Anomalous scaling exponents in nonlinear models of turbulence

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We construct, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by "statistically preserved structures" which are associated with exact conservation laws. The latter can be used, for example, to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the Navier-Stokes equations.     

Lagrangian chaos and the effect of drag on the enstrophy cascade in two-dimensional turbulence

We investigate the effect of drag force on the enstrophy cascade of two-dimensional Navier-Stokes turbulence. We find a power law decrease of the energy wave number (k) spectrum that is faster than the classical (no-drag) prediction of k(-3). It is shown that the enstrophy cascade with drag can be analyzed by making use of a previous theory for finite lifetime passive scalars advected by a Lagrangian chaotic fluid flow. Using this we relate the power law exponent of the energy wave number spectrum to the distribution of finite time Lyapunov exponents and the drag coefficient.     

A causal continuous-time stochastic model for the turbulent energy cascade in a helium jet flow

We discuss continuous cascade models and their potential for modelling the energy dissipation in a turbulent flow. Continuous cascade processes, expressed in terms of stochastic integrals with respect to Lévy bases, are examples of ambit processes. These models are known to reproduce experimentally observed properties of turbulence: the scaling and self-scaling of the correlators of the energy dissipation and of the moments of the coarse-grained energy dissipation. We compare three models: a normal model, a normal inverse Gaussian model, and a stable model. We show that the normal inverse Gaussian model is superior to both, the normal and the stable models, in terms of reproducing the distribution of the energy dissipation; and that the normal inverse Gaussian model is superior to the normal model and competitive with the stable model in terms of reproducing the self-scaling exponents. Furthermore, we show that the presented analysis is parsimonious in the sense that the self-scaling exponents are predicted from the one-point distribution of the energy dissipation, and that the shape of these distributions is independent of the Reynolds number.     

Inverse turbulent cascades and conformally invariant curves

We offer a new example of conformal invariance (local scale invariance) far from equilibrium-the inverse cascade of surface quasigeostrophic (SQG) turbulence. We show that temperature isolines are statistically equivalent to curves that can be mapped into a one-dimensional Brownian walk (called Schramm-Loewner evolution or SLEkappa). The diffusivity is close to kappa=4, that is, isotemperature curves belong to the same universality class as domain walls in the O(2) spin model. Several statistics of temperature clusters and isolines are shown to agree with the theoretical expectations for such a spin system at criticality. We also show that the direct cascade in two-dimensional Navier-Stokes turbulence is not conformal invariant. The emerging picture is that conformal invariance may be expected for inverse turbulent cascades of strongly interacting systems.     

Small-scale vorticity filaments and structure functions of the developed turbulence

We develop a theory of turbulence based on the Navier-Stokes equation, without using dimensional or phenomenological considerations. A small scale vortex filament is the main element of the theory. The theory allows to obtain the scaling law and to calculate the scaling exponents of Lagrangian and Eulerian velocity structure functions in the inertial range. The obtained results are shown to be in very good agreement with numerical simulations and experimental data. The introduction of stochasticity into the equations and derivation of scaling exponents are discussed in details. A weak dependence on statistical propositions is demonstrated. The relation of the theory to the multifractal model is discussed.     

Physical mechanism of the two-dimensional enstrophy cascade

In two-dimensional turbulence, irreversible forward transfer of enstrophy requires anticorrelation of the turbulent vorticity transport vector and the inertial-range vorticity gradient. We investigate the basic mechanism by numerical simulation of the forced Navier-Stokes equation. In particular, we obtain the probability distributions of the local enstrophy flux and of the alignment angle between vorticity gradient and transport vector. These are surprisingly symmetric and cannot be explained by a local eddy-viscosity approximation. The vorticity transport tends to be directed along streamlines of the flow and only weakly aligned down the fluctuating vorticity gradient. All these features are well explained by a local nonlinear model. The physical origin of the cascade lies in steepening of inertial-range vorticity gradients due to compression of vorticity level sets by the large-scale strain field.     

Universality and saturation of intermittency in passive scalar turbulence

The statistical properties of a scalar field advected by the nonintermittent Navier-Stokes flow arising from a two-dimensional inverse energy cascade are investigated. The universality properties of the scalar field are probed by comparing the results obtained with two types of injection mechanisms. Scaling properties are shown to be universal, even though anisotropies injected at large scales persist down to the smallest scales and local isotropy is not fully restored. Scalar statistics is strongly intermittent and scaling exponents saturate to a constant for sufficiently high orders. This is observed also for the advection by a velocity field rapidly changing in time, pointing to the genericity of the phenomenon.     

