arXiv subjects

Victor Yakhot

Publications and source records attributed to Victor Yakhot.

At least 19 recordsLinked to original sources

arXiv2022

The saturation of exponents and the asymptotic fourth state of turbulence

A recent discovery about the inertial range of homogeneous and isotropic turbulence is the saturation of the scaling exponents $ζ_n$ for large $n$, defined via structure functions of order $n$ as $S_{n}(r)=\overline{(δ_r u)^{n}}=A(n)r^{ζ_{n}}$. We focus on longitudinal structure functions for $δ_r u$ between two positions that are $r$ apart in the same direction. In a previous paper (Phys.\ Rev.\ Fluids 6, 104604, 2021), we developed a theory for $ζ_n$, which agrees with measurements for all $n$ for which reliable data are available, and shows saturation for large $n$. Here, we derive expressions for the probability density functions of $δ_r u$ for four different states of turbulence, including the asymptotic fourth state corresponding to the saturation of exponents for large $n$. This saturation means that the scale separation is violated in favor of a strongly-coupled quasi-ordered flow structures, which take the form of long and thin (worm-like) structures of length $L$ and thickness $l=O(L/Re)$.

physics.flu-dyn↗

arXiv2020

Transitions and Multi-Scaling in Rayliegh-Benard Convection. Small-Scale Universality

Asymptotically large Reynolds number hydrodynamic turbulence is characterized by multi-scaling of moments of velocity increments and spatial derivatives. With decreasing Reynolds number toward $R_λ=R^{tr}_λ\approx 9.0$, the anomalous scaling disappears in favor of the "normal" one and close-to-Gaussian probability densities [Yakhot \& Donzis, {\bf 119}, 044501 (2017)]. The nature of this transition and its universality are subjects of this work. Here we consider Benard convection ( Prandtl number $Pr=1$) between infinite horizontal plates. It is shown that in this system the "competition" between Bolgiano and Kolmogorov processes, results in small-scale velocity fluctuations driven by effective "large-scale" Gaussian random temperature field. Therefore, the intermittent dynamics of velocity derivatives are similar or even identical to that in homogeneous and isotropic turbulence generated by the large-scale random forcing. It is shown that low-Rayleigh number instabilities make the problem much more involved and may lead to transition from Gaussian to exponential PDF of the temperature field. The developed {\it mean-field theory} yielded dimensionless heat flux $Nu\propto Ra^β$ with $β\approx 15/56\approx 0.27$, close to the outcome of Chicago experiment. These results point to an unusual small-scale universality of turbulent flows. It is also shown that at $R_λ\leq 9.0$, a flow "remembers" its laminar background and, therefore, cannot be universal.

physics.flu-dyn↗

arXiv2018

Transition to turbulence scaling in Rayleigh-Bénard convection

If a fluid flow is driven by a weak Gaussian random force, the nonlinearity in the Navier-Stokes equations is negligibly small and the resulting velocity field obeys Gaussian statistics. Nonlinear effects become important as the driving becomes stronger and a transition occurs to turbulence with anomalous scaling of velocity increments and derivatives. This process has been described by V. Yakhot and D. A. Donzis, Phys. Rev. Lett. 119, 044501 (2017) for homogeneous and isotropic turbulence (HIT). In more realistic flows driven by complex physical phenomena, such as instabilities and nonlocal forces, the initial state itself, and the transition to turbulence from that initial state, are much more complex. In this paper, we discuss the Reynolds-number-dependence of moments of the kinetic energy dissipation rate of orders 2 and 3 obtained in the bulk of thermal convection in the Rayleigh-Bénard system. The data are obtained from three-dimensional spectral element direct numerical simulations in a cell with square cross section and aspect ratio 25 by A. Pandey et al., Nat. Commun. 9, 2118 (2018). Different Reynolds numbers $1 \lesssim {\rm Re}_{\ell} \lesssim 1000$ which are based on the thickness of the bulk region $\ell$ and the corresponding root-mean-square velocity are obtained by varying the Prandtl number Pr from 0.005 to 100 at a fixed Rayleigh number ${\rm Ra}=10^5$. A few specific features of the data agree with the theory but the normalized moments of the kinetic energy dissipation rate, ${\cal E}_n$, show a non-monotonic dependence for small Reynolds numbers before obeying the algebraic scaling prediction for the turbulent state. Implications and reasons for this behavior are discussed.

