arXiv subjects

Alexandros Alexakis

Publications and source records attributed to Alexandros Alexakis.

At least 37 records · Page 2Linked to original sources

arXiv2021

$λ$-Navier-Stokes turbulence

We investigate numerically the model proposed in Sahoo et al [Phys. Rev. Lett. 118, 164501, (2017)] where a parameter $λ$ is introduced in the Navier-Stokes equations such that the weight of homochiral to heterochiral interactions is varied while preserving all original scaling symmetries and inviscid invariants. Decreasing the value of $λ$ leads to a change in the direction of the energy cascade at a critical value $λ_c \sim 0.3$. In this work, we perform numerical simulations at varying $λ$ in the forward energy cascade range and at changing the Reynolds number $\mathrm{Re}$. We show that for a fixed injection rate, as $λ\to λ_c$, the kinetic energy diverges with a scaling law $\mathcal{E} \propto (λ-λ_c)^{-2/3}$. The energy spectrum is shown to display a larger bottleneck as $λ$ is decreased. The forward heterochiral flux and the inverse homochiral flux both increase in amplitude as $λ_c$ is approached while keeping their difference fixed and equal to the injection rate. As a result, very close to $λ_c$ a stationary state is reached where the two opposite fluxes are of much higher amplitude than the mean flux and large fluctuations are observed. Furthermore, we show that intermittency as $λ_c$ is approached is reduced. The possibility of obtaining a statistical description of regular Navier-Stokes turbulence as an expansion around this newly found critical point is discussed.

physics.flu-dyn↗

arXiv2021

Geometric microcanonical theory of two-dimensional Truncated Euler flows

This paper presents a geometric microcanonical ensemble perspective on two-dimensional Truncated Euler flows, which contain a finite number of (Fourier) modes and conserve energy and enstrophy. We explicitly perform phase space volume integrals over shells of constant energy and enstrophy. Two applications are considered. In a first part, we determine the average energy spectrum for highly condensed flow configurations and show that the result is consistent with Kraichnan's canonical ensemble description, despite the fact that no thermodynamic limit is invoked. In a second part, we compute the probability density for the largest-scale mode of a free-slip flow in a square, which displays reversals. We test the results against numerical simulations of a minimal model and find excellent agreement with the microcanonical theory, unlike the canonical theory, which fails to describe the bimodal statistics. This article is part of the theme issue "Mathematical problems in physical fluid dynamics".

physics.flu-dyn↗

arXiv2021

Lévy on-off intermittency

We present a new form of intermittency, Lévy on-off intermittency, which arises from multiplicative $α$-stable white noise close to an instability threshold. We study this problem in the linear and nonlinear regimes, both theoretically and numerically, for the case of a pitchfork bifurcation with fluctuating growth rate. We compute the stationary distribution analytically and numerically from the associated fractional Fokker-Planck equation in the Stratonovich interpretation. We characterize the system in the parameter space $(α,β)$ of the noise, with stability parameter $α\in (0,2)$ and skewness parameter $β\in[-1,1]$. Five regimes are identified in this parameter space, in addition to the well-studied Gaussian case $α=2$. Three regimes are located at $1<α<2$, where the noise has finite mean but infinite variance. They are differentiated by $β$ and all display a critical transition at the deterministic instability threshold, with on-off intermittency close to onset. Critical exponents are computed from the stationary distribution. Each regime is characterized by a specific form of the density and specific critical exponents, which differ starkly from the Gaussian case. A finite or infinite number of integer-order moments may converge, depending on parameters. Two more regimes are found at $0<α\leq 1$. There, the mean of the noise diverges, and no critical transition occurs. In one case the origin is always unstable, independently of the distance $μ$ from the deterministic threshold. In the other case, the origin is conversely always stable, independently of $μ$. We thus demonstrate that an instability subject to non-equilibrium, power-law-distributed fluctuations can display substantially different properties than for Gaussian thermal fluctuations, in terms of statistics and critical behavior.

