Aaron Towne

We present a resolvent-based framework for estimating turbulent velocity fluctuations in the wake of a spanwise-periodic NACA0012 airfoil at Mach 0.3, Reynolds number 23 000, and an angle of attack of $6^{\circ }$ . Building on the methodology of Jung et al. (2025, J. Fluid Mech. 1016, A41), we extend the approach to the more complex regime of a turbulent wake, which involves three primary challenges: (i) globally unstable modes in the linearised Navier–Stokes operator, (ii) multi-scale turbulent structures and (iii) high-dimensional datasets. To address these challenges, we employ a data-driven approach that constructs causal resolvent-based estimation kernels from cross-spectral densities obtained via large-eddy simulations. These kernels are derived using the Wiener–Hopf method, which optimally enforces causality, thereby enhancing real-time estimation accuracy. The framework captures the spectral signatures of coherent structures and, through the empirically determined cross-spectral densities, implicitly accounts for the coloured statistics of the nonlinear forcing acting on the linear system. To handle the computational demands of the high-dimensional estimation problem, we utilise parallel algorithms developed within the same framework. We further investigate sensor placement by analysing single-sensor estimation error and coherence with target flow quantities. Results demonstrate accurate causal estimation of streamwise velocity for the spanwise-averaged, spanwise-Fourier-transformed and mid-span flow using limited shear-stress measurements on the surface of the airfoil. This study underscores the potential of the resolvent-based framework for efficient estimation in compressible, turbulent environments.

We use a series of global models to investigate the linear and nonlinear interactions between shock cells, Kelvin-Helmholtz waves, guided jet modes, and other fluctuations in a screeching jet. First, we identify a set of lightly damped global eigenmodes of the Navier-Stokes operator linearized about the mean flow and show that they result from interactions with different shock-cell wavenumbers. Second, we use resolvent analysis to study the linear input-output behavior of the jet and obtain a time-periodic representation of the screech mode, which compares favorably with experimental data. Third, we use harmonic resolvent analysis to study triadic interactions, including inter-frequency energy transfer, between the screech mode determined from resolvent analysis and other fluctuations in the jet. The components of the optimal harmonic resolvent mode at harmonics of the screech frequency match experimental observations that have not been previously predicted by global models. Fourth, we leverage a novel bilinear formulation of harmonic resolvent analysis to study the impact of the screech mode's nonlinear self-interaction on other fluctuations in the jet. We show that the forcing provided by this nonlinear self-interaction of the screech mode, along with its triadic interactions with other frequencies embedded within the harmonic resolvent operator, is sufficient to explain the redistribution of energy to other frequencies and the associated experimental observations. In aggregate, these findings underscore the critical role of triadic and nonlinear interactions in shaping screech dynamics and offer a promising workflow for studying similar interactions in other flows dominated by periodic motions.

We use a series of global models to investigate the linear and nonlinear interactions between shock cells, Kelvin-Helmholtz waves, guided jet modes, and other fluctuations in a screeching jet. First, we identify a set of lightly damped global eigenmodes of the Navier-Stokes operator linearized about the mean flow and show that they result from interactions with different shock-cell wavenumbers. Second, we use resolvent analysis to study the linear input-output behavior of the jet and obtain a time-periodic representation of the screech mode, which compares favorably with experimental data. Third, we use harmonic resolvent analysis to study triadic interactions, including inter-frequency energy transfer, between the screech mode determined from resolvent analysis and other fluctuations in the jet. The components of the optimal harmonic resolvent mode at harmonics of the screech frequency match experimental observations that have not been previously predicted by global models. Fourth, we leverage a novel bilinear formulation of harmonic resolvent analysis to study the impact of the screech mode's nonlinear self-interaction on other fluctuations in the jet. We show that the forcing provided by this nonlinear self-interaction of the screech mode, along with its triadic interactions with other frequencies embedded within the harmonic resolvent operator, is sufficient to explain the redistribution of energy to other frequencies and the associated experimental observations. In aggregate, these findings underscore the critical role of triadic and nonlinear interactions in shaping screech dynamics and offer a promising workflow for studying similar interactions in other flows dominated by periodic motions.

