arXiv subjects

Victor Yakhot

Publications and source records attributed to Victor Yakhot.

At least 37 records · Page 2Linked to original sources

arXiv2005

Probability Densities in Strong Turbulence

According to modern developments in turbulence theory, the "dissipation" scales (u.v. cut-offs) $η$ form a random field related to velocity increments $δ_ηu$. In this work we, using Mellin's transform combined with the Gaussain large -scale boundary condition, calculate probability densities (PDFs) of velocity increments $P(δ_{r}u,r)$ and the PDF of the dissipation scales $Q(η, Re)$, where $Re$ is the large-scale Reynolds number. The resulting expressions strongly deviate from the Log-normal PDF $P_{L}(δ_{r}u,r)$ often quoted in the literature. It is shown that the probability density of the small-scale velocity fluctuations includes information about the large (integral) scale dynamics which is responsible for deviation of $P(δ_{r}u,r)$ from $P_{L}(δ_{r}u,r)$. A framework for evaluation of the PDFs of various turbulence characteristics involving spatial derivatives is developed. The exact relation, free of spurious Logarithms recently discussed in Frisch et al (J. Fluid Mech. {\bf 542}, 97 (2005)), for the multifractal probability density of velocity increments, not based on the steepest descent evaluation of the integrals is obtained and the calculated function $D(h)$ is close to experimental data. A novel derivation (Polyakov, 2005), of a well-known result of the multi-fractal theory [Frisch, "Turbulence. {\it Legacy of A.N.Kolmogorov}", Cambridge University Press, 1995)), based on the concepts described in this paper, is also presented.

physics.flu-dyn↗

arXiv2005

Anomalous Scaling of Structure Functions and Dynamic Constraints on Turbulence Simulations

The connection between anomalous scaling of structure functions (intermittency) and numerical methods for turbulence simulations is discussed. It is argued that the computational work for direct numerical simulations (DNS) of fully developed turbulence increases as $Re^{4}$, and not as $Re^{3}$ expected from Kolmogorov's theory, where $Re$ is a large-scale Reynolds number. Various relations for the moments of acceleration and velocity derivatives are derived. An infinite set of exact constraints on dynamically consistent subgrid models for Large Eddy Simulations (LES) is derived from the Navier-Stokes equations, and some problems of principle associated with existing LES models are highlighted.

nlin.CD↗

arXiv2005

Does multifractal theory of turbulence have logarithms in the scaling relations?

The multifractal theory of turbulence uses a saddle-point evaluation in determining the power-law behaviour of structure functions. Without suitable precautions, this could lead to the presence of logarithmic corrections, thereby violating known exact relations such as the four-fifths law. Using the theory of large deviations applied to the random multiplicative model of turbulence and calculating subdominant terms, we explain here why such corrections cannot be present.

nlin.CD↗

arXiv2004

Generalized Boltzmann Equation: Slip-No -Slip Dynamic Transition in Flows of Strongly Non-Linear Fluids

The Navier-Stokes equations, are understood as the result of the low-order expansion in powers of dimensionless rate of strain $η_{ij}=τ_{0}S_{ij}$, where $τ_{0}$ is the microscopic relaxation time of a close-to- thermodynamic equilibrium fluid. In strongly sheared non-equilibrium fluids where $|η_{ij}|\geq 1$, the hydrodynamic description breaks down. According to Bogolubov's conjecture, strongly non-equlibrium systems are characterized by an hierarchy of relaxation times corresponding to various stages of the relaxation process. A "hydro-kinetic" equation with the relaxation time involving both molecular and hydrodynamic components proposed in this paper, reflects qualitative aspects of Bogolubov's hierarchy. It is shown that, applied to wall flows, this equation leads to qualitatively correct results in an extremely wide range of parameter $η$-variation. Among other features, it predicts the onset of slip velocity at the wall as an instability of the corresponding hydrodynamic approximation.

physics.flu-dyn↗

arXiv2004

Large-Scale Coherence and Law of Decay of Two-Dimensional Turbulence

At the short times, the enstrophy $Ω$ of a two-dimensional flow, generated by a random Gaussian initial condition decays as $Ω(t)\propto t^{-γ}$ with $γ\approx 0.7$. After that, the flow undergoes transition to a different state characterized by the magnitude of the decay exponent $γ\approx 0.4$ (Yakhot, Wanderer, Phys.Rev.Lett.{\bf 93}, 154502 \~(2004)). It is shown that the very existence of this transition and various characteristics of evolving flow crucially depend upon phase correlation between the large-scale modes containing only a few percent of total enstrophy.

