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 predictions, which are generated after a considerable number of simulation steps, exhibit predicted variances. The autoregressive training and prediction of a posteriori states are deemed crucial steps towards the development of more complex data-driven turbulence models and simulations. The highly nonlinear and chaotic dynamics of turbulent flows can be handled by the proposed model, and accurate predictions over long time horizons can be generated. Overall, the potential of using deep learning techniques to improve the accuracy and efficiency of turbulence modeling and simulation is demonstrated by this approach. The proposed model can be further optimized and extended to incorporate additional physics and boundary conditions, paving the way for more realistic simulations of complex fluid dynamics.
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…
Recurrent Neural Networks (RNNs) have become an integral part of modeling and forecasting frameworks in areas like natural language processing and high-dimensional dynamical systems such as turbulent fluid flows. To improve the accuracy of predictions, RNNs are trained using the Backpropagation Through Time (BPTT) method to minimize prediction loss. During testing, RNNs are often used in autoregressive scenarios where the output of the network is fed back into the input. However, this can lead to the exposure bias effect, as the network was trained to receive ground-truth data instead of its o…
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…
University of Cambridge（剑桥大学）；Delft University of Technology（代尔夫特理工大学）；Imperial College London（帝国理工学院）；The Alan Turing Institute（艾伦·图灵研究所）
The spatiotemporal dynamics of turbulent flows is chaotic and difficult to predict. This makes the design of accurate and stable reduced-order models challenging. The overarching objective of this paper is to propose a nonlinear decomposition of the turbulent state for a reduced-order representation of the dynamics. We divide the turbulent flow into a spatial problem and a temporal problem. First, we compute the latent space, which is the manifold onto which the turbulent dynamics live (i.e., it is a numerical approximation of the turbulent attractor). The latent space is found by a series of …
An extension of Proper Orthogonal Decomposition is applied to the wall layer of a turbulent channel flow (Re τ = 590), so that empirical eigenfunctions are defined in both space and time. Due to the statistical symmetries of the flow, the igenfunctions are associated with individual wavenumbers and frequencies. Self-similarity of the dominant eigenfunctions, consistent with wall-attached structures transferring energy into the core region, is established. The most energetic modes are characterized by a fundamental time scale in the range 200-300 viscous wall units. The full spatio-temporal dec…
Streaming Dynamic Mode Decomposition (sDMD) (Hemati et al., Phys. Fluids 26(2014)) is a low-storage version of Dynamic Mode Decomposition (DMD) (Schmid, J. Fluid Mech. 656 (2010)), a data-driven method to extract spatio-temporal flow patterns. Streaming DMD avoids storing the entire data sequence in memory by approximating the dynamic modes through incremental updates with new available data. In this paper, we use sDMD to identify and extract dominant spatio-temporal structures of different turbulent flows, requiring the analysis of large datasets. First, the efficiency and accuracy of sDMD ar…
The spatio-temporal statistical structure of the turbulent dissipation field and its stochastic representation as a Gaussian Multiplicative Chaos
Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique（里昂大学，巴黎高等师范学院里昂分校，克洛德·贝尔纳大学，法国国家科学研究中心，物理实验室）；CNRS, ICJ UMR5208, Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, Université Jean Monnet（法国国家科学研究中心，里昂中央理工学院，里昂国立应用科学学院，里昂克洛德·贝尔纳大学，让·莫内大学）
Numerical simulations of a model of plane Couette flow focusing on its in-plane spatio-temporal properties are used to study the dynamics of turbulent spots.
We present a new turbulent data reconstruction method with supervised machine learning techniques inspired by super resolution and inbetweening, which can recover high-resolution turbulent flows from grossly coarse flow data in space and time. For the present machine learning based data reconstruction, we use the downsampled skip-connection/multi-scale model based on a convolutional neural network to incorporate the multi-scale nature of fluid flows into its network structure. As an initial example, the model is applied to a two-dimensional cylinder wake at $Re_D$ = 100. The reconstructed flow…
The spatio-temporal dynamics of separation bubbles induced to form in a fully-developed turbulent boundary layer (with Reynolds number based on momentum thickness of the boundary layer of 490) over a flat plate are studied via direct numerical simulations. Two different separation bubbles are examined: one induced by a suction-blowing velocity profile on the top boundary and the other, by a suction-only velocity profile. The latter condition allows reattachment to occur without an externally imposed favourable pressure gradient and leads to a separation bubble more representative of those occu…
A Physics-Informed Spatiotemporal Deep Learning Framework for Turbulent Systems
Department of Applied and Computational Mathematics and Statistics, University of University of Notre Dame（应用与计算数学与统计学系，诺特大学）；Department of Environmental and Civil Engineering, University of Notre Dame（环境与土木工程系，诺特大学）
We present a reduced model for the transition to turbulence in shear flow that is simple enough to admit a thorough numerical investigation while allowing spatio-temporal dynamics that are substantially more complex than those allowed in previous modal truncations. Our model allows a comparison of the dynamics resulting from initial perturbations that are localised in the spanwise direction with those resulting from sinusoidal perturbations. For spanwise-localised initial conditions the subcritical transition to a `turbulent' state (i) takes place more abruptly, with a boundary between laminar…
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…
We investigate the spatio-temporal quantity of coherence for turbulent velocity fluctuations at spatial distances of the order or larger than the integral length scale $l_{0}$. Using controlled laboratory experiments, an exponential decay as a function of distance is observed with a decay rate which depends on the flow properties. The same law is observed in two different flows indicating that it can be a generic property of turbulent flows.
