When time and velocities are dynamically rescaled relative to the instantaneous turnover time, the Sabra shell model acquires another (hidden) form of scaling symmetry. It has been previously shown that this symmetry is statistically restored in the inertial interval of developed turbulence, thereby establishing a self-similarity property derived from first principles and replacing the broken $1/3$-scaling of the K41 theory. Multifractal intermittency follows from the restored hidden symmetry, in which the anomalous scaling exponents $ζ_p$ are identified as Perron-Frobenius eigenvalues. In this paper, we use the hypothesis of restored hidden symmetry to address the multi-time statistics of turbulent fluctuations. The central result is the self-similarity rule stating that any observable that is time-scale homogeneous of degree $p$ is self-similar with the Hölder exponent $h = ζ_p/p$. As a particular case, it yields the scaling laws for decorrelation times of fluctuations obtained previously within the phenomenological multifractal approach. As further applications, we formulate self-similarity rules for multi-time structure functions and multi-time Kolmogorov multipliers and verify them numerically.
We show that, at low inertia and large elasticity, shell models of viscoelastic fluids develop a chaotic behaviour with properties similar to those of elastic turbulence. The low dimensionality of shell models allows us to explore a wide range both in polymer concentration and in Weissenberg number. Our results demonstrate that the physical mechanisms at the origin of elastic turbulence do not rely on the boundary conditions or on the geometry of the mean flow.
It is known that scale invariance is broken in the developed hydrodynamic turbulence due to intermittency, substantiating complexity of turbulent flows. Here we challenge the concept of broken scale invariance by establishing a hidden self-similarity in intermittent turbulence. Using a simplified (shell) model, we derive a nonlinear spatiotemporal scaling symmetry of inviscid equations, which are reformulated in terms of intrinsic times introduced at different scales of motion. Numerical analysis persuasively confirms that this symmetry is restored in a statistical sense within the inertial in…
In this paper we study analytically the viscous `sabra' shell model of energy turbulent cascade. We prove the global regularity of solutions and show that the shell model has finitely many asymptotic degrees of freedom, specifically: a finite dimensional global attractor and globally invariant inertial manifolds. Moreover, we establish the existence of exponentially decaying energy dissipation range for the sufficiently smooth forcing.
In this paper we present a unified shell model for stably stratified and convective turbulence. Numerical simulation of this model for stably stratified flow shows Bolgiano-Obukhbov scaling in which the kinetic energy spectrum varies as $k^{-11/5}$. We also observe a dual scaling ($k^{-11/5}$ and $k^{-5/3}$) for a limited range of parameters. The shell model of convective turbulence yields Kolmogorov's spectrum. These results are consistent with the energy flux and energy feed due to buoyancy, and are in good agreement with direct numerical simulations of Kumar {\em et al.} [Phys. Rev. E {\bf …
We analyze the blowup (finite-time singularity) in inviscid shell models of convective turbulence. We show that the blowup exists and its internal structure undergoes a series of bifurcations under a change of shell model parameter. Various blowup structures are observed and explained, which vary from self-similar to periodic, quasi-periodic and chaotic regimes. Though the blowup takes sophisticated forms, its asymptotic small-scale structure is independent of initial conditions, i.e., universal. Finally, we discuss implications of the obtained results for the open problems of blowup in invisc…
Laminar-turbulent intermittency is intrinsic to the transitional regime of a wide range of fluid flows including pipe, channel, boundary layer and Couette flow. In the latter turbulent spots can grow and form continuous stripes, yet in the stripe-normal direction they remain interspersed by laminar fluid. We carry out direct numerical simulations in a long narrow domain and observe that individual turbulent stripes are transient. In agreement with recent observations in pipe flow we find that turbulence becomes sustained at a distinct critical point once the spatial proliferation outweighs the…
Although the current Reply by Grebenev et al. (2021a) makes their original analysis in Grebenev et al. (2017) more transparent, the actual problem remains. Their claim to have analytically proven conformal invariance in 2D turbulence for a zero-vorticity characteristic equation is not true. We refuted this claim in Frewer & Khujadze (2021a,b), which we will briefly summarize here again with respect to the presented Reply. In particular our proof on the symmetry-breaking property of the integral normalization constraint is misrepresented and misconstrued, especially in their second Reply (Grebe…
