A Gaussian approximation and steepest-descent estimation

# A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows

## Résumé

A phenomenological theory of the fluctuations of velocity occurring in a fully developed homogeneous and isotropic turbulent flow is presented. The focus is made on the fluctuations of the spatial (Eulerian) and temporal (Lagrangian) velocity increments. The universal nature of the intermittency phenomenon as observed in experimental measurements and numerical simulations is shown to be fully taken into account by the multi scale picture proposed by the multifractal formalism, and its extensions to the dissipative scales and to the Lagrangian framework. The article is devoted to the presentation of these arguments and to their comparisons against empirical data. In particular, explicit predictions of the statistics, such as probability density functions and high order moments, of the velocity gradients and acceleration are derived. In the Eulerian framework, at a given Reynolds number, they are shown to depend on a single parameter function called the singularity spectrum and to a universal constant governing the transition between the inertial and dissipative ranges. The Lagrangian singularity spectrum compares well with its Eulerian counterpart by a transformation based on incompressibility, homogeneity and isotropy and the remaining constant is shown to be difficult to estimate on empirical data. It is finally underlined the limitations of the increment to quantify accurately the singular nature of Lagrangian velocity. This is confirmed using higher order increments unbiased by the presence of linear trends, as they are observed on velocity along a trajectory.

Résumé Une théorie phénoménologique des fluctuations de vitesse Eulérienne et Lagrangienne dans un écoulement turbulent Nous présentons une théorie phénoménologique des fluctuations de vitesse dans un écoulement turbulent pleinement développé isotrope et homogène. Nous mettons l’accent sur les fluctuations des incréments spatiaux (Eulérien) et temporels (Lagrangien) de vitesse. La nature universelle du phénomène d’intermittence observé sur les mesures expérimentales et les simulations numériques est complètement pris en compte par les arguments développés par le formalisme multifractal, et ses extensions aux échelles dissipatives et au cadre Lagrangien. Cet article présente les prédictions de cette description multifractale et les compare aux données empiriques. En particulier, des prédictions explicites sont obtenues pour des grandeurs statistiques, comme les fonctions de densité de probabilité et les moments d’ordres supérieurs, des gradients de vitesse et de l’accélération. Dans le cadre Eulérien, à un nombre de Reynolds donné, nous montrons que ces prédictions ne dépendent que d’une fonction à paramètres, appelée spectre de singularités, et d’une constante régissant la transition entre les régimes inertiels et dissipatifs. Le spectre des singularités Lagrangien est relié à son homologue Eulérien par une transformation basée sur la nature incompressible, homogène et isotrope de l’écoulement, alors que la constante restante est difficile à estimer à partir des données. Nous montrons finalement que l’incrément est inadapté à quantifier précisément la nature singulière de la vitesse Lagrangienne. Cela est confirmé par l’utilisation d’incréments d’ordres supérieurs non biaisés par la présence de comportements linéaires, comme nous l’observons sur la vitesse le long d’une trajectoire. Pour citer cet article : L. Chevillard et al., C. R. Physique volume (année).

###### keywords:
Turbulence; Intermittency; Eulerian and Lagrangian Mots-clés : Turbulence; Intermittence ; Eulérien et Lagrangien

Received *****; accepted after revision +++++

## 1 Introduction

As a long-standing challenge in classical physics [1, 2, 3, 4, 5, 6, 7, 8, 9], fully developed turbulence is an archetypical non linear and non local phenomenon. When a flow is stirred at a large scale , typically the mesh size of a grid in a wind tunnel or the width of the blades in a von Karman washing machine, the input energy cascades towards the small scales without being dissipated according to the classical picture of Richardson and Kolmogorov [1, 2]. One of the most important objectives in turbulence research is to understand the processes that lead to this very peculiar cascading structure of energy. As far as we know, recent theoretical progress made in this long lasting field comes from the systematic analysis and description of empirical (experimental and numerical) velocity data.

The first experimental measurements of turbulent velocity were performed in the Eulerian framework and focussed on the longitudinal velocity profile. In this context, the Taylor hypothesis allowed to interpret the time dependence of the measurements obtained in a wind tunnel behind a grid or in an air jet as a spatial dependence [7]. These experiments gave access to the longitudinal velocity increments, i.e.

 δℓu(x)≡(u(x+\boldmathℓ)−u(x)).\boldmathℓℓ , (1)

where u is the velocity vector and a vector of norm , and the respective structure functions are given by

 Mn(ℓ)=⟨|δℓu|n⟩ . (2)

For such flows, the Reynolds number defined as

 Re=σLν , (3)

where is a characteristic velocity at the so-called integral length scale and the kinematic viscosity, can be considered as very large compared to unity. At these high Reynolds numbers, Kolmogorov showed, in a first seminal article [2] using a dimensional analysis, that the second order structure function behaves as a power law, i.e.