Estimation of the initial state of turbulent channel flow from limited data is investigated using an adjoint-variational approach. The data are generated from a reference direct numerical simulation (DNS) which is sub-sampled at different spatiotemporal resolutions. When the velocity data are at 1/4096 the spatiotemporal resolution of DNS, the correlation coefficient between the true and adjoint-variational estimated state exceeds 99 percent. The robustness of the algorithm to observation noise is demonstrated. In addition, the impact of the spatiotemporal density of the data on estimation qua…
Super-resolution is an innovative technique that upscales the resolution of an image or a video and thus enables us to reconstruct high-fidelity images from low-resolution data. This study performs super-resolution analysis on turbulent flow fields spatially and temporally using various state-of-the-art machine learning techniques like ESPCN, ESRGAN and TecoGAN to reconstruct high-resolution flow fields from low-resolution flow field data, especially keeping in mind the need for low resource consumption and rapid results production/verification. The dataset used for this study is extracted fro…
Turbulent-laminar patterns are ubiquitous near transition in wall-bounded shear flows. Despite recent progress in describing their dynamics in analogy to non-equilibrium phase transitions, there is no theory explaining their emergence. Dynamical-system approaches suggest that invariant solutions to the Navier--Stokes equations, such as traveling waves and relative periodic orbits in pipe flow, act as building blocks of the disordered dynamics. While recent studies have shown how transient chaos arises from such solutions, the ensuing dynamics lacks the strong fluctuations in size, shape and sp…
The outward propagation of asymmetries introduced to originally axisymmetric turbulent flows is investigated, where 3D Batchelor vortices at high Reynolds numbers and with arbitrary swirl numbers are chosen as test cases. It is well known that disturbances (asymmetries) added to a 2D axisymmetric flow propagate radially outward, in order to re-axisymmetrize the vortex, but cease to travel at a critical distance, known as the stagnation radius. We utilize direct numerical simulations (DNS) and an inviscid model developed by linearizing the momentum and vorticity transport equations around the b…
We introduce a new variational method for finding periodic orbits of flows and spatio-temporally periodic solutions of classical field theories, a generalization of the Newton method to a flow in the space of loops. The feasibility of the method is demonstrated by its application to several dynamical systems, including the Kuramoto-Sivashinsky system.
The Sun provides an excellent natural laboratory for nonlinear phenomena. We use motions of magnetic bright points on the solar surface, at the smallest scales yet observed, to study the small scale dynamics of the photospheric plasma. The paths of the bright points are analyzed within a continuous time random walk framework. Their spatial and temporal scaling suggest that the observed motions are the walks of imperfectly correlated tracers on a turbulent fluid flow in the lanes between granular convection cells.
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.
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.
