PHYSICAL REVIEW E 90, 043019 (2014)

Reynolds number of transition and self-organized criticality of strong turbulence Victor Yakhot Department of Mechanical Engineering, Boston University, Boston, Massachusetts 02215, USA (Received 29 August 2013; revised manuscript received 15 July 2014; published 27 October 2014) A turbulent flow is characterized by velocity fluctuations excited in an extremely broad interval of wave numbers k > f , where f is a relatively small set of the wave vectors where energy is pumped into fluid by external forces. Iterative averaging over small-scale velocity fluctuations from the interval f < k  0 , where η = 2π/0 is the dissipation scale, leads to an infinite number of “relevant” scale-dependent coupling constants (Reynolds numbers) Ren (k) = O(1). It is shown that in the infrared limit k → f , the Reynolds numbers Re(k) → Retr , where Retr is the recently numerically and experimentally discovered universal Reynolds number of “smooth” transition from Gaussian to anomalous statistics of spatial velocity derivatives. The calculated relation Re(f ) = Retr “selects” the lowest-order nonlinearity as the only relevant one. This means that in the infrared limit k → f , all high-order nonlinearities generated by the scale elimination sum up to zero. DOI: 10.1103/PhysRevE.90.043019

PACS number(s): 47.27.Ak, 47.27.Gs

I. INTRODUCTION

“The turbulence problem” can be formulated as follows: consider the Navier-Stokes (NS) equations driven by the large-scale force F(f ) where f denotes a relatively small set of wave vectors |k| ≈ 2π/L. We fix both force F = O(1) and the integral scale L = 2π/f = O(1) independent upon Reynolds number, and by decreasing kinematic viscosity ν, vary the Reynolds number Re = uL/ν, where u(r,t) is a solution to the NS equations of motion. As long as ν > νtr (Re < Retr ) the flow is laminar; i.e., u = u(k) with k ≈ f . At Re = Retr (ν = νtr ), the solution u = u0 (f ) becomes unstable and at Re > Retr , the velocity field can be written as u(k) = u0 (f ) + v(k,t), where k > f . Formation of the small-scale time-dependent velocity components v(k,t) is the main manifestation of transition to turbulence. We would like to stress a relatively trivial, but extremely important for what follows, statement: at Re  Retr the laminar flow pattern u0 (f ) is a solution to the Navier-Stokes equations characterized by a single coupling constant, which is a properly chosen Reynolds number. When the Reynolds number Re  Retr , the flow is characterized by velocity fluctuations v(k) excited in a broad interval of scales f  k  0 , and in the limit Re → ∞, the ratio f /0 → 0. Below we set the UV cutoff 0 equal 3 to Kolmogorov’s dissipation scale 0 ≈ f Re 4 . In this case, as will be shown below, the equation for turbulent fluctuations is similar to the Navier-Stokes equation with the broad-band “force” fj = −vi ∂i u0,j in the right side. Following K. G. Wilson [1], we can average the governing NS equations over velocity fluctuations from a thin “slice” in the wave-vector space (r) = 0 e−r  k  0 with r → 0. In this case this exact procedure leads to equations for the remaining long-wave-length modes v(k) from the interval k < 0 e−r . The main problem is that the derived equation, in addition to corrections to viscosity [ν(r) → ν + ν(r)] and driving force f, includes an infinite number of coupling constants (see below), which, unlike in the theory of critical phenomena, are relevant when r  1. Since ν(r) > 0, the r-dependent Reynolds number Re[(r)] < Re(0 ). Iterating this procedure one can derive equations for the modes with k  0 belonging to the so-called inertial range where molecular 1539-3755/2014/90(4)/043019(8)

