Energy Cascade Research Articles

Forgetfulness of turbulent energy cascade associated with different mechanisms

The small scales of high Reynolds number turbulence exhibit the statistical universality by forgetting about the large scales during the energy cascade. In this study, we revisit this schematic description with an information-theoretic point of view. Our objective is to understand the importance of different cascade mechanisms in this process. To this end, we compute the “information flux”, the information-theoretic measure of causality formulated by one of the authors, of the decomposed inter-scale energy transfer. For the kinetics, the ratio of the inertial-range inter-scale energy transfer between the strain self-amplification (SSA) and the vortex stretching (VS) mechanisms is 5 : 3. However, for the information-flux, the VS contribution is roughly twice as large as the SSA. We also find several deviations between the kinetics and the informatics, highlighting the nontrivial nature of information distributed among different scales and mechanics in turbulence. This result infers that the SSA is a less efficient mechanism than the VS in terms of information transfer, and contributes to losing the large-scale information.

Universal relative scaling of longitudinal structure functions in shear-dominated turbulence

Shear significantly influences turbulence in the energy-containing range of shear-dominated flows, and the longitudinal structure functions do not have a universal form as they do in homogeneous isotropic turbulence. Despite this, the relative scaling of structure functions exhibits universal sub-Gaussian behaviour in shear-dominated flows, in particular for turbulent boundary layers, channels and Taylor–Couette flows. Our investigation of a turbulent vertical buoyancy layer at $Pr = 0.71$ using direct numerical simulation shows this universality even in moderate-Reynolds-number buoyancy-driven but shear-dominated boundary layers. It is demonstrated that the universality is related to the energy density of the eddies, which attains a hierarchical equilibrium in the energy-containing range of shear-dominated turbulence. We conjecture that the universal sub-Gaussian behaviour of the energy density of the energy-containing range, which was considered to be non-trivial in prior studies, is related to the universal anomalous scaling exponents of the inertial subrange turbulence. Based on this conjecture, we propose a hypothesis that relates large-scale eddies and the intermittent dissipation field in shear-dominated turbulence, highlighting a relationship between large and small scales. A phenomenological model is also developed to predict the scaling, which is verified using data from a turbulent boundary layer, half-channel and vertical buoyancy layer at friction Reynolds numbers spanning four orders of magnitude. Excellent agreement is observed.

Data-driven nonlinear turbulent flow scaling with Buckingham Pi variables

Nonlinear machine learning for turbulent flows can exhibit robust performance even outside the range of training data. This is achieved when machine-learning models can accommodate scale-invariant characteristics of turbulent flow structures. This study presents a data-driven approach to reveal scale-invariant vortical structures across Reynolds numbers that provide insights for supporting nonlinear machine-learning-based studies of turbulent flows. To uncover conditions for which nonlinear models are likely to perform well, we use a Buckingham-Pi-based sparse nonlinear scaling to find the influence of the Pi groups on the turbulent flow data. We consider nonlinear scalings of the invariants of the velocity gradient tensor for an example of three-dimensional decaying isotropic turbulence. The present scaling not only enables the identification of vortical structures that are interpolatory and extrapolatory for the given flow field data but also captures non-equilibrium effects of the energy cascade. As a demonstration, the present findings are applied to machine-learning-based super-resolution analysis of three-dimensional isotropic turbulence. We show that machine-learning models reconstruct vortical structures well in the interpolatory space with reduced performance in the extrapolatory space revealed by the nonlinearly scaled invariants. The present approach enables us to depart from labelling turbulent flow data with a single parameter of Reynolds number and comprehensively examine the flow field to support training and testing of nonlinear machine-learning techniques.

