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Closure and order-reduction of turbulent dynamical systems

Turbulent dynamical systems are characterized by both a large dimensional phase space and a large dimension of persistent or intermittent instabilities (i.e., a large number of positive Lyapunov exponents on the attractor). They are ubiquitous in many complex systems with fluid flow such as, engineering turbulence at high Reynolds numbers, confined plasmas, as well as atmospheric and oceanic turbulence (movie below).

The scope of this work is i) to design effective turbulent closure schemes that respect the synergistic activity between unstable linear dynamics, nonlinear energy transfers, and stable dynamics, which is apparent in turbulent systems, and ii) to develop order-reduction techniques that take into account the nonlinear energy transfers associated with the omitted modes. 

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Quantification and Prediction of Extreme Events in Stochastic Systems

Data-driven reduced order prediction of extreme waves
Will Cousins, Themistoklis P. Sapsis

Rogue or freak waves are ocean waves whose height is abnormally large for a particular sea state. Often described an enormous "wall of water," (e.g. the New Year wave in Fig. 1) such waves have caused catastrophic damage to ships and coastal structures. For example, in 1978 the German super-tanker München vanished, along with her 26 crew members. Searches for the ship recovered little, but a lifeboat was recovered whose attachment pins showed evidence of being subjected to a great force. As this lifeboat was stowed 20m above the water line, some have conjectured that the München may have been struck by an extremely large wave [Liu, Geofizika 24, 2007]. Here we describe our method for reliably predicting these rogue waves before they occur while expending minimal computational effort, which we term Reduced Order Prediction of Extremes (ROPE).

The large, steep nature of these rogue waves, combined with recent evidence that they can occur more likely than Gaussian statistics would suggest, imply that nonlinear models are necessary to fully understand their dynamics. Thus, we focus our attention on models that incorporate this nonlinearity while remaining simple enough to be tractable. Two such examples are the equation of Majda, McLaughlin, and Tabak (MMT) and the Nonlinear Schrodinger Equation (NLS). The MMT model is similar to NLS, differing in that it includes a "fractional" differential operator in place of the 2nd derivative in NLS [Majda et. al., J. Nonlinear Sci. 6, 1997]. Although NLS and MMT are quite simple, they have attracted attention due to an inherent mechanism for generating rogue waves. Specifically, simple periodic solutions to both equations are unstable, with small perturbations initiating "focusing" events, where the nearby wave field is soaked up to produce a large, localized wave. This process, known as the Benjamin-Feir instability, is well understood theoretically and has been reproduced in experiment [Chabchoub et. al. PRL 106, 2011].

extreme time series

Figure 1: The New Year wave recorded at the Draupner platform in the North Sea on January 1, 1995.

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Transport and mixing of finite-size particles

Inertial (or finite-sized) particles in fluid flows are commonly encountered in natural phenomena and industrial processes. Examples of inertial particles include dust, impurities, droplets, and air bubbles, with applications in pollutant transport in the ocean and atmosphere, rain initiation, coexistence between plankton species in the hydrosphere, and even planet formation by dust accretion in the solar system. Finite-size or inertial particle dynamics in fluid flows can differ markedly from infinitesimal particle dynamics: both clustering and dispersion are well-documented phenomena in inertial particle motion, while they are absent in the incompressible motion of infinitesimal particles. 

  • A pollutants
  • A pred prey
  • A plankton
  • A clouds
  • Atmospheric and oceanic transportation of pollutants
  • Predator - prey interactions in jellyfish feeding
  • Plankton dynamics and formation of filamentary structures
  • Clustering of droplets in clouds and rain formation

Systems that can be modeled as finite-size particles in fluid flows.

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Order reduction and UQ of chaotic flows with low-dimensional attractors - DO equations

It is often the case that infinite-dimensional dynamical systems such as Navier-Stokes equations posses global attractors with low dimensionality, i.e. the set that contains all the possible outcomes of the chaotic system "lives" in a low-dimensional space. For such case it is beneficial to perform order-reduction of the original equation and this is usually done by selecting a set of basis elements or fields (usually based on energetic criteria) and perform a Galerkin projection of the dynamics. 

In many situations of practical interest this approach gives powerful results. However, in cases where the problem is strongly transient, even if the number of instabilities is small, it is very hard to choose the appropriate set of modes in order to perform effective order-reduction.

In this work we derive an exact, closed set of evolution equations (the Dynamically Orthogonal (DO) equations - read more...) that allow us to simultaneously evolve i) the basis elements that capture the modes of the flow that have important energy and ii) the statistical structure of the coefficients for these modes (i.e. the shape of the attractor). This new probabilistic framework allows for detailed understanding of the dynamical mechanisms (energy transfers, statistics, instabilities) but also for the efficient uncertainty quantification (UQ) of transient flows as long as low-dimensional attractors characterize these.

DO stochastic solution for the flow behind a cylinder (Re=100)

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Dynamical systems excited by correlated noise

Dynamical systems subjected to correlated noise is in most engineering applications the rule rather than the exception (in contrast to white-noise). While for white-noise excitation the input spectrum is essentially flat and there are probabilistic tools to analyze the response, correlated noise is characterized in most cases by broadband spectra with non-uniform distribution of energy across harmonics and for this case the determination of the response statistics is not straightforward. 

An important class of engineering systems modeled as dynamical systems under correlated noise is ships and ocean structures subjected to water waves, e.g. ship rolling motion. The random character of the environment that these systems operate in makes it essential to characterize their dynamics in a probabilistic framework (i.e. determine statistics for their motion rather than individual trajectories).

extreme rolling

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Targeted energy transfer in nonlinear oscillators

Our goal is the design of nonlinear mechanical oscillators, for targeted (guided) energy transfer applications from broad-band sources, which will go beyond the current state of the art of energy transfers which consists of linear resonance ideas or nonlinear systems excited by just a monochromatic source. These novel mechanical configurations will be able to harvest energy from broad-band sources by mimicking the robust and adaptive nonlinear-energy-transfer mechanism occurring across different scales in turbulent flows resulting in enhanced energy harvesting without the need for resonance conditions. Such properties will allow for the design of robust harvesting devices under stochastic excitations as well as the effective passive protection of structures subjected to random external loads.

Figure1 Energy harvesting

Figure 1: Vorticity and energy spectrum of the flow in a laminar and a turbulent jet – turbulent flow dissipates orders of magnitude larger amounts of energy because energy is nonlinearly distributed along many modes – each one of these dissipates energy.