Fractional and Fractal Derivatives Modeling of Turbulence

Fractional and fractal derivatives modeling of turbulence W. Chen Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Division Box 26, Beijing 100088, China This study makes the first attempt to use the 2/3-order fractional Laplacian modeling of enhanced diffusing movements of random turbulent particle resulting from nonlinear inertial interactions. A combined effect of the inertial interactions and the molecule Brownian diffusivities is found to be the bi-fractal mechanism behind multifractal scaling in the inertial range of scales of moderate Reynolds number turbulence. Accordingly, a stochastic equation is proposed to describe turbulence intermittency. The 2/3-order fractional Laplacian representation is also used to construct a fractional Reynolds equation for nonlinear interactions of fluctuating velocity components, underlying turbulence spacetime fractal structures of Lévy 2/3 stable distribution. The new perspective of this study is that the fractional calculus is an effective approach modeling of chaotic fractal phenomena induced by nonlinear interactions. PACS: 52.35.Ra, 47.53.+n, 92.10.Lq, 05.40.Fb 1. Introduction The Kolmogorov -5/3 scaling characterizes the statistical similarity of turbulent motion at small scales based on the argument of local homogeneous isotropy 1. To some extent, the scaling law has been validated by numerous experimental and numerical data of sufficiently high Reynolds number turbulence1,2. However, a clear departure from -5/3 scaling exponent is also often observed in various turbulence experiments at finite Reynolds numbers, namely, the so-called intermittency. The consensus is that the intermittent property of turbulence calls for a power law of energy spectrum having exponent -5/3-c (c≥0). There are a few theories in the derivation of the correction exponent c. For instance, the β model, various multifractal model3,4, and Kolmogorov himself also refined his original -5/3 scaling by assuming that the kinetic energy dissipation rate ε is scale-dependent and obeys a lognormal distribution leading to the so-called intermittency correction1. A school of researchers consider that the non-Gaussian distribution of turbulence velocity leads to the violation of the original Kolmogorov scaling and intermittency manifests in fact a non-Gaussian velocity distribution4. This argument has been controversial since many regard that the Kolmogorov theory does not assume the velocity Gaussianity. In section 2, we revisit this issue and show that the Kolmogorov scaling indeed underlies the Gaussian distribution of velocity increments while does not require Gaussianity of the displacement and acceleration fields. There exist quite a few statistical models of turbulence intermittency. To my knowledge, little, however, has been achieved in the partial differential equation modeling of intermittency. This study proposes a fractional Laplacian stochastic equation to describe the intermittency, whose one-dimension solution is given in appendix A. In section 3, by representing the nonlinear interactions of fluctuating velocity components with the fractional Laplacian, we obtain the fractional Reynolds equation underlying the Lévy 2/3 stable distribution of random turbulence displacements. Finally, section 4 concludes this paper with some remarks. In appendix B, the Richardson and Hausdorff fractal derivatives are used to model turbulence as alternative approaches to fractional derivatives. In appendix C, the elastic turbulence of non-Newtonian fluids is analyzed with the fractional time derivative. Appendix D proposes a revised cascade picture of turbulence energy transport. The profound understanding of turbulence is up to now regarded as an unsolved problem. We consider that one major reason of this long-standing difficulty is the lacking of an appropriate mathematical devise. In this study, the innovative fractional calculus modeling is attempted to describe the complicated random phenomena of turbulence. 2. Intermittent statistical equation of turbulent diffusion In the Kolmogorov’s view of local homogeneous isotropic turbulence, the second-order structure function of velocity increments ∆u = u(x + r) − u(x) over a distance r within the inertial range of scales is considered a stochastic variable and obeys a scaling law5 2 ˆ 2 3 2 3 (∆u) = Cε r , for η ≤ r ≤ L0 , (1) where Cˆ is a universal dimensionless constant, the brackets represent the mean value of random variable ensemble, ε denotes the kinetic energy dissipation rate per 1 4 unit mass and is considered scale-independent, and η = (ε 3 υ) is the Kolmogorov dissipation length. The corresponding Kolmogorov scaling of turbulence kinetic energy transportation is E(k) = Cε 2 3k −5 3 , (2) where E(k) is the energy spectrum in terms