Session F31: Nonlinear Dynamics: Model Reduction & Turbulence II

SwarmDMD: A Data-driven Method for Swarm Modeling and Analysis

Biological and engineering swarms often exhibit fluid-like behaviors that are challenging to model due to the high-dimensional dynamics, despite the emergence of low-dimensional patterns. Most existing swarm modelling approaches are based on first principles and result in swarm-specific parameterizations that do not generalize to a broad range of applications. In this work, we adapt the dynamic mode decomposition (DMD) from fluid mechanics to (1) learn approximate local interactions of homogeneous swarms through observation data and (2) generate similar swarming behavior using the learned model. The proposed swarmDMD algorithm is developed and tested on a canonical swarm model, where we show that (1) swarmDMD can faithfully reconstruct the swarm dynamics; and (2) swarmDMD allows for the prediction of nonlinear swarm dynamics from different initial conditions. We believe swarmDMD approach will be useful for studying multi-agent systems found in biology, physics, and engineering, and may provide additional insights into the understanding and control in the collective dynamics of vortices in multiscale turbulence.     

Symmetry-Aware Autoencoders for Model Reduction

Nonlinear principal component analysis (nlPCA) via autoencoders has attracted attention in the dynamical systems community due to its larger compression rate when compared to linear principal component analysis (PCA). These model reduction methods  experience an increase in the dimensionality of the latent space when the dataset these are applied to exhibits globally invariant samples due to the presence of symmetries. In this study, we introduce a novel machine learning embedding, which uses spatial transformer networks and siamese networks to account for continuous and discrete symmetries, respectively. This embedding can be employed with both linear and nonlinear methods, which we term symmetry-aware PCA and symmetry-aware nlPCA. We apply the proposed framework to datasets generated by the viscous Burgers' equation, the simulation of the flow through a stepped diffuser and the Kolmogorov Flow to showcase the capabilities for cases exhibiting only continuous symmetries, only discrete symmetries or a combination of both, respectively.   

3D Dual Basis Proper Orthogonal Decomposition

Conventional implementations of the proper orthogonal decomposition (POD) produce a basis spanning the velocity field optimally with respect to its kinetic energy content, allowing formulation of reduced order models (ROMs) by projecting transport equations onto the truncated basis. The truncation may leave out dynamically relevant modes, leading to less accurate models. This issue is important to turbulent flows where a decisive role is played by interactions between multiple scales potentially associated with different mode orders.

Lee and Dowell [1] showed that supplementing an energy-optimized POD basis with a POD basis optimized with respect to the velocity gradient norm enhances the accuracy of ROMs at high Reynolds numbers in 2D. Given the different dynamics governing turbulence in 2D vs 3D it is not given that similar results would apply in 3D. We analyze the effect of using a dual-basis ROM for a 3D flow. This facilitates quantification of the dynamical importance of different modes and their interactions, yielding insight to the fundamental dynamics of the flow.

References:

[1] M.W. Lee and E.H.Dowell. "Improving the predictable accuracy of fluid Galerkin reduced-order models using two POD bases". In: Nonlinear Dynamics 101.2 (2020), pp. 1457-1471.   

Scale-dependence of Error Growth in Turbulence

We examine the rate of error growth in turbulence simulations as a function of the length scales at which the uncertainties in the initial fluid state reside.To this end, we carry out large-eddy and direct numerical simulations of the sinusoidally-driven three-dimensional fluid under periodic boundary conditions in all directions. In several settings, we estimate the rate of divergence of fluid states that are initially close to one another with deviations confined to a subset of length scales in the system. We find that smaller-scale fluctuations in the initial condition translate to faster-growing errors, resulting in finite-time predictions that deteriorate quicker. We discuss the implications of our results for weather forecasting, where it is well known that atmospheric phenomena change the faster the smaller are their spatial scales.   

Uncovering causality in isotropic turbulence by massive machine manipulation

    Flow structures important for the dynamics of isotropic turbulence are identified by running an unprecedentedly large number of direct numerical simulations (DNS) seeded with localized perturbations. The perturbations zero either the local velocity fluctuations or the local vorticity within a region of a given size $L$. Some perturbations grow up to $10^3$ times more than others after one turnover, in terms of the global $L_2$-norm of the perturbation field. Because perturbations are local in space, the procedure classifies regions of the flow according to how much they grow. The most `reactive' regions are studied, showing that they tend to contain strong events, either strong vortices or strong velocity perturbations. Large perturbations $(L \gtrsim 60\eta)$ grow or not based on the contribution of the region to the turbulent kinetic energy, whereas smaller perturbations $(L \lessim 60 \eta)$ do so based on the vorticity the local flow induces in the rest of the flow field. Differences found between the most and the least growing perturbations hint that while both of them affect the small scales of the flow, only the former contaminate the integral scales, resulting in a much stronger effect on the overall flow.   

Machine identification of causally important events in turbulent channel flow

Flow structures that are highly causal to the future state of wall turbulence are studied using a large number of numerical simulations of turbulent channel flow at Reτ = 600. The domain is divided into small cells, and the velocity fluctuations with respect to the local mean within each cell are removed before the flow is allowed to temporally develop. The causal effect of this operation after a given time is measured by the L₂ norm of the difference between the perturbed and unperturbed velocity or vorticity vectors. Cells resulting in large perturbations are considered highly causal. To avoid pre-assumptions as much as possible, the simulations are repeated many times changing the location and size of the cells. The analysis is then conducted on an ensemble of many realizations. The perturbation effect is found to propagate in the wall-normal direction with a celerity of the order of the friction velocity, although somewhat faster than the advection velocity of conventional turbulent sweeps and ejections. Comparison of different wall-normal cell heights shows that the causal effect of the flow structures becomes more intermittent as the perturbation is applied farther away from the wall. The structure of highly causal cells depends on whether the perturbation is characterized by its magnitude or by its amplification rate. Large magnitudes tend to be associated with velocity interfaces, whereas large amplifications are associated with sweeps. Low amplifications are associated with ejections.