We develop a conceptual framework for turbulent scaling laws across length scales extending beyond the inertial range. Classically, fully developed turbulence is defined as occurring on length scales in which energy transfer is dominated by inertial forces. Partially developed turbulence is defined as turbulent (i.e., non-laminar) ranges with significant dissipation or significant stirring. Thus, length scales with significant dissipation and stirring forces are included within partially developed turbulence. We propose new models with supporting verification which subsume and extend inertial and non-inertial range models of others. From our models and data analysis, we explain phenomena not previously observed and phenomena previously observed but not explained.

The inertial range is defined as an intermediate range of length scales, l, which are far from the Kolmogorov scale η and integral scale L, i.e., η ≪ l ≪ L. The dissipation range is the range of length scales between the laminar range and the inertial range. The stirring range can be at any length scale, but in common experiments, observations and simulations, stirring is at large scales. The influence of stirring decreases (in a controlled and scaling measurable manner) outside for scales smaller than those actually stirred. This line of investigation, while observed by the authors, has not been developed systematically in the present paper and could be addressed in the future.

Structure functions give a precise meaning to clustering of bursts of turbulent intensity and compound clustering (i.e., clustering of clusters) etc. They measure the dependence of these compound clustering rates on the length scale set by the observational size of the cluster. Two families of structure functions are studied here: one characterizes the powers of the energy dissipation rate, ϵ, and the other characterizes powers of the velocity difference, δu.

The turbulent dissipation rate can be derived from the turbulent velocity gradients as: (1)ϵ(x→,t)=ν2(∂ui∂xj+∂uj∂xi)2 , where ν is the kinematic viscosity of the fluid and u is the fluctuating velocity, and the summation convention is applied to repeated indices represented the three Cartesian directions by following the definition by [1].

We define the coarse grained, local average of the dissipation rate ϵ_l as, (2)ϵl(x→)=1|Vl|∭Vl(x→)ϵ(y→,t) dy→ , where Vl(x→) is a volume with a diameter l centered around x→ in 3D. The coarse grained averaging means that ϵ_l reflects properties occurring on the length scale l.

The structure functions of δ_lu and ϵ_l satisfy asymptotic scaling relations as a power of the length scale l. The structure functions and the associated scaling exponents τ_p and ζ_p are given by the expectation relations, (3)〈(ϵl)p〉~lτp and 〈|δlu|p〉~lζp. The length scale dependent scaling exponents τ_p and ζ_p are obtained from logarithmic local slopes, i.e., (4)τp=d(ln〈ϵlp〉)d(ln l) and ζp=d(ln〈|δlu|p〉)d(ln l) , as suggested by [2].

The main thrust of the theory is that turbulent scaling laws remain in effect across all length scale ranges, but with modified length dependent scaling exponents.

A key step in the parameterization of the introduced scaling models is that τ_p and ζ_p in Equation (4) are linear in ln l in the dissipation range. Hence, the slopes of τ_p and ζ_p are constructed, (5)Tp=d(τp)d(ln l) and Zp=d(ζp)d(ln l) , which are modeled across all length scales as piecewise constant.

The piecewise constant values of T_p and Z_p appear to be a new discovery. These piecewise constant values over the 4 length scale ranges are described as follows: (6)Tp={0,laminar range (LR)Tpdr,dissipation range (DR)0,inertial range (IR)Tpsr,stirring range (SR)Zp={0,laminar range (LR)Zpdr,dissipation range (DR)0,inertial range (IR)Zpsr,stirring range (SR) In the inertial range, we take T_p = 0. We do not observe T_p = 0 from our data in this range, but include this to be consistent with the classical model by [3], denoted SL, in lieu of a more comprehensive theory.

Tpdr and Zpdr are constant in the dissipation range, and Tpsr and Zpsr are constant in the stirring range for the problems we study. These T_p and Z_p values are verified in the JHTDB and the T_p values in the UMA data shown in section 4.1 addressing the laminar region where the eddy length scale is smaller than the Kolmogorov microscale η. In addition, we observe that T_p is linear in p, and Z_p is given by an explicit p dependent formula in the dissipation range.

In the dissipation range, the logarithmic dissipation rate is proportional to ln (l). In contrast, the dissipation defined in the Navier-Stokes equation itself occurs in the laminar range at a rate proportional to the length scale l. For p = 2, the laminar energy dissipation rate ϵ includes the classical viscosity ν within its definition. The increase of both τ_p and ζ_p for all p in a limit as l approaches η from above leads to negative constant values of T_p and Z_p in their respective dissipation ranges.

The key property of constant slopes T_p and Z_p is also satisfied in the stirring range and observed in the JHTDB data. In principle, stirring forces can be added in an arbitrary manner at any length scale. Parameterization of the stirring range is not addressed in this paper.