Carbon efficiency evaluation method for urban energy system with multiple energy complementary

Urban energy systems (UESs) play a pivotal role in the consumption of clean energy and the promotion of energy cascade utilization. In the context of the construction and operation strategy of UESs with multiple complementary energy resources, a comprehensive assessment of the energy efficiency is of paramount importance. First, a multi-dimensional evaluation system with four primary indexes of energy utilization, environmental protection, system operation, and economic efficiency and 21 secondary indexes is constructed to comprehensively portray the UES. Considering that the evaluation system may contain a large number of indexes and that there is overlapping information among them, an energy efficiency evaluation method based on data processing, dimensionality reduction, integration of combined weights, and gray correlation analysis is proposed. This method can effectively reduce the number of calculations and improve the accuracy of energy efficiency assessments. Third, a demonstration project for a UES in China is presented. The energy efficiency of each scenario is assessed using six operational scenarios. The results show that Scenario 5, in which parks operate independently and investors build shared energy-storage equipment, has the best results and is best suited for green and low-carbon development. The results of the comparative assessment methods show that the proposed method provides a good energy efficiency assessment. This study provides a reference for the optimal planning, construction, and operation of UESs with multiple energy sources.

A deep-learning super-resolution reconstruction model of turbulent reacting flow

Super-resolution reconstruction using deep-learning techniques is a valuable tool for enhancing computational and experimental visualization of flow fields to recover detailed features from coarse results. In this study, we propose the Pyramid Res U Super Resolution (PRUSR) network as a flow reconstruction model that upgrades the basic architecture of the Downsampled Skip-Connection Multi-Scale (DSC/MS) model with the pre-activated residual block (PRB). To improve the model accuracy, we also implement a new loss function called Mean Square Structure Similarity (MSSSIM), which combines the Mean Square Error (MSE) and Structure Similarity (SSIM). We test the model on a practical problem of visualizing the three-dimensional turbulent jet flame and find that the PRUSR model accurately reconstructs the super-resolution flow field, particularly in large downsampling ratio conditions, outperforming the original DSC/MS model and traditional interpolation algorithms such as bilinear and bicubic even with fewer input data. Additionally, the PRUSR model is capable of reconstructing sub-grid turbulent flow and flame thickness by inferring high-frequency information in the inertial subrange, making it a powerful post-processing tool for turbulent flow visualization and analysis.

Jovian Magnetosheath Turbulence Driven by Whistler

Jupiter’s magnetosheath is a natural yet complex laboratory for analyzing compressible plasma turbulence. Recent observations by the Juno mission provide a promising opportunity for the first time to reckon the energy cascade rate in the magnetohydrodynamic scales in the vicinity of Jupiter’s space. In the present work, a two-dimensional model is constructed for a whistler wave that is nonlinearly coupled with a wave magnetic field via ion density perturbation. The dynamics of whistler wave propagating in the direction of the magnetic field are derived within the limit of the two-fluid modeling approach. The magnetic field localization along with magnetic field spectra and spectral slope variations are estimated to realize the turbulence generation and energy cascade from large to small scales in the Jovian magnetosheath region. The simulated magnetic field spectrum in the wave number (in the unit of ion inertial length ρ i ) consists of turbulence in the inertial range with a spectral slope of −1.4 and a spectral knee at k ρ i = 1. Subsequently, the spectral slope increases to −2.6 and the spectrum becomes steeper. The simulated magnetic field spectrum in the wave number is further translated into the frequency domain using the whistler wave dispersion relation and by considering the Taylor frozen-in condition. The analytically estimated magnetic field spectrum slopes, i.e., −1.8 and −4.2 at low and high frequencies are further compared with recent Juno mission observations. The comparison further affirms the existence of Kolmogorov scaling, a spectral knee, and steepening in the spectrum at high frequencies. Furthermore, it is found that the two-fluid model can reasonably simulate the turbulence effects in Jovian magnetosheath in terms of magnetic field spectral distribution in wave number and frequency domains.

