In previous papers I have argued that the \emph{fusion rules hypothesis}, which was originally introduced by L'vov and Procaccia in the context of the problem of three-dimensional turbulence, can be used to gain a deeper insight in understanding the enstrophy cascade and inverse energy cascade of two-dimensional turbulence. In the present paper we show that the fusion rules hypothesis, combined with \emph{non-perturbative locality}, itself a consequence of the fusion rules hypothesis, dictates the location of the boundary separating the inertial range from the dissipation range. In so doing, the hypothesis that there may be an anomalous enstrophy sink at small scales and an anomalous energy sink at large scales emerges as a consequence of the fusion rules hypothesis. More broadly, we illustrate the significance of viewing inertial ranges as multi-dimensional regions where the fully unfused generalized structure functions of the velocity field are self-similar, by considering, in this paper, the simplified projection of such regions in a two-dimensional space, involving a small scale $r$ and a large scale $R$, which we call, in this paper, the $(r, R)$-plane. We see, for example, that the logarithmic correction in the enstrophy cascade, under standard molecular dissipation, plays an essential role in inflating the inertial range in the $(r, R)$ plane to ensure the possibility of local interactions. We have also seen that increasingly higher orders of hyperdiffusion at large scales or hypodiffusion at small scales make the predicted sink anomalies more resilient to possible violations of the fusion rules hypothesis.
Freely decaying two-dimensional Navier--Stokes turbulence is studied. The conservation of vorticity by advective nonlinearities renders a class of Casimirs that decays under viscous effects. A rigorous constraint on the palinstrophy production by nonlinear transfer is derived, and an upper bound for the enstrophy dissipation is obtained. This bound depends only on the decaying Casimirs, thus allowing the enstrophy dissipation to be bounded from above in terms of initial data of the flows. An upper bound for the enstrophy dissipation wavenumber is derived and the new result is compared with the…
We derive for the Navier-Stokes equation an exact equation satisfied by the dissipation rate correlation function. In the equal time limit, in the inertial range, for the homogeneous, isotropic state of fully-developed turbulence, we show that the correlation is characterized by two exponents, one of which depends on the sixth-order static structure function exponent and the other involves the behavior of the fourth-order dynamical structure function. This provides a direct derivation of the intermittency exponent of the dissipation-rate correlation.
In helical turbulence a linear cascade of helicity accompanying the energy cascade has been suggested. Since energy and helicity have different dimensionality we suggest the existence of a characteristic inner scale, $ξ=k_H^{-1}$, for helicity dissipation in a regime of hydrodynamic fully developed turbulence and estimate it on dimensional grounds. This scale is always larger than the Kolmogorov scale, $η=k_E^{-1}$, and their ratio $η/ ξ$ vanishes in the high Reynolds number limit, so the flow will always be helicity free in the small scales.
A new transient regime in the relaxation towards absolute equilibrium of the conservative and time-reversible 3-D Euler equation with high-wavenumber spectral truncation is characterized. Large-scale dissipative effects, caused by the thermalized modes that spontaneously appear between a transition wavenumber and the maximum wavenumber, are calculated using fluctuation dissipation relations. The large-scale dynamics is found to be similar to that of high-Reynolds number Navier-Stokes equations and thus to obey (at least approximately) Kolmogorov scaling.
The Reynolds number dependence of the statistics of energy dissipation is investigated in a shell model of fully developed turbulence. The results are in agreement with a model which accounts for fluctuations of the dissipative scale with the intensity of energy dissipation. It is shown that the assumption of a fixed dissipative scale leads to a different scaling with Reynolds which is not compatible with numerical results.
We study two-dimensional turbulence in a doubly periodic domain driven by a monoscale-like forcing and damped by various dissipation mechanisms of the form $ν_μ(-Δ)^μ$. By ``monoscale-like'' we mean that the forcing is applied over a finite range of wavenumbers $k_\min \leq k \leq k_\max$, and that the ratio of enstrophy injection $η\geq 0$ to energy injection $ε\geq 0$ is bounded by $k_\min^2 ε\leq η\leq k_\max^2 ε$. It is shown that for $μ\geq 0$ the asymptotic behaviour satisfies (eqnarray) \norm u_1^2&\leq&k_\max^2\norm u^2,(eqnarray) where $\norm u^2$ and $\norm u_1^2$ are the energy and …
We carry out a self consistent calculation of the structure functions in the dissipation range using Navier Stokes equation. Combining these results with the known structures in the inertial range, we actually propose crossover functions for the structure functions that takes one smoothly from the inertial to the dissipation regime. In the process the success of the extended self similarity is explicitly demonstrated.