 M2(ℓ)=⟨(δℓu)2⟩=σ2(ℓL)2/3=cK⟨ϵ⟩2/3ℓ2/3 , (4)

where is the Kolmogorov constant of order unity [10, 11] and the averaged dissipation that will be properly defined latterly. Equivalently, the power spectrum, i.e. the Fourier Transform of the velocity autocorrelation function, follows the celebrated Kolmogorov law,

 E(k)∝cK⟨ϵ⟩2/3k−5/3 , (5)

where the proportionality constant can be calculated [10, 11]. As stated by Kolmogorov himself [2], the laws predicted (Eqs. 4 and 5) are only valid in a range of scales, called the inertial range (or equivalently ) delimitated by the Kolmogorov length scale under which dissipative effects dominate the physics. These laws have been successfully compared to empirical data [7]. If velocity fluctuations were Gaussian, these predictions could be easily generalized to higher order structure functions. As observed experimentally, the probability density functions (PDFs) of velocity increments undergo a continuous shape deformation, starting from the integral length scale at which statistics can be considered as Gaussian, down to the dissipative scales where the PDF is highly non Gaussian [12, 13, 14]. This phenomenon is a manifestation of the intermittent nature of turbulence, as first underlined by Kolmogorov and Obukhov [15, 16]. We show in Fig. 1(a) the estimation of the PDFs of longitudinal velocity increments obtained in the giant wind tunnel of Modane [17] at a high Taylor-based Reynolds number (Eq. 26 provides a link between the large-scale Reynolds number and ). The curves are arbitraly shifted vertically for the sake of clarity. This continuous shape deformation of these PDFs motivated Castaing et al. to build a statistical description of these longitudinal velocity fluctuations [12]. Moreover, by a simple visual inspection, we can remark that the PDFs are not symmetric. This is related to the Skewness phenomenon associated with the mean energy transfer of energy towards the small scales that takes place in the inertial range. We will come back to this point latterly.

The first part of this article is devoted to review the predictions that can be made using both the so-called multifractal formalism [7] and the propagator approach [12]. This description, that will be shown to depend only on a parameter function and a universal constant (independent on the flow geometry and the Reynolds number), accurately reproduces the non Gaussian features formerly presented.

More recently, experimental [18, 19, 20, 21, 22, 23, 24, 25, 26, 27]) and numerical [21, 22, 24, 28, 29, 30] data have revealed a similar phenomenon in the Lagrangian framework (see recent review articles [31, 33, 34]). Lagrangian velocity is defined as the Eulerian velocity of a fluid particle at the position , initially at the position via the following identity

 v(X(t0),t)=u(X(t),t) . (6)

In this framework, the study of the Lagrangian velocity fluctuations focusses on the Lagrangian time increment defined as

 δτv(t)=v(t+τ)−v(t) , (7)

with a component of the Lagrangian velocity vector v (Eq. 6). A similar dimensional analysis à la Kolmogorov would give a linear dependence of the second order structure function

 M2(τ)=⟨(δτv)2⟩=σ2(τT)=cLK⟨ϵ⟩τ , (8)

where is the integral time scale and the respective Lagrangian Kolmogorov constant [40, 41]. This corresponds to a Lagrangian power spectrum of the form

 E(ω)∝cLK⟨ϵ⟩ω−2 . (9)

Once again, these laws are valid only in the respective inertial range (or ). Unfortunately, these laws cannot be easily generalized to higher order structure functions because of the fundamental non Gaussian nature of the velocity fluctuations. We show in Fig. 1(b) the estimation of the experimental velocity increments PDFs at various scales obtained at ENS Lyon [21] and the acceleration PDF obtained at the university of Cornell [25]. Let us first remark that the acceleration can be seen as a Lagrangian velocity increment at a scale much smaller than the Kolmogorov dissipative time scale. Again we observe a continuous shape deformation from Gaussian statistics at large scale, to long-tail acceleration statistics at vanishing scale. This is again a manifestation of the intermittency phenomenon.

We will show in this article that a similar statistical description can be developed in a Lagrangian context. The free parameters are again the corresponding singularity spectrum and a constant, as in the Eulerian framework. The Lagrangian singularity spectrum will be shown consistent with the prediction derived from its Eulerian counterpart using a transformation, presented latterly.

This article is devoted to the presentation of a phenomenological theory of the statistics of the Eulerian and Lagrangian velocity increments, from the integral length scale (or from the integral time scale ), down to the far dissipative scales based on the so-called multifractal formalism [7]. In this approach, statistical properties in the inertial range are assumed and compared to empirical data. This includes the classical K41 predictions “” (Eq. 5) and “” (Eq. 9), but also the intermittent (or multifractal) corrections. From this descriptive analysis, we predict the statistical properties of the (Eulerian) velocity gradients and (Lagrangian) acceleration as functions of the Reynolds number and of the corresponding singularity spectra. In this sense, we can consider the multifractal formalism, properly generalized to the dissipative range, as a phenomenological theory, i.e. statistical properties are assumed in the inertial range (given some free parameters fully encoded in the so-called singularity spectrum), and are predicted in the dissipative range. This was already recognized in Ref. [35]. To this regard, multifractal formalism can be viewed as a standard model of turbulence, and, as far as we know, it is the only formalism able to reproduce accurately higher order statistics of velocity increments. The most important perspective would be to establish a link between this formalism (in particular the existence of a singularity spectrum) and the equations of motion (i.e. the Navier-Stokes equations). This is out of the scope of the present article. Let us also mention some alternative statistical formalisms that are able to reproduce the non-Gaussian nature of the underlying statistics, such as, among others, “superstatistics” [36], continuous-time random walk models [37], vortex filament calculations [38] and kinetic equations approach [39]. As an interesting application of this multifractal formalism, we reinterpret former measurements by the group of Tabeling [44, 45], concerning the velocity gradients flatness as a function of the Reynolds number. We end with a general discussion of the skewness phenomenon and of its implications on the modeling of velocity increments in the inertial range. In the Lagrangian framework, we introduce higher order velocity increments designed to quantify accurately the singular nature of velocity and justify their use.