We provide a brief literature review in section 2. The numerical verification data from JHTDB and UMA are described in section 3 as well as the methods used for data analysis. The scaling law results, which are the technical core of the paper, are presented in sections 4, 6. The extended and refined scaling analytical methods are discussed in section 5. A comment on the asymptotics of the viscous limit can be found in section 7. Conclusions are summarized in section 8.

Kolmogorov [4], denoted K41, postulated universal laws to govern the statistics on all such length scales in the inertial range in which the flow is statistically self similar. Dimensional analysis in K41, based on the self similarity hypothesis, led to the −5/3 scaling law. However, because the energy dissipation for turbulent flows is intermittent, this model has been refined in various ways over the years to yield a multifractal scaling law. Summarized in [5], the K41 exponent ζ_p is modified to capture the compound clustering of turbulent structure using a multifractal analysis. Kolmogorov [6] refined his similarity hypothesis, denoted K62, added the influence of the large flow structure and included the influence of the intermittency. The refined similarity hypothesis in [6] for the classical inertial range links the scaling exponent ζ_p of the longitudinal velocity structure and the scaling exponent τ_p of the energy dissipation rate as (7)〈|δlu|p〉~〈ϵlp/3〉lp/3 , where 〈ϵlp/3〉~lτp/3, or equivalently (8)ζp=p3+τp3. The log-normal model from SL defines a theoretical model for the PDF of the coarse-grained energy dissipation in the inertial range. SL studied the quantity ϵl(p) defined as the ratio: (9)ϵl(p)=〈ϵlp+1〉〈ϵlp〉. where p can be any non-negative integer. The ϵl(0) and ϵl(∞) are related to the mean fluctuation structure ϵ¯ and the filamentary structure. The scaling law for p → ∞ is ϵl(∞)~l-2/3. As p → ∞, the definition of τ_p stating that, (10)τp+1-τp→-23 , or τp=-23·p+C. The codimension C is evaluated as C = 3 − 1 = 2 based on the assumption that the elementary filamentary structures have dimension 1. The expectation 〈ϵlp〉 has an l dependence which is not a pure exponential, but a mixture of exponentials, i.e., the scaling exponents for 〈ϵlp〉 are defined as a weighted average of exponentials.

From the assumption of the interaction between structures of different order, SL proposed the following relation between structures of adjacent order: (11)ϵl(p+1)=Apϵl(p)βϵl(∞)1-β. Based on Equations (9, 11), SL derives a two step recursion for τ_p. This recursion relation implies that τp=-23·p+2+f(p), where f(∞) = 0 is assumed. The equation for f(p) has the solution f(p) = αβ^p and with the boundary conditions τ₀ = τ₁ = 0, the solution becomes (12)τSL(p)=-23·p+2[1-(23)p]. Substituting Equation (12) into the relation shown in (8) yields (13)ζp=p9+2[1-(23)p3]. Novikov [7] suggested that the −2/3 on the RHS of Equation (10) should be replaced by −1 based on the theory of infinitely divisible distributions applied to the scaling of the locally averaged energy dissipation rate ϵ_l. Chen and Cao [8], denoted CC, accepted Novikov's suggestions and derived the formula: (14)τCC(p)=-p+[(1+τ2)p-1]/τ2 , which uses the classical value τ₂ ≈ −0.22, which is derived from simulations, observations, experiments, and theory (SL) all in approximate agreement.

Kolmogorov proposed that ζ₃ = 1 for incompressible, isotropic and homogeneous turbulence. Frick et al. [9] showed that in the case of non-homogeneous shell models with ζ₃ ≠ 1, the scaling of velocity structure functions in incompressible turbulence from SL still holds as ζp/ζ3=p/9+2[1-(2/3)p3].

Boldyrev et al. [10] predicted a new scaling law for the scaling exponent ζ_p of velocity structure functions as (15)ζp/ζ3=p/9+1-(1/3)p3 , in supersonic turbulence for star formation based on a Kolmogorov-Burgers model. The same behavior is observed by [11] in incompressible MHD.

The SL model has generated a considerable interest in the hierarchical nature of turbulence. Experiments and simulations have been conducted to evaluate the velocity and energy dissipation structures. Chavarria et al. [12–14] demonstrated experimental variables for the hierarchical structure assumption for the function ζ_p in Equation (13). Experimental studies on a turbulent pipe flow and a turbulent mixing layer by [15] verified the SL hierarchical symmetry. Cao et al. [16] showed agreement between the SL scaling exponents and high-resolution direct numerical simulations (DNS) of 3D Navier-Stokes turbulence.

