Wavenumber-frequency spectra of the streamwise velocity component obtained from large-eddy simulations are presented. Following a recent paper [Wilczek et al., J. Fluid. Mech., 769:R1, 2015] we show that the main features, a Doppler shift and a Doppler broadening of frequencies, are captured by an advection model based on the Tennekes-Kraichnan random-sweeping hypothesis with additional mean flow. In this paper, we focus on the height-dependence of the spectra within the logarithmic layer of the flow. We furthermore benchmark an analytical model spectrum that takes the predictions of the random-sweeping model as a starting point and find good agreement with the LES data. We also quantify the influence of LES grid resolution on the wavenumber-frequency spectra.
Identification and extraction of vortical structures and of waves in a disorganised flow is a mayor challenge in the study of turbulence. We present a study of the spatio-temporal behavior of turbulent flows in the presence of different restitutive forces. We show how to compute and analyse the spatio-temporal spectrum from data stemming from numerical simulations and from laboratory experiments. Four cases are considered: homogeneous and isotropic turbulence, rotating turbulence, stratified turbulence, and water wave turbulence. For homogeneous and isotropic turbulence, the spectrum allows id…
We present evidence of Kelvin excitations in space-time resolved spectra of numerical simulations of quantum turbulence. Kelvin waves are transverse and circularly polarized waves that propagate along quantized vortices, for which the restitutive force is the tension of the vortex line, and which play an important role in theories of superfluid turbulence. We use the Gross-Pitaevskii equation to model quantum flows, letting an initial array of well-organized vortices develop into a turbulent bundle of intertwined vortex filaments. By achieving high spatial and temporal resolution we are able t…
Motivated by specific data and accuracy requirements for building numerical databases of turbulent flows, data compression using spatio-temporal sub-sampling and local re-simulation is proposed. Numerical re-simulation experiments for decaying isotropic turbulence based on sub-sampled data are undertaken. The results and error analyses are used to establish parameter choices for sufficiently accurate sub-sampling and sub-domain re-simulation.
Motivated by the need to characterize the spatio-temporal structure of turbulence in wall-bounded flows, we study wavenumber-frequency spectra of the streamwise velocity component based on large-eddy simulation (LES) data. The LES data are used to measure spectra as a function of the two wall-parallel wavenumbers and the frequency in the equilibrium (logarithmic) layer. We then reformulate one of the simplest models that is able to reproduce the observations: the random sweeping model with a Gaussian large-scale fluctuating velocity and with additional mean flow. Comparison with LES data shows…
In incompressible and periodic statistically stationary turbulence, exchanges of turbulent energy across scales and space are characterised by very intense and intermittent spatio-temporal fluctuations around zero of the time-derivative term, the spatial turbulent transport of fluctuating energy, and the pressure-velocity term. These fluctuations are correlated with each other and with the intense intermittent fluctuations of the interscale energy transfer rate. These correlations are caused by the sweeping effect, the link between non-linearity and non-locality, and also relate to geometrical…
For wall turbulence, moments of velocity fluctuations are known to be logarithmic functions of the height from the wall. This logarithmic scaling is due to the existence of a characteristic velocity and to the nonexistence of any characteristic height in the range of the scaling. By using mathematics of random variables, we obtain its necessary and sufficient conditions. They are compared with characteristics of a phenomenological model of eddies attached to the wall and also with those of the logarithmic scaling of the mean velocity.
We show the skills of a data-driven low-dimensional linear model in predicting the spatio-temporal evolution of turbulent Rayleigh-Bénard convection. The model is based on dynamic mode decomposition with delay-embedding, which provides a data-driven finite-dimensional approximation to the system's Koopman operator. The model is built using vector-valued observables from direct numerical simulations, and can provide accurate predictions. Similar high prediction skills are found for the Kuramoto-Sivashinsky equation in the strongly-chaotic regimes.