(“bare”) viscosity ν is irrelevant and all coupling constants can rms (r) . depend on the “dressed” Reynolds number Re(r) = 2πv ν(r)(r) By dimensional reasoning, ν(r) ≈ vrms (r)/(r) and we see that in the inertial range Re(r) = O(1). This means that the resulting equations include infinite number of nonlinearities generated by the scale-eliminating procedure. In the inertial range, all these terms are relevant and are responsible for anomalous scaling and nonanalyticity of velocity increments. All past theories of turbulence dealt with this problem using various closures, i.e., ad hoc truncations of renormalized perturbation expansions. In his direct interaction approximation (DIA), resulting from the lowest order of a diagrammatic perturbation expansion, Kraichnan derived 3 the energy spectrum of developed turbulence E(k) ∝ k − 2 , which contradicted experimentally observed E(k) ∝ k −γ with γ ≈ 5/3 [2]. Also, this result contradicted Kolmogorov’s phenomenology stating that neither integral scale nor viscosity can enter the inertial range dynamics. The problem with Kraichnan’s closure was first explained by Kadomtsev who attributed it to the infrared divergence of the DIA leading to gross overestimation of the role of the large-scale fluctuations in the small-scale dynamics [3]. Later, to regularize the theory, Kraichnan proposed his Lagrangian description (LHDIA), which removed kinematic effect of advection of small-scale “eddies” by the large-scale ones, thus leading to Kolmogorov’s energy spectrum [4]. This success stimulated subsequent efforts based on the assumption that the infrared divergencies result exclusively in simple kinematic effects, which can be removed by Galileo transformation. Still, the divergencies appearing in high-order terms remained not understood and unaccounted for. Following Orszag’s work, Fournier and Frisch considered a small-time (t → 0) expansion of the d-dimensional Navier-Stokes equations and, associating the cutoff in the time integration with Kolmogorov’s eddy turnover time (EDQNM approximation), they got rid of kinematic effects and, naturally, derived Kolmogorov’s spectrum [5]. Attempts to account for the second-order contributions to the series (“two-loop theory”) did not lead to any improvement and new insights. More recent application of the dynamic renormalization group to turbulence problem was based on a mathematically unjustified ad hoc truncation of the series in powers of the O(1) parameter ( expansion). The procedure

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led to various dimensionless amplitudes and turbulence models widely used in engineering. However, unlike in the theory of critical point [1], the validity of truncation of an infinite series in powers of = 4 remained a mystery. (For a more detailed discussion of different closures, see review by U. Frisch [5].) The theory developed below is based on a new understanding of the physical meaning of parameter appearing in the old renormalization group approach to turbulence. It will be shown below that in the infrared limit k → f , the expansion parameter = 4, leading to accurate derivation of dimensionless amplitudes, is indeed a small parameter, which justifies low-order truncations of perturbation series. According to the picture described above, the integral scale L = 2π/f is the largest scale of turbulence, i.e., (r)  f . It will be shown below that as (r) → f the effective viscosity ν[(r)] → νtr and Re[(r)] → Retr . Since at this Reynolds number the marginally stable flow, described by the NavierStokes equations with ν = νtr is laminar, we may conclude that all high-order terms, additional to the NS equations, must disappear and only the lowest-order quadratic nonlinearity survives. This way transition to turbulence “selects” the only relevant nonlinearity. Assuming validity of Landau’s theory of transition, we will be able to conclude that in the limit (r) → f , high-order nonlinearities generated by the scale√ elimination procedure are O( Re((r)) − Retr ) → 0. A. Transition to turbulence

There is an abundance of literature on this topic, which, together with the theory of dynamical systems, evolved into a separate field of research. Typically, one searches for instabilities in laminar flow u0 manifested by exponential growth of perturbations u(k,t). We will loosely identify laminar flow as a pattern u0 formed by a small set of excited modes supported in the range of wave numbers k ≈ f . All modes with k > f are strongly overdamped; i.e., u(k) = 0 for both k  f and k  f . 1. Landau’s theory

