Session Q21: Nonlinear Dynamics: Resolvents, Coherent Structures and Transition

Simultaneous shadowing of multiple Exact Coherent Structures in experimental Taylor-Couette flow

Eberhard Hopf first envisioned turbulent flow as being comprised of a sequence of special solutions of the governing equations, often referred to as Exact Coherent Structures (ECS). In this picture, turbulent flow co-evolves with, or "shadows", one such solution for a time; then a different ECS; and so on. This qualitative picture has recently been confirmed quantitatively in experimental Taylor-Couette flow. We demonstrate that this three-dimensional turbulent flow shadows, episodically but repeatedly, the spatial and temporal structure of multiple ECSs. We also unexpectedly observe that more than one ECS may be shadowed simultaneously by the turbulent flow. We explain this phenomenon in part through the relation of certain ECSs via bifurcations, as well as the similarity of segments of unrelated ECSs.   

Exploring the connections between exact coherent structures and attractors in active nematic channel flow

Exact Coherent Structures (ECS) are stationary, periodic, quasiperiodic, or traveling wave solutions of the governing equations that, together with their invariant manifolds, serve as an organizing template of the dynamics of a fluid system. Active nematics is a class of active fluids that exhibits rich dynamical behavior, including spontaneous flows, periodic defect dynamics, and chaotic `active turbulence'. In this work, we compute over 100 ECSs in a 2D active nematic channel flow system, and provide evidence that these ECSs are shadowed by typical trajectories of the system. We also discuss the presence of robust heteroclinic cycles in the system that cycle between the two opposite flowing unidirectional flow states, as well as several other exotic trajectories that can be explained using the ECS framework.   

Global resolvent analysis of three-dimensional jets using randomized linear algebra and time stepping

Resolvent analysis is a powerful tool for modeling coherent structures in turbulent flows. Resolvent forcing and response modes are defined in terms of the singular value decomposition (SVD) of the resolvent operator, but the computational cost of computing these modes scales poorly with problem size and becomes computationally expensive for large systems, typically limiting resolvent analysis to flows with one or two inhomogeneous directions. Recently, we have developed an improved algorithm (called RSVDt) by combining randomized SVD with an optimized direct and adjoint time-stepping routine, which achieves linear cost scaling with problem size and enables efficient resolvent analysis of three-dimensional flows. Time stepping constitutes the majority of the cost of our algorithm, and we propose a new approach to minimize this cost by eliminating undesired transients that otherwise lengthen the time interval to be computed. We use a series of jets to demonstrate our algorithm. First, we use two jet models to show that RSVDt decreases memory and CPU cost by two and three orders of magnitude, respectively, compared to other state-of-the-art algorithms. Second, we use RSVDt to compute resolvent modes for a previously intractable three-dimensional jet and use this new capability to explore the impact of steady streaks on Kelvin-Helmholtz wavepackets.   

From resolvent to Gramians: forcing and response modes for control

During the last decade, forcing and response modes produced by resolvent analysis have promised to have an impact on flow control applications due to their potential to guide sensor and actuator placement and design. However, resolvent modes are frequency-dependent, which, although responsible for their success in identifying scale interactions in turbulence, complicates their use for control purposes. In this work, we seek orthogonal bases of forcing and response modes that are the most responsive and receptive, respectively, across all frequencies. We show that these frequency-independent bases of representative resolvent modes are given by the eigenvectors of the observability and controllability Gramians of the system considering full state inputs and outputs. We present several numerical examples where we leverage these eigenbases by building orthogonal or interpolatory projectors. Forcing modes are used to identify dynamically relevant disturbances, to place point sensors to measure disturbances, and to design spatially-distributed and spatially-localized actuators for feedforward control. Response modes are used for point sensor placement aiming at flow reconstruction.   

Information-theoretic quantification of causality in turbulent flows

Understanding causality among quantities of interest in turbulent flows is essential for physical understanding, modeling, and control. In this talk, we discuss a new method to quantify causality in turbulence based on information fluxes. The main properties of the method and its suitability for quantifying causality are highlighted. The method is non-intrusive and only requires the time history of the flow, which is convenient both computationally and experimentally. We leveraged the approach to investigate the causality of the energy cascade in isotropic turbulence. The results are validated against (the more expensive) causality with interventions, in which the system is modified, and the consequences are measured.   