## 2 The Eulerian longitudinal velocity fluctuations

### 2.1 Behavior of the flatness of velocity increments from experimental investigation

We show in Fig. 2 the estimation of the flatness of the longitudinal velocity increments in various flow configurations and various Reynolds numbers. The first set of data has been obtained in a Helium jet [46] at four different Reynolds numbers: , where the Taylor-based Reynolds number is proportional to the square-root of the large-scale Reynolds number (see Eq. 26). Also are displayed the results in an air jet experiment at the ENS Lyon at of C. Baudet and A. Naert. The higher Reynolds number () comes from the wind tunnel experiment in Modane [17]. For comparisons are also reported the results coming from a classical direct numerical simulation (DNS) of E. Lévêque [42] at a moderate Reynolds number based on grid points in a periodic domain.

The flatness as a function of the scale is represented in a logarithmic representation. Flatness is divided by 3, i.e. by the value of the flatness of a Gaussian random variable. The scales are renormalized by the length scale estimated such that the power-laws that are observed in the inertial range are indistinguishable. Then, is itself normalized by the Reynolds number, i.e. , where is a universal constant, that will be shown to be linked to the Kolmogorov constant (subsection 2.1.2). A similar procedure is applied to .

In this representation, the Kolmogorov length scale can be seen directly as . It delimits the inertial range, i.e. , and the dissipative range, i.e. .

#### The inertial range

Let us first focus on the inertial range. Over this range, we observe a universal (i.e. independent on both the Reynolds number and the flow geometry) power law of exponent . In the same range, the second order structure function behaves as a power law, i.e. (data not shown). Furthermore, we can see that the statistics at scales larger than the integral length scale are consistent with a Gaussian process since the Flatness is very close to 3. Indeed the flatness is slightly smaller than 3. This can be justified theoretically from PDF closures [47, 48]. We will neglect this effect in the sequel. At this stage, for , a probabilistic description of the velocity increments is straightforward, i.e. , where is a zero average unit variance Gaussian noise and , independent on the scale . Henceforth, the symbol stands for an equality in law, i.e. in probability. It means the the PDF and all the moments of the random variables on the two sides of the equality are equal.

In the inertial range, we need a probabilistic formulation of this power law behavior. This is clear at this stage that a simple Gaussian modeling is not enough since it would predict a scale-independent flatness of constant value 3. Furthermore, the parameters that we will use must be universal since the observed power-law is universal. The main idea to build up a probabilistic description is to mix Gaussian variables and, for example, to use a Gaussian random variable with a fluctuating variance. This was proposed in the so-called propagator approach [12]. The form of the fluctuations of this stochastic variance will be given by the standard arguments of the multifractal formalism [7]. The corresponding non-Gaussian modeling consists in writing:

 δℓulaw=σ(ℓL)hδ , (10)

with a fluctuating variable, independent on the unit-variance zero-average Gaussian noise , and characterized by its distribution function

 P(ℓ)h(h)=(ℓL)1−DE(h)∫hmaxhmin(ℓL)1−DE(h)dh . (11)

In the limit of vanishing values of , and gain the mathematical status of the Hölder exponent and the singularity spectrum respectively. Let us stress that the -distribution (Eq. 11) is indeed normalized, and the range of integration [;] depends on the precise shape of the singularity spectrum. One of the main hypothesis of the multifractal formalism is to assume that is independent on the scale [7]. In this case, assuming , a steepest-descent calculation (see Appendix A) shows that (recall that and are assumed independent)

 ⟨|δℓu|p⟩=σp⟨|δ|p⟩∫hmaxhmin(ℓL)phP(ℓ)h(h)dh≈ℓ→0σp⟨|δ|p⟩(ℓL)minh[ph+1−DE(h)] , (12)

where and is the Gamma function. Hence, the form of the density (Eq. 11) implies that the structure functions behave as power-laws, with a set of exponents linked to via a Legendre transform [7]

 ζp=minh[ph+1−DE(h)] . (13)

It gives in a straightforward manner the behavior of the flatness in the inertial range, i.e. . Let us first mention that in a K41 framework, without any intermittency corrections, the singularity spectrum is equal to and for . In this case, using Eq. 13, we can easily show that , and because of the linearity of the function, the flatness is independent on the scale . Clearly, the universal power-law behavior of the flatness in the inertial range as shown in Fig. 2, is the signature of the presence of intermittency in turbulence.

In the literature, several models for have been proposed. One of the most widely used is the lognormal approximation, giving the simplest quadratic form of the spectrum including intermittency corrections, as proposed by Kolmogorov and Oboukhov [15, 16]:

 DE(h)=1−(h−c1)22c2 . (14)

In this case, the proposed stochastic modeling of the velocity increments (Eq. 10) has a simple probabilistic interpretation: with and , the velocity increments are modeled as a Gaussian noise multiplied by a lognormal multiplicator . This was the proposition made in the propagator approach [12]. It is easily seen that , leading to . In the sequel, we will choose and because (i) a numerical integration of Eq. 12 shows no difference with the indefinite case, (ii) as we will see, the extension to the dissipative range implies (see section 2.1.2) and (iii) rigorously, only Hölder exponents greater than 0 and smaller than 1 are accessible when using increments. Experimental and numerical data as displayed in Fig. 2 show that the flatness behaves as a universal power-law of exponent . This corresponds to , and is called the intermittency coefficient. Also seen on empirical data (data not shown), [49]. Thus, in the sequel, we will take , very close to its K41 prediction . This defines completely the quadratic singularity spectrum (Eq. 14).

