Turbulent Dynamics Research Articles

Turbulent energy budget analysis based on coherent wind lidar observations

Abstract. The turbulent kinetic energy (TKE) budget terms, which collectively are a key physical quantity for describing the generation and dissipation processes of turbulence, are crucial for revealing the essence and characteristics of turbulence. Due to limitations in current observational methods, the generation and dissipation mechanisms of atmospheric turbulent energy are mainly based on ground or tower-based observations, and studies on the budget terms of TKE of vertical structures are lacking. We propose a new method for detecting TKE budget terms based on coherent wind lidar and compare it with data obtained with a three-dimensional ultrasonic anemometer. The results indicate that the error in the buoyancy generation term estimated by the wind lidar is relatively small, less than 0.00014 m2 s−3, which verifies the accuracy and reliability of our method. We explore the generation and dissipation mechanisms of turbulence under different weather conditions, and find that the buoyancy generation term plays a role in dissipating TKE under low-cloud and light-rain conditions. During the day, turbulent transport and the dissipation rate are the main dissipation terms, while buoyancy generation is the main dissipation term at night. The results show that the proposed method can accurately capture the vertical distribution of TKE, the dissipation rate, shear generation, turbulent transport, and buoyancy generation terms in the boundary layer and can comprehensively reflect the influence of each budget term on the vertical structure of turbulent energy. This research provides a new perspective and method for studies of atmospheric turbulence, which can be further applied to fine observations of the vertical structure and dynamics of turbulence.

Automatic Mitigation of Dynamic Atmospheric Turbulence Using Optical Phase Conjugation for Coherent Free-Space Optical Communications

Automatic Mitigation of Dynamic Atmospheric Turbulence Using Optical Phase Conjugation for Coherent Free-Space Optical Communications

A MODELING FRAMEWORK FOR FLOCCULATED COHESIVE SEDIMENT TRANSPORT IN THE CURRENT BOTTOM BOUNDARY LAYER

Cohesive sediment transport, where its settling velocity is controlled by the flocculation process, is a crucial component in determining biochemical cycles, fate of pollutants, and morphodynamics in many aquatic ecosystems. In this study, a modeling framework is presented to investigate how flocculation influences cohesive sediment transport in the current bottom boundary layer in dilute conditions, consistent with the calibration range of the flocculation model. From a local analysis of floc dynamics in homogenous turbulence, we identify that the floc size distribution is mainly controlled by floc cohesion and yield strength. The uncertainty in fractal dimension plays a minor role for the floc size but it influences the resulting floc density and settling velocity. The transport analysis in the current boundary layer shows that the flocculation process alters the vertical distribution of the settling velocity and hence the sediment concentration with a strong dependence on cohesion, floc yield strength, and floc structure. When the flocs are more susceptible to breaking, a well-mixed concentration profile is obtained. In contrast, for flocs with higher cohesion or yield strength, higher concentration with a sharp gradient is observed close to the bed. Overall, the settling velocity exhibits a low vertical variability within 20% of the depth-averaged value except near the bed. This suggests that using a depth-averaged settling velocity yields acceptable predictions of the sediment concentration profiles, especially for flocs with lower cohesion.

Turbulent Energy Conversion Associated With Kinetic Microinstabilities in Earth's Magnetosheath

AbstractPlasma in Earth's magnetosheath rarely experiences interparticle collisions, so kinetic microinstabilities are thought to contribute to regulating the plasma thermodynamics. Instabilities excite waves and redistribute free energy in velocity space, reducing free energy in the velocity distribution function (VDF). Using 24 hr of data spread over 163 intervals of in situ magnetosheath observations by Magnetospheric Multiscale (MMS), we investigate signatures of energy conversion where the turbulent dynamics have locally distorted the VDFs into non‐Maxwellian shapes, in the context of electron and ion temperature anisotropy driven instabilities. We find enhanced average energy conversion into the particles along instability boundaries, suggesting turbulence plays a role in redistributing free energy. In so doing, we quantify the energetics associated with unstable conditions for both species. This work provides insight into the open question of how specific plasma processes couple into the turbulent dynamics, ultimately leading to energy dissipation and particle energization in collisionless plasmas.