Chevillard et al. [17] clearly frames the problem we address as the major theme of our paper. Their model and ours study the ζ_p and τ_p exponents in the turbulent dissipative regime. Their predictions are limited to the structure of order 4 or less. The validation results are complicated by differences in the experimental data, but each of their plots shows agreement with some of the data. Briefly, our results go beyond this work in several respects. We model the scaling exponent ζ_p and τ_p scaling asymptotics, each with two parameters, giving both the scaling exponent and the multiplying coefficient. We set these (4 in total, p dependent) parameters for values up to p = 10 on ζ_p and up to p = 30 on τ_p, with consistent validation against two independent sets of simulation data.

We relate our modeling to a variety of other and more recent studies of the setting of parameters for the She Levegue model. Traditionally, stirring is introduced at the large length scales and becomes less significant at the smaller ones as has been studied [18–20] previously. We also observed the same effect, and in our study of the JHTDB, found scaling laws similar to that reported in the dissipative regime, as is suggested in the drawing of Figures 2–4.

The CC model defined by [8] shown in Equation (14) has a dependence on inertial constant τ2=τ2SL≈-0.22 value from SL. We extend the model beyond the inertial range, where a constant τ2SL is no longer an accurate value for length scale dependent τ₂. To calculate τ_p for p > 2, we substitute the τ₂ data that are measured across all length scales into Equation (14). The extended model is denoted as the CC₂ model. Figure 10 shows this model for τ₃ (solid line) in comparison to the observed τ₃ value (data points). The same comparison for p ≤ 10 is shown in Figure 8. The data and model values agree for small p ≤ 5, but diverge at higher p values. We propose a modification of CC₂, which we denote as CC_q model. Instead of taking a measured τ₂, as in our extended CC₂ model, we substitute the measured τ_q data that is available across all measurable length scales, and solve numerically for τ^2 shown as below: (20)τq=-q+[(1+τ^2)q-1]/τ^2. Then, the calculated τ^2 can be substituted back to Equation (14) to calculate τ_p for any p. From this point, all τ_p values can be modeled as a function of p and the measured τ_q using the CC_q model.

Figure 11 shows the linear fit T₂ of τ₂ that is calculated from the CC_q model using measured τ_q values in the dissipation range, for q = 2, 3, …, ∞. This sequence of T₂ values are fitted by: (21)T2=0.229/q+0.001 for any q value. T₂ decreases and converges asymptomatically to 0.001 as q → ∞. Hence, the CC_∞ model is defined as CC∞=limq→∞CCq.

SL discusses the large p asymptotes of ϵlp in developing parameters for their τ_p inertial range methodology. They predicted dominance by vortices in this regime. We come to the same conclusion in the large p asymptotes more directly through an analysis of ζ_p, which reflects the intensity of the vortical structures. We find in Figure 13 and Equation (24) that ζ_p is increasing in p as the length l/η moves toward the dissipation range.

We interpret the length scale l/η limit in terms of Taylor-Green vortices continued past the instability point. Assuming the CC model remains applicable in this range, Figure 11 analyzes the CC_∞ model and shows a non-zero residual value for T₂ and similarly for all T_p. Line vortices do not dissipate energy and are described by the vanishing τ_p for all p. The presence of non-zero T_p for all p indicates that the line vortices occur in an unstable state, as occurs in a Taylor-Green vortex, continued past its singular value.

We have extended scaling laws for longitudinal velocity increments and energy dissipation rate structure functions from the inertial range to all length scales by modifying their exponential scaling exponents. We verify the complete parameterization models for τ_p and ζ_p in the dissipation range, i.e., from the Kolmogorov scale to the Taylor scale, limited to p ≤ 10 for ζ_p. Our major model feature, linearity of the log scale dissipation processes, is verified through comparison to the JHTDB and UMA data.

In local regions, even within a fully developed turbulent flow, the turbulence is not isotropic nor scale invariant due to the influence of larger turbulent structures (or their absence). For this reason, turbulence that is not fully developed is an important issue which the present analysis addresses.

In Kolmogorov theory and in advanced multifractal scaling law theories, ζ₂ has a range in which it is a flat line of constant value in log length scale, as observed. It has been noted that τ_p is not constant in the inertial range that is defined by ζ₂. Our data analysis and data of others show no clearly defined inertial range for the energy dissipation rate. The τ_p peak is p dependent, but occurs approximately at the Taylor micro-scale. The transition from the dissipation range to the inertial range takes place near the Taylor micro-scale.

We find that the Chen and Cao model for τ_p can be extended across all length scales for small p moments. Our refined CC_q model describes the relation between any τ_p and τ_q exponents. The CC_∞ model complements the SL analysis of vortices in the l/η → 1 limit are given.

Publicly available datasets were analyzed in this study. This data can be found here: http://turbulence.pha.jhu.edu/.

AH took the lead in the numerical analysis of the DNS data and in some of the discovery of the analysis algorithms. RK took the lead in designing the numerical algorithms had a major role in the interpolation of the data. JG lead in the overall planning and formulation of the study outline. All authors contributed to all aspects of the study, numerics, analysis, and conceptual modeling.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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