Another widely used singularity spectrum is the She-Lévêque spectrum [50],

 DE(h)=−1+3[1+ln(ln(3/2))ln(3/2)−1](h−1/9)−3ln(3/2)(h−1/9)ln(h−1/9) . (15)

This spectrum is based on log-Poisson statistics [51] and yields . This implies that , a power-law behavior in good agreement with the empirical data shown in Fig. 2. More sophisticated methods based on wavelets [52, 53] agree on the difficulty to discriminate between log-normal and log-Poisson statistics, and none of these approximations, for the moment, have been derived rigorously from first principles (i.e. the Navier-Stokes equations). For these reasons, we will use, in the sequel, the simplest lognormal approximation (Eq. 14) that compares well with empirical data, at least for low order moments with and that gives a Gaussian distribution for the -exponent, that is easy to manipulate (see in particular in the stochastic modeling proposed in section 2.2.3).

Finally, the proposed stochastic modeling (Eqs. 10 and 11) includes a functional form for the probability density functions of the velocity increments. A formal derivation of these PDFs, starting from the product of two independent random variables is reported in Appendix B. In simple words, we can see that the random variable (Eq. 10) is fully defined when is given the law of (Eq. 11) and (let us say ), that we assume, at this stage, independent. To simplify the derivation of the PDF of (see Appendix B for a more rigorous derivation), let us consider only the absolute value of the velocity increment. We get

 ln|δℓu|=lnσ+hlnℓL+ln|δ|.

Given that the random variables and are independent, the PDF of is the convolution product of the PDFs of and , as it was noticed in Ref. [12], namely

 Pln|δℓu|σ(ln|δℓu|σ) =∫Pln|δ|(ln|δℓu|σ−hlnℓL)PhlnℓL(hlnℓL)d(hlnℓL) =∫Pln|δ|(ln|δℓu|σ−hlnℓL)Ph(h)dh .

Noticing that and , we finally get

 Extra open brace or missing close brace (16)

which shows the expression of the PDF of the absolute value of the velocity increment. Without further assumptions, this derivation cannot be generalized to derive the PDF of the signed velocity increment, since we are taking at one point a logarithm, but it clarifies the fact that defining in probability a random variable (such as in Eq. 10) allows to derive the associated PDF.

Actually, Eq. 16 is also true for the signed velocity increment, as shown in Appendix B, and we get for the velocity increment PDF at the scale the following form:

 Pδℓu(δℓu)=∫hmaxhmin1σ(ℓL)−hPδ[δℓuσ(ℓL)−h]P(ℓ)h(h)dh , (17)

where is the PDF of a unit-variance zero-mean Gaussian variable. A numerical investigation (data not shown) of the PDF (Eq. 17) shows the characteristic continuous shape deformation associated to the intermittency phenomenon, in a similar way than observed in Fig. 1(a). Once again, this PDF is symmetric, i.e. and thus fails to describe the skewness phenomenon. We invite the reader to have a look at the sections 2.2.1 and 2.2.3 that provide a formalism able to reproduce the asymmetry of the PDFs, as observed in empirical data. We will see in the following section how to introduce the dissipative effects in order to obtain predictions on the statistics of the velocity gradients.

#### The dissipative range

The flatness of velocity increments, shown in Fig. 2, behaves in a very different way in the dissipative range, i.e. for scales . First of all, no power-law is observed. Then, we can see a strong Reynolds-number dependence. This is very different from the behavior observed in the inertial range. In the limit of vanishing scales, the velocity increments can be Taylor expanded and we obtain , i.e. a linear behavior as a function of the scale. This implies that the flatness tends to a Reynolds-number dependent function, independent on the scale , i.e. . The aim of this section is to understand and model the transition from the observed power-law behavior of the flatness in the inertial range, to this scale-independent behavior in the dissipative range. But first of all, let us have a look at the empirical data.

In Fig. 2, we see that not all of the measurements exhibit at very small scales a scale-independent flatness. Only the DNS (), the air-jet () and the Helium-jet () have succedeed in resolving scales smaller than the Kolmogorov length scale. Indeed, the measurements are difficult because the Hot-wire size is usually of the order of the Kolmogorov length scale. If the Reynolds number is too high, dissipative scales are too small to be resolved (as for , and ). In the Modane’s wind tunnel, the integral length scale is very large, and despite the large Reynolds number (), should have been accessible (which is of the order of the millimeter). Unfortunately, the noisy local environment of the giant wind tunnel and the high temperatures reached ( Celsius) implied by the strong level of turbulence, prevented electronics from working properly at high frequency.