The quasi-periodic doubling cascade in the transition to weak turbulence

The quasi-periodic doubling cascade is shown to occur in the transition from regular to weakly turbulent behaviour in simulations of incompressible Navier–Stokes flow on a three-periodic domain. Special symmetries are imposed on the flow field in order to reduce the computational effort. Thus we can apply tools from dynamical systems theory such as continuation of periodic orbits and computation of Lyapunov exponents. We propose a model ODE for the quasi-period doubling cascade which, in a limit of a perturbation parameter to zero, avoids resonance related problems. The cascade we observe in the simulations is then compared to the perturbed case, in which resonances complicate the bifurcation scenario. In particular, we compare the frequency spectrum and the Lyapunov exponents. The perturbed model ODE is shown to be in good agreement with the simulations of weak turbulence. The scaling of the observed cascade is shown to resemble the unperturbed case, which is directly related to the well known doubling cascade of periodic orbits.     

Numerical study of chaos based on a shell model

A shell model is introduced to study a turbulence driven by the thermal instability (Rayleigh-Benard convection). This model equation describes cascade and chaos in the strong turbulence with high Rayleigh number. The chaos is numerically studied based on this model. The characteristics of the turbulence are analyzed and compared with those of the Gledzer-Ohkitani-Yamada (GOY) model. Quantities such as a mean value of total fluctuation energy, it's standard deviation, time averaged wave spectrum, probability distribution function, frequency spectrum, the maximum instantaneous Lyapunov exponent, distribution of instantaneous Lyapunov exponents, are evaluated. The dependences of these quantities on the error of numerical integration are also examined. There is not a clear correlation between the numerical accuracy and the accuracy of these quantities, since the interaction between a truncation error and an intrinsic nonlinearity of the system exists. A finding is that the maximum Lyapunov exponent is insensitive to a truncation error. (c) 1999 American Institute of Physics.     

Differential model for 2D turbulence

We present a phenomenological model for 2D turbulence in which the energy spectrum obeys a nonlinear fourth-order differential equation. This equation respects the scaling properties of the original Navier-Stokes equations, and it has both the −5/3 inverse-cascade and the −3 direct-cascade spectra. In addition, our model has Raleigh-Jeans thermodynamic distributions as exact steady state solutions. We use the model to derive a relation between the direct-cascade and the inverse-cascade Kolmogorov constants, which is in good qualitative agreement with the laboratory and numerical experiments. We discuss a steady state solution where both the enstrophy and the energy cascades are present simultaneously, and we discuss it in the context of the Nastrom-Gage spectrum observed in atmospheric turbulence. We also consider the effect of the bottom friction on the cascade solutions and show that it leads to an additional decrease and finite-wavenumber cutoffs of the respective cascade spectra, which agrees with the existing experimental and numerical results. The text was submitted by the authors in English.     

Lagrangian statistical theory of fully developed hydrodynamical turbulence

The Lagrangian velocity structure functions in the inertial range of fully developed fluid turbulence are for the first time derived based on the Navier-Stokes equation. For time tau much smaller than the correlation time, the structure functions are shown to obey the scaling relations K_{n}(tau) proportional, varianttau;{zeta_{n}}. The scaling exponents zeta_{n} are calculated analytically without any fitting parameters. The obtained values are in amazing agreement with the experimental results of the Bodenschatz group. A new relation connecting the Lagrangian structure functions of different orders analogously to the extended self-similarity ansatz is found.     

Quasi-Gaussian statistics of hydrodynamic turbulence in 4 / 3 +epsilon dimensions

The statistics of two-dimensional turbulence exhibit a riddle: the scaling exponents in the regime of inverse energy cascade agree with the K41 theory of turbulence far from equilibrium, but the probability distribution functions are close to Gaussian-like in equilibrium. The skewness S identical with S3(R)/S(3/2)(2)(R) was measured as S (exp) approximately 0.03. This contradiction is lifted by understanding that two-dimensional turbulence is not far from a situation with equipartition of enstrophy, which exists as true thermodynamic equilibrium with K41 exponents in space dimension of d= 4 / 3. We evaluate the skewness S( d) for 4 / 3 < or =d< or =2, showing that S(d)=0 at d= 4 / 3, and that it remains as small as S (exp) in two dimensions.     

