Turbulent Energy Spectrum Research Articles

Scaling Transition of Active Turbulence from Two to Three Dimensions.

Turbulent flows are observed in low-Reynolds active fluids, which display similar phenomenology to the classical inertial turbulence but are of a different nature. Understanding the dependence of this new type of turbulence on dimensionality is a fundamental challenge in non-equilibrium physics. Real-space structures and kinetic energy spectra of bacterial turbulence are experimentally measured from two to three dimensions. The turbulence shows three regimes separated by two critical confinement heights, resulting from the competition of bacterial length, vortex size and confinement height. Meanwhile, the kinetic energy spectra display distinct universal scaling laws in quasi-2D and 3D regimes, independent of bacterial activity, length, and confinement height, whereas scaling exponents transition in two steps around the critical heights. The scaling behaviors are well captured by the hydrodynamic model wedevelop, which employs image systems to represent the effects of confining boundaries. The study suggests a framework for investigating the effect of dimensionality on non-equilibrium self-organizedsystems.

Response of turbulent energy spectrum and flow structures when vortical motion of a certain scale is suppressed by artificial forcing

Response of turbulent energy spectrum and flow structures when vortical motion of a certain scale is suppressed by artificial forcing

Numerical simulations of particle concentration and size effects in a slurry bubble column with a CFD‐PBM coupled model

AbstractThe combined effects of particle concentration (0–20 vol%) and particle size on the hydrodynamics of a slurry bubble column were studied by computational fluid dynamics coupled with population balance model (CFD‐PBM) in a wide range of superficial gas velocity (0.02–0.20 m/s). This CFD‐PBM coupled model included multiple effects of particles, namely increased slurry viscosity, reduced drag force, promoted bubble coalescence, and attenuated liquid turbulence, among which turbulence attenuation was crucial. To quantify liquid turbulence attenuation, a mechanistic model was established by equating the energy dissipated by particle motion to the attenuated liquid turbulence energy calculated from turbulent energy spectrum. The turbulent energy dissipation rate was reduced by over 60% at a solid concentration of 20%. Using this model, the effects of particle concentration and particle size on overall gas holdup were well predicted. This model advanced our understanding of how particles affect the gas holdup in a slurry bubble column.

An artificial neural network model for recovering small-scale velocity in large-eddy simulation of isotropic turbulent flows

Small-scale motions in turbulent flows play a significant role in various small-scale processes, such as particle relative dispersion and collision, bubble or droplet deformation, and orientation dynamics of non-sphere particles. Recovering the small-scale flows that cannot be resolved in large eddy simulation (LES) is of great importance for such processes sensitive to the small-scale motions in turbulent flows. This study proposes a subgrid-scale model for recovering the small-scale turbulent velocity field based on the artificial neural network (ANN). The governing equations of small-scale turbulent velocity are linearized, and the pressure gradient and the nonlinear convection term are modeled with the aid of the ANN. Direct numerical simulation (DNS) and filtered direct numerical simulation (FDNS) provide the data required for training and validating the ANN. The large-scale velocity and velocity gradient tensor are selected as inputs for the ANN model. The linearized governing equations of small-scale turbulent velocity are numerically solved by coupling the large-scale flow field information. The results indicate that the model established by the ANN can accurately recover the small-scale velocity lost in FDNS due to filtering operation. With the ANN model, the flow fields at different Reynolds numbers agree well with the DNS results regarding velocity field statistics, flow field structures, turbulent energy spectra, and two-point, two-time Lagrangian correlation functions. This study demonstrates that the proposed ANN model can be applied to recovering the small-scale velocity field in the LES of isotropic turbulent flows at different Reynolds numbers.

Evaluation of seedless wavelet-based optical flow velocimetry for schlieren images

In harsh flow environments, traditional particle-based velocimetry methods face challenges. This study explores the use of seedless schlieren images for velocimetry through a novel algorithm, namely, wavelet-based optical flow velocimetry (wOFV). Various data term constraints for wOFV were examined. It is found that the data term derived from the integrated continuity equation (ICE) outperformed the conventional displaced frame difference constraint and the schlieren-tailored constraints (SE and SSE). Evaluation based on the root mean square error (RMSE) and turbulence energy spectrum (TES) reveals that the choice of wavelet becomes insignificant for the optimal estimated velocity field when the wavelet support length is sufficiently long. In addition, the implementation of a proper truncation in wOFV shows little dependence of the RMSE on the weighting coefficient, therefore alleviating the uncertainty associated with selecting an appropriate weighting coefficient. It is found that the retrieved flow field from schlieren images approximates a down-sampled result based on available structural scales in images. Considering the prevalence of under-resolved velocity field in practical applications, schlieren-based wOFV offers a reasonable alternative to particle-based velocimetry, particularly in harsh flow environments.