nlin.CD↗

arXiv2004

Towards a dynamical theory of multifractals in turbulence

Making use of the exact equations for structure functions, supplemented by the equations for dissipa tive anomaly as well as an estimate for the Lagrangian acceleration of fluid particles, we obtain a main result of the multifractal theory of turbulence. The central element of the theory is a dissipation cut-off that depends on the order of the structure function. An expression obtained for the exponents $s_{n}$ in the scaling relations $\bar{(\frac{\partial u}{\partial x})^{n}}/\bar{(\frac{\partial u}{\partial x})^{2}}^{\frac{n}{2}} \propto Re^{s_{n}}$, between the velocity gradients $\frac{\partial u}{\partial x}$ and the Reynolds number $Re$, agrees well with experimental data.

nlin.CD↗

arXiv2004

A Cross-Over in the Enstrophy Decay in Two-Dimensional Turbulence in a Finite Box

The numerical simulation of two-dimensional decaying turbulence in a large but finite box presented in this paper uncovered two physically different regimes of enstrophy decay. During the initial stage, the enstrophy, generated by a random Gaussian initial condition, decays as t^{-gamma} with gamma approximately 0.7-0.8. After that, the flow undergoes a transition to a gas or fluid composed of distinct vortices. Simultaneously, the magnitude of the decay exponent crosses over to gamma approximately 0.4. An exact relation for the total number of vortices, N(t), in terms of the mean circulation of an individual vortex is derived. A theory predicting that N(t) is proportional to t^{-xi} and the magnitudes of exponents gamma=2/5 and xi=4/5 is presented and the possibility of an additional very late-time cross-over to gamma=1/3 and xi=2/3 is also discussed.

nlin.CD↗

arXiv2002

Self-sustained oscillations in homogeneous shear flow

Generation of the large-scale coherent vortical structurs in homogeneous shear flow couples dynamical processes of energy and enstrophy production. In the large rate of strain limit, the simple estimates of the contributions to the energy and enstrophy equations result in a dynamical system, describing experimentally and numerically observed self-sustained non-linear oscillations of energy and enstrophy. It is shown that the period of these oscilaltions is independent upon the box size and the energy and enstrophy fluctuations are strongly correlated.

nlin.CD↗

arXiv2002

On a puzzle posed by the Kurien-Sreenivasan boundary layer experiment

The Kolmogorov inertial range ratio of the mixed- to- longitudinal third-order structure functions of isotropic and homogeneous turbulence is $\frac{S_{1,2}}{S_{3,0}}=1/3$. Recent measurements by Kurien and Sreenivasan (Phys. Rev. E {\bf 64}, 056302 (2001)), showed that, while the longitudinal structure function was extremely close to Kolmogorov's, the measured ratio was very far: $\frac{S_{1,2}}{S_{3,0}}\approx 0.43$. Explanation of this puzzle is presented in this letter.

nlin.CD↗

arXiv2001

Mean- Field Approximation and Extended Self-Similarity in Turbulence

Recent experimental discovery of extended self-similarity (ESS) was one of the most interesting developments, enabling precise determination of the scaling exponents of fully developed turbulence. Here we show that the ESS is consistent with the Navier-Stokes equations, provided the pressure -gradient contributions are expressed in terms of velocity differences in the mean field approximation (Yakhot, Phys.Rev. E{\bf 63}, 026307, (2001)). A sufficient condition for extended self-similarity in a general dynamical system

nlin.CD↗

arXiv2001

A Critical "Dimension" in a Shell Model for Turbulence

We investigate the GOY shell model within the scenario of a critical dimension in fully developed turbulence. By changing the conserved quantities, one can continuously vary an ``effective dimension'' between $d=2$ and $d=3$. We identify a critical point between these two situations where the flux of energy changes sign and the helicity flux diverges. Close to the critical point the energy spectrum exhibits a turbulent scaling regime followed by a plateau of thermal equilibrium. We identify scaling laws and perform a rescaling argument to derive a relation between the critical exponents. We further discuss the distribution function of the energy flux.

nlin.CD↗

arXiv2000

Mean- Field Approximation and a Small Parameter in Turbulence Theory

Numerical and physical experiments on two-dimensional (2d) turbulence show that the differences of transverse components of velocity field are well described by a gaussian statistics and Kolmogorov scaling exponents. In this case the dissipation fluctuations are irrelevant in the limit of small viscosity. In general, one can assume existence of critical space-dimensionality $d=d_{c}$, at which the energy flux and all odd-order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant. At $d 0$ and $r/L\to 0$ in three-dimensional flows in close agreement with experimental data. In addition, some new exact relations between correlation functions of velocity differences are derived. It is also predicted that the single-point pdf of transverse velocity difference in developing as well as in the large-scale stabilized two-dimensional turbulence is a gaussian.

nlin.CD↗

arXiv1999

Symmetry Breaking, Anomalous Scaling and Large-Scale Flow Generation in a Convection Cell