physics.flu-dyn↗

arXiv2018

Anomalous Exponents in Strong Turbulence

To characterize fluctuations in a turbulent flow, one usually studies different moments of velocity increments and dissipation rate, $\overline{(v(x+r)-v(x))^{n}}\propto r^{ζ_{n}}$ and $\overline{{\cal E}^{n}}\propto Re^{d_{n}}$, respectively. In high Reynolds number flows, the moments of different orders cannot be simply related to each other which is the signature of anomalous scaling, one of the most puzzling features of turbulent flows. High-order moments are related to extreme, rare events and our ability to quantitatively describe them is crucially important for meteorology, heat, mass transfer and other applications. In this work we present a solution to this problem in the particular case of the Navier-Stokes equations driven by a random force. A novel aspect of this work is that, unlike previous efforts which aimed at seeking solutions around the infinite Reynolds number limit, we concentrate on the vicinity of transitional Reynolds numbers $Re^{tr}$ where the first emergence of anomalous scaling is observed out of a low-$Re$ Gaussian background. The obtained closed expressions for anomalous scaling exponents $ζ_{n}$ and $d_{n}$, which depend on the transition Reynolds number, agree well with experimental and numerical data in the literature and, when $n\gg 1$, $d_{n}\approx 0.19n \ln(n)$. The theory yields the energy spectrum $E(k)\propto k^{-ζ_{2}-1}$ with $ζ_{2}\approx 0.699$, different from the outcome of Kolmogorov's theory. It is also argued that fluctuations of dissipation rate and those of the transition point itself are responsible for both, deviation from Gaussian statistics and multiscaling of velocity field.

physics.flu-dyn↗

arXiv2017

Multiscaling in Strong Turbulence Driven by a Random Force

Turbulence problem is often considered as "the last unsolved problem of classical physics". It is due to strong interaction between velocity and/or velocity gradient fluctuations, a high Reynolds number flow is a fascinating mixture of purely random, close to Gaussian, fields and coherent structures where substantial fraction of kinetic energy is dissipated into heat. To evaluate intensity of fluctuations, one usually studies different moments of velocity increments and/or dissipation rate, characterized by scaling exponents $ζ_{n}$ and $d_{n}$, respectively. In high Reynolds number flows, the moments of different orders with $n\neq m$ cannot be simply related to each other, which is the signature of anomalous scaling, making this problem "the last unsolvable". No perturbative treatment can lead to quantitative description of this feature. In this work the expressions for the moments of dissipation rate $e_{n}=\overline{{\cal E}^{n}}\propto Re^{d_{n}}$ and those of velocity derivatives $M_{2n}=\overline{(\partial_{x}u_{x})^{2n}}\propto \frac{v_{o}^{2n}}{L^{2n}}Re^{ρ_{2n}}$ are derived for an infinite fluid stirred by a white-in-time Gaussian random force supported in the vicinity of the wave number $k_{f}\approx \frac{2π}{L}=O(1)$, where $v_{0}$ and $L$ are characteristic velocity and integral scale, respectively. A novel aspect of this work is that unlike previous efforts which aimed at seeking solutions around the infinite Reynolds number limit, we concentrate on the vicinity of transitional Reynolds numbers $Re^{tr}$ of the first emergence of anomalous scaling out of Low-Re Gaussian background. The obtained closed expressions for anomalous scaling exponents $d_{n}$ and $ρ_{n}$ agree well with available in literature experimental and numerical data and, when $n\gg 1$, $d_{n}\approx 0.3n \ln(n)$.