Turbulence is an ubiquitous phenomenon in natural and industrial flows. Since the celebrated work of Kolmogorov in 1941, understanding the statistical properties of fully developed turbulence has remained a major quest. In particular, deriving the properties of turbulent flows from a mesoscopic description, that is from Navier-Stokes equation, has eluded most theoretical attempts. Here, we provide a theoretical prediction for the {\it space and time} dependent velocity-velocity correlation function of homogeneous and isotropic turbulence from the field theory associated to Navier-Stokes equati…
We examine the steady state of turbulent flows in thin layers using direct numerical simulations. It is shown that when the layer thickness is smaller than a critical height, an inverse cascade arises which leads to the formation of a steady state condensate where most of the energy is concentrated in the largest scale of the system. For layers of thickness smaller than a second critical height, the flow at steady state becomes exactly two-dimensional. The amplitude of the condensate is studied as a function of layer thickness and Reynolds number. Bi-stability and intermittent bursts are found…
We consider linear feedback flow control of the largest scales in an incompressible turbulent channel flow at a friction Reynolds number of Re$_τ$ = 2000. A linear model is formed by linearizing the Navier-Stokes equations about the turbulent mean and augmenting it with an eddy viscosity. Velocity perturbations are then generated by stochastically forcing the linear operator. The objective is to reduce the kinetic energy of these velocity perturbations at the largest scales using body forces. It is shown that a control set-up with a well-placed array of sensors and actuators performs comparabl…
Turbulent flows within and over sparse canopies are investigated using direct numerical simulations. We focus on the effect of the canopy on the background turbulence, the part of the flow that remains once the element-induced flow is filtered out. In channel flows, the distribution of the total stress is linear with height. Over smooth walls, the total stress is only the `fluid stress' $τ_f$, the sum of the viscous and the Reynolds shear stresses. In canopies, in turn, there is an additional contribution from the canopy drag, which can dominate within. We find that, for sparse canopies, the r…
We consider general infinite-dimensional dynamical systems with the Galilean and spatiotemporal scaling symmetry groups. Introducing the equivalence relation with respect to temporal scalings and Galilean transformations, we define a representative set containing a single element within each equivalence class. Temporal scalings and Galilean transformations do not commute with the evolution operator (flow) and, hence, the equivalence relation is not invariant. Despite of that, we prove that a normalized flow with an invariant probability measure can be introduced on the representative set, such…
Using direct numerical simulations (DNSs), the interaction between linear waves and turbulence under the compressible magnetohydrodynamic (CMHD) approach was studied. A set of DNSs in three dimensions for a spatial resolution of $128^3$ and $256^3$ were performed. A parametric study was carried out varying the sonic Mach number, the mean magnetic field and the compressibility amplitude of the forcing. Spatio-temporal spectra of the magnetic energy were built and analyzed, allowing for direct identification of all wave modes in a CMHD turbulent system and quantification of the amount of energy …
Bayesian Conditional Diffusion Models for Versatile Spatiotemporal Turbulence Generation
University of Notre Dame（圣母大学）；Harvard University（哈佛大学）；Tufts University（塔夫茨大学）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；The Ohio State University（俄亥俄州立大学）
Using a reduced model focusing on the in-plane dependence of plane Couette flow, it is shown that the turbulent-to-laminar relaxation process can be understood as a nucleation problem similar to that occurring at a thermodynamic first-order phase transition. The approach, apt to deal with the large extension of the system considered, challenges the current interpretation in terms of chaotic transients typical of temporal chaos. The study of the distribution of the sizes of laminar domains embedded in turbulent flow proves that an abrupt transition from sustained spatiotemporal chaos to laminar…
Temporal and spatio-temporal (turbulence) distributed chaos in Bénard-Marangoni and Rayleigh-Bénard convection have been studied using results of laboratory experiments and direct numerical simulations in the terms of effective chaotic diffusivity (viscosity) and action. It is shown that for the both cases decaying part of the power spectra has stretched exponential form - for temporal spectrum $E(f) \propto \exp-(f/f_β)^{1/2}$ and for spatial spectrum $E(k) \propto \exp-(k/k_β)^{2/3}$, where the $f_β$ and $k_β$ represent low-frequency (large-scale) coherent oscillations.