A strong local form of the ``4/3-law'' in turbulent flow has been proved recently by Duchon and Robert for a triple moment of velocity increments averaged over both a bounded spacetime region and separation vector directions, and for energy dissipation averaged over the same spacetime region. Under precisely stated hypotheses, the two are proved to be proportional, by a constant 4/3, and to appear as a nonnegative defect measure in the local energy balance of singular (distributional) solutions of the incompressible Euler equations. Here we prove that the energy defect measure can be represent…
The Weak Turbulence Theory has been applied to waves in thin elastic plates obeying the Föppl-Von Kármán dynamical equations. Subsequent experiments have shown a strong discrepancy between the theoretical predictions and the measurements. Both the dynamical equations and the Weak Turbulence Theory treatment require some restrictive hypotheses. Here a direct numerical simulation of the Föppl-Von Kármán equations is performed and reproduces qualitatively and quantitatively the experimental results when the experimentally measured damping rate of waves $γ_\mathbf{k}= a + bk^2$ is used. This confi…
We revisit the issue of Lagrangian irreversibility in the context of recent results [Xu, et al., PNAS, 111, 7558 (2014)] on flight-crash events in turbulent flows and show how extreme events in the Eulerian dissipation statistics are related to the statistics of power-fluctuations for tracer trajectories. Surprisingly, we find that particle trajectories in intense dissipation zones are dominated by energy gains sharper than energy losses, contrary to flight-crashes, through a pressure-gradient driven take-off phenomenon. Our conclusions are rationalised by analysing data from simulations of th…
We study stationary solutions to the continuity equation for weakly compressible flows. These describe non-equilibrium steady states of weakly dissipative dynamical systems. Compressibility is a singular perturbation that changes the steady state density from a constant "microcanonical" distribution into a singular multifractal measure supported on the "strange attractor". We introduce a representation of the latter and show that the space-averaged properties are described universally by a log-normal distribution determined by a single structure function. The spectrum of fractal dimensions is …
We generalize the wavelet transform modulus maxima (WTMM) method to multifractal analysis of 3D random fields. This method is calibrated on synthetic 3D monofractal fractional Brownian fields and on 3D multifractal singular cascade measures as well as their random function counterpart obtained by fractional integration. Then we apply the 3D WTMM method to the dissipation field issue from 3D isotropic turbulence simulations. We comment on the need to revisiting previous box-counting analysis which have failed to estimate correctly the corresponding multifractal spectra because of their intrinsi…
A logarithmic scaling for structure functions, in the form $S_p \sim [\ln (r/η)]^{ζ_p}$, where $η$ is the Kolmogorov dissipation scale and $ζ_p$ are the scaling exponents, is suggested for the statistical description of the near-dissipation range for which classical power-law scaling does not apply. From experimental data at moderate Reynolds numbers, it is shown that the logarithmic scaling, deduced from general considerations for the near-dissipation range, covers almost the entire range of scales (about two decades) of structure functions, for both velocity and passive scalar fields. This n…
Maxima of the scalar dissipation rate in turbulence appear in form of sheets and correspond to the potentially most intensive scalar mixing events. Their cross-section extension determines a locally varying diffusion scale of the mixing process and extends the classical Batchelor picture of one mean diffusion scale. The distribution of the local diffusion scales is analysed for different Reynolds and Schmidt numbers with a fast multiscale technique applied to very high-resolution simulation data. The scales take always values across the whole Batchelor range and beyond. Furthermore, their dist…
We relate the intermittent fluctuations of velocity gradients in turbulence to a whole range of local dissipation scales generalizing the picture of a single mean dissipation length. The statistical distribution of these local dissipation scales as a function of Reynolds number is determined in numerical simulations of forced homogeneous isotropic turbulence with a spectral resolution never applied before which exceeds the standard one by at least a factor of eight. The core of the scale distribution agrees well with a theoretical prediction. Increasing Reynolds number causes the generation of…
We propose and verify a wave-vector-space version of generalized extended self similarity and broaden its applicability to uncover intriguing, universal scaling in the far dissipation range by computing high-order ($\leq 20\/$) structure functions numerically for: (1) the three-dimensional, incompressible Navier Stokes equation (with and without hyperviscosity); and (2) the GOY shell model for turbulence. Also, in case (2), with Taylor-microscale Reynolds numbers $4 \times 10^{4} \leq Re_λ \leq 3 \times 10^{6}\/$, we find that the inertial-range exponents ($ζ_{p}\/$) of the order - $p\/$ struc…
The very small scales of isotropic, Navier-Stokes turbulence at Reynolds number ${\cal R}_λ\approx 15$ are studied by high-resolution direct numerical simulation (DNS) and by integration of the direct-interaction (DIA) equations. The DNS follows the tail of the energy spectrum over more than thirty decades of magnitude. The energy spectrum in the far-dissipation range $5k_d < k < 10k_d$ is well-fitted by $k^α\exp(-ck/k_d)$, where $k_d$ is the Kolmogorov dissipation wavenumber, $α\approx 3.3$ and $c\approx 7.1$. For values of $m$ that emphasize the far-dissipation range, the fields $(-\nabla^2)…
Probability density function (PDF) of passive scalar dissipation ${\cal P} (ε)$ is found analytically in the limit of large Peclet and Prandtl numbers (Batchelor-Kraichnan regime) in two dimensions. The tail of PDF at $ε\gg<ε>$ is shown to be stretched exponent $\ln{\cal P}(ε)\proptoε^{1/3}$. At $ε\ll<ε>$, ${\cal P}\propto 1/\sqrtε$.