The Generation of 150 km Echoes Through Nonlinear Wave Mode Coupling

AbstractA fundamental problem in plasma turbulence is understanding how energy cascades across multiple scales. In this paper, a new weak turbulence theory is developed to explain how energy can be transferred from Langmuir and Upper‐Hybrid waves to ion‐acoustic waves. A kinetic approach is used where the Boltzmann equation is Fourier‐Laplace transformed, and the nonlinear term is retained. A unique feature of this approach is the ability to calculate power spectra at low frequencies, for any wavelength or magnetic aspect angle. The results of this theory explain how the predominant type of 150‐km radar echoes are generated in the ionosphere. First, peaks in the suprathermal electron velocity distribution drive a bump‐on‐tail like instability that excites the Upper‐Hybrid mode. This excited wave then couples nonlinearly to the ion‐acoustic mode, generating the ∼10 dB enhancement observed by radars. This theory also explains why higher frequency radars like ALTAIR do not observe these echoes.

Bubble size distribution and turbulence characterization in a bubbly flow in the presence of surfactant

This study aimed to quantify the less explored complex multiphase hydrodynamics of a bubble swarm in the presence of surfactant which is the key to flotation process widely used in the resources and environmental engineering applications.Experiments were conducted in a rectangular column (cross-section: 100 mm × 100 mm) in batch mode by varying the gas flux (0.02 to and 0.08 cm/s) in the presence of an anionic surfactant sodium dodecyl sulphate (5 to 15 ppm). High-speed imaging was then used to visualise the unsteady bubble plume dispersion behaviour and estimate the bubble size distribution (BSD) which showed a reduction in the mean bubble size with the increasing gas flux and surfactant concentration.Next, particle image velocimetry (PIV) was utilised to measure the instantaneous velocity field which was used to determine the turbulence characteristics. It was shown that the bubble plume contributes to significant anisotropy in the flow field which increased in the higher surfactant concentration and gas flux cases. The energy containing turbulence length scale was characterized by the integral length scale, which was observed to increase linearly with both the gas flux and surfactant concentration. Also, the local turbulence energy dissipation rate exhibited a strong linear correlation with the bubble surface area flux parameter. In the presence of surfactant, the turbulence energy spectrum of the system exhibited a less steep slope in the inertial subrange regime compared to the Kolmogorov −5/3 slope. The spectrum also showed a leftward shift indicating energy addition to the larger turbulence length scales which was reflected in the formation of large recirculation zones around the bubble plume.

A Combined Experimental and Turbulence-Resolved Modeling Approach for Aeroengine Turbine Rim Seals

Abstract Ingress is the penetration of hot mainstream gas into the rotor–stator wheel-space formed between adjacent disks; a rim seal is installed at the periphery of the wheel-space. Purge flow is bled from the compressor and re-introduced in the turbine to reduce, or in the limit prevent, ingress. This study presents a unique, concomitant experimental and turbulence-resolved numerical investigation of ingress in an aeroengine rim seal, with leakage flow. Experimental modeling is conducted in the University of Bath's 1-stage turbine test facility. Measurements of gas concentration, pressure and swirl were used to assess the performance of the rim seal. A parallel study using improved delayed detached eddy simulations (IDDES) was used to generate time-averaged and time-resolved flow-fields, enabling direct comparison with experimental data. The modeled geometry included realistic features typical of aeroengine architectures, including a contoured stator undershroud and an omega-seal cover plate. Such features were shown to locally distort the flow field, highlighting the limitation when modeling simplified geometry. The circumferential distribution of sealing effectiveness was nonaxisymmetric and synchronized in accordance with the local radial velocity field. Utilization of a detached eddy simulation (DES) turbulent kinetic energy (TKE) dissipation multiplier demonstrated regions where increased turbulence resolution was required to resolve the appropriate scale of turbulent eddies. IDDES computations were found to accurately capture the radial distributions of pressure, swirl and effectiveness, both in the absence and presence of a superposed leakage flow, provided that the mesh was sufficiently refined so as to resolve ≥50% of the energy cascade. The IDDES approach exhibited significantly superior agreement with experiments when compared to previous studies that employed the unsteady Reynolds-averaged Navier–Stokes (URANS) methodology.

