Scaling Of Energy Spectrum Research Articles

Freely Decaying Saffman Turbulence Experimentally Generated by Magnetic Stirrers.

We investigate experimentally the decay of three-dimensional hydrodynamic turbulence, initially generated by the erratic motions of centimeter-size magnetic stirrers in a closed container. Such zero-mean-flow homogeneous isotropic turbulence is well suited to test Saffman's model and Batchelor's model of freely decaying turbulence. Here, we report a consistent set of experimental measurements (temporal decay of the turbulent kinetic energy, of the energy dissipation rate, and growth of the integral scale) strongly supporting the Saffman model. We also measure the conservation of the Saffman invariant at early times of the decay and show that the energy spectrum scales as k^{2} at large scales and keeps its self-similar shape during the decay. This Letter thus presents the first experimental evidence of the validity of the connection between the Saffman invariant and the k^{2}-energy spectrum of the large scales. The final decay regime closely corresponds to Saffman's model when the container size is sufficiently large.

Inertial range scaling of inhomogeneous turbulence

We investigate how inhomogeneity influences the $k^{-5/3}$ inertial range scaling of turbulent kinetic energy spectra (with $k$ the wavenumber). For weak statistical inhomogeneity, the energy spectrum can be described as an equilibrium spectrum plus a perturbation. Theoretical arguments suggest that this latter contribution scales as $k^{-7/3}$ . This prediction is assessed using direct numerical simulations of three-dimensional Kolmogorov flow.

Intermittency, fluctuations and maximal chaos in an emergent universal state of active turbulence

The phenomenon of active turbulence, a complex organization of matter driven at the scale of its constituent agents, is puzzling. Specifically, the lack of scale-separation in low-Reynolds-number active flows breaks away from the familiar notions of the energy cascade and approximate scale-invariance of inertial turbulence. Here, using a generalized hydrodynamic model developed for bacterial turbulence, we provide compelling analytical and numerical evidence that, beyond a critical drive, active turbulence indeed attains universality akin to inertial turbulence. In this asymptotic state, the energy spectrum scales as k−3/2, reminiscent of some classes of inertial turbulence. The flow also exhibits spatio-temporal intermittency beyond the transition, as seen from non-Gaussian fluctuations in velocity differences. With these tell-tale fingerprints, active turbulence is placed closer in phenomenology to inertial turbulence than previously held. We show, however, that as a consequence of a finite range of scales, the degree of chaoticity and hence mixing efficiency saturates to a maximum in the asymptotic regime, unlike unbounded chaos in inertial turbulence. We conclude that active turbulence, depending on the level of drive, can switch between fundamentally distinct non-universal and universal states. Active fluids exhibit regimes with a complex spatio-temporal structure reminiscent of inertial turbulence. Now, inertial and active turbulence are theoretically shown to be closely related indeed.

The case of SU(3) criticality in spin-2 chains

It was proposed in [(https://doi.org/10.1103/PhysRevLett.114.145301){Chen et al., Phys. Rev. Lett. $\mathbf{114}$, 145301 (2015)}] that spin-2 chains display an extended critical phase with enhanced SU$(3)$ symmetry. This hypothesis is highly unexpected for a spin-2 system and, as we argue, would imply an unconventional mechanism for symmetry emergence. Yet, the absence of convenient critical points for renormalization group perturbative expansions, allied with the usual difficulty in the convergence of numerical methods in critical or small-gapped phases, renders the verification of this hypothetical SU$(3)$-symmetric phase a non-trivial matter. By tracing parallels with the well-understood phase diagram of spin-1 chains and searching for signatures robust against finite-size effects, we draw criticism on the existence of this phase. We perform non-Abelian density matrix renormalization group studies of multipolar static correlation function, energy spectrum scaling, single-mode approximation, and entanglement spectrum to shed light on the problem. We determine that the hypothetical SU$(3)$ spin-2 phase is, in fact, dominated by ferro-octupolar correlations and also observe a lack of Luttinger-liquid-like behavior in correlation functions that suggests that is perhaps not critical. We further construct an infinite family of spin-$S$ systems with similar ferro-octupolar-dominated quasi-SU$(3)$-like phenomenology; curiously, we note that the spin-3 version of the problem is located in a subspace of exact G$_2$ symmetry, making this a point of interest for search of Fibonacci topological properties in magnetic systems.