Given the experimental limitations, we see that the flatness underlies a rapid increase. This was the subject of Ref. [42]. The main underlying idea is the differential action of the viscosity. The multiplicator fluctuates in the inertial range. In the dissipative range, up to a (random) coefficient, the multiplicator becomes linear with respect to the scale in order to be consistent with the Taylor’s expansion. The viscosity will regularize this inertial-range singular behavior at a scale that depends on the strength of the singularity: the bigger is the multiplicator, corresponding to smaller -exponents, the smaller is the scale of regularization. Purely kinematic arguments, proposed in Ref. [42], show that the width of the so-called intermediate dissipative range [35], where and are respectively the smallest and biggest dissipative scales, indeed depends weakly on the Reynolds number

 ln(η+η−)∼√lnRe , (18)

the Kolmogorov length scale lying in between. In the representation chosen in Fig. 2, this prediction (Eq. 18) implies that the observed rapid increase occuring in the intermediate dissipative range should be steeper and steeper as the Reynolds number increases. This is what is qualitatively observed as long as experimental technics were able to reach these very small scales.

This can be fully modeled in the context of the multifractal formalism. To do so, one has to come up with a dissipative scale that depends explicitly on the strength of the multiplicator, or in a more straightforward manner, on the exponent . Paladin and Vulpiani [54] first proposed such a -dependent cut-off. Their reasoning was based on the local Reynolds number. For a scale lying in the inertial range, a fluctuating Reynolds number can be defined using a fluctuating characteristic velocity , leading to the local Reynolds number . We can see that at the integral length scale, the associated Reynolds number is unique, does not fluctuate, and is given by . This allows to define unambiguously a dissipative length scale: a scale at which the local Reynolds number is of order unity . The order one constant , is a free parameter of the formalism, a priori universal. It has been introduced phenomenologically in Refs. [46, 55, 56]. We will take it to be and we will show that it is related to the Kolmogorov constant . From there, one obtains directly the main result of Ref. [54], namely

 η(h)=L(ReR∗)−1h+1 . (19)

Based on this result, Nelkin [57] showed the implication of a fluctuating dissipative scale (Eq. 19) on the modeling of the velocity increments for scales lying in the dissipative range. Within our approach summarized by Eq. 10, the velocity increment at a scale smaller than the dissipative scale will be modeled in the following way:

 δℓu% law=ℓ≤η(h)σℓL(η(h)L)h−1δ , (20)

where , as in Eq. 10, is a zero-mean unit variance Gaussian random variable, , independent of the random exponent . This probabilistic modeling (Eq. 20) is consistent with the Taylor expansion of the velocity increment. It implies a model for the velocity gradient [57], namely:

 ∂xulaw=σL(η(h)L)h−1δ . (21)

Also we see that the proposed modeling for (Eq. 10) and for (Eq. 20) is continuous at the dissipative length scale .

As in Eqs. 10 and 11, to fully characterize in a probabilistic manner the random variable (Eq. 20), we need to define the distribution of the exponent . The proposition of Ref. [57] is to take for the distribution of at the dissipative scales a scale-independent distribution (up to a normalizing function ) that also has to be continuous at the transition , namely

 P(ℓ)h(h)=ℓ≤η(h)1Z(ℓ)(η(h)L)1−DE(h) , (22)

where is the same universal function entering in the -distribution of the inertial range (Eq. 11), that we will approximate to be quadratic (Eq. 14).

As a general remark, as previously underlined in Ref. [35], the multifractal formalism becomes predictive. Indeed, given the description of the inertial range (Eqs. 10 and 11), and in particular given the singularity spectrum , we can predict the behavior of the velocity increments in the dissipative range (Eqs. 20 and 22) by a simple continuity argument. As a consequence, using Eqs. 19, 21, and 22, the -order moment of the velocity gradients is given by following function of the Reynolds number:

 ⟨(∂xu)2p⟩=⟨δ2p⟩(σL)2p1Z(0)∫hmaxhmin(ReR∗)−2p(h−1)+1−DE(h)h+1dh , (23)

where , and . The distribution of a Gaussian variable is even, so all the odd moments of both velocity increments (Eqs. 10, 20) and gradients (Eq. 21) are vanishing at this stage. To include the skewness phenomenon, that requires further probabilistic modeling, we invite the reader to take a look at section 2.2.1.

The explicit form of even-order moments of velocity gradients (Eq. (23)) allows us to derive two important predictions: the computation of the average dissipation and the dependence on the Reynolds number of the flatness of the velocity derivatives. The dissipation is a key quantity in turbulence theory [7]. In an isotropic and homogeneous flow, the average dissipation is related to the second order moment of the velocity gradients as . Using Eq. (23), one obtains

 ⟨ϵ⟩=15νσ21L2Z(0)∫hmaxhmin(ReR∗)−2(h−1)+1−DE(h)h+1dh . (24)

In the limit of large Reynolds number, a steepest-descent calculation (see Appendix A) shows that, up to an order one multiplicative constant, the former integral, is dominated by the term , where . Because by construction, and for a wide class of singularity spectrum such that (via a Legendre transform, c.f. Eq. 13) , which is the case with the quadratic approximation (Eq. 14) with , we can show [7] that . This implies that is independent on the Reynolds number. This verifies a basic hypothesis of Kolmogorov, namely the finiteness of at infinite Reynolds number. A precise estimation of the integrals entering in Eq. 24 (see Appendix A) shows that

 ⟨ϵ⟩≈Re→+∞15R∗σ3L . (25)