Energy exchange in fast optical soliton collisions as a random cascade model

We study the dynamics of a probe soliton propagating in an optical fiber and exchanging energy in fast collisions with a random sequence of pump solitons. The energy exchange is induced by Raman scattering or by cubic nonlinear loss/gain. We show that the equation describing the dynamics of the probe soliton's amplitude has the same form as the equation for the local space average of energy dissipation in random cascade models in turbulence. We characterize the statistics of the probe soliton's amplitude by the τq exponents from multifractal theory and by the Cramér function S(x). We find that the nth moment of the two-time correlation function and the bit-error-rate contribution from amplitude decay exhibit power-law behavior as functions of propagation distance, where the exponents can be expressed in terms of τq or S(x).     

Inverse cascade regime in shell models of two-dimensional turbulence

We consider shell models that display an inverse energy cascade similar to two-dimensional turbulence (together with a direct cascade of an enstrophylike invariant). Previous attempts to construct such models ended negatively, stating that shell models give rise to a "quasiequilibrium" situation with equipartition of the energy among the shells. We show analytically that the quasiequilibrium state predicts its own disappearance upon changing the model parameters in favor of the establishment of an inverse cascade regime with Kolmogorov scaling. The latter regime is found where predicted, offering a useful model to study inverse cascades.     

On numerical modeling of the dynamics of turbulent wake behind a towed body in linearly stratified medium

Two numerical models of the dynamics of a turbulent wake behind a towed body in a linearly stratified medium are compared, namely, the model based on direct numerical integration of Navier-Stokes equations in the Oberbeck-Boussinesq approximation and the mathematical model with applying a semiempirical turbulence model of the second order. The calculation results of the two models are similar to the known experimental data and are in good agreement.     

叶栅内定常及非定常粘性流动数值模拟

本文给出了一个模拟叶栅内准三维定常和非定常粘性流动的数值方法。对于定常流动,采用ＴＶＤ Ｌａｘ－Ｗｅｎｄｒｏｆｆ格式和代数湍流模型求解雷诺平均Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程,使用当地时间步长和多网格技术使计算加速收敛到定常状态;对于非定常流动,使用双时间步长和全隐式离散,采用与求解定常流动相似的多网格方法求解隐式离散方程。文中给出了ＶＫＩ透平叶栅内的定常流结果和１．５级透平叶栅内的非定常数值结果。     

Turbulent cascades

Turbulent cascades at high Reynolds numbers are explained briefly in terms of multipliers and multiplier distributions. Two properties of the multipliers ensure the existence of power laws for locally averaged energy dissipation rate: (a) the existence of a multiplier probability density function that is independent of the level of the cascade, and (b) the statistical independence of multipliers at one level on those at previous levels. Under certain conditions described in the paper, the same two properties of multipliers guarantee that velocity increments over inertial-range separation distances also possess power laws. This work is specifically motivated by the need to understand the influence on scaling of the experimental observations that property (a) is true for turbulence, but property (b) is not; and additional motivation is the need to relate cascade models to intermittent vortex stretching (and folding). This effect has been modeled by allowing the multiplier distribution to depend on the magnitude of the local strain rate, and it is demonstrated that this rate-dependent model accounts for the statistical dependence observed in experiments. It is also shown that this model is consistent with the uncorrelated cascade models except for very weak singularity strengths (or for negative moments below a certain order), leading to the conclusion that, for all practical purposes, the uncorrelated level-independent multipliers abstract the essence of the breakdown process in turbulence.     

Pipe Flow and Wall Turbulence Using a Modified Navier-Stokes Equation

We use a derived incompressible modified Navier-Stokes equation to model pipe flow and wall turbulence.We reproduce the observed flattened paraboloid velocity profiles of turbulence that cannot be obtained directly using standard incompressible Navier-Stokes equation.The solutions found are in harmony with multi-valued velocity fields as a definition of turbulence.Repeating the procedure for the flow of turbulent fluid between two parallel flat plates we find similar flattened velocity profiles.We extend the analysis to the turbulent flow along a single wall and compare the results with experimental data and the established controversial von Karman logarithmic law of the wall.     

Multifractal dimension of Lagrangian turbulence

We report experimental measurements of the Lagrangian multifractal dimension spectrum in an intensely turbulent laboratory water flow by the optical tracking of tracer particles. The Legendre transform of the measured spectrum is compared with measurements of the scaling exponents of the Lagrangian velocity structure functions, and excellent agreement between the two measurements is found, in support of the multifractal picture of turbulence. These measurements are compared with three model dimension spectra. When the nonexistence of structure functions of order less than -1 is accounted for, the models are shown to agree well with the measured spectrum.