physics.flu-dyn↗

arXiv2017

Emergence of Multi-Scaling in a Random Force-Stirred Fluid

We consider transition to strong turbulence in an infinite fluid stirred by a gaussian random force. The transition is {\bf defined} as a first appearance of anomalous scaling of normalized moments of velocity derivatives (dissipation rates) emerging from the low-Reynolds-number Gaussian background. It is shown that due to multi-scaling, strongly intermittent rare events can be quantitatively described in terms of an infinite number of different "Reynolds numbers" reflecting multitude of anomalous scaling exponents. The theoretically predicted transition disappears at $R_λ\leq 3$. The developed theory, is in a quantitative agreement with the outcome of large-scale numerical simulations.

physics.flu-dyn↗

arXiv2017

Generalized Knudsen Number for Unsteady Fluid Flow

We explore the scaling behavior of an unsteady flow that is generated by an oscillating body of finite size in a gas. If the gas is gradually rarefied, the Navier-Stokes equations begin to fail and a kinetic description of the flow becomes more appropriate. The failure of the Navier-Stokes equations can be thought to take place via two different physical mechanisms: either the continuum hypothesis breaks down as a result of a finite size effect; or local equilibrium is violated due to the high rate of strain. By independently tuning the relevant linear dimension and the frequency of the oscillating body, we can experimentally observe these two different physical mechanisms. All the experimental data, however, can be collapsed using a single dimensionless scaling parameter that combines the relevant linear dimension and the frequency of the body. This proposed Knudsen number for an unsteady flow is rooted in a fundamental symmetry principle, namely Galilean invariance.

physics.flu-dyn↗

arXiv2014

Reynolds number of transition and large-scale properties of strong turbulence

A turbulent flow is characterized by velocity fluctuations excited in an extremely broad interval of wave numbers $k> Λ_{f}$ where $Λ_{f}$ is a relatively small set of the wave-vectors where energy is pumped into fluid by external forces. Iterative averaging over small-scale velocity fluctuations from the interval $Λ_{f}< k\leq Λ_{0}$, where $η=2π/Λ_{0}$ is the dissipation scale, leads to an infinite number of "relevant" scale-dependent coupling constants ( Reynolds numbers ) $Re_{n}(k)=O(1)$. It is shown that in the i.r. limit $k\rightarrow Λ_{f}$, the Reynolds numbers $Re(k)\rightarrow Re_{tr}$ where $Re_{tr}$ is the recently numerically and experimentally discovered universal Reynolds number of "smooth" transition from Gaussian to anomalous statistics of spatial velocity derivatives. The calculated relation $Re(Λ_{f})=Re_{tr}$ "selects" the lowest - order non-linearity as the only relevant one. This means that in the infra-red limit $k\rightarrow Λ_{f}$ all high-order nonlinearities generated by the scale-elimination sum up to zero.

physics.flu-dyn↗

arXiv2013

Reynolds Number of Transition as a Dynamic Constraint on Statistical Theory of Turbulence

Iterative coarse-graining procedure based on Wyld's perturbation expansion is applied to the problem of Navier-Stokes turbulence. It is shown that the low-order calculation gives the fixed-point Reynolds number $ Re_{fp}$ (coupling constant) almost identical to the Reynolds number of the recently discovered transition to anomalous scaling of the moments of {\bf "velocity derivatives"}. Using this result as a dynamic constraint, it is argued that in the vicinity of the fixed point (integral scale) the high-order non-linearities, generated by the procedure, are irrelevant. The infra-red divergencies do not disappear but are are contained in the derived equations for the symmetry-breaking large-scale flows (turbulence models or "condensates"), which are source of the small-scale turbulence.

physics.flu-dyn↗

arXiv2011

Anomalous scaling of structure functions and sub-grid models for large eddy simulations of strong turbulence

The original goal of Large Eddy Simulations of fully developed turbulent flows was to accurately describe large-scale flow features ${\bf u}(Δ)$ at the scales $r\geq Δ$ where $Δ$ is a size of computational mesh. The effect of small-scale velocity fluctuations ($r<Δ$) was to be accounted for by effective transport coefficients (subgrid models) in the coarse-grained Navier-Stokes equations. It is shown in this paper that, due to anomalous inertial range scaling (intermittency) of the moments of velocity difference, the existing subgrid models are intrinsically incapable of quantitatively describing flow features at the scales $r<NΔ$ with $N\approx 10$. This increases computational work approximately by a factor $10^{3}-10^{4}$. The breakdown of the widely used Smagorinsky relation for the subgrid viscosity on the scales $Δ/L<1$ is demonstrated and a modification accounting for intermittency of the filtered out small-scale fluctuations is proposed.

