Resolvent Modeling of Turbulent Jets

Optimal control of turbulent flows requires a detailed prediction of the unsteady, three-dimensional turbulent structures that govern quantities of interest like noise, drag, and mixing efficiency. There is a need for physics-based, reduced-order models of turbulent structure for those cases where direct simulation of the flow would be computationally prohibitive. In this thesis, we explore resolvent analysis as a framework for such models. Based on a linearization about the turbulent mean flow field, the resolvent finds optimal (highest gain) forcing functions that give rise, through linear amplification mechanisms, to energetic coherent structures. The forcing functions represent the nonlinear interactions between the coherent structures as well as with background incoherent turbulence. While the high-gain structures capture many characteristics of the observed turbulent coherent structures in both wall-bounded and free-shear flows, closures for the forcing function are required to make these models predictive and thus utilize them for flow control. In the first part of this thesis, we examine a linear model for the resolvent forcing by adapting the concept of a turbulent (eddy) viscosity from classical Reynolds-Averaged Navier--Stokes (RANS) turbulence modeling. We present a data-driven approach to identify an optimal eddy-viscosity field that best matches the resolvent prediction to the most energetic coherent structure educed via spectral proper orthogonal decomposition (SPOD) of data from high-fidelity simulations. We analyze the specific case of turbulent jets spanning a range of Mach numbers from subsonic to supersonic. We find the optimal eddy-viscosity field to be effective at matching both the shape and energy distribution of structures. More importantly, we find that calibrated eddy-viscosity fields predicted using standard eddy-viscosity models (utilizing only quantities available from RANS) yield results that are close to optimal. We use the resulting resolvent model together with the high-fidelity data to investigate the full spectrum of amplification mechanisms and coherent structures present in turbulent jets. The addition of a turbulence model provides a clear separation between two established mechanisms in turbulent jets (Kelvin-Helmholtz and Orr) and leads to the identification of a third mechanism known as lift-up. Lift-up becomes the dominant mechanism at low-frequency limits for nonzero azimuthal wavenumbers, generating elongated, streaky structures. We find these streaks to be the most energetic structures in the jet, and that their presence has implications for altering the mean flow and controlling noise. Finally, we extend resolvent analysis to that of an acoustic analogy that relates the near-field forcing to the far-field acoustics 100 diameters from the nozzle. We again leverage high-fidelity data to produce an ensemble of realizations of the acoustic field and find that only a few resolvent modes are necessary for reconstruction. Ultimately, we find that a resolvent model based solely upon RANS quantities can reconstruct and predict the peak acoustic field at rank-1 to within 2 decibels for both the supersonic and transonic jets.</p

Discovering and forecasting extreme events via active learning in neural operators

Extreme events in society and nature, such as pandemic spikes or rogue waves, can have catastrophic consequences. Characterizing extremes is difficult as they occur rarely, arise from seemingly benign conditions, and belong to complex and often unknown infinite-dimensional systems. Such challenges render attempts at characterizing them as moot. We address each of these difficulties by combining novel training schemes in Bayesian experimental design (BED) with an ensemble of deep neural operators (DNOs). This model-agnostic framework pairs a BED scheme that actively selects data for quantifying extreme events with an ensemble of DNOs that approximate infinite-dimensional nonlinear operators. We find that not only does this framework clearly beat Gaussian processes (GPs) but that 1) shallow ensembles of just two members perform best; 2) extremes are uncovered regardless of the state of initial data (i.e. with or without extremes); 3) our method eliminates "double-descent" phenomena; 4) the use of batches of suboptimal acquisition points compared to step-by-step global optima does not hinder BED performance; and 5) Monte Carlo acquisition outperforms standard minimizers in high-dimensions. Together these conclusions form the foundation of an AI-assisted experimental infrastructure that can efficiently infer and pinpoint critical situations across many domains, from physical to societal systems.Comment: 19 pages, 7 figures, Submitted to Nature Computational Scienc