Scale-to-scale energy flux in the oceanic global circulation models

Lewis Fry Richardson proposed his famous picture of turbulent flows in 1922, where the kinetic energy is transferred from large-scale to small-scale structures until the viscosity converts it into heat. This cascade idea, also known as the forward energy cascade, is now widely accepted and is treated as the cornerstone of not only turbulent modeling, but also global circulation models of the ocean and atmosphere. In this work, the Filter-Space-Technique is applied to the oceanic flow field provided by the CMEMS reanalysis model to quantify the scale-to-scale energy flux. A rich dynamical pattern associated with different scales is observed. More precisely, either positive or negative fluxes are observed, indicating the direction of the energy cascade, where the energy is transferred from large-scale structures to small-scale ones or vice versa. High-intensity energy exchange is found mainly in the Western Boundary Current Systems and Equatorial Counter Currents. For the latter case, a wavelike pattern is observed on the westward travel. Moreover, strong seasonal variation is evident for some scales and regions. These results confirm the existence of forward and inverse cascades and rich regional dynamics.

Pattern formation by turbulent cascades

Fully developed turbulence is a universal and scale-invariant chaotic state characterized by an energy cascade from large to small scales at which the cascade is eventually arrested by dissipation1–6. Here we show how to harness these seemingly structureless turbulent cascades to generate patterns. Pattern formation entails a process of wavelength selection, which can usually be traced to the linear instability of a homogeneous state7. By contrast, the mechanism we propose here is fully nonlinear. It is triggered by the non-dissipative arrest of turbulent cascades: energy piles up at an intermediate scale, which is neither the system size nor the smallest scales at which energy is usually dissipated. Using a combination of theory and large-scale simulations, we show that the tunable wavelength of these cascade-induced patterns can be set by a non-dissipative transport coefficient called odd viscosity, ubiquitous in chiral fluids ranging from bioactive to quantum systems8–12. Odd viscosity, which acts as a scale-dependent Coriolis-like force, leads to a two-dimensionalization of the flow at small scales, in contrast with rotating fluids in which a two-dimensionalization occurs at large scales4. Apart from odd viscosity fluids, we discuss how cascade-induced patterns can arise in natural systems, including atmospheric flows13–19, stellar plasma such as the solar wind20–22, or the pulverization and coagulation of objects or droplets in which mass rather than energy cascades23–25.

Energy Cascade in Halide Perovskite-Multiple Chromophore Films: Direct versus Mediated Transfer

Energy Cascade in Halide Perovskite-Multiple Chromophore Films: Direct versus Mediated Transfer

Multi-scale invariant solutions in plane Couette flow: a reduced-order model approach

Plane Couette flow at Reynolds number $Re=1200$ (based on the channel half-height and half the velocity difference between the top and bottom plates) is investigated with a spatial domain designed to retain only two spanwise integral length scales. In this system, the computation of invariant solutions that are physically representative of the turbulent state has been understood to be challenging. To address this challenge, our approach is to employ an accurate reduced-order model with 600 degrees of freedom (Cavalieri & Nogueira, Phys. Rev. Fluids, vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal cross-correlation of key observables, it is first demonstrated that the model contains most of the multi-scale physical processes identified recently (Doohan et al., J. Fluid Mech., vol. 913, 2021, A8); i.e. the large- and small-scale self-sustaining processes, the energy cascade for turbulent dissipation, and an energy-cascade mediated small-scale production mechanism. Invariant solutions of the reduced-order model are subsequently computed, including 96 equilibria and 43 periodic orbits. It is found that none of the computed equilibrium solutions are able to reproduce an accurate energy balance associated with the multi-scale dynamics of the turbulent state. Incorporation of unsteadiness into invariant solutions is seen to be essential for a sensible description of the multi-scale turbulent dynamics and the related energetics, at least in this type of flow, as periodic orbits with a sufficiently long period are mainly able to describe the complex spatio-temporal dynamics associated with the known multi-scale phenomena.