We provide a detailed comparison of the two-dimensional, temporal and the spatiotemporal linearized analyses of the viscoelastic free shear flows in the limit of low to moderate Reynolds number and Elasticity number obeying four different types of stress-strain constitutive equations: Oldroyd-B, Upper Convected Maxwell, Johnson-Segalman (JS) and linear Phan-Thien Tanner (PTT). The resulting fourth-order Orr-Sommerfeld Equation is transformed into a set of six auxiliary equations that are numerically integrated via the Compound Matrix Method. The temporal stability analysis suggest (a) elastic …
We report experiments on the full space and time resolved statistics of capillary wave turbulence at the air-water interface. The three-dimensional shape of the free interface is measured as a function of time by using the optical method of Diffusing Light Photography associated with a fast camera. Linear and nonlinear dispersion relations are extracted from the spatio-temporal power spectrum of wave amplitude. When wave turbulence regime is reached, we observe power-law spectra both in frequency and in wave number, whose exponents are found in agreement with the predictions of capillary wave …
We report the observation of the cross-over between gravity and capillary wave turbulence on the surface of mercury. The probability density functions of the turbulent wave height are found to be asymmetric and thus non Gaussian. The surface wave height displays power-law spectra in both regimes. In the capillary region, the exponent is in fair agreement with weak turbulence theory. In the gravity region, it depends on the forcing parameters. This can be related to the finite size of the container. In addition, the scaling of those spectra with the mean energy flux is found in disagreement wit…
Within wall turbulence, there is a sublayer where the mean velocity and the variance of velocity fluctuations vary logarithmically with the height from the wall. This logarithmic scaling is also known for the mean concentration of a passive scalar. By using heat as such a scalar in a laboratory experiment of a turbulent boundary layer, the existence of the logarithmic scaling is shown here for the variance of fluctuations of the scalar concentration. It is reproduced by a model of energy-containing eddies that are attached to the wall.
Synchronization of turbulence in channel flow is investigated using continuous data assimilation. The flow is unknown within a region of the channel. Beyond this region the velocity field is provided, and is directly prescribed in the simulation, while the pressure is unknown throughout the entire domain. Synchronization takes place when the simulation recovers the full true state of the flow, or in other words when the missing region is accurately re-established, spontaneously. Successful synchronization depends on the orientation, location and size of the missing layer. For friction Reynolds…
Wall-bounded turbulent flows can be challenging to measure within experiments due to the breadth of spatial and temporal scales inherent in such flows. Instrumentation capable of obtaining time-resolved data (e.g., Hot-Wire Anemometers) tends to be restricted to spatially-localized point measurements; likewise, instrumentation capable of achieving spatially-resolved field measurements (e.g., Particle Image Velocimetry) tends to lack the sampling rates needed to attain time-resolution in many such flows. In this study, we propose to fuse measurements from multi-rate and multi-fidelity sensors w…
Elasto-inertial turbulence is a new state of turbulence found in flows with polymer additives . The dynamics of turbulence generated and controlled by such additives is investigated from the perspective of the coupling between polymer dynamics and flow structures. Direct numerical simulations of channel flow with Reynolds numbers ranging from 1000 to 6000 (based on the bulk and the channel height) are used to study the formation and dynamics of elastic instabilities and their effects on the flow. The resulting mechanism of interactions between polymer dynamics and the flow helps resolve a long…
While in classical turbulence helicity depletes nonlinearity and can alter the evolution of turbulent flows, in quantum turbulence its role is not fully understood. We present numerical simulations of the free decay of a helical quantum turbulent flow using the Gross-Pitaevskii equation at high spatial resolution. The evolution has remarkable similarities with classical flows, which go as far as displaying a dual transfer of incompressible kinetic energy and helicity to small scales. Spatio-temporal analysis indicates that both quantities are dissipated at small scales through non-linear excit…
The present study investigates the behaviour of the far-field sound radiated by low Mach number tip clearance flow induced by placing a stationary cambered airfoil adjacent to a stationary wall. The tip clearance heights ranged from 14% to 30% of the incoming, undisturbed boundary layer thickness and the clearance heights based Reynolds numbers were between 2,600 and 16,000. The far-field sound measured using a microphone array was beamformed to reveal the dominant noise sources and how they behave when the flow Mach number, angle of attack and the clearance height were varied. The near-field …
Pipe flow and many other shear flows show a transition to turbulence at flow rates for which the laminar profile is stable against infinitesimal perturbations. In this brief review the recent progress in the understanding of this transition will be summarized, with a focus on the linear and nonlinear states that drive the transitions, the extended and localized patterns that appear, and on the spatio-temporal dynamics and their relation to directed percolation.