Particle Image Velocimetry (PIV) systems are often limited in their ability to fully resolve the spatiotemporal fluctuations inherent in turbulent flows due to hardware constraints. In this study, we develop models based on Rapid Distortion Theory (RDT) and Taylor's Hypothesis (TH) to reconstruct the time evolution of a turbulent flow field in the intermediate period between consecutive PIV snapshots obtained using a non-time resolved system. The linear governing equations are evolved forwards and backwards in time using the PIV snapshots as initial conditions. The flow field in the intervenin…
Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows
School of Science and Engineering University of Missouri-Kansas City（科学与工程学院，密苏里大学-堪萨斯城分校）；Department of Mathematics Emory University（数学系，埃默里大学）；Center for Applied Scientific Computing Lawrence Livermore National Laboratory（应用科学计算中心，劳伦斯利弗莫尔国家实验室）
Target Parameterization in Diffusion Models for Nonlinear Spatiotemporal System Identification
Université Marie et Louis Pasteur, SUPMICROTECH, CNRS, institut FEMTO-ST（玛丽-路易·巴斯蒂安大学，SUPMICROTECH，法国国家科学研究中心，FEMTO-ST研究所）；LLF, CNRS, Université Paris Cité（LLF，法国国家科学研究中心，巴黎cité大学）
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…
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…
Turbulent-laminar intermittency, typically in the form of bands and spots, is a ubiquitous feature of the route to turbulence in wall-bounded shear flows. Here we study the idealised shear between stress-free boundaries driven by a sinusoidal body force and demonstrate quantitative agreement between turbulence in this flow and that found in the interior of plane Couette flow -- the region excluding the boundary layers. Exploiting the absence of boundary layers, we construct a model flow that uses only four Fourier modes in the shear direction and yet robustly captures the range of spatiotempor…
Turbulent and vortical flows are ubiquitous and their characterization is crucial for the understanding of several natural and industrial processes. Among different techniques to study spatio-temporal flow fields, complex networks represent a recent and promising tool to deal with the large amount of data on turbulent flows and shed light on their physical mechanisms. The aim of this review is to bring together the main findings achieved so far from the application of network-based techniques to study turbulent and vortical flows. A critical discussion on the potentialities and limitations of …
It is shown, using results of direct numerical simulations, laboratory experiments, measurements in the atmospheric boundary layer, satellite infrared radiances data, the planetary-scale surface-air and tropospheric data, that the temperature fluctuations in atmospheric free convection can be well described by the spatio-temporal distributed chaos approach based on the Bolgiano-Obukhov phenomenology. A possibility of analogous consideration of the atmospheric dynamics on other planets (Mars, Jupiter and Saturn) has been briefly discussed using the data obtained with the Phoenix lander and the …
On its way to turbulence, plane Couette flow - the flow between counter-translating parallel plates - displays a puzzling steady oblique laminar-turbulent pattern. We approach this problem via Galerkin modelling of the Navier-Stokes equations. The wall-normal dependence of the hydrodynamic field is treated by means of expansions on functional bases fitting the boundary conditions exactly. This yields a set of partial differential equations for the spatiotemporal dynamics in the plane of the flow. Truncating this set beyond lowest nontrivial order is numerically shown to produce the expected pa…
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）
Active matter systems display a fascinating range of dynamical states, including stationary patterns and turbulent phases. While the former can be tackled with methods from the field of pattern formation, the spatio-temporal disorder of the active turbulence phase calls for a statistical description. Borrowing techniques from turbulence theory, we here establish a quantitative description of correlation functions and spectra of a minimal continuum model for active turbulence. Further exploring the parameter space, we also report on a surprising type of turbulence-driven pattern formation far b…
Uni-Flow: a unified autoregressive-diffusion model for complex multiscale flows
Centre for Computational Science, University College London, London, UK；Department of Earth Science；Engineering, Imperial College London, London, UK；Department of Chemical Engineering, University College London, London, UK；Department of Physics, Eindhoven University of Technology, Eindhoven, Netherlands；School of Civil \& Environmental Engineering, Queensland University of Technology, Brisbane, Australia；Australian Centre for Water；Environmental Biotechnology, The University of Queensland, Brisbane, Australia；Institute for Mechanics, Computational Mechanics Group, Technical University of Darmstadt, Germany；Centre for Advanced Research Computing, University College London, London, UK
Using various techniques from dynamical systems theory, we rigorously study an experimentally validated model by [Barkley et al., Nature, 526:550-553, 2015], which describes the rise of turbulent pipe flow via a PDE system of reduced complexity. The fast evolution of turbulence is governed by reaction-diffusion dynamics coupled to the centerline velocity, which evolves with advection of Burgers' type and a slow relaminarization term. Applying to this model a spatial dynamics ansatz and geometric singular perturbation theory, we prove the existence of a heteroclinic loop between a turbulent and…
Microswimmer suspensions self-organize into complex spatio-temporal flow patterns, including vortex lattices and mesoscale turbulence. Here we explore the consequences for the motion of passive tracers, based on a continuum model for the microswimmer velocity field. We observe two qualitatively different regimes distinguished via the dimensionless Kubo number $K$. At advection strengths right above the transition to turbulence, the flow field evolves very slowly ($K \gg 1$) and the spatial vortex structures lead to dominant trapping effects. In contrast, deep in the turbulent state, much faste…
In absence of advection, reaction-diffusion systems are able to organize into spatiotemporal patterns, in particular spiral and target waves. Whenever advection is present and can be parameterised in terms of effective or turbulent diffusion $D_{*}$, these patterns should be attainable on much greater, boosted lengthscale. However, so far, experimental evidence of these boosted patterns in turbulent flow was lacking. Here, we report the first experimental observation of boosted target and spiral patterns in an excitable chemical reaction in a quasi two-dimensional turbulent flow. The wave patt…