Linear growth of disturbances plays a crucial role in the laminar-to-turbulent transition in boundary layers as well as other wall-bounded flows. Though the linearized Navier-Stokes operator may be stable in such flows, the non-normality of the operator can lead to significant transient growth of disturbances, which can ultimately lead to transition. The importance of this transition mechanism is usually quantified by the maximum energy growth undergone by any disturbance to the boundary layer. Clearly, this is an upper bound for the (linear) transient growth experienced by real disturbances, and a statistical framework that studies the average, rather than extreme, disturbance behavior has recently been proposed. The framework is based on the assumption that disturbances evolve linearly, and it gives an exact relation between the two-point correlation tensor of the incipient disturbances and the turbulent intensity of downstream ones. The primary aims of this work are to implement this framework for a transitional Blasius boundary layer and to conduct a direct numerical simulation (DNS) of the same flow in order to test the predictions of the statistical framework. Applying the framework to spatial transient growth requires the use of a spatial marching method, and we use the one-way Navier-Stokes equations (OWNS). We also develop a reduced-basis technique to approximate formulae in the statistical framework that would otherwise be intractable for this flow. To model the statistics of the free-stream turbulence, the inflow disturbances are prescribed using the von K\'arm\'an energy spectrum. Finally, we present a comparison of the kinetic energy growth obtained from the DNS with that predicted by the statistical framework.

A novel statistical framework developed by Frame and Towne (2024) is employed and extended to examine how initial conditions influence the linear evolution of the Rayleigh-Taylor instability. The analysis focuses on the evolution of the mean energy for two representative classes of initial perturbations. The first class, based on a Gaussian spatial correlation, reveals a time delay in the onset of instability growth and a non-monotonic dependence of this delay on the perturbation correlation length. This behavior suggests a low-pass-filter-like selection of modes governing the evolution of the mean energy. The second class consists of perturbations formed as linear combinations of Fourier modes with random phase shifts and amplitudes drawn from a prescribed spectrum. For this case, an analytical expression for the mean energy growth is derived, linking the spectral content of the initial perturbations to the temporal evolution of the energy through the initial spectral amplitudes. This formulation enables direct comparison with ensembles of three-dimensional direct numerical simulations, demonstrating both the framework's predictive capability and the limitations imposed by the problem's non-autonomous nature. Finally, the effects of viscosity and density stratification are analyzed, showing how these factors modify the time-delay mechanism and influence the early-stage evolution of the instability.

High-speed-jet turbulent mixing noise remains a challenging problem, and here we aim to reduce it using a wavepacket-cancellation strategy. This approach is enabled by the recently developed resolvent-based estimation and control framework, which uses near-nozzle sensors to detect noise-generating wavepackets and suppress them via actuation. This paper presents three main results toward this larger goal: (i) data-driven estimation for a Mach 1.5 supersonic jet using large-eddy simulations to identify coherent structures and inform sensor-target placement; (ii) resolvent-based estimation for the linearized jet, which achieves reasonable accuracy in reconstructing relevant flow features from limited sensor data; and (iii) preliminary resolvent-based control for the linearized jet, demonstrating a 34% reduction in the root mean square of streamwise-momentum fluctuations using only one sensor and one actuator. These findings demonstrate the potential of the resolvent-based framework for mitigating noise-generating wavepacket structures in supersonic jets and provide an important foundation for future computational and experimental investigations.

We introduce a community challenge designed to facilitate direct comparisons between data-driven methods for compression, forecasting, and sensing of complex aerospace flows. The challenge is organized into three tracks that target these complementary capabilities: compression (compact representations for large datasets), forecasting (predicting future flow states from a finite history), and sensing (inferring unmeasured flow states from limited measurements). Across these tracks, multiple challenges span diverse flow datasets and use cases, each emphasizing different model requirements. The challenge is open to anyone, and we invite broad participation to build a comprehensive and balanced picture of what works and where current methods fall short. To support fair comparisons, we provide standardized success metrics, evaluation tools, and baseline implementations, with one classical and one machine-learning baseline per challenge. Final assessments use blind tests on withheld data. We explicitly encourage negative results and careful analyses of limitations. Outcomes will be disseminated through an AIAA Journal Virtual Collection and invited presentations at AIAA conferences.