Numerical study of vortex–bubble interaction mechanism in bubble breaker using the volume of fluid and large eddy simulation method

Effectively removing or reducing the number of bubbles doped in a material is of great significance to stabilize product characteristics and improve production safety. In this paper, the volume of fluid method and large eddy simulation method are used to numerically simulate the bubble dynamics in unsteady turbulence near a four-blade propeller in a bubble breaker at different speeds. The results show that the final rupture position of the bubble occurs in the main vortex region. The interaction mechanism between the vortex and the bubble involves not only the impact of the small vortex on the bubble but also the influence of the tail vortex pair, which exerts two strong torsional stresses in opposing directions. This interaction ultimately results in the breakup of the bubble. In addition, three different fracture modes were observed: Ring Broken Mode (RBM), Dented pull out Broken Mode (DBM), and Tear Broken Mode (TBM). The bubble breaking process is quantitatively described in terms of vorticity change rate and critical Weber number (We). It is found that DBM does not occur at high rotational speed, and TBM occurs faster and more violently with increase in vorticity change rate. After multiple cycles, the vorticity change in RBM is similar to that at medium speed, and when the change rate reaches its maximum, the size of the fragmentation bubble becomes smaller.

Hydrodynamic analysis of fish swimming behavior in turbulent river confluences

This study focuses on selecting the most appropriate turbulence model for simulating fish swimming behavior in river confluences. To achieve this, three numerical models—k-ε, k-ω, and large eddy simulation—were compared by running simulations under identical flow conditions and evaluating the results against biological experimental data. Among the models, the k-ω model demonstrated the smallest relative error, consistently within 5% of the experimental results, confirming its superior accuracy and reliability for this application. The k-ω model's ability to capture boundary layer turbulence and near-wall flow dynamics proved essential for studying fish swimming in complex turbulent environments. Simulations revealed that both the flow velocity ratio between the main stream and tributary and the confluence angle are critical factors influencing the flow structure. At higher flow velocity ratios (R = 1/3 and 3/1) or large confluence angles (α ≥ 90°), turbulence intensity increased, leading to more complex vortex formations that significantly impacted fish swimming speed. When the flow velocity ratio (R) is 1/3, the fish can achieve a maximum swimming speed of 2.75 L/s, which is significantly higher than the swimming speed of 1.18 L/s observed when R is 3/1. Additionally, fish closer to the center of the flow field experienced greater turbulence, resulting in higher energy expenditure. The findings provide crucial insights into the hydrodynamic mechanisms driving fish swimming behavior in dynamic aquatic environments.

Deep learning-enhanced aerodynamics design of high-load compressor cascade at low Reynolds numbers

This study addresses the challenges of designing high-efficiency compressors for long-endurance, high-altitude unmanned aerial vehicles (UAVs) under low Reynolds number (Re) and high load conditions, exploring the aerodynamic performance limits of compressor blades under extreme conditions. The research integrates advanced numerical simulations, experimental methods, and deep learning technologies to minimize profile losses and deviation angle on compressor blades. An orthogonal experimental design systematically explored the impact of key geometric factors on aerodynamic performance. A deep learning model incorporating neural networks with spatial attention mechanisms was developed to significantly enhance the accuracy of aerodynamic predictions. This model adeptly captured the complex nonlinear interactions between aerodynamic and geometric parameters. Sobol sensitivity analysis revealed that the dimensionless maximum thickness is the most critical factor, accounting for 44 % of the total variance in total pressure loss and deviation angle. The position of maximum thickness and the aspect ratio of the elliptical leading edge also significantly influenced performance. The optimized high-load compressor blade profile was validated through experimental data and detailed computational fluid dynamics analysis using large eddy simulation methods. This analysis revealed flow separation and reattachment mechanisms, shedding light on turbulence and vortex dynamics critical to performance. This research deepens the theoretical understanding of compressor cascade fluid dynamics and provides practical insights for designing more efficient compressors, especially for micro axial compressors in high-altitude UAVs.