Here we mention just one work that is relevant for the considerations presented below [6]. Assuming that in the vicinity of a transition point imaginary part of complex frequency is much smaller than the real one, Landau considered the linearized Navier-Stokes equations for incompressible fluid. Denoting the velocity field at a transition point u0 and introducing an infinitesimal perturbation u1 , he wrote u = u0 + u1 with u1 = A(t)f (r). Based on general qualitative considerations, Landau proposed d|A|2 = 2γ |A|2 − α|A|4 , dt where in the vicinity of transition point γ = c(Re − Retr ) and α > 0. In principle, |A|2 must be considered as time-averaged. Landau noted, however, that u1 (k) is a slow mode and, since the averaging is taken over relatively short time intervals, the averaging sign in the above equations is not necessary. At small times the solution √exponentially grows and then reaches the maximum Amax ∝ Re − Retr . When γ = Re − Retr < 0, any initial perturbation decays. In this theory, the magnitude of transitional Reynolds number is a free parameter and since

the large-scale field u0 strongly depends on geometry, external forces, and stresses, the transition Reynolds number Retr is not expected to be a universal constant. Landau assumed that further increase of the Reynolds number leads to instability of first unstable mode generating next two excited modes with the wave vectors k2 > k1 , etc. In modern lingo, this process can be perceived as an onset of the energy cascade toward small- scale excitations with k > f . This leads to formation of “inertial range” and strongly intermittent small-scale dissipation rate E. 2. Transition to turbulence: A new angle

A new way of looking at phenomenon of transition to turbulence was introduced in numerical simulations of a flow at  5 a relatively low Reynolds number Rλ = 3Eν u2rms  4.0 [7]. In this approach, transition to turbulence is identified with the first appearance of non-Gaussian anomalous fluctuations of velocity derivatives, including those of the dissipation rate E. As will be shown below, in a sense, it is a transition between two different states: Gaussian (structureless) and anomalous (structured), resembling those observed in experiments on Benard convection. On a first glance, the transition is smooth, meaning that no “jumps” in velocity field were detected. However, the precise nature of this transition is yet to be investigated. All we can state at this point is that the transformation happens in a narrow range of the Reynolds number variation. The homogeneous and isotropic turbulence (HIT) was generated in a periodic box by a force in the right δ , side of the Navier-Stokes equation F(k,t) = P  u(k,t) |u(k,t)|2 k,k where summation is carried over kf = (1,1,2); (1,2,2). It is easy to see that the model with this forcing generates flows ∂ui 2 with constant energy flux P = E = ν( ∂x ) = const and the j variation of the Reynolds number is achieved by variation of viscosity. The results of Ref. [7] can be briefly summarized as follows: (1) Extremely well-resolved simulations of the low-Reynolds number flows at Rλ  9 − 10 revealed a clear scaling range n 2

∂u n ∂u 2 Mn = ( ∂x ) /( ∂x ) ∝ Reρn with anomalous scaling exponents ρn consistent with the inertial range exponents typically observed only in very high Reynolds number flows Re  Retr . Identical scaling exponents ρn were later obtained in isotropic turbulence generated by a different forcing [8], channel and pipe flows [9], and, more recently, in Benard convection [10], indicating possibility of a broad universality. (2) For Rλ < 9 − 10 all flows were sub-Gaussian, indicating a dynamical system consisting of a small number of modes with the small-scale fluctuations strongly overdamped. This flow can be called “quasilaminar” or coherent. (3) At a transition point Rλ,tr ≈ 9–10 the fluctuating velocity derivatives obey gaussian statistics and at Rλ > 9–10 a strongly anomalous scaling of the moments, typical of high-Reynolds number turbulence, is clearly seen. (4) It has also been noticed that transition was − + smooth, i.e., velocity field at u(Rλ,tr ) − u(Rλ,tr ) → 0.

On Fig. 1 the normalized moments EE n of the dissipation rate computed in Ref. [7] are combined with the data obtained by Donzis et al. [8] in HIT generated by a completely different large-scale forcing. We can see that the scaling exponents,

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that at Rλ = 9–10 a relatively sharp transformation from a sub-Gaussian at Rλ < 9–10 to anomalous scaling of the dissipation rate moments occurs independently on the driving force. This surprising result will be used below as a constraint on development of turbulence models.