We introduce a community challenge designed to facilitate direct comparisons between data-driven methods for compression, forecasting, and sensing of complex aerospace flows. The challenge is organized into three tracks that target these complementary capabilities: compression (compact representations for large datasets), forecasting (predicting future flow states from a finite history), and sensing (inferring unmeasured flow states from limited measurements). Across these tracks, multiple challenges span diverse flow datasets and use cases, each emphasizing different model requirements. The challenge is open to anyone, and we invite broad participation to build a comprehensive and balanced picture of what works and where current methods fall short. To support fair comparisons, we provide standardized success metrics, evaluation tools, and baseline implementations, with one classical and one machine-learning baseline per challenge. Final assessments use blind tests on withheld data. We explicitly encourage negative results and careful analyses of limitations. Outcomes will be disseminated through an AIAA Journal Virtual Collection and invited presentations at AIAA conferences.

Most model reduction methods reduce the state dimension and then temporally evolve a set of coefficients that encode the state in the reduced representation. In this paper, we instead employ an efficient representation of the entire trajectory of the state over some time interval of interest and then solve for the static coefficients that encode the trajectory on the interval. We use spectral proper orthogonal decomposition (SPOD) modes, which are provably optimal for representing long trajectories and substantially outperform any representation of the trajectory in a purely spatial basis (e.g., POD). We develop a method to solve for the SPOD coefficients that encode the trajectories for forced linear dynamical systems given the forcing and initial condition, thereby obtaining the accurate prediction of the dynamics afforded by the SPOD representation of the trajectory. The method, which we refer to as spectral solution operator projection (SSOP), is derived by projecting the general time-domain solution for a linear time-invariant system onto the SPOD modes. We demonstrate the new method using two examples: a linearized Ginzburg-Landau equation and an advection-diffusion problem. In both cases, the error of the proposed method is orders of magnitude lower than that of POD-Galerkin projection and balanced truncation. The method is also fast, with CPU time comparable to or lower than both benchmarks in our examples. Finally, we describe a data-free space-time method that is a derivative of the proposed method and show that it is also more accurate than balanced truncation in most cases.

We develop an optimal resolvent-based estimator and controller to predict and attenuate unsteady vortex-shedding fluctuations in the laminar wake of a NACA 0012 airfoil at an angle of attack of 6.5°, chord-based Reynolds number of 5000 and Mach number of 0.3. The resolvent-based estimation and control framework offers several advantages over standard methods. Under equivalent assumptions, the resolvent-based estimator and controller reproduce the Kalman filter and LQG controller, respectively, but at substantially lower computational cost using either an operator-based or data-driven implementation. Unlike these methods, the resolvent-based approach can naturally accommodate forcing terms (nonlinear terms from Navier–Stokes) with coloured-in-time statistics, significantly improving estimation accuracy and control efficacy. Causality is optimally enforced using a Wiener–Hopf formalism. We integrate these tools into a high-performance-computing-ready compressible flow solver and demonstrate their effectiveness for estimating and controlling velocity fluctuations in the wake of the airfoil immersed in clean and noisy free streams, the latter of which prevents the flow from falling into a periodic limit cycle. Using four shear–stress sensors on the surface of the airfoil, the resolvent-based estimator predicts a series of downstream targets with approximately $3\,\%$ and $30\,\%$ error for the clean and noisy free stream conditions, respectively. For the latter case, using four actuators on the airfoil surface, the resolvent-based controller reduces the turbulent kinetic energy in the wake by $98\,\%$ .

We develop an optimal resolvent-based estimator and controller to predict and attenuate unsteady vortex shedding fluctuations in the laminar wake of a NACA 0012 airfoil at an angle of attack of 6.5 degrees, chord-based Reynolds number of 5000, and Mach number of 0.3. The resolvent-based estimation and control framework offers several advantages over standard methods. Under equivalent assumptions, the resolvent-based estimator and controller reproduce the Kalman filter and LQG controller, respectively, but at substantially lower computational cost using either an operator-based or data-driven implementation. Unlike these methods, the resolvent-based approach can naturally accommodate forcing terms (nonlinear terms from Navier-Stokes) with colored-in-time statistics, significantly improving estimation accuracy and control efficacy. Causality is optimally enforced using a Wiener-Hopf formalism. We integrate these tools into a high-performance-computing-ready compressible flow solver and demonstrate their effectiveness for estimating and controlling velocity fluctuations in the wake of the airfoil immersed in clean and noisy freestreams, the latter of which prevents the flow from falling into a periodic limit cycle. Using four shear-stress sensors on the surface of the airfoil, the resolvent-based estimator predicts a series of downstream targets with approximately 3% and 30% error for the clean and noisy freestream conditions, respectively. For the latter case, using four actuators on the airfoil surface, the resolvent-based controller reduces the turbulent kinetic energy in the wake by 98%.