We present a spatio-temporal analysis of a statistically stationary rotating turbulence experiment, aiming to extract a signature of inertial waves, and to determine the scales and frequencies at which they can be detected. The analysis uses two-point spatial correlations of the temporal Fourier transform of velocity fields obtained from time-resolved stereoscopic particle image velocimetry measurements in the rotating frame. We quantify the degree of anisotropy of turbulence as a function of frequency and spatial scale. We show that this space-time-dependent anisotropy is well described by th…
While studies of active nematics in two dimensions have shed light on various aspects of the flow regimes and topology of active matter, three-dimensional properties of topological defects and chaotic flows remain unexplored. By confining a film of active nematics between two parallel plates, we use continuum simulations and analytical arguments to demonstrate that the crossover from quasi-2D to 3D chaotic flows is controlled by the morphology of the disclination lines. For small plate separations, the active nematic behaves as a quasi-2D material, with straight topological disclination lines …
We study the existence and property of Fast magnetosonic modes in 3D compressible MHD turbulence by carrying out a number of simulations with compressible and incompressible driving conditions. We use two approaches to determine the presence of Fast modes: mode decomposition based on spatial variations only and spatio-temporal 4D-FFT analysis of all fluctuations. The latter method enables us to quantify fluctuations that satisfy the dispersion relation of Fast modes with finite frequency. Overall, we find that the fraction of Fast modes identified via spatio-temporal 4D FFT approach in total f…
In active nematic liquid crystals activity is able to drive chaotic spatiotemporal flows referred to as active turbulence. Active turbulence has been characterized through theoretical and experimental work as a low Reynolds number phenomenon. We show that, in two-dimensions, the active forcing alone is able to trigger hydrodynamic turbulence leading to the coexistence of active and inertial turbulence. This type of flows develops for sufficiently active and extensile flow-aligning nematics. We observe that the combined effect of an extensile nematic and large values of the flow-aligning parame…
Transitional pipe flow is modeled as a one-dimensional excitable and bistable medium. Models are presented in two variables, turbulence intensity and mean shear, that evolve according to established properties of transitional turbulence. A continuous model captures the essence of the puff-slug transition as a change from excitability to bistability. A discrete model, that additionally incorporates turbulence locally as a chaotic repeller, reproduces almost all large-scale features of transitional pipe flow. In particular it captures metastable localized puffs, puff splitting, slugs, a continuo…
Laboratory experiments point out the existence of patterns made of alternately laminar and turbulent oblique bands in plane Couette flow in its way to/from turbulence as the Reynolds number R is varied. Many previous theoretical and numerical works on the problem have considered small aspect-ratio systems subjected to periodic boundary conditions, while experiments correspond to the opposite limit of large aspect-ratio. Here, by means of fully resolved direct numerical simulations of the Navier-Stokes equations at decreasing R, we scrutinize the transition from temporal to spatiotemporal behav…
We use a data-driven approach to model a three-dimensional turbulent flow using cutting-edge Deep Learning techniques. The deep learning framework incorporates physical constraints on the flow, such as preserving incompressibility and global statistical invariants of velocity gradient tensor. The accuracy of the model is assessed using statistical and physics-based metrics. The data set comes from Direct Numerical Simulation of an incompressible, statistically stationary, isotropic turbulent flow in a cubic box. Since the size of the dataset is memory intensive, we first generate a low-dimensi…
A convolutional encoder-decoder-based transformer model is proposed for autoregressively training on spatio-temporal data of turbulent flows. The prediction of future fluid flow fields is based on the previously predicted fluid flow field to ensure long-term predictions without diverging. A combination of convolutional neural networks and transformer architecture is utilized to handle both the spatial and temporal dimensions of the data. To assess the performance of the model, a priori assessments are conducted, and significant agreements are found with the ground truth data. The a posteriori …