Characterizing Turbulence and Wave Dynamics Under Rotation: Enhancing Semiconductor Processes through CFD Analysis and High-Speed Imaging

IntroductionIn semiconductor manufacturing, precision in wafer processing is crucial for advancing device technology. The spinning system, essential for wafer cleaning and etching, significantly impacts device quality by facilitating dynamic air-liquid interactions, affecting surface chemistry. This study utilizes Computational Fluid Dynamics (CFD) and high-speed camera analysis to explore turbulence and wave dynamics, confronting Fast Fourier Transform (FFT) analysis against the Kolmogorov cascade to improve manufacturing outcomes by understanding these dynamics more thoroughly.Experimental SetupOur approach integrates high-speed imaging with advanced CFD simulations to capture and analyze the wave phenomena in spinning liquid films. High-speed cameras are strategically positioned to provide comprehensive side and top views, enabling detailed observation of wave formation and interaction with the air. Concurrently, CFD simulations offer a granular view of the fluid dynamics, focusing on the generation and propagation of turbulence within the liquid film. The application of FFT analysis to CFD data allows us to delve into the turbulent wave dynamics, specifically investigating the energy cascade, per the Kolmogorov theory [1]. This methodology not only deepens our understanding of wave behavior but also highlight its potential impacts on key factors defining the exchange kinetics of the liquid-air interface.DiscussionFor a single wafer, a gas molecule, characterized by a diffusion coefficient D, will diffuse through the liquid film of thickness h to reach the wafer surface in a time approximated by t ~ h²/D. Specifically for oxygen, (D~2×10⁻⁹ m²/s) and 100µm thin film, the process takes about 5 seconds, while the coverage of the liquid takes less than 0.5s so an order of magnitude lower [2]. The Peclet number (Pe), defined by Pe=Uh/D, represents the advection to diffusion ratio. Here, Pe~10-4 >> 1 indicating advection dominance. This suggests wave and hydrodynamics (laminar vs. turbulence) primarily transport air into the liquid. Near the wafer edge, where the flow speeds up and the liquid film thins out, gas molecules reach the wafer faster than at its center.Advection rates are higher on wavy surfaces due to complex streamlines shaped by the undulating contours of the interface. This will enhance mixing transport, while the increased surface area boosts heat and mass exchange. Depending on wave celerity, recirculation atop the waves further boost local mixing [3,4].In Figure 1, the FFT analysis indicates that turbulent dynamics align with the Kolmogorov cascade, highlighting a spectrum of energy dissipation that could significantly impact oxygen diffusion and heat transfer. These findings suggest that the wave-induced turbulence could be leveraged to optimize the distribution of reactive species and thermal management during the spinning process. It highlights that the energy is injected by the swirling and concentric waves near the center of the wafer.In Figure 2, the Iribaren number, a dimensionless metric for classifying wave breaking patterns [5], is extracted from the liquid thickness distribution on wafer. It compares the slope steepness to the wave steepness, predicting whether waves will spill, plunge, or surge. Spilling waves, associated with low Iribarren numbers below 0.5 as found in our study, gently dissipate energy over wide areas. These spilling waves dissipate energy over long distances, breaking into smaller wavelets and introducing turbulence showing complex interactions between wave dynamics and the wafer.Figure 3 shows the dynamics of spilling waves with concentric waves collapsing towards the edge using a high-speed camera. This phenomenon is observed in a side view of a centrally dispensed fluid on a rotating wafer. The captured speed range is notably high, exceeding 50 km/h. The size and number of waves are affected by the rotation speed. A wave will collapse faster at high rpm because of a thinner film and higher centrifugal forces.ConclusionThis research reveals the relationship between wave dynamics in spinning liquid films and their potential impact on semiconductor manufacturing efficiency. The synergy of high-speed imaging, CFD analysis, and FFT insights not only enhances our understanding of fluid dynamics but also paves the way for optimizing wafer cleaning and etching techniques. This study lays the groundwork for future technological advancements, promising to elevate the precision, efficiency, and quality of semiconductor production.

Effect of an incoming Gaussian wave packet on underlying turbulence

We present a simulation-based study of the effect of a passing wave packet on underlying fully developed turbulence. We propose a novel wave-phase-resolved simulation method inspired by Helmholtz decomposition to directly couple the turbulence simulation with instantaneous wave orbital motions without wave-phase averaging. We also introduce a boundary condition treatment for the turbulence at the wave surface, which allows the turbulence simulation to be conducted in a rectangular domain while retaining the wave-phase effect. The results obtained from the proposed method reveal considerable variations in turbulence statistics, including the enstrophy and Reynolds normal stresses, during wave packet passage. Most changes occur rapidly when the narrow bandwidth around the wave packet core passes. Further analyses of the energy spectra indicate that the enhancement of turbulence occurs across a wide range of scales, with the near-surface small-scale motions experiencing the most significant intensification. Meanwhile, large-scale motions with scales comparable to the boundary layer depth are also enhanced. The mechanisms underlying the Reynolds normal stress variation at different length scales are related to the energy transfer from the wave orbital straining to turbulence through production, the pressure–strain effect, the pressure diffusion and the wave advection. By assessing the turbulence statistics and dynamics impacted by a wave packet in detail, this study provides an improved understanding of the response of a developed turbulent flow to a transient wave field. The proposed simulation method also proves to be a promising phase-resolved approach for efficiently modelling the wave effect on turbulence.