cond-mat.stat-mech↗

arXiv2021

Intermittency of three-dimensional perturbations in a point-vortex model

Three-dimensional (3D) instabilities on a (potentially turbulent) two-dimensional (2D) flow are still incompletely understood, despite recent progress. Here, based on known physical properties of such 3-D instabilities, we propose a simple, energy-conserving model describing this situation. It consists of a 2D point-vortex flow coupled to localized 3D perturbations (ergophages), such that ergophages can gain energy by altering vortex-vortex distances through an induced divergent velocity field, thus decreasing point-vortex energy. We investigate the model in three distinct stages of evolution: (i) The linear regime, where the ergophage amplitude grows or decays exponentially on average, with a randomly fluctuating instantaneous growth rate. The growth rate has a small auto-correlation time, and follows a probability distribution featuring a power-law tail with exponent between -2 and -5/3 (up to a cut-off), depending on the point-vortex base flow. Consequently, the logarithmic ergophage amplitude performs a free Lévy flight. (ii) The passive-nonlinear regime of the model, where the 2D flow evolves independently of the ergophage amplitudes, which saturate by non-linear self-interactions without affecting the 2D flow. In this regime the system exhibits a new type of on-off intermittency that we name Lévy on-off intermittency, and which we study in a companion paper. We compute the bifurcation diagram for the mean and variance of the perturbation amplitude, as well as the probability density of the perturbation amplitude. (iii) Finally, we characterize the the fully nonlinear regime, where ergophages feed back on the 2D flow, and study how the vortex temperature is altered by the interaction with ergophages. It is shown that when the amplitude of the ergophages is sufficiently large, the 2D flow saturates to a zero-temperature state. Given the limitations of existing theories ...

physics.flu-dyn↗

arXiv2021

Do extreme events trigger turbulence decay? - a numerical study of turbulence decay time in pipe flows

Turbulence locally created in laminar pipe flows shows sudden decay or splitting after a stochastic waiting time. In laboratory experiments, the mean waiting time was observed to increase double-exponentially as the Reynolds number (Re) approaches its critical value. To understand the origin of this double-exponential increase, we perform many pipe flow direct numerical simulations of the Navier-Stokes equations, and measure the cumulative histogram of the maximum axial vorticity field over the pipe (turbulence intensity). In the domain where the turbulence intensity is not small, we observe that the histogram is well-approximated by the Gumbel extreme-value distribution. The smallest turbulence intensity in this domain roughly corresponds to the transition value between the locally stable turbulence and a meta-stable (edge) state. Studying the Re dependence of the fitting parameters in this distribution, we derive that the time scale of the transition between these two states increases double-exponentially as Re approaches its critical value. On the contrary, in smaller turbulence intensities below this domain, we observe that the distribution is not sensible to the change of Re. This means that the decay time of the meta-stable state (to the laminar state) is stochastic but Re-independent in average. Our observation suggests that the conjecture made by Goldenfeld et al. to derive the double-exponential increase of turbulence decay time is approximately satisfied in the range of Re we studied. We also discuss using another extreme-value distribution, Fréchet distribution, instead of the Gumbel distribution to approximate the histogram of the turbulence intensify, which reveals interesting perspectives.

physics.flu-dyn↗

arXiv2021

Symmetry breaking in a turbulent environment

In this work we investigate symmetry breaking in the presence of a turbulent environment. The transition from a symmetric state to a symmetry-breaking state is demonstrated using two examples: (i) the transition of a two-dimensional flow to a three dimensional flow as the fluid layer thickness is varied and (ii) the dynamo instability in a thin layer flow as the magnetic Reynolds number is varied. We show that these examples have similar critical exponents that differ from the mean-field predictions. The critical behavior can be related to the multiplicative nature of the fluctuations and can be predicted in certain limits using results from the statistical properties of random interfaces. Our results indicate the possibility of existence of a new class of out-of-equilibrium phase transition controlled by the multiplicative noise.

physics.flu-dyn↗

arXiv2020

Critical transition in fast-rotating turbulence within highly elongated domains

We study rapidly rotating turbulent flows in a highly elongated domain using an asymptotic expansion at simultaneously low Rossby number $Ro\ll 1$ and large domain height compared to the energy injection scale, $h=H/\ell_{in}\gg 1$. We solve the resulting equations using an extensive set of direct numerical simulations for different parameter regimes. As the parameter $λ= (h Ro)^{-1}$ is increased beyond a threshold $λ_c$, a transition is observed from a state without an inverse energy cascade to a state with an inverse energy cascade. } For large Reynolds number and large horizontal box size, we provide evidence for criticality of the transition in terms of the large-scale energy dissipation rate.