Sign-Indefinite Invariants Shape Turbulent Cascades.

We highlight a noncanonical yet natural choice of variables for an efficient derivation of a kinetic equation for the energy density in nonisotropic systems, including internal gravity waves on a vertical plane, inertial, and Rossby waves. The existence of a second quadratic invariant simplifies the kinetic equation and leads to extra conservation laws for resonant interactions. We analytically determine the scaling of the radial turbulent energy spectrum. Our findings suggest the existence of an inverse energy cascade of internal gravity waves, from small to large scales, in practically relevant scenarios.

Numerical study of separation flows in a U-duct using DDES method

Separation flow in a curved duct is a common phenomenon in engineering applications, and it highly contributes to the performance of fluid machinery. Accurate prediction of curved duct flows using the computational fluid dynamics method remains a challenge due to the limitations of turbulence modeling. Hence, the high-fidelity method of the delayed detached eddy simulation (DDES) approach is employed to simulate the U-duct flow with a Reynolds number of 105. The DDES results are compared with experimental data from the study by Monson et al. (1990) and analyzed in detail. The Q-criterion is defined to analyze the vortex structures and study the mechanism in the flow separation region. Discussions are made on turbulence characteristics, including turbulence energy spectra, helicity density, and turbulence anisotropy in the U-duct flow. Results indicate that the regions near the wall and within flow separation are highly anisotropic. The turbulence near the wall region is in a two-dimensional state, and the turbulence within the flow separation region is in a “rod-like” state.

What is the Role of Gravity, Turbulence and Magnetic Fields in High-mass Star Formation Clouds?

To explore the potential role of gravity, turbulence and magnetic fields in high-mass star formation in molecular clouds, this study revisits the velocity dispersion–size (σ–L) and density–size (ρ–L) scalings and the associated turbulent energy spectrum using an extensive data sample. The sample includes various hierarchical density structures in high-mass star formation clouds, across scales of 0.01–100 pc. We observe σ ∝ L 0.26 and ρ ∝ L −1.54 scalings, converging toward a virial equilibrium state. A nearly flat virial parameter–mass (α vir−M) distribution is seen across all density scales, with α vir values centered around unity, suggesting a global equilibrium maintained by the interplay between gravity and turbulence across multiple scales. Our turbulent energy spectrum (E(k)) analysis, based on the σ–L and ρ–L scalings, yields a characteristic E(k) ∝ k −1.52. These findings indicate the potential significance of gravity, turbulence, and possibly magnetic fields in regulating dynamics of molecular clouds and high-mass star formation therein.

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.

Joint-mode diffusion analysis of discontinuous Galerkin methods: Towards superior dissipation estimates for nonlinear problems and implicit LES

We present a new linear eigensolution analysis technique that provides superior estimates of dissipation distribution in wavenumber space for the discontinuous Galerkin (DG) method. The technique builds upon traditional dispersion-diffusion analyses that have been applied to spectral/hp element methods, but in particular is an improvement upon the non-modal eigenanalysis approach proposed by Fernandez et al. in [1]. The present technique takes into account the indirect effects that dispersion may have on dissipation, as recently discussed by Moura et al. in [2], in order to better represent dissipation itself. Also, a concept often used with dynamic mode decomposition (DMD) techniques is invoked to weight the relative contribution of the multiple diffusion curves that stem from temporal eigenanalysis. This allows for obtaining a single dissipation profile in wavenumber space, so that the proposed technique is named joint-mode analysis. Although the non-modal approach also provides a single diffusion curve, the joint-mode dissipation curve is shown to correlate significantly better with the energy spectrum of Burgers' turbulence at large and intermediate scales, which is particularly relevant for implicit large-eddy simulation (LES). The proposed technique is readily extensible to other spectral/hp element methods.