Large-scale characteristics of a stably stratified turbulent shear layer

Implicit large eddy simulation is performed to investigate large-scale characteristics of a temporally evolving, stably stratified turbulent shear layer arising from the Kelvin–Helmholtz instability. The shear layer at late time has two energy-containing length scales: the scale of the shear layer thickness, which characterizes large-scale motions (LSM) of the shear layer; and the larger streamwise scale of elongated large-scale structures (ELSS), which increases with time. The ELSS forms in the middle of the shear layer when the Richardson number is sufficiently large. The contribution of the ELSS to velocity and density variances becomes relatively important with time although the LSM dominate the momentum and density transport. The ELSS have a highly anisotropic Reynolds stress, to a degree similar to the near-wall region of turbulent boundary layers, while the Reynolds stress of the LSM is as anisotropic as in the outer region. Peaks in the spectral energy density associated with the ELSS emerge because of the slow decay of turbulence at very large scales. A forward interscale energy transfer from large to small scales occurs even at a small buoyancy Reynolds number. However, an inverse transfer also occurs for the energy of spanwise velocity. Negative production of streamwise velocity and density spectra, i.e. counter-gradient transport of momentum and density, is found at small scales. These behaviours are consistent with channel flows, indicating similar flow dynamics in the stratified shear layer and wall-bounded shear flows. The structure function exhibits a logarithmic law at large scales, implying a $k^{-1}$ scaling of energy spectra.

Inertial range statistics of the entropic lattice Boltzmann method in three-dimensional turbulence.

We present a quantitative analysis of the inertial range statistics produced by entropic lattice Boltzmann method (ELBM) in the context of three-dimensional homogeneous and isotropic turbulence. ELBM is a promising mesoscopic model particularly interesting for the study of fully developed turbulent flows because of its intrinsic scalability and its unconditional stability. In the hydrodynamic limit, the ELBM is equivalent to the Navier-Stokes equations with an extra eddy viscosity term. From this macroscopic formulation, we have derived a new hydrodynamical model that can be implemented as a large-eddy simulation closure. This model is not positive definite, hence, able to reproduce backscatter events of energy transferred from the subgrid to the resolved scales. A statistical comparison of both mesoscopic and macroscopic entropic models based on the ELBM approach is presented and validated against fully resolved direct numerical simulations. Besides, we provide a second comparison of the ELBM with respect to the well-known Smagorinsky closure. We found that ELBM is able to extend the energy spectrum scaling range preserving at the same time the simulation stability. Concerning the statistics of higher order, inertial range observables, ELBM accuracy is shown to be comparable with other approaches such as Smagorinsky model.

Scaling of the maximum-entropy turbulence energy spectra

The log-normal type of turbulence energy spectral function, derived from the maximum entropy principle, is shown to be parameterizable in terms of root turbulence variables including the Reynolds number. The spectral function is first compared with a number of experimental data sets, showing a very close agreement across the entire energy and length (wavenumber) scales. The peak wavenumber (m) and the width parameter (C2) prescribe the spectral location and broadening, respectively, when the Reynolds number increases, where C2 has ∼1/Rem dependence. The energy scale is adjusted with a multiplicative factor. In this perspective, the inertial scaling from k−3 to k−5/3 when the flow transitions from two- to three-dimensions is explained as the increase in spectral width since the range of scales varies as Re1∕2 and Re3/4 for two and three-dimensional turbulence, respectively. Energy spectra at various locations in channel flows are also reproduced using the same function, indicating applicability wherever local equilibrium is achieved. Therefore, based on a small number of scaling parameters the full energy spectra can be prescribed using the maximum-entropy formalism.

Dynamic Phase Alignment in Inertial Alfvén Turbulence.

In weakly collisional plasma environments with sufficiently low electron beta, Alfvénic turbulence transforms into inertial Alfvénic turbulence at scales below the electron skin depth, k_{⊥}d_{e}≳1. We argue that, in inertial Alfvénic turbulence, both energy and generalized kinetic helicity exhibit direct cascades. We demonstrate that the two cascades are compatible due to the existence of a strong scale dependence of the phase alignment angle between velocity and magnetic field fluctuations, with the phase alignment angle scaling as cosα_{k}∝k_{⊥}^{-1}. The kinetic and magnetic energy spectra scale as ∝k_{⊥}^{-5/3} and ∝k_{⊥}^{-11/3}, respectively. As a result of the dual direct cascade, the generalized helicity spectrum scales as ∝k_{⊥}^{-5/3}, implying progressive balancing of the turbulence as the cascade proceeds to smaller scales in the k_{⊥}d_{e}≫1 range. Turbulent eddies exhibit a phase-space anisotropy k_{∥}∝k_{⊥}^{5/3}, consistent with critically balanced inertial Alfvén fluctuations. Our results may be applicable to a variety of geophysical, space, and astrophysical environments, including the Earth's magnetosheath and ionosphere, solar corona, and nonrelativistic pair plasmas, as well as to strongly rotating nonionized fluids.