physics.flu-dyn↗

arXiv2020

On the local energy flux of turbulent flows

We investigate the local energy flux rate $Π_\ell(\bf x)$ towards small scales in isotropic turbulent flows using direct numerical simulations and applying different low-pass filters. Two different filters are examined, a sharp Fourier filter and a Gaussian filter. The probability density function (pdf) of the local energy flux is calculated for the different filters and for different filtering scales. It is shown that the local energy flux is a largely fluctuating quantity taking both negative and positive values and this is more pronounced for the sharp filter. The variance, the skewness and the kurtosis of these fluctuations are shown to increase as the filtering scale is decreased. Furthermore we calculate the joint pdf of $Π_\ell(\bf x)$ with the local filtered strain rate $S_\ell$ and the enstrophy $Ω_\ell$. The flux shows a good correlation with the strain but not with the enstrophy. It is shown that its conditional mean value scales like $<{Π_\ell}>_S \propto \ell^2 S_\ell^{3} $ in support to the Smagorinsky eddy viscosity model. Nonetheless strong fluctuations exist around this value that also need to be modeled. We discuss the implications of our results for subgrid scale models, and propose new modelling directions.

physics.flu-dyn↗

arXiv2020

Role of the forcing dimensionality in thin-layer turbulent energy cascades

We investigate the transition from forward to inverse energy cascade in turbulent flows in thin layers, varying the functional form of the forcing and the thickness of the layer. We show that, as the forcing function becomes more three-dimensional, the inverse cascade is suppressed and the critical height hc, where the transition occurs, is decreased. We study the dependence of this critical height on a parameter r which measures the dimensionality of the forcing and thus construct a phase space diagram in the parameter space r-h. We discuss the effect of Reynolds number and domain size.

physics.flu-dyn↗

arXiv2020

Abrupt transition between three and two-dimensional quantum turbulence

We present numerical evidence of a critical-like transition in an out-of-equilibrium mean-field description of a quantum system. By numerically solving the Gross-Pitaevskii equation we show that quantum turbulence displays an abrupt change between three-dimensional (3D) and two-dimensional (2D) behavior. The transition is observed both in quasi-2D flows in cubic domains (controlled by the amplitude of a 3D perturbation to the flow), as well as in flows in thin domains (controlled by the domain aspect ratio) in a configuration that mimics systems realized in laboratory experiments. In one regime the system displays a transfer of the energy towards smaller scales, while in the other the system displays a transfer of the energy towards larger scales and a coherent self-organization of the quantized vortices.

cond-mat.quant-gas↗

arXiv2019

Turbulent cascade, bottleneck and thermalized spectrum in hyperviscous flows

In many simulations of turbulent flows the viscous forces $ν\nabla^2 {\bf u}$ are replaced by a hyper-viscous term $-ν_p(-\nabla^2)^{p}{\bf u}$ in order to suppress the effect of viscosity at the large scales. In this work we examine the effect of hyper-viscosity on decaying turbulence for values of $p$ ranging from $p=1$ (regular viscosity) up to $p=100$. Our study is based on direct numerical simulations of the Taylor-Green vortex for resolutions from $512^3$ to $2048^3$. Our results demonstrate that the evolution of the total energy $E$ and the energy dissipation $ε$ remain almost unaffected by the order of the hyper-viscosity used. However, as the order of the hyper-viscosity is increased ,the energy spectrum develops a more pronounced bottleneck that contaminates the inertial range. At the largest values of $p$ examined, the spectrum at the bottleneck range has a positive power-law behavior $E(k)\propto k^α$ with the power-law exponent $α$ approaching the value obtained in flows at thermal equilibrium $α=2$. This agrees with the prediction of Frisch et al. [Phys. Rev. Lett. 101, 144501 (2008)] who suggested that at high values of $p$, the flow should behave like the truncated Euler equations (TEE). Nonetheless, despite the thermalization of the spectrum, the flow retains a finite dissipation rate up to the examined order, which disagrees with the predictions of the TEE system implying suppression of energy dissipation. We reconcile the two apparently contradictory results, predicting the value of $p$ for which the hyper-viscous Navier-Stokes goes over to the TEE system and we discuss why thermalization appears at smaller values of $p$.

