring cooperation. I also thank T. Nakano, D. Fukayama and T. Gotoh for sharing
their data and discussions.
[1] G. Comte-Bellot and S. Corrsin, J. Fluid Mech., 48, 273 (1971).
[2] J. Schumacher, K. R. Sreenivasan, and V. Yakhot, New Journal of Physics 9, 89 (2007).
[3] R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioloi, and S. Succi, Phys. Rev. E., 48, R29 (1993).
[4] T. Gotoh and D. Fukayama, Phys. Rev. Lett. 86, 3775 (2001).
[5] G. Falkovich, Phys. Fluids 6, 1411 (1994).
[6] D. Lohse and A. Muller-Groeling, Phys. Rev. Lett., 74, 1747 (1995); Phys. Rev. E 54, 395 (1996).
[7] M.K. Verma and D. Donzis, J. Phys. A, 40, 4401 (2007).
[8] A. Bershadskii, J. Stat. Phys., online first: DOI 10.1007/s10955-007-9322-0 (see also arXiv:nlin.CD/0603070).
[9] J.A. Domaradzki, Phys. Fluids A, 4, 2037 (1992).
[10] K.R. Sreenivasan and A. Bershadskii, J. Fluid. Mech. 554, 477 (2006).
[11] P.R. Woodward, D.H. Porter, B. K. Edgar, S. E. Anderson, and G. Basset, Comput. Appl. Math. 14, 97 (1995).
[12] T. Gotoh, D. Fukayama and T. Nakano, Phys. Fluids, 14, 1065 (2002).
[13] K.R. Sreenivasan, J. Fluid Mech., 151, 81 (1985).
[14] C. Foias, O. Manley, and L. Sirovich, Phys. Fluids, A 2, 464 (1990).
[15] Z-S. She and E. Jackson, Phys. Fluids A, 5 1526 (1993).
[16] M. Nelkin, Adv. Phys. 43, 143 (1994).
[17] B.B. Kadomtsev, Plasma Turbulence (Academic Press, New York, 1965).
[18] S. Nazarenko and J.-P. Laval, J. Fluid Mech., 408, 301 (2000).
[19] J-P. Laval, B. Dubrulle and S. Nazarenko, Phys. Fluids, 13: 1995 (2001).
[20] V. Yakhot and V. Zakharov, Physica D, 64 379 (1993).
[21] S.V. Nazarenko, Physica D, 102 343 (1997).
http://arxiv.org/abs/nlin/0603070
[22] A.C. Newell, private communication.
[23] G.I. Barrenblatt, Scaling, self-similarity, and intermediate asymptotics (Plenum Press, New York/London, 218 p., 1979).
[24] A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics, Vol. 2, (MIT Press, Cambridge 1975).
[25] F. Toschi, G. Amati, S. Succi, R. Benzi, and R. Piva, Phys. Rev. Lett., 82, 5044 (1999).
[26] B. Dubrulle and J. Graner, J. Phys. II, 6, 797 (1996).
[27] A. Pocheau, Europhys. Lett, 35, 183 (1996).
[28] K. R. Sreenivasan, Phys. Fluids 7, 2778 (1995).
[29] T. Ishihara, Y. Kaneda, M. Yokokawa, K. Itakura and A. Uno, J. Phys. Soc. of Japan, 74, 1464 (2005).
[30] T. Watanabe and T. Gotoh, New J. Phys. 6: Art. No. 40. (2004)
	Introduction
	 Perturbations to scaling
	 Nonlocal interactions
	 Strain and viscosity
	Passive scalar
	Acknowledgments
	References