Lattice Boltzmann Simulation for two-dimensional bacterial turbulence

The minimal numerical simulation of two-dimensional bacterial turbulence is investigated by using the Lattice Boltzmann method (LBM). The dynamic behavior of incompressible 2D meso-scale turbulence is described by Toner–Tu theory and the Swift–Hohenberg-type fourth-order term. The double scaling of energy spectrum is obtained. At larger scale, the scaling behaviors of energy spectrum E(k)∼k5∕3. The energy spectrum E(k)∼k−8∕3 at smaller scale, which is corresponding to experiment and simulation results. The collection behavior of self-organized pattern is described. By a coarse-graining approach, the energy cascades to large scale and the enstrophy cascades to small scale at l<3R. The intermittency exists in two scaling region of energy spectrum. The measured scaling exponents ζ(p) are determined by a lognormal formula. The measured intermittency parameter is μ=0.26 which denotes the more intermittency in small scale range of 2D meso-scale turbulence.

Spectral signatures of recursive magnetic field reconnection

ABSTRACT We use 2.5D magnetohydrodynamic simulations to investigate the spectral signatures of the non-linear disruption of a tearing unstable current sheet via the generation of multiple secondary current sheets and magnetic islands. During the non-linear phase of tearing mode evolution, there develops a regime in which the magnetic energy density shows a spectrum with a power law close to B(k)2 ∼ k−0.8. Such an energy spectrum is found in correspondence of the neutral line, within the diffusion region of the primary current sheet, where energy is conveyed towards smaller scales via a ‘recursive’ process of fast tearing-type instabilities. Far from the neutral line, we find that magnetic energy spectra evolve towards slopes compatible with the ‘standard’ Kolmogorov spectrum. Starting from a self-similar description of the non-linear stage at the neutral line, we provide a model that predicts a reconnecting magnetic field energy spectrum scaling as k−4/5, in good agreement with numerical results. An extension of the predicted power law to generic current sheet profiles is also given and possible implications for turbulence phenomenology are discussed. These results provide a step forward to understand the ‘recursive’ generation of magnetic islands (plasmoids), which has been proposed as a possible explanation for the energy release during flares, but which, more in general, can have an impact on the subsequent turbulent evolution of unstable sheets that naturally form in the high Lundquist number and collisionless plasmas found in most of the astrophysical environments.

Numerical and experimental analysis of shallow turbulent flow over complex roughness beds

ABSTRACTA set of shallow-water equations (SWEs) based on a Reynold stress model is established to simulate the turbulent flows over a complex roughness bed. The fundamental equations are discretized by the second-order finite-difference method (FDM), in which spatial and temporal discretization are conducted by staggered-grid and leap-frog schemes, respectively. The turbulent model in this study stems from the standard model, but is enhanced by replacing the conventional vertical production with a more rigorous and precise generation derived from the energy spectrum and turbulence scales. To verify its effectiveness, the model is applied to compute the turbulence in complex flow surroundings (including a rough bed) in an abrupt bend and in a natural waterway. The comparison of the model results against experimental data and other numerical results shows the robustness and accuracy of the present model in describing hydrodynamic characteristics, especially turbulence features on the complex roughness bottom.

Modelling of mass transfer for gas-liquid two-phase flow in bubble column reactor with a bubble breakage model considering bubble-induced turbulence

A quantitative description of the interaction between the turbulence eddies and bubble groups is crucial for the prediction of the bubble size distribution in bubble columns when adopting the population balance model (PBM) to estimate the heat and mass transfer across the interfaces between bubbles and carrier liquid. The majority of breakage kernels that are currently adopted focus on the shear turbulence in the liquid phase and model the kinetic energy contained in the arrival eddies that hit the bubbles by using the classical single-phase turbulence Kolmogorov −5/3 scaling law. In bubble columns, eddies that collide with bubbles and cause the bubble breakage are mainly generated by bubble-induced turbulence (BIT). The present work focuses on the influence of κ−3 scaling of the bubble-induced turbulence energy spectrum on the bubble breakage due to the eddy-bubble collision in bubble columns. A modified breakage model accounting for the bubble-induced turbulence is proposed. The proposed breakage model considers the mean turbulent velocity of eddies under the influence of bubble-induced turbulence and the characteristic wavenumber/length scale that corresponds to BIT. A comparison of the simulation results with the experimental data clearly demonstrate that the modified breakage model appropriately describes the bubble breakage events occurring under the influence of bubble-induced turbulence in bubble columns. It is also shown that the interaction of bubbles with the bubble-induced turbulence eddies has an impact on prediction of the mass transfer in bubble column flows.