Recent direct numerical simulations of the FENE-P model of non-Newtonian hydrodynamics revealed that the phenomenon of drag reduction by polymer additives exists (albeit in reduced form) also in homogeneous turbulence. We introduce here a simple shell model for homogeneous viscoelastic flows that recaptures the essential observations of the full simulations. The simplicity of the shell model allows us to offer a transparent explanation of the main observations. It is shown that the mechanism for drag reduction operates mainly on the large scales. Understanding the mechanism allows us to predic…
In this paper we continue the analytical study of the sabra shell model of energy turbulent cascade initiated in \cite{CLT05}. We prove the global existence of weak solutions of the inviscid sabra shell model, and show that these solutions are unique for some short interval of time. In addition, we prove that the solutions conserve the energy, provided that the components of the solution satisfy $|{u_n}| \le C k_n^{-1/3} (\sqrt{n} \log(n+1))^{-1}$, for some positive absolute constant $C$, which is the analogue of the Onsager's conjecture for the Euler's equations. Moreover, we give a Beal-Kato…
Complex nonlinear turbulent dynamical systems are ubiquitous in many areas. Recovering unobserved state variables is an important topic for the data assimilation of turbulent systems. In this article, an efficient continuous in time data assimilation scheme is developed, which exploits closed analytic formulae for updating the unobserved state variables. Therefore, it is computationally efficient and accurate. The new data assimilation scheme is combined with a simple reduced order modeling technique that involves a cheap closure approximation and a noise inflation. In such a way, many complic…
Applications of the shell model of turbulence to the case of rapidly rotating bodies are considered. Starting from the classical GOY model we introduce the Coriolis force and obtain a $\sim k^{-2}$ spectrum for 3D hydrodynamical turbulence for the free decay regime as well for the regime with external forcing. Additional modifications of the GOY model providing a realistic form of the helicity are proposed.
A data-driven framework for formulation of closures of the Reynolds-Average Navier--Stokes (RANS) equations is presented. In recent years, the scientific community has turned to machine learning techniques to distill a wealth of highly resolved data into improved RANS closures. While the body of work in this area has primarily leveraged Neural Networks (NNs), we alternately leverage a sparse regression framework. This methodology has two important properties: (1) The resultant model is in a closed, algebraic form, allowing for direct physical inferences to be drawn and naive integration into e…
Shell models of turbulence have a finite-time blowup in the inviscid limit, i.e., the enstrophy diverges while the single-shell velocities stay finite. The signature of this blowup is represented by self-similar instantonic structures traveling coherently through the inertial range. These solutions might influence the energy transfer and the anomalous scaling properties empirically observed for the forced and viscous models. In this paper we present a study of the instantonic solutions for a set of four shell models of turbulence based on the exact decomposition of the Navier-Stokes equations …
In this work we derive a lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing and for sufficiently small viscosity term $ν$, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to $\log_λν^{-1}$ for all values of the governing parameter $ε$, except for $ε=1$. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. More…
We introduce a shell model of turbulence featuring intermittent behaviour with anomalous power-law scaling of structure functions. This model is solved analytically with the explicit derivation of anomalous exponents. The solution associates the intermittency with the hidden symmetry for Kolmogorov multipliers, making our approach relevant for real turbulence.
We study conformal invariance of vorticity clusters in weakly compressible two-dimensional turbulence at low Mach numbers. On the basis of very high resolution direct numerical simulation we demonstrate the scaling invariance of the inverse cascade with scaling close to Kolmogorov prediction. In this range of scales, the statistics of zero-vorticity isolines are found to be compatible with those of critical percolation, thus generalizing the results obtained in incompressible Navier-Stokes turbulence.