physics.flu-dyn↗

arXiv2019

Energy fluxes in quasi-equilibrium flows

We examine the relation between the absolute equilibrium state of the spectrally truncated Euler equations (TEE) predicted by Kraichnan (1973) to the forced and dissipated flows of the spectrally truncated Navier-Stokes (TNS) equations. In both of these idealized systems a finite number of Fourier modes is kept contained inside a sphere of radius $k_{max}$ but while the first conserves energy in the second energy is injected by a body-force $\bf{f}$ and dissipated by the viscosity $ν$. For the TNS system stochastically forced with energy injection rate $\mathcal{I}_\mathcal{E}$ we show, using an asymptotic expansion of the Fokker-Planck equation, that in the limit of small $k_{max}η$ (where $η=(ν^3/\mathcal{I}_\mathcal{E})^{1/4}$ the Kolmogorov lengthscale) the flow approaches the absolute equilibrium solution of Kraichnan with such an effective "temperature" so that there is a balance between the energy injection and the energy dissipation rate. Numerical simulations verify the predictions of the model for small values of $k_{max}η$. For intermediate values of $k_{max}η$ a transition from the quasi-equilibrium "thermal" state to Kolmogorov turbulence is observed. If the forcing is applied at small scales and the dissipation acts only at large scales then the equipartition spectrum appears at all scales for all values of $ν$. In both cases a finite forward or inverse flux is present even for the cases where the flow is close to the equilibrium state solutions. However, unlike the classical turbulence where an energy cascade develops with a mean energy flux that is large compared to its fluctuations, the quasi-equilibrium state has a mean flux of energy that is subdominant to the large flux fluctuations observed.

physics.flu-dyn↗

arXiv2019

Rare transitions to thin-layer turbulent condensates

Turbulent flows in a thin layer can develop an inverse energy cascade leading to spectral condensation of energy when the layer height is smaller than a certain threshold. These spectral condensates take the form of large-scale vortices in physical space. Recently, evidence for bistability was found in this system close to the critical height: depending on the initial conditions, the flow is either in a condensate state with most of the energy in the two-dimensional (2-D) large-scale modes, or in a three-dimensional (3-D) flow state with most of the energy in the small-scale modes. This bistable regime is characterised by the statistical properties of random and rare transitions between these two locally stable states. Here, we examine these statistical properties in thin-layer turbulent flows, where the energy is injected by either stochastic or deterministic forcing. To this end, by using a large number of direct numerical simulations (DNS), we measure the decay time $τ_d$ of the 2-D condensate to 3-D flow state and the build-up time $τ_b$ of the 2-D condensate. We show that both of these times $τ_d,τ_b$ follow an exponential distribution with mean values increasing faster than exponentially as the layer height approaches the threshold. We further show that the dynamics of large-scale kinetic energy may be modeled by a stochastic Langevin equation. From time-series analysis of DNS data, we determine the effective potential that shows two minima corresponding to the 2-D and 3-D states when the layer height is close to the threshold.

physics.flu-dyn↗

arXiv2019

Condensates in thin-layer turbulence

We examine the steady state of turbulent flows in thin layers using direct numerical simulations. It is shown that when the layer thickness is smaller than a critical height, an inverse cascade arises which leads to the formation of a steady state condensate where most of the energy is concentrated in the largest scale of the system. For layers of thickness smaller than a second critical height, the flow at steady state becomes exactly two-dimensional. The amplitude of the condensate is studied as a function of layer thickness and Reynolds number. Bi-stability and intermittent bursts are found close to the two critical points. The results are interpreted based on a mean field three-scale model that reproduces some of the basic features of the numerical results.