Kinetic-magnetic energy exchanges in rotating magnetohydrodynamic turbulence

ABSTRACTWe use direct numerical simulations to study the dynamics of incompressible homogeneous turbulence subjected to a uniform magnetic field in a rotating frame with rotation vector . We consider two cases: and . The initial state is homogeneous isotropic hydrodynamic turbulence with Reynolds number The magnetic Prandtl number and the Elsasser number , 0.9 or 2. For both the cases and the total energy decays as for and 0.9, and as for . In the spectral range 2 < k < 20, the kinetic energy spectrum scales as where p increases with time (). This scaling is similar to that observed in quasi-static MHD. The two rotating MHD flow cases differ mainly in how kinetic and magnetic fluctuations exchange energy, with a mechanism mostly driven by the dynamics of the spectral buffer layer around . At the inertial and Alfvén waves frequencies vanish when , but only the inertial waves frequency vanishes when . When , rotation results in an increased reduction of magnetic fluctuations generation. In terms of anisotropy, we show that the elongated structures occurring in rapidly non-magnetised rotating flows are distorted or inhibited for , and weakened for .

Spectral energy scaling in precessing turbulence.

We study precessing turbulence, which appears in several geophysical and astrophysical systems, by direct numerical simulations of homogeneous turbulence where precessional instability is triggered due to the imposed background flow. We show that the time development of kinetic energy K occurs in two main phases associated with different flow topologies: (i) an exponential growth characterizing three-dimensional turbulence dynamics and (ii) nonlinear saturation during which K remains almost time independent, the flow becoming quasi-two-dimensional. The latter stage, wherein the development of K remains insensitive to the initial state, shares an important common feature with other quasi-two-dimensional rotating flows such as rotating Rayleigh-Bénard convection, or the large atmospheric scales: in the plane k_{∥}=0, i.e., the plane associated to an infinite wavelength in the direction parallel to the principal rotation axis, the kinetic energy spectrum scales as k_{⊥}^{-3}. We show that this power law is observed for wave numbers ranging between the Zeman "precessional" and "rotational" scales, k_{S}^{-1} and k_{Ω}^{-1}, respectively, at which the associated background shear or inertial timescales are equal to the eddy turnover time. In addition, an inverse cascade develops for (k_{⊥},k)<k_{S}, and the spherically averaged kinetic energy spectrum exhibits a k^{-2} inertial scaling for k_{S}<k<k_{Ω}.

Structure of wall-bounded flows at transcritical conditions

Direct numerical simulations of channel flow at transcritical conditions with density change in the flow field are conducted. We find a regime with useful semi-local scaling as well as streamwise velocity and energy spectrum scaling which support the attached eddy model at transcritical conditions.

Fractal dimension of turbulent black holes

We present measurements of the fractal dimension of a turbulent asymptotically anti--de Sitter black brane reconstructed from simulated boundary fluid data at the perfect fluid order using the fluid-gravity duality. We argue that the boundary fluid energy spectrum scaling as $E(k)\ensuremath{\sim}{k}^{\ensuremath{-}2}$ is a more natural setting for the fluid-gravity duality than the Kraichnan-Kolmogorov scaling of $E(k)\ensuremath{\sim}{k}^{\ensuremath{-}5/3}$, but we obtain fractal dimensions $D$ for spatial sections of the horizon $H\ensuremath{\cap}\mathrm{\ensuremath{\Sigma}}$ in both cases: $D=2.584(1)$ and $D=2.645(4)$, respectively. These results are consistent with the upper bound of $D=3$, thereby resolving the tension with the recent claim in Adams et al. [Phys. Rev. Lett. 112, 151602 (2014)] that $D=3+1/3$. We offer a critical examination of the calculation which led to their result, and show that their proposed definition of the fractal dimension performs poorly as a fractal dimension estimator on one-dimensional curves with known fractal dimension. Finally, we describe how to define and in principle calculate the fractal dimension of spatial sections of the horizon $H\ensuremath{\cap}\mathrm{\ensuremath{\Sigma}}$ in a covariant manner, and we speculate on assigning a ``bootstrapped'' value of fractal dimension to the entire horizon $H$ when it is in a statistically quasisteady turbulent state.