We consider a convection process in a thin loop. At $Ra=Ra^{\prime}_{cr}$ a first transition leading to the generation of corner vortices is observed. At higher $Ra$ a coherent large-scale flow, which persists for a very long time, sets up. The mean velocity $\bar{v}$, mass flux $\dm$, and the Nusselt number $Nu$ in this flow scale with $Ra$ as $\bar{v} \propto \dot{m} \propto Ra^{0.45}$ and $Nu\propto Ra^{0.9}$, respectively. The time evolution of the coherent flow is well described by the Landau amplitude equation within a wide range of $Ra$-variation. The anomalous scaling of the mean velocity, found in this work, resembles the one experimentally observed in the ``hard turbulence'' regime of Benard convection. A possible relation between the two systems is discussed.

chao-dyn↗

arXiv1999

Statistics of Transverse Velocity Differences in Turbulence

An unusual symmetry of the equation for the generating function of transverse velocity differences $Δv=v(x+r)-v(x)$ is used to obtain a closed equation for the probability density function $P(Δv,r)$ in strong three-dimensional turbulence. It is shown that the terms, mixing longitudinal and transverse components of velocity field, dominate the pressure contributions. The dissipation terms are closed on qualitative grounds. The resulting equation gives the shape of the pdf, anomalous scaling exponents and amplitudes of the moments of transversevelocity differences.

chao-dyn↗

arXiv1999

Two-Dimensional turbulence in the inverse cascade range

Numerical and physical experiments on the forced two-dimensional Navier-Stokes equations show that transverse velocity differences are described by ``normal'' Kolmogorov scaling $<(Δv)^{2n}> \propto r^{2n/3}$ and obey a gaussian statistics. Since the non-trivial scaling is a sign of a strong non-linearity of the problem, these two results seem to contradict each other. A theory, explaining this result is presented in this paper. Strong time-dependence of the large-scale features of the flow ($\bar{u^2}\propto t$) results in decoupling of the large-scale dynamics from statistically steady-state small-scale random processes. This time-dependence is also a reason for the localness of the pressure-gradient terms in the equations governing the small-scale velocity difference PDF's. The derived self-consistent expression for the pressure gradient contributions lead to the conclusion that the small-scale transverse velocity differences are governed by a linear Langevin-like equation, strirred by a non-local, universal, solution-depending gaussian random force. This explains the experimentally observed gaussian statistics of transverse velocity differences and their Kolmogorov scaling. The solution for the PDF of longitudinal velocity differences is based on a smallness of the energy flux in two-dimensional turbulence. The theory makes a few quantitative predictions which can be tested experimentally.

chao-dyn↗

arXiv1998

Strong Turbulence in d-Dimensions

In the limit $d\to\infty$ the role of pressure gradients and that of the incompressibility constraint decreases, thus blurring the difference between transverse and longitudinal velocity correlation functions. Using Polyakov's expression for the dissipation anomaly the closed equation for the probability density function is obtained. This model for the dissipation terms is the only one satisfying both equations of motion and a set of dynamical constraints. The resulting equations show that when $d\to\infty$, the predictions of Kolmogorov theory are exact. It is also shown that the $O(1/d)$ pressure effects, producing the regularization of the equations of motion for the PDF are responsible for the distinction between the Navier-Stokes and Burgers dynamics.

chao-dyn↗

arXiv1997

Probability density and scaling exponents of the moments of longitudinal velocity difference in strong turbulence

We consider a few cases of homogeneous and isotropic turbulence differing by the mechanisms of turbulence generation. The advective terms in the Navier-Stokes and Burgers equations are similar. It is proposed that the longitudinal structure functions $S_{n}(r)$ in homogeneous and isotropic three- dimensional turbulence are governed by a one-dimensional equation of motion, resembling the 1D-Burgers equation, with the strongly non-local pressure contributions accounted for by galilean-invariance-breaking terms. The resulting equations, not involving parameters taken from experimental data, give both scaling exponents and amplitudes of the structure functions in an excellent agreement with experimental data. The derived probability density function $P(Δu,r)\neq P(-Δu,r)$ but $P(Δu,r)=P(-Δu,-r)$, in accord with the symmetry properties of the Navier-Stokes equations. With decrease of the displacement $r$, the probability density, which cannot be represented in a scale-invariant form, shows smooth variation from the gaussian at the large scales to close-to-exponential function, thus demonstrating onset of small-scale intermittency. It is shown that accounting for the sub-dominant contributions to the structure functions $S_{n}(r)\propto r^{ξ_{n}}$ is crucial for derivation of the amplitudes of the moments of velocity difference.

chao-dyn↗

arXiv1996

Scaling Exponents of Strong Turbulence in the Eddy Viscosity Approximation

A dynamic equation for velocity structure functions $S_{n}(r) = <(u(x)-u(x'))^{n}>$ in strong turbulence is derived in the one-loop (eddy viscosity) approximation. This homogemeous differential equation yields scaling exponents $ξ_{n}$ in the relations $S_{n}(r)\propto r^{ξ_{n}}$ which are in a very good agreement with experimental data.

chao-dyn↗