The Influence of Rotation on the Preservation of Heterogeneities in Magma Oceans

AbstractUnderstanding the composition of lavas erupted at the surface of the Earth is key to reconstruct the long‐term history of our planet. Recent geochemical analyses of ocean island basalt samples indicate the preservation of ancient mantle heterogeneities dating from the earliest stages of Earth's evolution (Péron & Moreira, 2018, https://doi.org/10.7185/geochemlet.1833), when a global magma ocean was present. Such observations contrast with fluid dynamics studies which demonstrated that in a magma ocean the convective motions, primarily driven by buoyancy, are extremely vigorous (Gastine et al., 2016, https://doi.org/10.1017/jfm.2016.659) and are therefore expected to mix heterogeneities within just a few minutes (Thomas et al., 2023, https://doi.org/10.1093/gji/ggad452). To elucidate this paradox we explored the effects of the Earth's rapid rotation on the stirring efficiency of a magma ocean, by performing state‐of‐the‐art fluid dynamics simulations of low‐viscosity, turbulent convective dynamics in a spherical shell. We found that rotational effects drastically affect the convective structure and the associated stirring efficiency. Rotation leads to the emergence of three domains with limited mass exchanges, and distinct stirring and cooling efficiencies. Still, efficient convective stirring within each region likely results in homogenization within each domain on timescales that are short compared with the solidification timescales of a magma ocean. However, the lack of mass exchange between these regions could lead to three or four large‐scale domains with internally homogeneous, but distinct compositions. The existence of these separate regions in a terrestrial magma ocean suggests a new mechanism to preserve distinct geochemical signatures dating from the earliest stages of Earth's evolution.

Significance of the smaller scales for hypersonic turbulent boundary layers with focused laser differential interferometry

While designed to suppress disturbances located away from its beam foci, focused laser differential interferometry (FLDI) is known to have some integration of signals along its optical axis. This is especially true for longer-wavelength signals, although smaller-scale flow structures also have a non-infinitesimal sensitivity length. This study investigates the performance of FLDI in a hypersonic turbulent boundary layer to better understand its behavior. FLDI simulations were conducted using both full-scale direct numerical simulation (DNS) and spatially averaged DNS inputs to directly assess the influence of smaller flow structures on FLDI measurements. The full-scale FLDI results indicate that integration along the optical axis likely results in lower FLDI amplitudes than for true point measurements. Comparison with the spatially averaged FLDI simulations reveals the significance of small-scale structures in FLDI signal roll-off and root mean square amplitudes. Further, the influence of FLDI setup parameters on the response across the frequency spectrum are analyzed. Circular, Gaussian beams with smaller widths are verified to present increased performance relative to elliptical or uniform-intensity beams. Also, measurements using distinct differentiation directions suggest an experimental way to measure turbulence isotropy scales. These results have notable implications for understanding hypersonic turbulent boundary-layer dynamics and interpreting experimental data. Distribution Statement A: Approved for Public Release; Distribution is Unlimited. PA# AFRL-2024-4678; Cleared 08/23/2024.

Dynamic instabilities and turbulence of merged rotating Bose–Einstein condensates

We present the simulation results of merging harmonically confined rotating Bose–Einstein condensates in two dimensions. Merging of the condensate is triggered by positioning the rotation axis at the trap minima and moving both condensates toward each other while slowly ramping their rotation frequency. We analyze the dynamics of the merged condensate by letting them evolve under a single harmonic trap. We systematically investigate the formation of solitonic and vortex structures in the final, unified condensate, considering both nonrotating and rotating initial states. In both cases, merging leads to the formation of solitons that decay into vortex pairs through snake instability, and subsequently, these pairs annihilate. Soliton formation and decay-induced phase excitations generate sound waves, more pronounced when the merging time is short. We witness no sound wave generation at sufficiently longer merging times that finally leads to the condensate reaching its ground state. With rotation, we notice off-axis merging (where the rotation axes are not aligned), leading to the distortion and weakening of soliton formation. The incompressible kinetic energy spectrum exhibits a Kolmogorov-like cascade [E(k)∼k−5/3] in the initial stage for merging condensates rotating above a critical frequency and a Vinen-like cascade [E(k)∼k−1] at a later time for all cases. Our findings hold potential significance for atomic interferometry, continuous atomic lasers, and quantum sensing applications.