physics.flu-dyn↗

arXiv2011

Porous Superhydrophobic Membranes: Hydrodynamic Anomaly in Oscillating Flows

We have fabricated and characterized a novel superhydrophobic system, a mesh-like porous superhydrophobic membrane with solid area fraction $Φ_s$, which can maintain intimate contact with outside air and water reservoirs simultaneously. Oscillatory hydrodynamic measurements on porous superhydrophobic membranes as a function of $Φ_s$ reveal surprising effects. The hydrodynamic mass oscillating in-phase with the membranes stays constant for $0.9\leΦ_s\le1$, but drops precipitously for $Φ_s < 0.9$. The viscous friction shows a similar drop after a slow initial decrease proportional to $Φ_s$. We attribute these effects to the percolation of a stable Knudsen layer of air at the interface.

cond-mat.soft↗

arXiv2010

Skin Friction in Simple Wall - Bounded Shear Flows in Large Reynolds Number Limit

A global approach to analysis of fully developed turbulent flows in pipes/channels and zero pressure gradient boundary layers is proposed. A new dynamic definition of the boundary layer thickness $δ(x)$, where $x$ is the distance to the plate origin, is proposed. The Coles - Fernholtz empirical correlation for skin friction $λ=\frac{2τ_{w}}{ρU_{0}^{2}}\propto 1/\ln^{2}δ(x)$ and $δ(x)\propto x/\ln^{2}(\frac{x}{x_{0}})$ are derived from the Navier-Stokes equations in the limit $Re_{x}\to \infty$. Here $τ_{w}$ and $U_{0}$ are the wall shear stress and free stream velocity, respectively. The theory is formulated as an expansion in powers of a small dimensionless parameter $\frac{dδ(x)}{dx}\to 0$ in the limit $x\to \infty$.

physics.flu-dyn↗

arXiv2009

Lagrangian Structure Functions in Turbulence: Scaling Exponents and Universality

In this paper, the approach for investigation of asymptotic ($Re\to \infty$) scaling exponents of Eulerian structure functions (J. Schumacher et al, New. J. of Physics {\bf 9}, 89 (2007)) is generalized to studies of Lagrangian structure functions in turbulence. The novel "bridging relation" based on the derived expression for the fluctuating, moment-order - dependent dissipation time $τ_{η,n}$ enabled us to calculate scaling exponents ($κ_{n}$) of the moments of Lagrangian velocity differences $S_{n,L}(τ)=\bar{(u(t+τ)-u(t))^{n}}\propto τ^{κ_{n}}$ in a good agreement with experimental and numerical data.

physics.flu-dyn↗

arXiv2009

Measurement of local dissipation scales in turbulent pipe flow

Local dissipation scales are a manifestation of the intermittent small-scale nature of turbulence. We report the first experimental evaluation of the distribution of local dissipation scales in turbulent pipe flows for a range of Reynolds numbers, 2.4x10^4<=Re_D<=7.0x10^4. Our measurements at the nearly isotropic pipe centerline and within the anisotropic logarithmic layer show excellent agreement with distributions that were previously calculated from numerical simulations of homogeneous isotropic box turbulence and with those predicted by theory. The reported results suggest a universality of the smallest-scale fluctuations around the classical Kolmogorov dissipation length.

physics.flu-dyn↗

arXiv2009

Lattice Boltzmann Simulation of High-Frequency Flows: Electromechanical Resonators in Gaseous Media