Deformation of FTLE ridges under the action of control in unsteady fluid flows

Finite-time Lyapunov Exponent fields allow us to visualize transport barriers and regions of strong attraction or repulsion for passive particles in an unsteady fluid flow. It is also possible to extract FTLE from data generated by active agents that use actuation to move through the flow. The effect of actuation and control may be viewed as resulting in a new effective background vector field in which the active agent appears to be a passive particle. Therefore, when FTLE is computed for active agents, the resulting control FTLE (cFTLE) field deviates from the passive FTLE field. In our work, we investigate the deformation of the cFTLE field from the passive FTLE on agents using model predictive control (MPC) to move through the flow. In particular, we investigate how changes in the control parameters produce deformations in the cFTLE ridges; these parameters include time horizon, energy penalty, and location of the goal target. These results could be useful in the context of agents navigating in the ocean, for example to generate high-level control policies or to decide where to deploy agents.   

Efficient tensor-based sensor placement for turbulent flow reconstructions

This talk will address the question of determining sensor locations to optimally place a limited number of sensors, from which the collected data can be used to accurately recover the turbulent flow fields. Previous work has used a basis obtained using proper orthogonal decomposition to approximate the flow field and used row subset selection to obtain near-optimal interpolation points that determine sensor locations. However, this approach does not exploit the inherent multidimensional structure of the flow fields. To address this, we use a tensor-based approach to approximate the flow field and obtain near-optimal interpolation points along each tensor mode. The resulting approach has much lower storage requirements and is often more accurate for a comparable number of sensors. Numerical experiments on a variety of fluid problems, such as the Kolmogorov flow and seasurface temperature, will illustrate the performance of the proposed methods.

    

Resolvent analysis of turbulent pipe flow laden with low-inertia particles

We extend the resolvent framework to turbulent flows laden with low-inertia particles. The particle velocities are modelled using the equilibrium Eulerian model, which is assumed to be valid for Stokes numbers up to 1. We analyse a vertical turbulent pipe flow with Reynolds number equal to 5300 based on diameter and bulk velocity, Froude numbers F r = −4, −0.4, 0.4, 4 and Stokes numbers St+ = 0−1. A direct numerical simulation (DNS) for a pipe with a length of 7.5 diameters is performed with the particles released uniformly at the inlet. The resolvent formulation can reproduce the physical phenomena observed in inertial particle flows, such as localized high concentration due to the vortical centrifuge effect, turbophoresis and gravitational effects. It also reveals that upward flow increases particle concentration in the log layer while downward flow increases concentration near the centre of the pipe: both features have been observed in previous Lagrangian simulations as well as experiments. The main effect of Stokes number is the amplification of the gain for resolvent modes with smaller streamwise wavelengths, resulting in an increase of the local scale clustering of particles and turbophoresis.   

Large-scale circulation patterns rule the predictability of extreme events

Extreme events in geophysical flows have a strong impact in human life and in diverse economical activities, and their

prediction is crucial to mitigate their consequences. Particularly now, in the context of climate change, interest has

emerged to develop models and indicators that allows for early detection and warning of extreme events. Some of these

models have proved successful, but it is unclear if they can be improved because the predictability limit of extreme

events is unknown. In this talk, we show that it is now possible to measure the predictability of extreme events directly by probing phase

space with a computationally-intensive approach. This allows to assess the exact potential of predictive models and to

determine precisely the conditions under which they may be effective. We analysed the predictability of extreme bursts

of the dissipation in a two-dimensional Kolmogorov flow by producing massive ensembles of initial conditions perturbed

around independent base flows. We produced millions of realisations to cover the full attractor of the Kolmogorov flow,

and used the Kullback—Leibler divergence, an information-theoretical tool, to assess predictability. This analysis

shows that extreme bursts of the dissipation may be successfully predicted beyond a few Lyapunov times, but that their

predictability depends strongly on the phase-space region from which they emerge. Specifically, we reveal that predictable

and unpredictable events evolve from two distinctly different large-scale circulation patterns. These results

open the possibility of improving predictive models by tuning them to the large-scale dynamics. Our approach could

be adapted with the available compute power to more complex flows.