physics.flu-dyn↗

arXiv2018

On the thermal equilibrium state of large scale flows

In a forced three-dimensional turbulent flow the scales larger than the forcing scale have been conjectured to reach a thermal equilibrium state forming a $k^2$ energy spectrum. In this work we examine the properties of these large scales in turbulent flows with the use of numerical simulations. We show that the choice of forcing can strongly affect the behavior of the large scales. A spectrally-dense forcing (a forcing that acts on {\it all modes} inside a finite-width spherical shell) with long correlation times may lead to strong deviations from the $k^2$ energy spectrum, while a spectrally-sparse forcing (a forcing that acts only on a few modes) with short correlated time-scale can reproduce the thermal spectrum. It is shown that the spectrally-dense forcing allows for numerous triadic interactions that couple one large scale mode with two forced modes and this leads to an excess of energy input in the large scales. This excess of energy is then moved back to the small-scales by self-interactions of the large-scale modes and by interactions with the turbulent small-scales. The overall picture that arises from the present analysis is that the large scales in a turbulent flow resemble a reservoir that is in (non-local) contact with a second out-of equilibrium reservoir consisting of the smaller (forced and turbulent) scales. If the injection of energy at the large scales from the forced modes is relative weak then the large-scale spectrum remains close to a thermal equilibrium and the role of long-range interactions is to set the global energy (temperature) of the equilibrium state. If, on the other hand, the long-range interactions are dominant, the large-scale self-interactions cannot respond fast enough to bring the system into equilibrium. Then the large scales deviate from the equilibrium state with energy spectrum that may display exponents different from the $k^2$ spectrum.

physics.flu-dyn↗

arXiv2018

3D instabilities and negative eddy viscosity in thin-layer flows

The stability of flows in layers of finite thickness $H$ is examined against small scale three dimensional (3D) perturbations and large scale two-dimensional (2D) perturbations. The former provide an indication of a forward transfer of energy while the later indicate an inverse transfer and the possibility of an inverse cascade. The analysis is performed using a Floquet-Bloch code that allows to examine the stability of modes with arbitrary large scale separation. For thin layers the 3D perturbations become unstable when the layer thickness $H$ becomes larger than $H > c_1 (ν\ell_{_U}/U)^{1/2}= c_1 \ell_{_U} Re^{-1/2}$, where $U$ is the rms velocity of the flown, $\ell_{_U}$ is the correlation length scale of the flow, $ν$ the viscosity and $Re=\ell_{_U} U/ν$ is the Reynolds number. At the same time large scale 2D perturbations also become unstable by an eddy viscosity mechanism when $Re>c_2$, where $c_1,c_2$ are order one non-dimensional numbers. These relations define different regions in parameter space where 2D and 3D instabilities can (co-)exist and this allows to construct a stability diagram. Implications of these results for fully turbulent flows that display a change of direction of cascade as $H$ is varied are discussed.

physics.flu-dyn↗

arXiv2017

Method to measure efficiently rare fluctuations of turbulence intensity for turbulent-laminar transitions in pipe flows

The fluctuations of turbulence intensity in a pipe flow around the critical Reynolds number is difficult to study but important because they are related to turbulent-laminar transitions. We here propose a rare-event sampling method to study such fluctuations in order to measure the time-scale of the transition efficiently. The method is composed of two parts: (i) the measurement of typical fluctuations (the bulk part of an accumulative probability function) and (ii) the measurement of rare fluctuations (the tail part of the probability function) by employing dynamics where a feedback control of the Reynolds number is implemented. We apply this method to a chaotic model of turbulent puffs proposed by Barkley and confirm that the time-scale of turbulence decay increases super-exponentially even for high Reynolds numbers up to Re = 2500, where getting enough statistics by brute-force calculations is difficult. The method uses a simple procedure of changing Reynolds number that can be applied even to experiments.

physics.flu-dyn↗

arXiv2017

Transition to turbulent dynamo saturation

While the saturated magnetic energy is independent of viscosity in dynamo experiments, it remains viscosity-dependent in state-of-the-art 3D direct numerical simulations (DNS). Extrapolating such viscous scaling-laws to realistic parameter values leads to an underestimation of the magnetic energy by several orders of magnitude. The origin of this discrepancy is that fully 3D DNS cannot reach low enough values of the magnetic Prandtl number $Pm$. To bypass this limitation and investigate dynamo saturation at very low $Pm$, we focus on the vicinity of the dynamo threshold in a rapidly rotating flow: the velocity field then depends on two spatial coordinates only, while the magnetic field consists of a single Fourier mode in the third direction. We perform numerical simulations of the resulting set of reduced equations for $Pm$ down to $2\cdot 10^{-5}$. This parameter regime is currently out of reach to fully 3D DNS. We show that the magnetic energy transitions from a high-$Pm$ viscous scaling regime to a low-$Pm$ turbulent scaling regime, the latter being independent of viscosity. The transition to the turbulent saturation regime occurs at a low value of the magnetic Prandtl number, $Pm \simeq 10^{-3}$, which explains why it has been overlooked by numerical studies so far.

physics.flu-dyn↗