As shown in Refs. [42, 43], Eq. 25 gives a prediction of the Kolmogorov constant . Indeed, in the inertial range, neglecting intermittent corrections, the structure function is given by . Using Eq. 25, it gives , showing indeed that this constant is related to the Kolmogorov constant . Using , one finds , which is slightly bigger than what has be measured on empirical data [10, 11]. One of the reasons of this apparent discrepancy is related to the fact that the constant is clearly linked to the presence of intermittency and the implied extension of the intermediate dissipative range, whereas the definition of the assumes the absence of intermittency corrections. The prediction of the second order moment of the gradients allows also to link precisely the (large-scale) Reynolds number to the Taylor based Reynolds number in the following way:

 Re=4R∗R2λ . (26)

Another important prediction is the dependence of the flatness of derivatives on the Reynolds number [57]. From Eq. 23, using a steepest-descent argument, we can show that, up to multiplicative constants of order unity (given in Appendix A):

 ⟨(∂xu)4⟩⟨(∂xu)2⟩2≈Re→+∞3(ReR∗)χ4−2χ2 with χp=minh[−p(h−1)+1−DE(h)h+1] . (27)

Using the quadratic spectrum (Eq. 14), , and Eq. 26 for the Taylor-based Reynolds number dependence, we obtain:

 ⟨(∂xu)4⟩⟨(∂xu)2⟩2≈Re→+∞3(ReR∗)0.18=1.47R0.18e=0.93R0.36λ , (28)

which is consistent with empirical observations [58, 59, 60, 61]. More precisely, we compare in Fig. 3 the multifractal prediction (Eq. 28) with various experimental measurements and numerical simulations as compiled in Ref. [61]. The present theoretical prediction reproduces quantitatively the empirical observations. Let us stress that the quantitative dependence of the flatness of velocity derivatives on the Reynolds number (Eq. 28) is a genuine consequence of the intermittent nature of turbulence. In a K41 framework, this quantity would be Reynolds number independent.

#### Full multi scale description

In the preceding sections (2.1.1 and 2.1.2), we have seen how to model velocity fluctuations in respectively the inertial and far-dissipative ranges. It remains to write down a formalism, based on these two well known limiting ranges, that reproduces velocity statistics in the entire range of scales, including the intermediate dissipative range.

A first naive idea to gather both the inertial and dissipative ranges in a unified description is to consider, at a given scale and Reynolds number , the -exponents. Within the range , if , then the velocity increment lies in the inertial range; On the opposite, if , then the velocity increment lies in the dissipative range. The transition occurs at the exponent defined by , namely

 h∗(ℓ,Re)=−(1+ln(Re/R∗)ln(ℓ/L)) . (29)

Using both the laws of the velocity increments in the inertial (Eqs. 10 and 11) and dissipative (Eqs. 20 and 22) ranges, we then obtain

 ⟨|δℓu|p⟩=⟨|δ|p⟩Z(ℓ)⎡⎢⎣∫h∗hmin(ReR∗)−p(h−1)+1−D(h)h+1dh+∫hmaxh∗(ℓL)ph+1−D(h)dh⎤⎥⎦ , (30)

where normalizes the probability densities, namely

 Extra open brace or missing close brace (31)

Unfortunately, this model does not compare well with empirical data because the transition is too sharp (data not shown, see [63]). Furthermore, the present modeling gives a continuous but not differentiable modeling of the velocity increment.

A continuous and differentiable transition, inspired by the interpolation function of Batchelor to model the second-order structure function [65], has been proposed in Ref. [64] as the entire range of scales. In Ref. [43], a slight modification of the proposition of Ref. [64] has been introduced in order to make the transition compatible with the far-dissipative predictions of Ref. [57]. It reads

 δℓulaw=σβℓδ with βℓ=(ℓL)h[1+(ℓη(h))−2](1−h)/2 , (32)

where the random variable is again a zero-mean unit-variance Gaussian noise, , and

 P(ℓ)h(h)=1Z(ℓ)(ℓL)1−D(h)[1+(ℓη(h))−2](D(h)−1)/2 . (33)

The normalizing constant is again such that , namely

 Z(ℓ)=∫hmaxhmin(ℓL)1−D(h)[1+(ℓη(h))−2](D(h)−1)/2dh . (34)

Note that at a fixed given scale , the velocity increment (Eq. 32) and the -distribution (Eq. 33) tend to the inertial description (given by Eqs. 10 and 11) in the limit of infinite Reynolds number. In the same manner, at a given finite Reynolds number, this proposed description tends to the dissipative predictions (Eqs. 20 and 22)) in the limit of vanishing scales .

We show in Fig. 3(a) the theoretical predictions of the flatness as a function of the scales for the different Reynolds numbers previously investigated in Fig. 2. To do so, we integrated numerically Eqs. 32 and 33 using a quadratic singularity spectrum, with and , and and . The Reynolds number is obtained from the Taylor-based Reynolds number using Eq. 26. The proposed description based on Eqs. 32 and 33, reproduces the main characteristics shown in Fig. 2, namely, the universal power-law behavior in the inertial range and the rapid increase of the flatness in the dissipative range. Obviously, the theoretical predictions do not suffer from a lack of resolution and so with the proposed renormalization of the scales and of the flatness, all the curves tend to a universal plateau given by Eq. 28 when . The rapid increase that takes place in the intermediate dissipative range is consistent with the kinematic prediction given in Eq. 18, namely the slope of this increase behaves as in this representation. We can see indeed that the higher is the Reynolds number, the steepest is the increase.

