Laminar-Turbulent Patterning in Transitional Flows

Wall-bounded flows experience a transition to turbulence characterized by the coexistence of laminar and turbulent domains in some range of Reynolds number R, the natural control parameter. This transitional regime takes place between an upper threshold Rt above which turbulence is uniform (featureless) and a lower threshold Rg below which any form of turbulence decays, possibly at the end of overlong chaotic transients. The most emblematic cases of flow along flat plates transiting to/from turbulence according to this scenario are reviewed. The coexistence is generally in the form of bands, alternatively laminar and turbulent, and oriented obliquely with respect to the general flow direction. The final decay of the bands at Rg points to the relevance of directed percolation and criticality in the sense of statistical-physics phase transitions. The nature of the transition at Rt where bands form is still somewhat mysterious and does not easily fit the scheme holding for pattern-forming instabilities at increasing control parameter on a laminar background. In contrast, the bands arise at Rt out of a uniform turbulent background at a decreasing control parameter. Ingredients of a possible theory of laminar-turbulent patterning are discussed.Comment: 29 pages, 5 illustrations, written for special issue on "Complex Systems, Non-Equilibrium Dynamics and Self-Organisation" of journal Entropy edited by Dr. G. Pruessne

Study of wave chaos in a randomly-inhomogeneous oceanic acoustic waveguide: spectral analysis of the finite-range evolution operator

The proplem of sound propagation in an oceanic waveguide is considered. Scattering on random inhomogeneity of the waveguide leads to wave chaos. Chaos reveals itself in spectral properties of the finite-range evolution operator (FREO). FREO describes transformation of a wavefield in course of propagation along a finite segment of a waveguide. We study transition to chaos by tracking variations in spectral statistics with increasing length of the segment. Analysis of the FREO is accompanied with ray calculations using the one-step Poincar\'e map which is the classical counterpart of the FREO. Underwater sound channel in the Sea of Japan is taken for an example. Several methods of spectral analysis are utilized. In particular, we approximate level spacing statistics by means of the Berry-Robnik and Brody distributions, explore the spectrum using the procedure elaborated by A. Relano with coworkers (Relano et al, Phys. Rev. Lett., 2002; Relano, Phys. Rev. Lett., 2008), and analyze modal expansions of the eigenfunctions. We show that the analysis of FREO eigenfunctions is more informative than the analysis of eigenvalue statistics. It is found that near-axial sound propagation in the Sea of Japan preserves stability even over distances of hundreds kilometers. This phenomenon is associated with the presence of a shearless torus in the classical phase space. Increasing of acoustic wavelength degrades scattering, resulting in recovery of localization near periodic orbits of the one-step Poincar\'e map. Relying upon the formal analogy between wave and quantum chaos, we suggest that the concept of FREO, supported by classical calculations via the one-step Poincar\'e map, can be efficiently applied for studying chaos-induced decoherence in quantum systems

Sub-grid modelling for two-dimensional turbulence using neural networks

In this investigation, a data-driven turbulence closure framework is introduced and deployed for the sub-grid modelling of Kraichnan turbulence. The novelty of the proposed method lies in the fact that snapshots from high-fidelity numerical data are used to inform artificial neural networks for predicting the turbulence source term through localized grid-resolved information. In particular, our proposed methodology successfully establishes a map between inputs given by stencils of the vorticity and the streamfunction along with information from two well-known eddy-viscosity kernels. Through this we predict the sub-grid vorticity forcing in a temporally and spatially dynamic fashion. Our study is both a-priori and a-posteriori in nature. In the former, we present an extensive hyper-parameter optimization analysis in addition to learning quantification through probability density function based validation of sub-grid predictions. In the latter, we analyse the performance of our framework for flow evolution in a classical decaying two-dimensional turbulence test case in the presence of errors related to temporal and spatial discretization. Statistical assessments in the form of angle-averaged kinetic energy spectra demonstrate the promise of the proposed methodology for sub-grid quantity inference. In addition, it is also observed that some measure of a-posteriori error must be considered during optimal model selection for greater accuracy. The results in this article thus represent a promising development in the formalization of a framework for generation of heuristic-free turbulence closures from data

Data Driven Prognosis: A multi-physics approach verified via balloon burst experiment

A multi-physics formulation for Data Driven Prognosis (DDP) is developed. Unlike traditional predictive strategies that require controlled off-line measurements or training for determination of constitutive parameters to derive the transitional statistics, the proposed DDP algorithm relies solely on in situ measurements. It utilizes a deterministic mechanics framework, but the stochastic nature of the solution arises naturally from the underlying assumptions regarding the order of the conservation potential as well as the number of dimensions involved. The proposed DDP scheme is capable of predicting onset of instabilities. Since the need for off-line testing (or training) is obviated, it can be easily implemented for systems where such a priori testing is difficult or even impossible to conduct. The prognosis capability is demonstrated here via a balloon burst experiment where the instability is predicted utilizing only on-line visual observations. The DDP scheme never failed to predict the incipient failure, and no false positives were issued. The DDP algorithm is applicable to others types of datasets. Time horizons of DDP predictions can be adjusted by using memory over different time windows. Thus, a big dataset can be parsed in time to make a range of predictions over varying time horizons