In this paper, using Pao's conjecture [Y.-H. Pao, Phys. Fluids 11, 1371 (1968)], we derive expressions for the spectra and fluxes of kinetic energy and enstrophy for two-dimensional (2D) forced turbulence that extend beyond the inertial range. In these expressions, the fluxes and the spectra contain additional factors of the exponential form. To validate these model predictions, we perform numerical simulations of 2D turbulence with an external force applied at $k=k_f$ in the intermediate range. The numerical results match with the model predictions, except for the energy and enstrophy fluxes …
We investigate the transfer of energy from large scales to small scales in fully developed forced three-dimensional MHD-turbulence by analyzing the results of direct numerical simulations in the absence of an externally imposed uniform magnetic field. Our results show that the transfer of kinetic energy from the large scales to kinetic energy at smaller scales, and the transfer of magnetic energy from the large scales to magnetic energy at smaller scales, are local, as is also found in the case of neutral fluids, and in a way that is compatible with Kolmogorov (1941) theory of turbulence. Howe…
Shell models of hydrodynamic turbulence originated in the seventies. Their main aim was to describe the statistics of homogeneous and isotropic turbulence in spectral space, using a simple set of ordinary differential equations. In the eighties, shell models of magnetohydrodynamic (MHD) turbulence emerged based on the same principles as their hydrodynamic counter-part but also incorporating interactions between magnetic and velocity fields. In recent years, significant improvements have been made such as the inclusion of non-local interactions and appropriate definitions for helicities. Though…
We examine the multiscaling behavior of the normal- and superfluid-velocity structure functions in three-dimensional superfluid turbulence by using a shell model for the three-dimensional (3D) Hall-Vinen-Bekharevich-Khalatnikov (HVBK) equations. Our 3D-HVBK shell model is based on the Gledzer-Okhitani-Yamada (GOY) shell model. We examine the dependence of the multiscaling exponents on the normal-fluid fraction and the mutual-friction coefficients. Our extensive study of the 3D-HVBK shell model shows that the multiscaling behavior of the velocity structure functions in superfluid turbulence is …
We present experimental evidence and theoretical arguments showing that the time-evolution of freely decaying 2-d turbulence is governed by a {\it discrete} time scale invariance rather than a continuous time scale invariance. Physically, this reflects that the time-evolution of the merging of vortices is not smooth but punctuated, leading to a prefered scale factor and as a consequence to log-periodic oscillations. From a thorough analysis of freely decaying 2-d turbulence experiments, we show that the number of vortices, their radius and separation display log-periodic oscillations as a func…
Superfluid helium consists of two inter-penetrating fluids, a viscous normal fluid and an inviscid superfluid, coupled by a mutual friction. We develop a two-fluid shell model to study superfluid turbulence. We investigate the energy spectra and the balance of fluxes between the two fluids as a function of temperature in continuously forced turbulence, and, in the absence of forcing, the decay of turbulence. We furthermore investigate deviations from the $k^{-5/3}$ spectrum caused by the mutual friction force. We compare our results with experiments and existing calculations. We find that, at …
We study the statistical properties of helicity in direct numerical simulations of fully developed homogeneous and isotropic turbulence and in a class of turbulence shell models. We consider correlation functions based on combinations of vorticity and velocity increments that are not invariant under mirror symmetry. We also study the scaling properties of high-order structure functions based on the moments of the velocity increments projected on a subset of modes with either positive or negative helicity (chirality). We show that mirror symmetry is recovered at small-scales, i.e., chiral terms…
We investigate the scaling form of appropriate time-scales extracted from time-dependent correlation functions in rotating, turbulent flows. In particular, we obtain precise estimates of the dynamic exponents $z_p$, associated with the time-scales, and their relation with the more commonly measured equal-time exponents $ζ_p$. These theoretical predictions, obtained by using the multifractal formalism, are validated through extensive numerical simulations of a shell model for such rotating flows.
A lattice Boltzmann scheme simulating the dynamics of shell models of turbulence is developed. The influence of high order kinetic modes (ghosts) on the dissipative properties of turbulence dynamics is studied. It is analytically found that when ghost fields relax on the same time scale as the hydrodynamic ones, their major effect is a net enhancement of the fluid viscosity. The bare fluid viscosity is recovered by letting ghost fields evolve on a much longer time scale. Analytical results are borne out by high-resolution numerical simulations. These simulations indicate that the hydrodynamic …
We introduce a model for the turbulent energy cascade aimed at studying the effect of dynamical scaling on intermittency. In particular, we show that by slowing down the energy transfer mechanism for fixed energy flux, intermittency decreases and eventually disappears. This result supports the conjecture that intermittency can be observed only if energy is flowing towards faster and faster scales of motion.