Analysis of coupled energy and helicity spectra in stratified turbulence: Theory and balloon measurements

Analysis of coupled energy and helicity spectra in stratified turbulence: Theory and balloon measurements

Gust-induced flow perturbations on circular cylinder: Investigating the effects and characteristics

This paper delves into the fascinating realm of gust flow past a circular cylinder, unveiling the dynamic interactions and key features that influence the system's behavior. In this study, ANSYS Fluent is used to perform numerical simulations based on finite volume method. The aim is to investigate the temporal evolution of the aerodynamic forces at different Reynolds numbers (Re) at different gust frequencies. In this study, gusts are generated by harmonically changing the direction of the inlet velocity at different frequencies. Unlike uniform flow, vortices are shed simultaneously from the upper and lower surface of cylinder, sometimes with equal strengths and sometimes with varying strengths depending upon the gust frequency. Cl, Cd and Strouhal number values increase with gust frequency corresponding to any Reynolds number. However, the values remain less than no-gust case under similar flow situations. Fast Fourier transform analysis has been used to examine the energy spectrum and kinetic energy contours of vorticity and turbulence to gain further insight into fluid physics. The results of this study provide valuable insight into the physics of gust flow and circular cylinders.

Dissipation closure of fluid turbulence from energy spectrum

Direct method of the turbulence closure consists of obtaining expression for small-scale velocity and inserting this expression in the Reynolds tensor and in the transport equations for turbulent energy, dissipation, eddy and so on, implementing some averaging. Previously such approaches were used for Burgers-Lundgren vortex. Here we used ‘naïve’ perturbation method for the transport equation for small-scale velocity incompressible fluid - the equation suggested by one of the authors earlier. It is assumed that the energy spectrum of turbulence is given.

Amplification and Saturation of Turbulent Magnetic Fields in Collapsing Primordial Gas Clouds

Recent numerical studies suggest that magnetic fields play an important role in primordial star formation in the early Universe. However, the detailed evolution of the magnetic field in the collapse phase still has uncertainties because of the complicated physics associated with turbulence in a collapsing magnetized system. Here, we perform a suite of numerical MHD simulations that follow the collapse of magnetized, turbulent primordial gas clouds to investigate the evolution of the magnetic field associated with the turbulence, assuming a polytropic equation of state with exponent γ eff and with various numerical resolutions. In addition, we generalize the analytic theory of magnetic field growth/saturation so that it can deal with various exponents γ eff and turbulence energy spectra. We find that the numerical results are well reproduced by the theory for various γ eff through the collapse phase during the formation of the first stars. The magnetic field is eventually amplified by a factor of 1012–1015 due to kinematic and nonlinear turbulent dynamo effects and reaches 3%–100% of the equipartition level, depending on γ eff. We also find that the transition between the kinematic and nonlinear stages can be analytically estimated. These results indicate that the strong magnetic field accompanied by supersonic turbulence is a general property and suggest that it can play a crucial role in the formation of the first stars.

Space–time structure of weak magnetohydrodynamic turbulence

The two-time energy spectrum of weak magnetohydrodynamic turbulence is found by applying a wave-turbulence closure to the cumulant hierarchy constructed from the dynamical equations. Solutions are facilitated via asymptotic expansions in terms of the small parameter $\varepsilon$ , describing the ratio of time scales corresponding to Alfvénic propagation and nonlinear interactions between counter-propagating Alfvén waves. The strength of nonlinearity at a given spatial scale is further quantified by an integration over all possible delta-correlated modes compliant in a given set of three-wave interactions that are associated with energy flux through the said scale. The wave-turbulence closure for the two-time spectrum uncovers a secularity occurring on a time scale of order $\varepsilon ^{-2}$ , and the asymptotic expansion for the spectrum is reordered in a manner comparable to the one-time case. It is shown that for the regime of stationary turbulence, the two-time energy spectrum exponentially decays on a lagged time scale $(\varepsilon ^2 \gamma _k^s)^{-1}$ in proportion to the strength of the associated three-wave interactions, characterized by nonlinear decorrelation frequency $\gamma _k^s$ . The scaling of the form $k_{\perp } v_0 \chi _0$ exhibited by this frequency is reminiscent of random sweeping by the outer scale with characteristic fluctuation velocity $v_0$ that is modified due to competition with Alfvénic propagation (characterized by $\chi _0$ ) at the said scale. A brief calculation of frequency broadening of the power spectrum due to nonlinear interactions is also presented.