Lift-up, Kelvin-Helmholtz and Orr mechanisms in turbulent jets

Three amplification mechanisms present in turbulent jets, namely lift-up, Kelvin–Helmholtz and Orr, are characterized via global resolvent analysis and spectral proper orthogonal decomposition (SPOD) over a range of Mach numbers. The lift-up mechanism was recently identified in turbulent jets via local analysis by Nogueira et al. (J. Fluid Mech., vol. 873, 2019, pp. 211–237) at low Strouhal number ( St ) and non-zero azimuthal wavenumbers ( m ). In these limits, a global SPOD analysis of data from high-fidelity simulations reveals streamwise vortices and streaks similar to those found in turbulent wall-bounded flows. These structures are in qualitative agreement with the global resolvent analysis, which shows that they are a response to upstream forcing of streamwise vorticity near the nozzle exit. Analysis of mode shapes, component-wise amplitudes and sensitivity analysis distinguishes the three mechanisms and the regions of frequency–wavenumber space where each dominates, finding lift-up to be dominant as St/m→0 . Finally, SPOD and resolvent analyses of localized regions show that the lift-up mechanism is present throughout the jet, with a dominant azimuthal wavenumber inversely proportional to streamwise distance from the nozzle, with streaks of azimuthal wavenumber exceeding five near the nozzle, and wavenumbers one and two most energetic far downstream of the potential core

Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets

Response modes computed via linear resolvent analysis of the turbulent mean-flow field have been shown to qualitatively capture characteristics of the observed turbulent coherent structures in both wall-bounded and free shear flows. To make such models predictive, the nonlinear forcing term must be closed either by including a self-consistent set of triadic interactions or through turbulence modeling. For the latter, several investigators have proposed using the mean-field eddy viscosity acting linearly on the fluctuation field. In this study, a data-driven approach is taken to quantitatively improve linear resolvent models by deducing an optimal eddy-viscosity field that maximizes the projection of the dominant resolvent mode to the energy-optimal coherent structure educed using spectral proper orthogonal decomposition (SPOD) of data from high-fidelity simulations. We use large-eddy simulation databases for round isothermal jets at subsonic, transonic, and supersonic conditions and show that the optimal eddy viscosity substantially improves the alignment between resolvent and SPOD modes, reaching over 90% alignment at those frequencies where the jet exhibits a low-rank response. We then consider a fixed model for the eddy viscosity and show that with the calibration of a single constant, the results are generally close to the optimal one. In particular, the use of a standard Reynolds-Averaged-Navier-Stokes (RANS) eddy-viscosity resolvent model, with a single scaling coefficient, provides substantial agreement between SPOD and resolvent modes for three turbulent jets and across the most energetic wavenumbers and frequencies

Resolvent-based analysis of streaks in turbulent jets

Large scale, elongated structures, similar those ones widely studied in wall-bounded flows, are also present in turbulent jets. Several characteristics of these streaks can be identified via reduced order models such as resolvent analysis. The present work involves a resolvent-based study of these structures in turbulent jets. We focus on obtaining the optimal forcing that generates these energetic coherent structures. Results are compared with experimental data post-processed using spectral proper orthogonal decomposition, allowing us to draw conclusions about the nature of the non-linear forcing, since the two analyses should provide equivalent results if this term is modelled as spatially white. By identifying streaks in a global framework, we expect to better understand the mechanism by which they are generated

Neighborhood Racial Characteristics, Credit History, and Bankcard Credit in Indian Country

We examine whether concerns about lenders’ discrimination based on community racial characteristics can be empirically substantiated in the context of neighborhoods on and near American Indian reservations. Drawing on a large-scale dataset consisting of individual-level credit bureau records, we find that residing in a predominantly American Indian neighborhood is ceteris paribus associated with worse bankcard credit outcomes than residing in a neighborhood where the share of American Indian residents is low. While these results are consistent with the possibility of lenders’ discrimination based on community racial characteristics, we explain why our findings should not be readily interpreted as conclusive evidence thereof. We further find that consumer’s credit history is a robust and quantitatively more important predictor of bankcard credit outcomes than racial composition of the consumer’s neighborhood, and that the consumer’s location vis-à-vis a reservation exhibits no effect on bankcard credit outcomes