In Fig. 3(b), we compare more precisely the velocity increment flatness obtained in Modane’s wind tunnel [17] to the multifractal prediction. The model does reproduce both the inertial and intermediate dissipative ranges, given the experimental limitations to reach the far dissipative range.

#### Reinterpretation of the Tabeling’s data as a non trivial effect of the dissipative physics

As an example of the implications of the present theory, we reexamine in this section the observations of Tabeling and Willaime [45]. These authors investigated fully developed turbulence in a Von-Karman flow in gazeous helium at low temperature. Varying the pressure of the gas, they could span an unusually large range of Taylor scale based Reynolds numbers . They measured the local velocity using a hot wire probe. Through the Taylor “frozen turbulence” hypothesis, they could access to the longitudinal derivative of the velocity, and its flatness (Eq. 28). Up to , their results are in good agreement with previous literature and the present prediction (Eq. 28). Surprisingly, at , presents a maximum, and goes down up to , then raises again slowly (Fig. 4).Tabeling and Willaime interpreted their results as some evidence of a transition in turbulent flows. Some comments suggested that this behavior of could be due to the finite size of the probe. However, as remarked by Tabeling and Willaime, this size (about 10m) is much smaller than the Kolmogorov dissipation length at . Moreover, such a limitation was expected to yield a saturation of at a constant value, not a well pronounced maximum.

Indeed, following the present multifractal theory, identifying the “velocity derivative” with a finite difference at a constant length gives a maximum for the flatness as shown in Fig. 4. This is due to the rapid rise of the longitudinal velocity difference flatness in the intermediate dissipative range. This maximum occurs when the length coincides with the lowest scale of this intermediate range, which is much smaller than . Using Eqs. 32 and 33 with a quadratic singularity spectrum (Eq. 14), , stars in Fig. (4) show the behavior predicted for if , in reasonable agreement with the size of the sensor.

However, the present theory cannot predict a further rise of as observed experimentally. Also, the width of the predicted peak is much wider than observed. The present theory can explain some of the surprising features observed, not all.

### 2.2 Consistent description of the skewness phenomenon

#### General discussion on the skewness phenomenon

This section is devoted to the modeling of the skewness of the velocity increments. As we just saw, modeling the velocity increment as a Gaussian random variable multiplied by a random amplitude (see Eq. 32) cannot reproduce the asymmetric nature of the distribution of velocity increments since the Gaussian random variable , and its independence on the amplitude lead to vanishing odd-order moments, i.e. , . Nevertheless, keeping the same probabilistic description as in Eqs. 32 and 33 for the second order moment of velocity increments, allows us to predict in a consistent way the third-order moment of velocity increments if we use the Karman-Howarth-Kolmogorov equation:

 ⟨(δℓu)3⟩=−45⟨ϵ⟩ℓ+6νd⟨(δℓu)2⟩dℓ . (35)

In Ref. [43] (see also Ref. [66]), we compared experimental data to the predictions obtained for using the Karman-Howarth-Kolmogorov equation 35 and the second order structure function obtained from Eqs. 32 and 33. Predictions and emprical data compares well in the whole range of scales [43] without additional free parameters. In particular, the level of skewness in the inertial range is well reproduced and can be shown to be related to the universal constant . If we neglect dissipative effects in the exact relation Eq. 35, and the intermittent corrections on the second order structure function, it is easy to see that is independent on the scale and can be further approximated to , in excellent agreement with empirical data [43]. Moreover, applying a Taylor development on both Eq. 35 and on the multifractal predictions (Eqs. 32 and 33) for the second order structure function, in the limit of vanishing scales, we get the following multifractal prediction for the third order moment of the derivatives:

 Extra open brace or missing close brace (36)

where is a negligible additive term, coming from the Taylor’s development of the normalizing factor (Eq. 34) [43]. This prediction of the third order moment of the velocity gradient (Eq. 36) using a Batchelor-Meneveau type of transition between the inertial and dissipative ranges (Eqs. 32 and 33), depends on the singularity spectrum , measured in the inertial range on empirical data, and on the universal constant . We show in Fig. 3(d) the numerical estimation of the multifractal prediction of the Skewness of derivatives (using Eqs. 32, 33 and 36) as a function of the Taylor-based Reynolds number, using the quadratic singularity spectrum (Eq. 14) and . This prediction is compared to empirical data [60, 61, 62] as described in the figure caption and compiled in Ref. [61]. We observe some dispersion between the three different empirical skewnesses, although the dependence on the Reynolds number seems to be universal. The difference in amplitude could be due to a lack of experimental and numerical resolution, and/or to a lack of statistical convergence. Thus, if the multifractal approach fails to predict the value of the skewness, it does reproduce accurately the Reynolds number dependence. Indeed, using Eq. 36, a steepest-descent calculation shows that the skewness of the derivatives behaves as a power law of the Reynolds number, i.e.

 ln(−S(0))/ln(Re)∼χS−1 , (37)

with

 χS=minh[−2(h−2)+1−DE(h)h+1]−32minh[−2(h−1)+1−DE(h)h+1] . (38)