Resolvent analysis is a powerful tool for studying coherent structures in turbulent flows. However, its application beyond canonical flows with symmetries that can be used to simplify the problem to inherently three-dimensional flows and other large systems has been hindered by the computational cost of computing resolvent modes. In particular, the CPU and memory requirements of state-of-the-art algorithms scale poorly with the problem dimension, \ie the number of discrete degrees of freedom. In this paper, we present RSVD-$\Delta t$, a novel approach that overcomes these limitations by combining randomized singular value decomposition with an optimized time-stepping method for computing the action of the resolvent operator. Critically, the CPU cost and memory requirements of the algorithm scale linearly with the problem dimension. We develop additional strategies to minimize these costs and control errors. We validate the algorithm using a Ginzburg-Landau test problem and demonstrate RSVD-$\Delta t$'s low cost and improved scaling using a three-dimensional discretization of a turbulent jet. Lastly, we use it to study the impact of low-speed streaks on the development of Kelvin-Helmholtz wavepackets in the jet via secondary stability analysis, a problem that would have been intractable using previous algorithms.

We propose a space-time reduced-order model (ROM) for nonlinear dynamical systems, building upon previous work on linear systems. Whereas most ROMs are space-only in that they reduce only the spatial dimension of the state, the proposed method leverages an efficient encoding of the entire trajectory of the state on the time interval $[0,T]$, enabling significant additional reduction. Trajectories are encoded using SPOD modes, a spatial basis at each temporal frequency tailored to the structures that appear at that frequency. These modes have a number of properties that make them an ideal choice for space-time model reduction, including separability and near-optimality for long trajectories. We derive a system of algebraic equations involving the SPOD coefficients, forcing, and initial condition by projecting an implicit solution of the governing equations onto the set of SPOD modes in a space-time inner product. We therefore refer to the method as spectral solution operator projection (SSOP). The online phase of SSOP comprises solving this system for the SPOD coefficients, given the initial condition and forcing. We find that SSOP gives two orders of magnitude lower error than POD-Galerkin projection at the same number of modes and CPU time across a suite of tests, including ones that use out-of-sample forcings and affine parameter variation. In fact, the method is substantially more accurate even than the projection of the solution onto the POD modes, which is a lower bound for the error of any method based on a linear space-only encoding of the state.

Guided-jet waves have been shown to close resonance loops in a myriad of problems such as screech and impingement tones in jets. These discrete, upstream-travelling waves have long been identified in linear-stability models of jet flows, but in this work they are instead considered in the context of an acoustic-scattering problem. It is shown that the guided-jet mode results from total internal reflection and transmission of acoustic waves, arising from the shear layer behaving like a duct with some given wall impedance. After total reflection, only discrete streamwise wavenumbers may be supported by the flow, with these wavenumbers dictated by the fact that the standing wave formed inside of the jet must fit between the two shear layers. Close to the sonic line, the transmission of this mode to the outside is maximum, leading to a net-energy flux directed upstream, which dictates the direction of propagation of this mode, providing a clear connection to the better understood soft-duct mode (Towne et al. , J. Fluid Mech. , vol. 825, 2017, pp. 1113–1152). The model also indicates that these waves are generated in the core of the flow and can only be efficiently transmitted to the quiescent region under certain conditions, providing an explanation as to why screech is only observed at conditions where the discrete mode is supported by the flow. The present results explain, for the first time, the nature and characteristics of the guided-jet waves.

