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Register a new userRevealing the state space of turbulence using machine learning
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Despite the apparent complexity of turbulent flow, identifying a simpler description of the underlying dynamical system remains a fundamental challenge. Capturing how the turbulent flow meanders amongst unstable states (simple invariant solutions) in phase space, as envisaged by Hopf in 1948, using some efficient representation offers the best hope of doing this, despite the inherent difficulty in identifying these states. Here, we make a significant step towards this goal by demonstrating that deep convolutional autoencoders can identify low-dimensional representations of two-dimensional turbulence which are closely associated with the simple invariant solutions characterizing the turbulent attractor. To establish this, we develop latent Fourier analysis that decomposes the flow embedding into a set of orthogonal latent Fourier modes which decode into physically meaningful patterns resembling simple invariant solutions. The utility of this approach is highlighted by analysing turbulent Kolmogorov flow (flow on a 2D torus forced at large scale) at $Re=40$ where, in between intermittent bursts, the flow resides in the neighbourhood of an unstable state and is very low dimensional. Projections onto individual latent Fourier wavenumbers reveal the simple invariant solutions organising both the quiescent and bursting dynamics in a systematic way inaccessible to previous approaches.
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Symmetry reduction by the method of slices is applied to pipe flow in order to quotient the stream-wise translation and azimuthal rotation symmetries of turbulent flow states. Within the symmetry-reduced state space, all travelling wave solutions reduce to equilibria, and all relative periodic orbits reduce to periodic orbits. Projections of these solutions and their unstable manifolds from their $infty$-dimensional symmetry-reduced state space onto suitably chosen 2- or 3-dimensional subspaces reveal their interrelations and the role they play in organising turbulence in wall-bounded shear flows. Visualisations of the flow within the slice and its linearisation at equilibria enable us to trace out the unstable manifolds, determine close recurrences, identify connections between different travelling wave solutions, and find, for the first time for pipe flows, relative periodic orbits that are embedded within the chaotic attractor, which capture turbulent dynamics at transitional Reynolds numbers.
In this paper, we show that a revised convolutional recurrent neural network (CRNN) can decrease, by orders of magnitude, the time needed for the phase-resolved prediction of waves in a spatiotemporal domain of a nonlinear dispersive wave field. The problem of predicting such waves suffers from two major challenges that have so far hindered analytical or direct computational solutions in real time or faster: (i) the reconstruction problem, that is, how one can calculate from measurable wave amplitude data the state of the wave field (wave components, nonlinear couplings, etc.), and (ii) if such a reconstruction is in hand, how to integrate equations fast enough to be able to predict an upcoming rouge wave in a timely manner. Here, we demonstrate that these two challenges can be overcome at once through advanced machine learning techniques based on spatiotemporal patches of the time history of wave height data in the domain. Specifically, as a benchmark here we consider equations that govern the evolution of weakly nonlinear surface gravity waves such as those propagating on the surface of the oceans. For the case of oceanic surface waves considered here, we demonstrate that the proposed methodology, while maintaining a high accuracy, can make phase-resolved predictions more than two orders of magnitude faster than numerically integrating governing equations.
This article presents an original methodology for the prediction of steady turbulent aerodynamic fields. Due to the important computational cost of high-fidelity aerodynamic simulations, a surrogate model is employed to cope with the significant variations of several inflow conditions. Specifically, the Local Decomposition Method presented in this paper has been derived to capture nonlinear behaviors resulting from the presence of continuous and discontinuous signals. A combination of unsupervised and supervised learning algorithms is coupled with a physical criterion. It decomposes automatically the input parameter space, from a limited number of high-fidelity simulations, into subspaces. These latter correspond to different flow regimes. A measure of entropy identifies the subspace with the expected strongest non-linear behavior allowing to perform an active resampling on this low-dimensional structure. Local reduced-order models are built on each subspace using Proper Orthogonal Decomposition coupled with a multivariate interpolation tool. The methodology is assessed on the turbulent two-dimensional flow around the RAE2822 transonic airfoil. It exhibits a significant improvement in term of prediction accuracy for the Local Decomposition Method compared with the classical method of surrogate modeling for cases with different flow regimes.
This paper proposes a new data assimilation method for recovering high fidelity turbulent flow field around airfoil at high Reynolds numbers based on experimental data, which is called Proper Orthogonal Decomposition Inversion (POD-Inversion) data assimilation method. Aiming at the flows including shock wave discontinuities or separated flows at high angle of attack, the proposed method can reconstruct high-fidelity turbulent flow field combining with experimental distributed force coefficients. We firstly perform the POD analysis to the turbulent eddy viscosity fields computed by SA model and obtain the base POD modes. Then optimized the POD coefficients by global optimization algorithm coupling with the Navier-Stokes equations solver. The high-fidelity turbulent flied are recovered by several main modes, which can dramatically reduce the dimensions of the system. The effectiveness of the method is verified by the cases of transonic flow around the RAE2822 airfoil at high Reynolds numbers and the separated flow at high angles of attack. The results demonstrate that the proposed assimilation method can recover the turbulent flow field which optimally match the experimental data, and significantly reduce the error of pressure coefficients. The proposed data assimilation method can offer high-fidelity field data for turbulent model based on machine learning.
Predicting and simulating aerodynamic fields for civil aircraft over wide flight envelopes represent a real challenge mainly due to significant numerical costs and complex flows. Surrogate models and reduced-order models help to estimate aerodynamic fields from a few well-selected simulations. However, their accuracy dramatically decreases when different physical regimes are involved. Therefore, a method of local non-intrusive reduced-order models using machine learning, called Local Decomposition Method, has been developed to mitigate this issue. This paper introduces several enhancements to this method and presents a complex application to an industrial-like three-dimensional aircraft configuration over a full flight envelope. The enhancements of the method cover several aspects: choosing the best number of models, estimating apriori errors, improving the adaptive sampling for parallel issues, and better handling the borders between local models. The application is supported by an analysis of the model behavior, with a focus on the machine learning methods and the local properties. The model achieves strong levels of accuracy, in particular with two sub-models: one for the subsonic regime and one for the transonic regime. These results highlight that local models and machine learning represent very promising solutions to deal with surrogate models for aerodynamics.
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