Dynamic Modeling and Control of Supercritical Carbon Dioxide Power Cycle for Gas Turbine Waste Heat Recovery

The gas turbine is a crucial piece of equipment in the energy and power industry. The exhaust gas has a sufficiently high temperature to be recovered for energy cascade use. The supercritical carbon dioxide (S-CO2) Brayton cycle is an advanced power system that offers benefits in terms of efficiency, volume, and flexibility. It may be utilized for waste heat recovery (WHR) in gas turbines. This study involved the design of a 5 MW S-CO2 recompression cycle specifically for the purpose of operational control. The dynamic models for the printed circuit heat exchangers, compressors, and turbines were developed. The stability and dynamic behavior of the components were validated. The suggested control strategies entail utilizing the cooling water controller to maintain the compressor inlet temperature above the critical temperature of CO2 (304.13 K). Additionally, the circulating mass flow rate is regulated to modify the output power, while the exhaust gas flow rate is controlled to ensure that the turbine inlet temperature remains within safe limits. The simulations compare the performance of PI controllers tuned using the SIMC rule and ADRC controllers tuned using the bandwidth method. The findings demonstrated that both controllers are capable of adjusting operating conditions and effectively suppressing fluctuations in the exhaust gas. The ADRC controllers exhibit a superior control performance, resulting in a 55% reduction in settling time under the load-tracking scenario.

Turbulence in Magnetic Reconnection Jets from Injection to Sub-Ion Scales.

We investigate turbulence in magnetic reconnection jets in the Earth's magnetotail using data from the Magnetospheric Multiscale spacecraft. We show that signatures of a limited inertial range are observed in many reconnection jets. The observed turbulence develops on the timescale of a few ion gyroperiods, resulting in intermittent multifractal energy cascade from the characteristic scale of the jet down to the ion scales. We show that at sub-ion scales, the fluctuations are close to monofractal and predominantly kinetic Alfvén waves. The observed energy transfer rate across the inertial range is ∼10^{8} J kg^{-1} s^{-1}, which is the largest reported for space plasmas so far.

Near-integrable dynamics of the Fermi-Pasta-Ulam-Tsingou problem.

It is well known that the classic Fermi-Pasta-Ulam-Tsingou (FPUT) study of a chain of nonlinear oscillators is closely related to a number of completely integrable systems, including the Toda lattice. Here, we present a method that captures the departure of nonintegrable FPUT dynamics from those of a nearby integrable Toda lattice. Using initial long-wave data, we find that the former depart rather sharply from the latter near the predicted shock time of an asymptotic partial differential equation approximation, at which point energy cascades into higher lattice modes. Our method provides an appropriate frame of reference for one to distinguish the short-term dynamics of the two systems, whose macroscopic trajectories diverge noticeably only on a much longer timescale, when the FPUT dynamics thermalize.

Spectral energy transfer analysis of a forced homogeneous isotropic turbulence using triple decomposition of velocity gradient tensor

Triple decomposition of the velocity gradient tensor is exploited to investigate the role of the simple-shear, normal-strain, and rigid-body rotation, in the turbulent kinetic energy transfer of a forced, high-Reynolds-number homogeneous isotropic turbulent flow. Splitting the total energy flux into three elementary partial fluxes shows that the rigid-body rotation has a secondary contribution to turbulent kinetic energy transfer, while within 80% of the total energy flux is produced through the simple-shear and normal-strain processes. Energy is dominantly extracted through the shearing process from larger motion scales and injected into smaller scales by the normal-straining process. In addition, each partial energy flux contains a strong conservative part and a weak non-conservative part. Besides the total kinetic energy transfer, all partial energy fluxes also show scale-invariance behaviour over the inertial subrange. Statistically, total energy flux shows a strong correlation with the simple-shear and normal-strain processes. However, the relatively poor correlation between these two dominant processes, implies a fairly low efficiency for turbulent kinetic energy transfer. These results can provide new insights into sub-grid scale turbulence modelling based on the physical mechanism of simple-shear and normal-strain processes.