This work aims to provide a more complete understanding of the resonance mechanisms that occur in turbulent jets at high subsonic Mach number, as shown by Towne et al. (J. Fluid Mech., vol. 825, 2017, pp. 1113-1152). Resonance was suggested by that study to exist between upstream- and downstream-travelling guided waves. Five possible resonance mechanisms were postulated, each involving different families of guided waves that reflect in the nozzle exit plane and at a number of downstream turning points. However, that study did not identify which of the five resonance mechanisms underpin the observed spectral peaks. In this work, the waves underpinning resonance are identified via a biorthogonal projection of Large Eddy Simulation data on eigenbases provided by a locally parallel linear stability analysis. Two of the five scenarios postulated by Towne et al. are thus confirmed to exist in the turbulent jet. The reflection-coefficients in the nozzle exit and turning-point planes are, furthermore, identified. Such information is required as input for simplified resonance-modelling strategies such as developed in Jordan et al. (J. Fluid Mech., vol. 853, 2018, pp. 333-358) for jet-edge resonance, and in Mancinelli et al. (Exp. Fluids, vol. 60, 2019, pp. 1-9) for supersonic screech.

We aim to reduce the noise emitted by high-speed turbulent jets using recently developed resolvent-based estimation and control tools. Our approach relies on detecting noise-generating wavepackets and canceling them via actuation. This paper reports on our progress toward this objective in the form of (i) implementation and validation of these resolvent-based tools in a large-scale CFD solver and (ii) preliminary estimation results for a subsonic jet. We validate our implementation via comparisons to the literature for a laminar channel flow, the acoustic response to a monopole forcing in a freestream, a trailing-line vortex problem, an airfoil wake, and resolvent modes for a jet. The preliminary estimation study for the subsonic jet shows that operator-based and data-driven versions of the methods yield similar estimation kernels and results. Future work will focus on extending this study to a series of supersonic jets and systematically exploring the selection and placement of sensors, actuators, and targets to mitigate noise-generating wavepackets most effectively.

Upstream-travelling guided jet waves have been shown to be one of the key elements in many resonance processes underpinned in high-speed jets. Despite its importance, many of its characteristics, including how these waves are generated and how it can travel subsonically, have not been detailed in the literature. In this work, we aim to provide a clarification about the dynamics of this mode. With the aid of an acoustic scattering formulation, we are able to show that the guided-jet mode results from total-internal-reflection and transmission to decaying waves, arising from the shear layer behaving like a hard duct. After total reflection, only discrete streamwise wavenumbers may be supported by the flow, with these wavenumbers dictated by the fact that the standing wave formed inside of the jet must fit between the two shear layers. Close to the sonic line, the transmission of this mode to the outside is maximum, leading to a net-energy flux directed upstream, which dictates the direction of propagation of this mode in the eigenspectrum, providing a clear connection to the better understood soft-duct mode.

We present an extension of the RSVD-Δt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} algorithm initially developed for resolvent analysis of statistically stationary flows to handle harmonic resolvent analysis of time-periodic flows. The harmonic resolvent operator, as proposed by Padovan et al. (J Fluid Mech 900, 2020), characterizes the linearized dynamics of time-periodic flows in the frequency domain, and its singular value decomposition reveals forcing and response modes with optimal energetic gain. However, computing harmonic resolvent modes poses challenges due to (i) the coupling of all Nω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\omega }$$\end{document} retained frequencies into a single harmonic resolvent operator and (ii) the singularity or near-singularity of the operator, making harmonic resolvent analysis considerably more computationally expensive than a standard resolvent analysis. To overcome these challenges, the RSVD-Δt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} algorithm leverages time stepping of the underlying time-periodic linearized Navier–Stokes operator, which is Nω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\omega }$$\end{document} times smaller than the harmonic resolvent operator, to compute the action of the harmonic resolvent operator. We develop strategies to minimize the algorithm’s CPU and memory consumption, and our results demonstrate that these costs scale linearly with the problem dimension. We validate the RSVD-Δt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} algorithm by computing modes for a periodically varying Ginzburg–Landau equation and demonstrate its performance using the flow over an airfoil.

Most model reduction methods reduce the state dimension and then temporally evolve a set of coefficients that encode the state in the reduced representation. In this paper, we instead employ an efficient representation of the entire trajectory of the state over some time interval of interest and then solve for the static coefficients that encode the trajectory on the interval. We use spectral proper orthogonal decomposition (SPOD) modes, which are provably optimal for representing long trajectories and substantially outperform any representation of the trajectory in a purely spatial basis (e.g., POD). We develop a method to solve for the SPOD coefficients that encode the trajectories for forced linear dynamical systems given the forcing and initial condition, thereby obtaining the accurate prediction of the dynamics afforded by the SPOD representation of the trajectory. The method, which we refer to as spectral solution operator projection (SSOP), is derived by projecting the general time-domain solution for a linear time-invariant system onto the SPOD modes. We demonstrate the new method using two examples: a linearized Ginzburg-Landau equation and an advection-diffusion problem. In both cases, the error of the proposed method is orders of magnitude lower than that of POD-Galerkin projection and balanced truncation. The method is also fast, with CPU time comparable to or lower than both benchmarks in our examples. Finally, we describe a data-free space-time method that is a derivative of the proposed method and show that it is also more accurate than balanced truncation in most cases.