Applying a modified version of the Gledzer-Ohkitani-Yamada (GOY) shell model, the signatures of so-called two-dimensionalization effect of three-dimensional incompressible, homogeneous, isotropic fully developed unforced turbulence have been studied and reproduced. Within the framework of shell models we have obtained the following results: (i) progressive steepening of the energy spectrum with increased strength of the rotation, and, (ii) depletion in the energy flux of the forward forward cascade, sometimes leading to an inverse cascade. The presence of extended self-similarity and self-simi…
Using direct numerical simulation of hydrodynamic turbulence with helicity forcing applied at all scales, a near-maximum helical turbulent state is obtained, with an inverse energy cascade at scales larger than the energy forcing scale and a forward helicity cascade at scales smaller than the energy forcing scale. In contrast to previous studies using decimated triads, our simulations contain all possible triads. By computing the shell-to-shell energy fluxes, we show that the inverse energy cascade results from weakly non-local interactions among homochiral triads. Varying the helicity injecti…
In this work we construct and analyze continuous hydrodynamic models in one space dimension, which are induced by shell models of turbulence. After Fourier transformation, such continuous models split into an infinite number of uncoupled subsystems, which are all identical to the same shell model. The two shell models, which allow such a construction, are considered: the dyadic (Desnyansky--Novikov) model with the intershell ratio $λ= 2^{3/2}$ and the Sabra model of turbulence with $λ= \sqrt{2+\sqrt{5}} \approx 2.058$. The continuous models allow understanding various properties of shell model…
Since Kolmogorov proposed his phenomenological theory of hydrodynamic turbulence in 1941, the description of mechanism leading to the energy cascade and anomalous scaling remains an open problem in fluid mechanics. Soon after, in 1949 Onsager noticed that the scaling properties in inertial range imply non-differentiability of the velocity field in the limit of vanishing viscosity. This observation suggests that the turbulence mechanism may be related to a finite-time singularity (blowup) of incompressible Euler equations. However, the existence of such blowup is still an open problem too. In t…
We investigate the predictability aspects of rotating turbulent flows through extensive numerical simulations of a shell model of rotating turbulence. In particular, we measure the large-scale predictability time and find that it increases with rotation rate to satisfy a power law in Rossby number with a scaling exponent of $-2/3$. Intriguingly, we find that before entering the algebraic growth stage, the error dynamics freezes for a time period determined by the finite Rossby number. We further analyse the scale dependence of the predictability time and observe that it tends to become scale i…
We propose a new Eulerian turbulence theory to obtain a closed set of equations for homogeneous, isotropic turbulent velocity field correlations and propagator functions by incorporating constraints of random Galilean invariance. This incorporation generates a few different equations for propagator and the present theory suggests a way to utilize them into the closure solutions of two-time and single-time velocity correlations' equations so as to properly account for random sweeping phenomena. The present theory yields exact solutions when applied to simple model problem of random oscillator.
We study the intermittency properties of the energy and helicity cascades in two 1536^3 direct numerical simulations of helical rotating turbulence. Symmetric and anti-symmetric velocity increments are examined, as well as probability density functions of the velocity field and of the helicity density. It is found that the direct cascade of energy to small scales is scale invariant and non-intermittent, whereas the direct cascade of helicity is highly intermittent. Furthermore, the study of structure functions of different orders allows us to identify a recovery of isotropy of strong events at…
We discuss a theoretical framework to define an optimal sub-grid closure for shell models of turbulence. The closure is based on the ansatz that consecutive shell multipliers are short-range correlated, following the third hypothesis of Kolmogorov formulated for similar quantities for the original three-dimensional Navier-Stokes turbulence. We also propose a series of systematic approximations to the optimal model by assuming different degrees of correlations across scales among amplitudes and phases of consecutive multipliers. We show numerically that such low-order closures work well, reprod…
We investigate the turbulence-induced dissipation of the large scales in a statistically homogeneous flow using an "optimal closure," which one of us (BT) has recently exposed in the context of Hamiltonian dynamics. This statistical closure employs a Gaussian model for the turbulent scales, with corresponding vanishing third cumulant, and yet it captures an intrinsic damping. The key to this apparent paradox lies in a clear distinction between true ensemble averages and their proxies, most easily grasped when one works directly with the Liouville equation rather than the cumulant hierarchy. We…