Beam emission spectroscopy diagnostic design and capabilities for two-dimensional turbulence measurements on Wendelstein 7-X

A beam emission spectroscopy (BES) diagnostic is designed for studying two-dimensional turbulent dynamics by measuring the Doppler-shifted Balmer-Alpha emission (n = 3 → 2) from neutral heating beams on Wendelstein 7-X (W7-X) stellarator. The BES viewing geometry has been determined in the conceptual design previously. However, the small Doppler shifts and small optical throughput compared to a typical BES diagnostic demand dedicated efforts on the optical assemblies and the detector module for the BES system. We present the detailed opto-mechanical design and specifications of BES, including a customized neutral beam viewing optical system, a semi-telecentric optical assembly, and a detector module for electronic amplification. The point spread function is calculated using the pyFIDASIM code with experimental parameters and W7-X magnetic configurations to estimate the BES spatial resolution and beam intensity. The as-manufactured interference filter is applied for the spectral isolated beam radiance calculation. Result shows that the BES system is capable of measuring the ion-scale turbulence for k ⊥ ρ i ≤ 0.4 at r/a = 0.75 with reasonable spatial and wavenumber resolutions. An integrated detector module is fabricated where two 8×4 avalanche photodiode detectors (APD) are embedded into the custom-designed pre-amplifier circuit to gain signals to the desired level. The detector noise measurement is performed and the signal-to-noise ratio (SNR) is evaluated. A detectable fluctuation level can be achieved as low as ñ e /n e ≈ 0.5% at frequency f ≤ 400 kHz with a bandwidth of 1 MHz.

Scientific machine learning based reduced-order models for plasma turbulence simulations

This paper investigates non-intrusive Scientific Machine Learning (SciML) Reduced-Order Models (ROMs) for plasma turbulence simulations. In particular, we focus on Operator Inference (OpInf) to build low-cost physics-based ROMs from data for such simulations. As a representative example, we consider the (classical) Hasegawa–Wakatani (HW) equations used for modeling two-dimensional electrostatic drift-wave turbulence. For a comprehensive perspective of the potential of OpInf to construct predictive ROMs, we consider three setups for the HW equations by varying a key parameter, namely, the adiabaticity coefficient. These setups lead to the formation of complex and nonlinear dynamics, which makes the construction of predictive ROMs of any kind challenging. We generate the training datasets by performing direct numerical simulations of the HW equations and recording the computed state data and outputs over a time horizon of 100 time units in the turbulent phase. We then use these datasets to construct OpInf ROMs for predictions over 400 additional time units, that is, 400% more than the training horizon. Our results show that the OpInf ROMs capture important statistical features of the turbulent dynamics and generalize beyond the training time horizon while reducing the computational effort of the high-fidelity simulation by up to five orders of magnitude. In the broader context of fusion research, this shows that non-intrusive SciML ROMs have the potential to drastically accelerate numerical studies, which can ultimately enable tasks such as the design of optimized fusion devices.