An explanation for the origin and number of clumps along the equatorial ring of Supernova 1987A has eluded decades of research. Our linear analysis and hydrodynamic simulations of the expanding ring prior to the supernova reveal that it is subject to the Crow instability between vortex cores. The dominant wave number is remarkably consistent with the number of clumps, suggesting that the Crow instability stimulates clump formation. Although the present analysis focuses on linear fluid flow, future nonlinear analysis and the incorporation of additional stellar physics may further elucidate the remnant structure and the evolution of the progenitor and other stars.

The theory of transient growth describes how linear mechanisms can cause temporary amplification of disturbances even when the linearized system is asymptotically stable as defined by its eigenvalues. This growth is traditionally quantified by finding the initial disturbance that generates the maximum response at the peak time of its evolution. However, this can vastly overstate the growth of a real disturbance. In this paper, we introduce a statistical perspective on transient growth that models statistics of the energy amplification of the disturbances. We derive a formula for the mean energy amplification and spatial correlation of the growing disturbance in terms of the spatial correlation of the initial disturbance. The eigendecomposition of the correlation provides the most prevalent structures, which are the statistical analogue of the standard left singular vectors of the evolution operator. We also derive accurate confidence bounds on the growth by approximating the probability density function of the energy. Applying our analysis to Poiseuille flow yields a number of observations. First, the mean energy amplification is often drastically smaller than the maximum. In these cases, it is exceedingly unlikely to achieve near-optimal growth due to the exponential behaviour observed in the probability density function. Second, the characteristic length scale of the initial disturbances has a significant impact on the expected growth, with large-scale initial disturbances growing orders of magnitude more than small-scale ones. Finally, while the optimal growth scales quadratically with Reynolds number, the mean energy amplification scales only linearly for certain reasonable choices of the initial correlation.

The growth of perturbations subject to the Crow instability along two vortex rings of equal and opposite circulation undergoing a head-on collision is examined. Unlike the planar case for semi-infinite line vortices, the zero-order geometry of the flow (i.e. the ring radius, core thickness and separation distance) and by extension the growth rates of perturbations vary in time. The governing equations are therefore temporally integrated to characterize the perturbation spectrum. The analysis, which considers the effects of ring curvature and the distribution of vorticity within the vortex cores, explains several key flow features observed in experiments. First, the zero-order motion of the rings is accurately reproduced. Next, the predicted emergent wavenumber, which sets the number of secondary vortex structures emerging after the cores come into contact, agrees with experiments, including the observed increase in the number of secondary structures with increasing Reynolds number. Finally, the analysis predicts an abrupt transition at a critical Reynolds number to a regime dominated by a higher-frequency, faster-growing instability mode that may be consistent with the experimentally observed rapid generation of a turbulent puff following the collision of rings at high Reynolds numbers.

We use a resolvent-based approach to estimate turbulent fluctuations in the near-wake of a spanwise-periodic NACA0012 airfoil at Ma = 0.3, Re = 23,000, and an angle of attack of a=6◦. To circumvent the challenge posed by the global instability of the associated linearized Navier-Stokes operator, we use a data-driven approach for obtaining optimal resolvent-based kernels for flow estimation. The data are obtained from a large-eddy simulation and are used to compute the cross-spectra that appear in the estimation kernels. The Wiener-Hopf formalism is then used to optimally enforce causality in the kernels, improving their accuracy for real-time estimation compared to a naive truncation of the non-causal part. By construction, the kernels include the impact of the colored statistics of nonlinear terms from the Navier-Stokes equations that act as a forcing on the linear dynamics. Our results demonstrate the effectiveness of the resolvent-based approach for estimating turbulent fluctuations of the spanwise-averaged and three-dimensional flows.