Submesoscale Processes Fueled by the Kuroshio in the Tokara Strait

AbstractSubmesoscale processes are essential in balancing global oceanic energy cascades and stimulating air–sea communications. However, measurements of submesoscale turbulence are still not acquired easily. Based on long‐term (2003–2012) shipboard current observations, we obtained direct evidence of considerable submesoscale energization in the Kuroshio, implied by conspicuous spectral characteristics across multiple scales. The submesoscale motions (peak at ∼26 km) exhibited topographically dependent features and typical contrarotating vortex‐pair forms for island wakes, indicating footprints of upstream islands and seamounts. The vortex regimes and stability were analyzed using dimensionless parameters. Furthermore, the submesoscale motions radiated energy in the form of near‐inertial waves, which exhibited consistent shear structures, and spread widely across the entire strait. These results highlight the role of topography in modulating dynamics by continuously promoting energy cascades into submesoscales, which is expected to be universal for rough topography in the ocean.

Imprint of Chaos on the Ocean Energy Cycle from an Eddying North Atlantic Ensemble

Abstract We examine the ocean energy cycle where the eddies are defined about the ensemble mean of a partially air–sea coupled, eddy-rich ensemble simulation of the North Atlantic. The decomposition about the ensemble mean leads to a parameter-free definition of eddies, which is interpreted as the expression of oceanic chaos. Using the ensemble framework, we define the reservoirs of mean and eddy kinetic energy (MKE and EKE, respectively) and mean total dynamic enthalpy (MTDE). We opt for the usage of dynamic enthalpy (DE) as a proxy for potential energy due to its dynamically consistent relation to hydrostatic pressure in Boussinesq fluids and nonreliance on any reference stratification. The curious result that emerges is that the potential energy reservoir cannot be decomposed into its mean and eddy components, and the eddy flux of DE can be absorbed into the EKE budget as pressure work. We find from the energy cycle that while baroclinic instability, associated with a positive vertical eddy buoyancy flux, tends to peak around February, EKE takes its maximum around September in the wind-driven gyre. Interestingly, the energy input from MKE to EKE, a process sometimes associated with barotropic processes, becomes larger than the vertical eddy buoyancy flux during the summer and autumn. Our results question the common notion that the inverse energy cascade of wintertime EKE energized by baroclinic instability within the mixed layer is solely responsible for the summer-to-autumn peak in EKE and suggest that both the eddy transport of DE and transfer of energy from MKE to EKE contribute to the seasonal EKE maxima. Significance Statement The Earth system, including the ocean, is chaotic. Namely, the state to be realized is highly sensitive to minute perturbations, a phenomenon commonly known as the “butterfly effect.” Here, we run a sweep of ocean simulations that allow us to disentangle the oceanic expression of chaos from the oceanic response to the atmosphere. We investigate the energy pathways between the two in a physically consistent manner in the North Atlantic region. Our approach can be extended to robustly examine the temporal change of oceanic energy and heat distribution under a warming climate.

Large-scale self-organization in dry turbulent atmospheres.

How turbulent convective fluctuations organize to form larger-scale structures in planetary atmospheres remains a question that eludes quantitative answers. The assumption that this process is the result of an inverse cascade was suggested half a century ago in two-dimensional fluids, but its applicability to atmospheric and oceanic flows remains heavily debated, hampering our understanding of the energy balance in planetary systems. We show using direct numerical simulations with spatial resolutions of 122882 × 384 points that rotating and stratified flows can support a bidirectional cascade of energy, in three dimensions, with a ratio of Rossby to Froude numbers comparable to that of Earth's atmosphere. Our results establish that, in dry atmospheres, spontaneous order can arise through an inverse cascade to the largest spatial scales.