Guillaume Daviller

Cluster-based reduced-order modelling of a mixing layer

We propose a novel cluster-based reduced-order modelling (CROM) strategy of unsteady flows. CROM combines the cluster analysis pioneered in Gunzburgers group (Burkardt et al. 2006) and and transition matrix models introduced in fluid dynamics in Eckhardts group (Schneider et al. 2007). CROM constitutes a potential alternative to POD models and generalises the Ulam-Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron-Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Secondly, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.

Cluster-based reduced-order modelling of shear flows

Cluster-based reduced-order modelling (CROM) builds on the pioneering works of Gunzburgers group in cluster analysis [1] and Eckhardts group in transition matrix models [2] and constitutes a potential alternative to reduced-order models based on a proper-orthogonal decomposition (POD). This strategy frames a time-resolved sequence of flow snapshots into a Markov model for the probabilities of cluster transitions. The information content of the Markov model is assessed with a Kullback-Leibler entropy. This entropy clearly discriminates between prediction times in which the initial conditions can be inferred by backward integration and the predictability horizon after which all information about the initial condition is lost. This approach is exemplified for a class of fluid dynamical benchmark problems like the periodic cylinder wake, the spatially evolving incompressible mixing layer, the bi-modal bluff body wake, and turbulent jet noise. For these examples, CROM is shown to distil nontrivial quasi-attr...

Sparsity enabled cluster reduced-order models for control

Abstract Characterizing and controlling nonlinear, multi-scale phenomena are central goals in science and engineering. Cluster-based reduced-order modeling (CROM) was introduced to exploit the underlying low-dimensional dynamics of complex systems. CROM builds a data-driven discretization of the Perron–Frobenius operator, resulting in a probabilistic model for ensembles of trajectories. A key advantage of CROM is that it embeds nonlinear dynamics in a linear framework, which enables the application of standard linear techniques to the nonlinear system. CROM is typically computed on high-dimensional data; however, access to and computations on this full-state data limit the online implementation of CROM for prediction and control. Here, we address this key challenge by identifying a small subset of critical measurements to learn an efficient CROM, referred to as sparsity-enabled CROM. In particular, we leverage compressive measurements to faithfully embed the cluster geometry and preserve the probabilistic dynamics. Further, we show how to identify fewer optimized sensor locations tailored to a specific problem that outperform random measurements. Both of these sparsity-enabled sensing strategies significantly reduce the burden of data acquisition and processing for low-latency in-time estimation and control. We illustrate this unsupervised learning approach on three different high-dimensional nonlinear dynamical systems from fluids with increasing complexity, with one application in flow control. Sparsity-enabled CROM is a critical facilitator for real-time implementation on high-dimensional systems where full-state information may be inaccessible.

Flow decompositions for the study of source mechanisms

In this work we propose some model problems (computed by DNS) to test the performance and utility–in terms of the physical insight which is provided–of an energy corollary proposed by Doak 8 and Jenvey. 14 The energy corollary, which is based on Helmholtz decompositions of both linear momentum and velocity, aspires to provide a clearer interpretational framework for studying the local physical mechanisms which underpin the production of sound energy by turbulence. In this paper we compute the response of the two-dimensional Euler equations to a wavepacket body-force excitation whose amplitude and vorticity-level can be controlled, and we assess the resulting flow fields using the said framework. We look at linear and non-linear, irrotational and rotational scenarios, and we present the flux and source terms of the energy corollaries. Initial results show how it is possible to identify the different flow processes (source mechanisms) which lead to the transport of fluctuation energy, as well as the physical mechanisms by which that fluctuation energy is transported away from the source region as propagating sound energy. I. Introduction The question ‘How does turbulent fluid motion generate sound ?’ is only meaningful if we know what we mean by ‘sound’ within the confines of the turbulent field. A sound wave is generally defined as a smallamplitude, energy-conserving, irrotational fluctuation characterised by propagation at the speed of sound; and so the notion of a ‘sound-field’ is, at the heart of a turbulent flow, meaningless. This is probably the single greatest obstacle blocking our efforts to understand the dynamics by which subsonic turbulent jets drive sound waves in the farfield. Rayleigh’s three classes of fluctuating motion—vortical, entropic and acoustic—constitute an unambiguous system of distinct kinds of fluid motion. These are uncoupled in the simplest, uniform, linear flow systems; and they can be shown to interact quadratically in another class of slighthly more complex flows, such that these quadratic interactions constitute further sources of excitation for each of the modes. 5 This classification provides an interpretational framework which is invaluable for the obtention of a mechanistic understanding of the different flow process which underpin energy exchange in compressible turbulent flows. However, the three kinds of motion can not be so easily defined in the general case, and so, in the kinds of turbulent flows which are of practical interest, we struggle to arrive at a phenomenological understanding of the said flow processes. In this work we consider a framework which was considered by Doak 8 and Jenvey 14 to be useful for precisely this: both proposed the Helmholtz decomposition, applied, respectively, to the momentum and velocity variables, as a means by which to distinguish vortical motion (which they considered synonymous with turbulence) from irrotational motion (which they considered synonymous with acoustic or entropic fields). Such decompositions are not readily applicable experimentally, and so they have seen little use in aeroacoustic analysis. In this work we apply the decomposition to a number of model problems (computed by DNS), where we can carefully control the level of complexity: from potential flows with linear (smallamplitude) perturbations to rotational flows were the dynamics become non-linear. In the simplest flows there is no ambiguity regarding the quantities we study and the decompositions we apply–the sound-production problem can here be described analytically; the degree of ambiguity increases with increasing flow complexity. Our objective is to assess the Helmholtz decompositions and the energy corollaries which result,