6

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n=2 n=3 n=4 n=5 n=2 (DSY) n=3 (DSY) n=4 (DSY)

Re1.519 0.056

< n>/< >n

II. THE MODEL

4

10

Based on these results, we consider a flow generated by the Navier-Stokes equations with a force F(f ). Keeping the force F = const and the length-scale L = 2π/f = const, let us vary viscosity in the interval 0  ν  ∞. In the range ν > νtr or Re < Retr the flow is laminar in a sense that it is described by a relatively small number of modes with u(k) with k ≈ f . At the transition point Reλ,tr ≈ 9–10 the transitional pattern u0 (f ) is formed, so that

Re0.916 0.019

3

Re0.455

10

L(u0 ,νtr ) = Re0.139

D Here, Dt ≡ equations read

2

10 2 10

3

4

10

10

Re

FIG. 1. (Color online) Normalized moments of the dissipation n rate EE n in homogeneous and isotropic turbulence. References [7] and [8]. DSY stands for Donzis, Sreenivasan, and Yeung.

found in the range of very low Reynolds number in Ref. [7], hold in a much wider range of the Reynolds number variation. This means that in the range Rλ  10, turbulence can be considered as fully developed. On Fig. 2 the moments Mn of velocity derivatives are shown in the vicinity of a transition point. One can see

∂ ∂t

Du0 + ∇p − νtr ∇ 2 u0 − F(f ) ≡ 0. Dt + u0 · ∇. When ν  νtr , the Navier-Stokes

Du = −∇p + ν∇ 2 u + F(f ) = 0. Dt If we write u = u0 + v, the equation for the “turbulent” component (k > f ) of velocity field: ∂v + v · ∇v = −∇p + ν∇ 2 v + f, ∂t

(1)

with f = f1 + f2 + f3 = −u0 · ∇v − v · ∇u0 + (ν − νtr )∇ 2 u0 (f ).

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10

(b)

3

Mn

10

(c)

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0

10 −1 10

0

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R

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λ

FIG. 2. (Color online) Normalized moments of velocity derivatives in isotropic and homogeneous turbulence. Numerical simulations from δ (for more details, see Refs. [7] and [8]. Right: Schumacher et al. [ 7]: homogeneous turbulence driven by a force F(k,t) = P  u(k,t) |u(k,t)|2 k,k above). Left: results of Donzis, Yeung, and Sreenivasan [8]; HIT driven by a Gaussian large-scale noise. In the range Rλ  9.0, we see a clear Gaussian behavior of a few moments with S3 = 0, M4 ≈ 3, M6 ≈ 15, M8 ≈ 105, typical of Gaussian distribution. At Rλ  9 − 10, all moments obey anomalous scaling of fully developed turbulence. 043019-3

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Based on the above argument, the force Eq. (2) is extrapolated onto interval f < k   ≡ 0 , so that fi Fj = 0 and, by construction, fi (k  f ,t) = 0. A. The renormalization or coarse graining

FIG. 3. Length scales and turbulent kinetic energies. K and Kr denote energy contents in inertial and dissipation ranges, respectively. L and L ≈ η stand for integral scale, corresponding to the top of inertial range and the dissipation scale, respectively. Straight line: 2 5 E(k) ≈ 1.6E 3 k − 3 .