We carry out a direct numerical simulation (DNS) study that reveals the effects of polymers on statistically steady, forced, homogeneous, isotropic fluid turbulence. We find clear manifestations of dissipation-reduction phenomena: On the addition of polymers to the turbulent fluid, we obtain a reduction in the energy dissipation rate, a significant modification of the fluid energy spectrum, especially in the deep-dissipation range, a suppression of small-scale intermittency, and a decrease in small-scale vorticity filaments. We also compare our results with recent experiments and earlier DNS s…
Reduction of Triadic Interactions Suppresses Intermittency and Anomalous Dissipation in Turbulence
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心，塔塔基础研究所）；Department of Mechanical Engineering, Johns Hopkins University（约翰斯·霍普金斯大学机械工程系）；Complex Fluids and Flows Unit, Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学复杂流体与流动单元）
We investigate the connection between the inertial range and the dissipation range statistics of rotating turbulence through detailed simulations of a helical shell model and a multifractal analysis. In particular, by using the latter, we find an explicit relation between the (anomalous) scaling exponents of equal-time structure functions in the inertial range in terms of the generalised dimensions associated with the energy dissipation rate. This theoretical prediction is validated by detailed simulations of a helical shell model for various strengths of rotation from where the statistics of …
A dissipation rate, which grows faster than any power of the wave number in Fourier space, may be scaled to lead a hydrodynamic system {\it actually} or {\it potentially} converge to its Galerkin truncation. Actual convergence we name for the asymptotic truncation at a finite wavenumber $k_G$ above which modes have no dynamics; and, we define potential convergence for the truncation at $k_G$ which, however, grows without bound. Both types of convergence can be obtained with the dissipation rate $μ[cosh(k/k_c)-1]$ who behaves as $k^2$ (newtonian) and $\exp\{k/k_c\}$ for small and large $k/k_c$ …
The anomalous scaling phenomena of three-dimensional passive scalar turbulence are studied using high resolution direct numerical simulation. The inertial range scaling exponents of the passive scalar increment and the scalar dissipation are obtained. The connection between the intermittency structure and the scaling exponent is examined and the structure instability of the high amplitude scalar dissipation is used to clarify previous experimental results for the scaling exponents.
Procedures for time-ordering the covariance function, as given in a previous paper (K. Kiyani and W.D. McComb Phys. Rev. E 70, 066303 (2004)), are extended and used to show that the response function associated at second order with the Kraichnan-Wyld perturbation series can be determined by a local (in wavenumber) energy balance. These time-ordering procedures also allow the two-time formulation to be reduced to time-independent form by means of exponential approximations and it is verified that the response equation does not have an infra-red divergence at infinite Reynolds number. Lastly, si…
This paper describes a conceptual framework for understanding kinetic plasma turbulence as a generalized form of energy cascade in phase space. It is emphasized that conversion of turbulent energy into thermodynamic heat is only achievable in the presence of some (however small) degree of collisionality. The smallness of the collision rate is compensated by the emergence of small-scale structure in the velocity space. For gyrokinetic turbulence, a nonlinear perpendicular phase mixing mechanism is identified and described as a turbulent cascade of entropy fluctuations simultaneously occurring a…