CFD Simulation of Stirling Engines: A Review

Stirling engines (SEs) have long attracted the attention of renewable energy researchers due to their external combustion design and flexibility in operating with various heat sources. The mathematical analysis of these devices is conducted by using a broad range of models ranging from basic zero-order to highly detailed fourth-order models, which are implemented through Computational Fluid Dynamics (CFD) simulations. The unique features of this last approach, combined with the increase in computing power, have promoted the use of CFD as a tool for analyzing SEs in recent years, significantly reducing the costs associated with prototype construction. However, Stirling CFD simulations are sophisticated due to the variety of physical phenomena involved, such as volume change, conjugated heat transfer, turbulent compressible fluid dynamics, and flow through porous media in the regenerator. Furthermore, there is currently no comprehensive review of CFD simulations of SEs in the literature; therefore, this contribution aims to fill that gap. Emphasis has been placed on identifying the type of engine, the physical phenomena modeled, the simplifying assumptions, and specific numerical aspects, such as mesh type, spatial and temporal discretization, and the order of the numerical schemes used. As a result, it has been found that in many cases, CFD numerical reports lack sufficient detail to ensure the reproducibility of the simulations. This work proposes guidelines for reporting CFD studies on Stirling engines to address this issue. Additionally, the need for a sufficiently detailed experimental benchmark database to validate future CFD studies is stressed. Finally, the use of Large Eddy Simulations on coupled key engine components—such as compression and expansion spaces, pistons, displacer, and regenerator—is suggested to provide further insights into the specific flow and heat transfer characteristics in Stirling engines.

Unfolding wall turbulence

We present the project FLARE (Unfolding wall turbulence), funded by the Spanish Ministry of Science and FEDER. The main objective of this project is to contribute to understanding the dynamics of wall-bounded turbulence, exploring two techniques: Lie-Symmetry and Machine Learning. Further, we have compiled several high-quality databases of different flows using Direct Numerical Simulations of the Navier-Stokes equations with the code LISO to use these techniques. This data is available to the turbulence community.

Investigation of the impact of transient pressure gradients on turbulent channel flow dynamics

AbstractThe effect of transient pressure gradients, or transient pumping in turbulent channel flow configuration, is investigated. Employing a cost‐efficient reduced‐order stochastic method known as one‐dimensional turbulence (ODT) modeling, simulations explore different signal shapes for the modulation of the prescribed pressure gradient forcing, including sinusoidal modulation, step‐like modulation, and piecewise sinusoidal beating. Various cycle periods and active pumping times are investigated. The simulations are conducted at a frictional Reynolds number of and a molecular Prandtl number of . The study adopts a passive scalar formulation to investigate heat transfer properties. The study quantifies the effects of transient pressure gradients on heat transfer rate, drag, and pumping power. Preliminary ODT predictions suggest that all transient cases exhibit lower heat transfer rates and a higher pumping power requirement than the constant pressure gradient case, with the step‐like modulation yields superior skin‐friction drag and heat transfer rate reductions relative to other signals.

Stable reproducibility of turbulence dynamics by machine learning

Stable reproducibility of turbulence dynamics by machine learning

Key Issues and Application Prospects in High-Temperature Plasma Physics

High-temperature plasma physics is the core area of research into nuclear fusion energy, and it also has broad application potential in industry and astrophysics. The paper systematically reviews the fundamental definition, the formation mechanism, and research history of high-temperature plasma; analyzes focal problems including energy confinement, plasma instability, radiation cooling, material tolerance, turbulent dynamics; and discusses the technical progress and challenges of magnetic confinement and inertial confinement experimental devices. The comparison of tokamak against laser inertial confinement gives further illustration of the pros and disadvantages of the two fusion routes. Nuclear fusion energy, from an application viewpoint, represents a very ideal energy source for future sustainable development, and high-temperature plasma technology is also very promising in industrial processing, astrophysical experiments, and other fields. In conjunction with relevant recent research, suggestions are also given in this paper regarding high-temperature plasma in the future development direction. All this is done as an effort to offer reference and guidance for future developments in that field.

Turbulent cascade arrests and the formation of intermediate-scale condensates.

Energy cascades lie at the heart of the dynamics of turbulent flows. In a recent study of turbulence in fluids with odd viscosity X.M. de Wit et al. [Nature (London) 627, 515 (2024)0028-083610.1038/s41586-024-07074-z], the two dimensionalization of the flow at small scales leads to the arrest of the energy cascade and selection of an intermediate scale, between the forcing and the viscous scales. To demonstrate the generality of the phenomenon and its existence for a wide class of turbulent systems, we study a shell model that is carefully constructed to have three-dimensional turbulent dynamics at small wave numbers and two-dimensional turbulent dynamics at large wave numbers. The large scale separation that we can achieve in our shell model allows us to examine clearly the interplay between these dynamics, which leads to an arrest of the energy cascade at a transitional wave number and an associated accumulation of energy at the same scale. Such pile-up of energy around the transitional wave number is reminiscent of the formation of condensates in two-dimensional turbulence, but, in contrast, it occurs at intermediate wave numbers instead of the smallest wave number.