The first term f1 in this expression describes kinematic transfer of small “eddies” by the large ones. The second, f2 , is responsible for turbulence production due to interaction of small-scale fluctuations with the large-scale, quasicoherent, flow u0 . This effect is well-known in the turbulence modeling literature. Thus, the total energy production rate is:  j 2 ∂u0,i − 4  j 2 P = f2 · v = −vi vj ≈ νT u0,i ≈ f 3 u0,i . ∂xj The balance can be written for each scale l = 2π/(r), but 1 4 with “turbulent viscosity” ν(r) ≡ ν(l) ∝ E 3 l 3 and introducing the projection operator Plmn = km Pln (k) + kn Plm (k) with kk Pi,j = (δij − ki 2j ) we have with kˆ = (k,ω):  i ˆ 0,n (kˆ − q)d ˆ q, ˆ f2,l = − Plmn (k) v(q)u 2 so that f2 = 0 and  13 q − 3  j 2 ˆ ˆ ˆ δ(k − q)δ(ω − )d qdω ∝ k −3 . P(k) ∝ k 2 4 u0,i 2 + q 3 The v fluctuations are driven by a pumping with algebraic spectrum. The experimental data of Refs. [7–10] point to independence of small-scale features of turbulence on the nature of production mechanism. On Fig. 3 the energy spectrum in a flow past circular cylinder of diameter D is shown for the large-scale Reynolds number Re = U D/ν ≈ 106 − 107 . The onset of Kolmogorov’s inertial range can be clearly seen at the wave number k ≈ f = 2π/L, separating inertial and nonuniversal, geometry-dependent energy-containing range of scales. This interval separating these two asymptotic limits is quite narrow, which points to the smallness of subleading contributions to the inertial range scaling of the energy spectrum. Therefore, for ν < νtr we choose the wellknown and well-studied model Eq. (1), where the random force, mimicking small-scale fluctuations, is defined by the correlation function:

The renormalization group for fluid flows has been developed in Refs. [11] and [12] and was generalized to enable computations of various dimensionless amplitudes in the low order in the expansion in Refs. [13–16]. Introducing velocity  and length and time scales U =

D0 /(ν0 20 ), X = 1/0 ,

respectively, Eq. (1) can be written as (for and T = simplicity we do not change notations for dimensionless variables): tν0 20 ,

∂v f + λˆ 0 v · ∇v = −λˆ 0 ∇p + ∇ 2 v +  , ∂T D0 ν0 2 where the single dimensionless coupling constant (“bare” 2 0 Reynolds number) is λˆ0 = νD 3 .  0

0

1. Projecting Navier-Stokes equation onto domain k  0 e−r where r → 0

Technical details of all calculations presented below are best described in Ref. [14]. Formally introducing modes v < (k,t) and v > (k,t) with k from the intervals k  0 e−r and −r 0  k  0 , respectively, and averaging over small-scale fluctuations v> , leads to equation for the large-scale modes: ∂σij(2) ∂vi< < < + vj · ∇ j vi = −∇ i p + + ν∇ 2 vi< + fi + fi , ∂t ∂xj (3) where the second-order correction to the Reynolds stress σij = −vi vj is D σij(2) = λˆ 2 (r)ν(r)Sij − λˆ 41 (r)ν(r) [τ (r)Sij ] Dt  ∂ui ∂uj −λˆ 41 (r)ν(r)τ (r) β1 ∂xk ∂xk

∂uj ∂uk ∂ui ∂uk + β2 + ∂xk ∂xj ∂xk ∂xi   ∂uk ∂uk +β3 + O λˆ 61 + · · ·, ∂xi ∂xj

where λˆ 1 = O[λˆ 0 (e−r − 1)] is a coupling constant generated by the scale-elimination and the time-constant 1/τ (r) = ν(r)2 (r). In this limit, the coefficients βi can be explicitly calculated [17]. The “dressed” viscosity is denoted as ν(r) = ν + ν(r) with correction to “viscosity” written in the wavenumber space as 

2 ( +2)r  4 D0 e r − 1 −1 k e ˆ0 , ν = Ad 2 + O λ + O  0 +2 ν0  +2 0 (5) Sd 1 d −d ˆ where = 4 + y − d and Ad = Aˆd (2π) d ; Ad = 2 d(d+2) . On −r the interval k < 0 e , Eqs. (3–5) are equivalent to the original equations of motion defined on the interval k  0 . 2

fi (k,ω)fj (k ,ω ) = 2D0 (2π )d+1 k −y Pij (k)δ(k + k )δ(ω + ω ).