Enhancing thermal mixing in turbulent bubbly flow by inhibiting bubble coalescence

The presence of bubbles in a turbulent flow changes the flow drastically and enhances the mixing. Adding salt to the bubbly aqueous flow changes the bubble coalescence properties as compared to regular demineralized water. Here we provide direct experimental evidence that also the turbulent thermal energy spectra are strongly changed. Experiments were performed in the Twente Mass and Heat Transfer water tunnel, in which we can measure the thermal spectra in bubbly turbulence in salty water. We find that the mean bubble diameter decreases with increasing concentration of salt (NaCl), due to the inhibition of bubble coalescence. With increasing salinity, the transition frequency from the classical −5/3 scaling of the thermal energy spectrum to the bubble induced −3 scaling shifts to higher frequencies, thus enhancing the overall thermal energy. We relate this frequency shift to the smaller size of the bubbles for the salty bubbly flow. Finally we measure the heat transport in the bubbly flow, and show how it varies with changing void fraction and salinity: Increases in both result into increases in the number of extreme events.

Physical consistency and invariance in machine learning of turbulent signals

This paper concerns an investigation of the invariance and consistency of deep learning of turbulent pressure fluctuations. The long-short-memory model is employed to predict wall pressure fluctuations across physical regimes featuring turbulence, shock–boundary layer interaction, and separation. The model's sensitivity to the data inputs is examined using different input data sets. Training the deep learning model based on the raw signals from different flow regions leads to large inaccuracies. It is shown that the data must be appropriately pre-processed before training for the deep learning model predictions to become consistent. Removing the mean and using the normalized fluctuating component of the signal, the deep learning predictions not only greatly improved in accuracy but, most importantly, converged and became consistent, provided that the signal sparsity remains within the inertial sub-range of the turbulence energy spectrum cascade. The power spectra of the surface pressure fluctuations reveal that the model provides high accuracy up to a certain frequency for the fully turbulent flow. The deep learning model's consistency is evidenced by being transferable across the various probe positions on the wall despite the significant differences in the turbulent flow properties in the training data set, i.e., signals obtained before, after, and inside the shock–boundary layer interaction regions. The model's prediction consistency and invariance to the turbulent signal training location(s) are promising for applying deep learning models to various turbulent flows.

Energy Spectra of Nonlocal Internal Gravity Wave Turbulence.

Starting from the classical formulation of the weak turbulence theory in a density stratified fluid, we derive a simplified version of the kinetic equation of internal gravity wave turbulence. This equation allows us to uncover scaling laws for the spatial and temporal energy spectra of internal wave turbulence which are consistent with typical scaling exponents observed in the oceans. The keystone of our description is the assumption that the energy transfers are dominated by a class of nonlocal resonant interactions, known as the "induced diffusion" triads, which conserve the ratio between the wave frequency and vertical wave number. Our analysis remarkably shows that the internal wave turbulence cascade is associated to an apparent constant flux of wave action.

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.

Can IR Images of the Water Surface Be Used to Quantify the Energy Spectrum and the Turbulent Kinetic Energy Dissipation Rate?

Near-surface oceanic turbulence plays an important role in the exchange of mass, momentum, and energy between the atmosphere and the ocean. The climate modifying the air-sea CO2 transfer rate varies linearly with the surface turbulent kinetic energy dissipation rate to the 1/4 power in a range of systems with different types of forcing, such as coastal oceans, river estuaries, large tidal freshwater rivers, and oceans. In the first part of this paper, we present a numerical study of the near-surface turbulent kinetic energy spectra deduced from a direct numerical simulation (DNS) compared to turbulent kinetic energy spectra deduced from idealized infrared (IR) images. The DNS temperature fields served as a surrogate for IR images from which we have calculated the underlying kinetic energy spectra. Despite the near-surface flow region being highly anisotropic, we demonstrated that modeled isotropic and homogeneous turbulence spectra can serve as an approximation to observed near-surface spectra within the inertial and dissipation ranges. The second part of this paper validates our numerical observations in a laboratory experiment. In this experiment, we compared the turbulent kinetic energy spectra near the surface, as measured using a submerged shear sensor with the spectra derived from infrared images collected from above the surface. The energy dissipation measured by the shear sensor was found to be within 20% of the dissipation value derived from the IR images. Numerically and experimentally, we have demonstrated that IR-based and remote measurement techniques of the aquatic near surface offer a potentially accurate and non-invasive way to measure near-surface turbulence, which is needed by the community to improve models of oceanic air-sea heat, momentum, and gas fluxes.