Upon decreasing the Reynolds number, plane Couette flow first forms alternately turbulent and laminar oblique bands out of featureless turbulence below some upper threshold R_t. These bands exist down to a global stability threshold R_g below which laminar flow ultimately prevails. We study the fragmentation and decay of these bands in systems that are extended enough for several bands to exist. We use direct numerical simulations appropriately tailored to deal with such large systems during long enough durations. We point out a two-stage process involving the rupture of a band and next its sl…
The decay of turbulent and laminar oblique bands in the lower transitional range of plane Couette flow is studied by means of direct numerical simulations of the Navier--Stokes equations. We consider systems that are extended enough for several bands to exist, thanks to mild wall-normal under-resolution considered as a consistent and well-validated modelling strategy. We point out a two-stage process involving the rupture of a band followed by a slow regression of the fragments left. Previous approaches to turbulence decay in wall-bounded flows making use of the chaotic transient paradigm are …
Delft University of Technology（代尔夫特理工大学）；Technical University of Munich（慕尼黑工业大学）；University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）；Imperial College London（伦敦帝国学院）
We present an Auto-Encoded Reservoir-Computing (AE-RC) approach to learn the dynamics of a 2D turbulent flow. The AE-RC consists of an Autoencoder, which discovers an efficient manifold representation of the flow state, and an Echo State Network, which learns the time evolution of the flow in the manifold. The AE-RC is able to both learn the time-accurate dynamics of the flow and predict its first-order statistical moments. The AE-RC approach opens up new possibilities for the spatio-temporal prediction of turbulence with machine learning.
The understanding of turbulent flows is one of the biggest current challenges in physics, as no first-principles theory exists to explain their observed spatio-temporal intermittency. Turbulent flows may be regarded as an intricate collection of mutually-interacting vortices. This picture becomes accurate in quantum turbulence, which is built on tangles of discrete vortex filaments. Here, we study the statistics of velocity circulation in quantum and classical turbulence. We show that, in quantum flows, Kolmogorov turbulence emerges from the correlation of vortex orientations, while deviations…
The main part of this contribution to the special issue of EJM-B/Fluids dedicated to Patrick Huerre outlines the problem of the subcritical transition to turbulence in wall-bounded flows in its historical perspective with emphasis on plane Couette flow, the flow generated between counter-translating parallel planes. Subcritical here means discontinuous and direct, with strong hysteresis. This is due to the existence of nontrivial flow regimes between the global stability threshold Re_g, the upper bound for unconditional return to the base flow, and the linear instability threshold Re_c charact…
Meso-scale turbulence is an innate phenomenon, distinct from inertial turbulence, that spontaneously occurs at zero-Reynolds number in fluidized biological systems. This spatio-temporal disordered flow radically changes nutrient and molecular transport in living fluids and can strongly affect the collective behaviour in prominent biological processes, including biofilm formation, morphogenesis and cancer invasion. Despite its crucial role in such physiological processes, understanding meso-scale turbulence and any relation to classical inertial turbulence remains obscure. Here, we show how the…
Three-dimensional turbulence is usually studied experimentally by using a spatially localized forcing at large scales (e.g. via rotating blades or oscillating grids), often in a deterministic way. Here, we report an original technique where the fluid is forced in volume, randomly in space and time, using small magnetic particles remotely driven. Such a forcing generates almost no mean flow and is closer to those of direct numerical simulations of isotropic homogeneous turbulence. We compute the energy spectra and structure functions using local and spatiotemporal flow velocity measurements. Th…
Hydrodynamics of Dense Active Fluids: Turbulence-Like States and the Role of Advected Activity
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心，塔塔基础研究所）；Department of Mechanical Engineering, Indian Institute of Technology Kanpur（坎普尔印度理工学院机械工程系）
Active fluids exhibit spontaneous flows with complex spatiotemporal structure, which have been observed in bacterial suspensions, sperm cells, cytoskeletal suspensions, self-propelled colloids, and cell tissues. Despite occurring in the absence of inertia, chaotic active flows are reminiscent of inertial turbulence, and hence they are known as active turbulence. Here, we survey the field, providing a unified perspective over different classes of active turbulence. To this end, we divide our review in sections for systems with either polar or nematic order, and with or without momentum conserva…