(2)

(4)

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2. Iterating scale-elimination procedure

The relations Eqs. (3–5) are exact as long as the eliminated “slice” in the wave-vector space is very thin. As will be shown below, in the limit (r)  0 , the high-order contributions to the Reynolds stress are not small. The problem is that due to proliferation of tensorial indexes, these terms, while they can be qualitatively analyzed using Wyld’s diagrammatic expansion [18], are very hard to calculate. Eliminating the modes from the interval 0 e−r  k  0 , the equations can be formally written ∂vi< + vj< · ∇ j vi< ∂t = −∇ i p + (ν + ν)∇ 2 vi< + f + fi + H.O.T., (6) where, by Galileo invariance, high-order (n > 1) contributions generated by the scale elimination can be formally written ∞ 

2n n−1 < < n H.O.T. = λˆ1 (r)τ (r)(∂t v + v · ∇) v
Retr , are contained in the Navier-Stokes equations.

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Therefore, it is not surprising that the recently discovered universal Reλ,tr ≈ 9–10 can serve as a dynamic constraint on the theory. (1) The physical picture emerging from this work is that of self-organized criticality. First, a large-scale transitional pattern u0 is formed at Re = Retr , which becomes unstable at Re > Retr . This instability leads to the “leaking” of kinetic energy into small- scale fluctuations (“energy cascade”). The small-scale fluctuations “dress” bare viscosity ν(f ) = νtr bringing the large-scale Reynolds number back to Re(f ) = Retr and the velocity field to u0 . Thus, the almost-Reynoldsnumber-independent transitional field u0 is marginally stable. The large-scale Gaussaian fluctuations from the range k < f randomly advect (“shake”) this pattern in the physical space. (2) One of the most difficult, not yet understood properties of turbulence is anomalous scaling of velocity increments on the small scales η  l  L = 2π/f . It was shown above that in this range of scales the flow is described by an infinite number of O(1) coupling constants. At this point we do not know how to deal with them. (3) The selection of relevant variables is possible in the limit l → L. Keeping only the lowest-order contributions, the calculated fixed-point Reynolds number Ref ≈ Reλ,tr , where Reλ,tr = 9–10 is the numerically computed Reynolds number of transition to turbulence. Since the numerically discovered transition is “smooth,” i.e., u0 = ufp and (∇ i u0,j = ∇ i ufp,j ), at this point all high-order nonlinearities are irrelevant. (4) Comparison with Landau’s theory of transition to turbulence shows that in the vicinity √ of transition point, the neglected nonlinear terms are O( Re − Retr ) → 0. (5) The infrared divergencies appearing in the each term of the expansion do not disappear but are summed up into equations of motion for the large-scale features of the flow. To stress how accurate the derived transport approximation is, we would like to reproduce our old result on decay of isotropic and homogeneous turbulence [14]. Since in this flow S i,j = 0, the equations governing decay of kinetic energy are very simple:

neglecting of H.O.T. The above equations give − C 1 −1 −γ −1.47 ,2 t t t = ≈ . K/K0 = t0 t0 t0 This result has a long and difficult history. First, it was shown by Kolmogorov that γ = 10/7 ≈ 1.43, very close to the one shown above. Somewhat later, Kolmogorov’s construction has been reinterpreted by Landau as a consequence of conservation of the angular momentum [6]. This theory has been criticized by Batchelor et al. [19] and the early experiments, yielding γ ≈ 1.0 − 1.3 (see, for example, Ref.[ 20]), seemed to support Batchelor’s conclusions. A huge number of experimental, theoretical, and later numerical papers dealt with this subject [20]. As a consequence, the constant C ,2 in the K − E model, widely used for engineering simulations, was taken as C ,2 ≈ 1.92 corresponding γ ≈ 1.1. This led to the over-dissipated turbulent velocity field computed with this model. It took many years to realize that Kolmogorov’s theory was developed for a finite patch of turbulence in an infinite fluid and the exponent γ was very sensitive to the finite-size effects, geometry, etc. This longstanding confusion has recently been resolved by a remarkable (4 0963 ) numerical simulation by Ishida et al. [21], who showed that when the initially prepared flow satisfied constraints of Kolmogorov’s theory, the exponent of kinetic energy decay was indeed γ ≈ 10/7. (6) To conclude, I would like to mention that if, in general, a turbulent flow is generated by an instability of a large-scale quasicoherent flow pattern (dynamical system) u0 , then the equations of motion governing anomalous velocity fluctuations are given by Eq. (1) with f2 = v · ∇u0 . This may explain a broad universality of small-scale features of strong turbulence discovered in Refs. [7–10].