We show that a shell-model version of the three-dimensional Hall-magnetohydrodynamic (3D Hall-MHD) equations provides a natural theoretical model for investigating the multiscaling behaviors of velocity and magnetic structure functions. We carry out extensive numerical studies of this shell model, obtain the scaling exponents for its structure functions, in both the low-$k$ and high-$k$ power-law ranges of 3D Hall-MHD, and find that the extended-self-similarity (ESS) procedure is helpful in extracting the multiscaling nature of structure functions in the high-$k$ regime, which otherwise appear…
A variation principle is suggested to find self-similar solitary solutions (``solitons'') of shell model of turbulence. For the Sabra shell model the shape of the solitons is approximated by rational trial functions with relative accuracy of O(0.001). It is found how the soliton shape, propagation time $t_n$ and the dynamical exponent $z_0$ (which governs the time rescaling of the solitons in different shells) depend on parameters of the model. For a finite interval of $z$ the author discovered ``quasi-solitons'' which approximate with high accuracy corresponding self-similar equations for an …
The scale locality of energy fluxes for magnetohydrodynamics (MHD) is investigated numerically for stationary states of turbulence. Two types of forces are used to drive turbulence, a kinetic force that acts only on the velocity field and a kinetic-inductive forcing mechanism, which acts on the velocity and magnetic fields alike. The analysis is performed in spectral space, which is decomposed into a series of shells following a power law for the boundaries. The triadic transfers occurring among these shells are computed and the fluxes and locality functions are recovered by partial summation …
To understand the earliest stages of planet formation, it is crucial to be able to predict the rate and the outcome of dust grains collisions, be it sticking and growth, bouncing, or fragmentation. The outcome of such collisions depends on the collision speed, so we need a solid understanding of the rate and velocity distribution of turbulence-induced dust grain collisions. The rate of the collisions depends both on the speed of the collisions and the degree of clustering experienced by the dust grains, which is a known outcome of turbulence. We evolve the motion of dust grains in simulated tu…
We examine three--dimensional turbulent flows in the presence of solid-body rotation and helical forcing in the framework of stochastic Schramm-Löwner evolution curves (SLE). The data stems from a run on a grid of $1536^3$ points, with Reynolds and Rossby numbers of respectively 5100 and 0.06. We average the parallel component of the vorticity in the direction parallel to that of rotation, and examine the resulting $<ω_\textrm{z}>_\textrm{z}$ field for scaling properties of its zero-value contours. We find for the first time for three-dimensional fluid turbulence evidence of nodal curves being…
We systematise the study of dynamic multiscaling of time-dependent structure functions in different models of passive-scalar and fluid turbulence. We show that, by suitably normalising these structure functions, we can eliminate their dependence on the origin of time at which we start our measurements and that these normalised structure functions yield the same linear bridge relations that relate the dynamic-multiscaling and equal-time exponents for statistically steady turbulence. We show analytically, for both the Kraichnan Model of passive-scalar turbulence and its shell model analogue, and…
The current claim by Grebenev et al. [J. Phys. A: Math. Theor. 52, 335501 (2019)], namely that the inviscid and unclosed 2D Lundgren-Monin-Novikov (LMN) equations on a zero-vorticity Lagrangian path admit conformal invariance, is based on a flawed and misleading analysis published earlier by Grebenev et al. (2017). All false results and conclusions made before in the Eulerian picture were now extended by Grebenev et al. (2019) to the Lagrangian picture. Although we have already commented on these errors and consistently refuted their previous study (Frewer & Khujadze, 2018), we deem it necessa…
Fully developed homogeneous isotropic turbulence in 2D is fundamentally different from 3D. In 2D, the simultaneous conservation of both energy and enstrophy in the inertial ranges of scales leads to a forward cascade of enstrophy and a reverse cascade of energy. In 3D, helicity, the integral of the scalar product of velocity and vorticity, is also an inviscid flow invariant along with kinetic energy. Unlike enstrophy, however, helicity does not block the cascade of energy to small scales. Energy and helicity are not only globally conserved but also conserved within each non-linear triadic inte…