Scaling of energy spectra in weakly compressible turbulence

We find an asymptotic expression for the characteristic timescales of decorrelation processes in weakly compressible and isothermal turbulence. This result is used in the Eddy-Damped Quasi-Normal Markovian equation to derive the scalings of compressible energy spectra: (1) if the acoustic waves are dominant, the compressible energy spectra exhibit $$-7/3$$ scaling; (2) if local eddy straining is dominant, the compressible energy spectra are scaled as $$-3$$ . Meanwhile, the energy spectra of incompressible components display the same scaling of $$-5/3$$ as those in incompressible turbulence. The direct numerical simulations of weakly compressible turbulence are used to examine the scaling.

Phenomenology of two-dimensional stably stratified turbulence under large-scale forcing

ABSTRACTIn this paper, we characterise the scaling of energy spectra, and the interscale transfer of energy and enstrophy, for strongly, moderately and weakly stably stratified two-dimensional (2D) turbulence, restricted in a vertical plane, under large-scale random forcing. In the strongly stratified case, a large-scale vertically sheared horizontal flow (VSHF) coexists with small scale turbulence. The VSHF consists of internal gravity waves and the turbulent flow has a kinetic energy (KE) spectrum that follows an approximate k−3 scaling with zero KE flux and a robust positive enstrophy flux. The spectrum of the turbulent potential energy (PE) also approximately follows a k−3 power-law and its flux is directed to small scales. For moderate stratification, there is no VSHF and the KE of the turbulent flow exhibits Bolgiano–Obukhov scaling that transitions from a shallow k−11/5 form at large scales, to a steeper approximate k−3 scaling at small scales. The entire range of scales shows a strong forward enstrophy flux, and interestingly, large (small) scales show an inverse (forward) KE flux. The PE flux in this regime is directed to small scales, and the PE spectrum is characterised by an approximate k−1.64 scaling. Finally, for weak stratification, KE is transferred upscale and its spectrum closely follows a k−2.5 scaling, while PE exhibits a forward transfer and its spectrum shows an approximate k−1.6 power-law. For all stratification strengths, the total energy always flows from large to small scales and almost all the spectral indicies are well explained by accounting for the scale-dependent nature of the corresponding flux.

Space-Time Correlations and Dynamic Coupling in Turbulent Flows

Space-time correlation is a staple method for investigating the dynamic coupling of spatial and temporal scales of motion in turbulent flows. In this article, we review the space-time correlation models in both the Eulerian and Lagrangian frames of reference, which include the random sweeping and local straining models for isotropic and homogeneous turbulence, Taylor's frozen-flow model and the elliptic approximation model for turbulent shear flows, and the linear-wave propagation model and swept-wave model for compressible turbulence. We then focus on how space-time correlations are used to develop time-accurate turbulence models for the large-eddy simulation of turbulence-generated noise and particle-laden turbulence. We briefly discuss their applications to two-point closures for Kolmogorov's universal scaling of energy spectra and to the reconstruction of space-time energy spectra from a subset of spatial and temporal signals in experimental measurements. Finally, we summarize the current understanding of space-time correlations and conclude with future issues for the field.

Dynamos at extreme magnetic Prandtl numbers: insights from shell models

ABSTRACTWe present a magnetohydrodynamic (MHD) shell model suitable for computation of various energy fluxes of MHD turbulence for very small and very large magnetic Prandtl numbers Pm; such computations are inaccessible to direct numerical simulations. For small Pm, we observe that both kinetic and magnetic energy spectra scale as k−5/3 in the inertial range, but the dissipative magnetic energy scales as k−11/3exp (− k/kη). Here the kinetic energy at large length scale feeds the large-scale magnetic field that cascades to small-scale magnetic field, which gets dissipated by Joule heating. The large-Pm dynamo has a similar behaviour except that the dissipative kinetic energy scales as k−13/3. For this case, the large-scale velocity field transfers energy to the large-scale magnetic field, which gets transferred to small-scale velocity and magnetic fields; the energy of the small-scale magnetic field also gets transferred to the small-scale velocity field, and the energy thus accumulated is dissipated by the viscous force.