ACKNOWLEDGMENTS

∂K ∂E E = −E; = −C ,2 , ∂t ∂t K with C ,2 = 1.68 calculated at the integral scale in the lowest order of renormalized perturbation expansion [14]. The present paper justifies the approximation and procedure leading to this and all other constants calculated in Ref. [14] by an ad hoc

I am grateful to A. M. Polyakov, N. Goldenfeld, V. Lebedev, I. Kolokolov, Y. Sinai, U. Frisch, E. Titi, H. Chen, I. Staroselsky, and J. Wanderer for their interest in this work and numerous suggestions. Many ideas leading to this paper emerged from numerical investigations of transition conducted jointly with J. Schumacher, D. Donzis, and K. R. Sreenivasan. I am grateful to the late S. A. Orszag for close collaboration and significant contributions at the start of this work.

[1] K. G. Wilson, Rev. Mod. Phys. 12, 75 (1974). [2] R. H. Kraichnan, J. Fluid Mech. 5, 497 (1959). [3] B. B. Kadomtsev, Plasma Turbulence (Academic Press, London, 1965). [4] R. H. Kraichnan, Phys. Fluids 8, 575 (1965). [5] S. A. Orszag and M. D. Kruskal, Phys. Rev. Lett. 16, 441 (1966); U. Frisch and J. P. Fournier, Phys. Rev. A 17, 747 (1978); U. Frisch, Turbulence (Cambridge University Press, Cambridge, 1995). [6] L. D. Landau and E. M. Lifshits, Fluid Mechanics (Pergamon, New York, 1982). [7] J. Schumacher, K. R. Sreenivasan, and V. Yakhot, New J. Phys. 9, 89 (2007).

[8] D. A. Donzis, P. K. Yeung, and K. R. Sreenivasan, Phys. Fluids 20, 045108 (2008). [9] P. E. Hamlington, D. Krasnov, T. Boeck, and J. Schumacher, J. Fluid. Mech. 701, 419 (2012). [10] J. Schumacher, J. D. Scheel, D. Krasnov, D. A. Donzis, V. Yakhot, and K. R. Sreenivasan, Proc. Natl. Acad. Sci. USA 111, 10961 (2014). [11] D. Forster, D. R. Nelson, and M. J. Stephen, Phys. Rev. A 16, 732 (1977). [12] C. DeDominisis and P. C. Martin, Phys. Rev. A 19, 419 (1979). [13] V. Yakhot and S. A. Orszag, Phys. Rev. Lett. 57, 1722 (1986). [14] V. Yakhot and L. Smith, J. Sci. Comp. 7, 35 (1992).

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[15] V. Yakhot, S. A. Orszag, S. Thangam, T. B. Gatski, and C. G. Speciale, Phys. Fluids A 4, 1510 (1992). [16] W. P. Dannevik, V. Yakhot and S. A. Orszag, Phys. Fluids 30, 2021 (1987). [17] R. Rubinstein and M. Barton, Phys. Fluids A 12, 1472 (1990); H. Chen, S. A. Orszag, I. Staroselsky and S. Succi, J. Fluid Mech 519, 301 (2004). [18] H. W. Wyld, Ann. Phys. 14, 143 (1961).

[19] G. K. Batchelor and I. Proudman, Trans. R. Soc. London A 248, 369 (1956). [20] G. Compte-Bellot and S. Corrsin, J. Fluid Mech. 25, 657 (1966); A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, MA, 1975). [21] T. Ishida, P. A. Davidson and Y. Kaneda, J. Fluid. Mech. 564, 455 (2006).

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Reynolds number of transition and self-organized criticality of strong turbulence.

A turbulent flow is characterized by velocity fluctuations excited in an extremely broad interval of wave numbers k>Λf, where Λf is a relatively small...
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