of wavenumber k, and C denotes the Kolmogorov constant. On the other hand, it is well known that the diffusion of displacements in the Kolmogorov turbulence is consistent with the Richardson’s particle pair-distance superdiffusion5-9 (enhanced diffusion) of a fully developed homogeneous turbulence, namely, r 2 = Cε∆t 3 , (3) where ∆t denotes time interval and the experimental value of the dimensionless constant C is 0.5 given in ref. 6. (3) means particles move much faster than in normal diffusion ( r 2 ∝ ∆t ). Through a dimensional analysis of (1) and (3), we can derive (∆u)2 ∝ ∆t . (4) The above (3) and (4) show that the Kolmogorov turbulence in the inertial range of scales is of the normal diffusion of the velocity difference and the enhanced diffusion of displacements. Consequently, turbulence in the inertial range is considered to have Gaussian velocity field and non-Gaussian displacement field. Laboratory experiments and field observations have found that the statistics of the velocity increments in the inertial range is often close to Gaussian6. The displacement diffusion (4) can be restated as r 2 ∝ ∆t 2 α , α = 2 3. (5) The above formula (5) can be interpreted the displacement increments in turbulence obeys the Lévy α-stable distribution11, where α represents the stability index of the Lévy distribution. The rigorous mathematics proof shows that Lévy stability index α must be positive and not larger than 2 (0<α≤2) with the Gaussian distribution being its limiting α =2 case10,11. The non-Gaussian Lévy stable distribution of velocity difference has an algebraic decay tail. It is noted that the Gaussian distribution drastically underestimates the occurrence probability of the large events, while for heavy tailed statistics like the Lévy stable distribution the occurrence of extreme events is drastically enhanced. The Lévy distribution has long been used to describe strong long-range spatiotemporal correlation, featuring heavy tails, of anomalous diffusion in turbulence12-14. To the author’s knowledge, the corresponding differential equation model, however, has been missing. The fractional Laplacian has been a popular approach in recent years to model the Lévy statistical superdiffusion in a variety of physical master equations such as the Fokker-Planck equation15 and the anomalous diffusion equation10,16. Intuitively, we construct a linear phenomenological statistical equation within the inertial range of scales of fully developed isotropic homogeneous turbulence at sufficiently high Reynolds numbers ∂P + γ ()− ∆ α 2 P = 0 , α = 2 3, (6) ∂t where P(x,t) is the probability density function (pdf) to find a particle at x at time instant t, and (− ∆)α 2 represents the homogeneous symmetric (isotropic) fractional Laplacian17,18, and γ is turbulent diffusion coefficient. α can be understood the fractal dimension in this study. In terms of the generalized Einstein dissipation-fluctuation theorem19 and (3), we can derive γ = (Cε 2)1 3 . The Green function of the Cauchy problem of equation (6) results in the time-dependent Lévy pdf, which can naturally leads to Richardson’s turbulence superdiffusion (3) through the so-called Lévy walk mechanism7,12, while underlying the Gaussian velocity field and the Kolmogorov scaling in the inertial range of scales. It is known that the scalings (3) and (4) are obtained for fully developed homogeneous turbulence under sufficiently high Reynolds numbers and reflect the statistical self-similarity of eddy structures generated from nonlinear inertial interactions. Therefore, the superdiffusion diffusion equation (6) actually describes the enhanced diffusivity originating from coarse-grained average of the nonlinear inertial term in the Navier-Stokes equation. And there is no advective term in (6). It is also noted that (6) is a linear phenomenological model equation to characterize the fractal self-similarity of complicated nonlinear interactions. We call equation (6) inertial diffusion equation. On the other hand, for the finite Reynolds number turbulence, a clear deviation from Gaussian velocity field and t3 displacement superdiffusion at small scales has been observed in various turbulence experiments and numerical simulations, namely, turbulence intermittency20,21. In the absence of molecular diffusion, model equation (6) can not describe the intermittency. The addition of molecular diffusion will reflect intermittency for finite Reynolds number turbulence, namely, ∂P + γ ()− ∆ 1 3 P −υ∆P = 0 , (7) ∂t where ∆ represents the Laplacian operator, and υ molecular viscosity. (7) is called intermittent stochastic equation in this study. It is noted that the two diffusion terms in (7) are induced by the inertial interactions and molecular viscosity in the Navier-Stokes equation

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