In this work, we employ a kinetic theory based approach to predict the hydrodynamic forces on electromechanical resonators operating in gaseous media. Using the Boltzmann-BGK equation, we investigate the influence of the resonator geometry on the fluid resistance in the entire range of nondimensional frequency variation $0\leτω\le\infty$; here the fluid relaxation time $τ=μ/p$ is determined by the gas viscosity $μ$ and pressure $p$ at thermodynamic equilibrium, and $ω$ is the (angular) oscillation frequency. Our results support the experimentally observed transition from viscous to viscoelastic flow in simple gases at $τω\approx1$. They are also in remarkable agreement with the measured geometric effects in resonators in a broad linear dimension, frequency, and pressure range.

physics.flu-dyn↗

arXiv2007

Turbulence models generator

In this paper we explore a possibility that all transport turbulent models are contained in a coarse-grained kinetic equation. Building on a recent work by H.Chen et al (2004), we account for fluctuations of a single -point probability density in turbulence, by introducing a``two-level'' (${\bf c,v}$)-phase-space, separating microscopic (${\bf c'\equiv c_{micro}= c-v}$) and hydrodynamic (${\bf v'=v-V}$) modes. Unlike traditional kinetic theories, with hydrodynamic approximations derived in terms of small deviations from thermodynamic equilibrium, the theory developed in this work, is based on a far- from -equilibrium isotropic and homogeneous turbulence as an unperturbed state. The expansion in dimensionless rate of strain leads to a new class of turbulent models, including the well-known ${\cal K}-{\cal E}$, Reynolds stress and all possible nonlinear models. The role of interaction of the fluxes in physical space with the energy flux across the scales, not present in standard modeling, is demonstrated on example of turbulent channel flow. To close the system, neither equation for turbulent kinetic energy nor information on pressure-velocity correlations, contained in the derived coarse-grained kinetic equation, are needed.

nlin.CG↗

arXiv2007

Dissipation Scale Fluctuations and Chemical Reaction Rates in Turbulent Flows

Small separation between reactants, not exceeding $10^{-8}-10^{-7}cm$, is the necessary condition for various chemical reactions. It is shown that random advection and stretching by turbulence leads to formation of scalar-enriched sheets of {\it strongly fluctuating thickness} $η_{c}$. The molecular-level mixing is achieved by diffusion across these sheets (interfaces) separating the reactants. Since diffusion time scale is $τ_{d}\propto η_{c}^{2}$, the knowledge of probability density $Q(η_{c},Re)$ is crucial for evaluation of chemical reaction rates. In this paper we derive the probability density $Q(η_{c},Re,Sc)$ and predict a transition in the reaction rate behavior from ${\cal R}\propto \sqrt{Re}$ ($Re\leq 10^{4}$) to the high-Re asymptotics ${\cal R}\propto Re^{0}$. The theory leads to an approximate universality of transitional Reynolds number $Re_{tr}\approx 10^{4}$. It is also shown that if chemical reaction involves short-lived reactants, very strong anomalous fluctuations of the length-scale $η_{c}$ may lead to non-negligibly small reaction rates.

nlin.CD↗

arXiv2007

Asymptotic Exponents from Low-Reynolds-Number Flows

The high-order statistics of fluctuations in velocity gradients in the crossover range from the inertial to the Kolmogorov and sub-Kolmogorov scales are studied by direct numerical simulations (DNS) of homogeneous isotropic turbulence with vastly improved resolution. The derivative moments for orders 0 <= n <= 8 are represented well as powers of the Reynolds number, Re, in the range 380 <= Re <= 5725, where Re is based on the periodic box length L_x. These low-Reynolds-number flows give no hint of scaling in the inertial range even when extended self-similarity is applied. Yet, the DNS scaling exponents of velocity gradients agree well with those deduced, using a recent theory of anomalous scaling, from the scaling exponents of the longitudinal structure functions at infinitely high Reynolds numbers. This suggests that the asymptotic state of turbulence is attained for the velocity gradients at far lower Reynolds numbers than those required for the inertial range to appear. We discuss these findings in the light of multifractal formalism. Our numerical studies also resolve the crossover of the velocity gradient statistics from the Gaussian to non-Gaussian behaviour that occurs as the Reynolds number is increased.

nlin.CD↗