Using a quadratic approximation for the parameter function (Eq. 14), we get . This power-law dependence on the Reynolds number has been already obtained by Nelkin [57] using a different, although related, approach based on the asymptotically exact relationship . As shown is Fig. 3(d), further numerical estimations of relation (36), once rephrased in terms of Taylor-based Reynolds numbers using Eq. 26, leads to the following dependence of the velocity derivative skewness on :

 ⟨(∂xu)3⟩⟨(∂xu)2⟩3/2≈Re→+∞−0.175R0.134λ . (39)

#### Modeling the velocity increments probability density function

Indeed, consistent predictions for higher odd order structure functions are needed to predict the shape of the full velocity increment probability density function (PDF). Several propositions were made in the literature to account for the asymmetry of the PDF linked to the skewness phenomenon [12, 43]. To do so, we must modify the noise entering in the probabilistic formulation Eq. 32, and/or correlate this noise with the exponent . Asymmetric PDFs can be obtained if we change the Gaussian random variable to a non-Gaussian noise, still independent on the scale and on the multiplicator . More precisely, it was proposed in Ref. [12] to consider the random variable as being a variable of density that now reads

 Pδ(δ)∝exp[−δ22(1+aSδ√1+δ2)] , (40)

where is a universal constant, independent on both Reynolds number and scales. The main problem using this peculiar noise (Eq. 40) is that it leads to non zero average velocity increments. This could be fixed by introducing a scale dependent free parameter that centers the whole velocity increment PDF. Furthermore, to reproduce the non trivial behavior of the skewness in the dissipative range, we need to modify the parameter , and to make it dependent on both scale and Reynolds number. In this spirit, still based on the hypothesis of independence of the two random variables and , a general development of the PDF of on a basis made of the successive derivatives of a Gaussian, called the Edgeworth’s development, was proposed in Ref. [43]:

 Pδ(δ)=1√2π+∞∑n=0λn(ℓ)dndδne−δ22 , (41)

where the coefficients are functions of the scale . As previously shown, the symmetric part (even terms) is well described by a Gaussian noise, which means that and for . The coefficient is set to zero since, from Eq. 41, . Under these hypotheses, the third order moment is then given by , which fully determines the coefficient thanks to the Karman-Howarth-Kolmogorov equation (Eq. 35). Importantly, does depend on scale and Reynolds number. As Eq. 35 is the only available constraint on , it is tempting (as a first approximation) to restrict the expansion to : for . Additional statistical equations involving higher order odd moments of would be needed to give the next . This would require further modeling (primarily to get ride of pressure terms), which is outside the scope of the present work. Unfortunately, this crude approximation for the odd terms leads to severe pathologies, such as negative probability for rare large events, and is not consistent with higher order statistics such as hyperskewness (data not shown). To remedy for this weakness, Ref. [43] proposed to modify the variance of the Gaussian associated to the third term in the development (Eq. 41), in the following way:

 Pδ(δ)=1√2π[e−δ2/2−λ3(ℓ)δ(δ2−1)e−δ2/(2a2)] , (42)

where is fully determined by the exact relation 35, and an add-hoc free parameter, close to unity [43], aimed at describing higher order odd statistics. Then, from Eqs. 32, 33 and 42, the velocity increment PDF can be written, under the hypothesis of independence of and as (see Appendix B):

 Missing or unrecognized delimiter for \left (43)

Then, using a quadratic singularity spectrum (Eq. 14), and the add-hoc coefficient (see Ref. [43]), the predicted PDF (Eq. 42) successfully compares to empirical data [43] as shown in Fig. 1(a) for various scales. The shape of the experimental velocity increment PDFs, from the inertial, to the intermediate dissipative and far dissipative ranges is well captured by the present theoretical prediction (Eq. 42), consistently with the behaviors of the flatness and skewness of the velocity increments. Unfortunately, at this stage, it is not possible to motivate the choice of the additional free parameter . To avoid having recourse to this parameter, we are forced to abandon the hypothesis of independence of the singularity exponent and the noise . A formalism that takes into account possible correlations between these two random variables is presented in the following section.

#### Probabilistic modeling of the skewness in the inertial range

It remains to give a consistent description of the asymmetric part of velocity increment PDFs in both the inertial and dissipative ranges without invoking an additional free parameter in the PDF of the noise (Eq. 42). In this section, we propose such a formalism for the inertial range, and leave the extension to the dissipative range to future investigations. The main idea is to correlate the exponent and the noise in the multifractal description provided by Eq. 32.

Let us first assume that the velocity increment can be written as a product of a Gaussian noise and an amplitude : , as in Eq. 32. The main difference with former assumptions is to let possible a correlation between and . The simplest way to deal with such a probabilistic formalism is to assume and jointly Gaussian. In this case, see Appendix B, the velocity increment PDF can be written as

 Pδℓu(δℓu)=∫∞−∞1σ(ℓL)−hPδ,h[δℓuσ(ℓL)−h,h]dh , (44)

where is the joint probability of the random variables and given by:

 Pδ,h(δ,h)=12πσδσh√1−ρ2exp{−12(1−ρ2)[(δ−mδ)2σ2δ+(h−mh)2σ2h−2ρ(δ−mδ)(h−mh)σδσh]} . (45)

In Eq. 45, and (resp. and ) stand for the mean and variance of the random variable (resp. ). The correlation coefficient lies in the range . At large scale one has to recover . This implies .

Consistently with Eqs. 11 and