This paper has recently been accepted in the Journal of Fluid Mechanics.Abstract: Guided jet waves have been shown to close resonance loops in a myriad of problems such as screech and impingement tones in jets. These discrete, upstream-traveling waves have long been identified in linear-stability models of jet flows, but in this work they are instead considered in the context of an acoustic-scattering problem. It is shown that the guided-jet mode results from total internal reflection and transmission of acoustic waves, arising from the shear layer behaving like a duct with some given wall impedance. After total reflection, only discrete streamwise wavenumbers may be supported by the flow, with these wavenumbers dictated by the fact that the standing wave formed inside of the jet must fit between the two shear layers. Close to the sonic line, the transmission of this mode to the outside is maximum, leading to a net-energy flux directed upstream, which dictates the direction of propagation of this mode, providing a clear connection to the better understood soft-duct mode (Towne et al., J. Fluid Mech., vol 825, 2017, pp. 1113-1152). The model also indicates that these waves are generated in the core of the flow and can only be efficiently transmitted to the quiescent region under certain conditions, providing an explanation as to why screech is only observed at conditions where the discrete mode is supported by the flow. The present results explain, for the first time, the nature and characteristics of the guided jet waves.

This paper has recently been accepted in the Journal of Fluid Mechanics.Abstract: Guided jet waves have been shown to close resonance loops in a myriad of problems such as screech and impingement tones in jets. These discrete, upstream-traveling waves have long been identified in linear-stability models of jet flows, but in this work they are instead considered in the context of an acoustic-scattering problem. It is shown that the guided-jet mode results from total internal reflection and transmission of acoustic waves, arising from the shear layer behaving like a duct with some given wall impedance. After total reflection, only discrete streamwise wavenumbers may be supported by the flow, with these wavenumbers dictated by the fact that the standing wave formed inside of the jet must fit between the two shear layers. Close to the sonic line, the transmission of this mode to the outside is maximum, leading to a net-energy flux directed upstream, which dictates the direction of propagation of this mode, providing a clear connection to the better understood soft-duct mode (Towne et al., J. Fluid Mech., vol 825, 2017, pp. 1113-1152). The model also indicates that these waves are generated in the core of the flow and can only be efficiently transmitted to the quiescent region under certain conditions, providing an explanation as to why screech is only observed at conditions where the discrete mode is supported by the flow. The present results explain, for the first time, the nature and characteristics of the guided jet waves.

We present an extension of the RSVD-$Δt$ algorithm initially developed for resolvent analysis of statistically stationary flows to handle harmonic resolvent analysis of time-periodic flows. The harmonic resolvent operator, as proposed by \citet{Padovanetal20}, characterizes the linearized dynamics of time-periodic flows in the frequency domain, and its singular value decomposition reveals forcing and response modes with optimal energetic gain. However, computing harmonic resolvent modes poses challenges due to $(i)$ the coupling of all $N_ω$ retained frequencies into a single harmonic resolvent operator and $(ii)$ the singularity or near-singularity of the operator, making harmonic resolvent analysis considerably more computationally expensive than a standard resolvent analysis. To overcome these challenges, the RSVD-$Δt$ algorithm leverages time stepping of the underlying time-periodic linearized Navier-Stokes operator, which is $N_ω$ times smaller than the harmonic resolvent operator, to compute the action of the harmonic resolvent operator. We develop strategies to minimize the algorithm's CPU and memory consumption, and our results demonstrate that these costs scale linearly with the problem dimension. We validate the RSVD-$Δt$ algorithm by computing modes for a periodically varying Ginzburg-Landau equation and demonstrate its performance using the flow over an airfoil.

The origin of the clumps along the gaseous circumstellar ring surrounding the remnant of Supernova 1987A has puzzled scientists for decades.Twenty-thousand years prior to the supernova, the interaction between solar wind from the progenitor star and the ring likely generated vorticity conducive to the formation of a circular vortex dipole subject to the cylindrical Crow instability, as shown above.Our analysis predicts a dominant unstable wavenumber consistent with the number of clumps, and simulations reproduce both the clumping behavior and the thin annulus of mass, shed by the vortex dipole, recently observed with the James Webb Space Telescope, as shown below.

Resolvent analysis is a powerful tool for studying coherent structures in turbulent flows. However, its application beyond canonical flows with symmetries that can be used to simplify the problem to inherently three-dimensional flows and other large systems has been hindered by the computational cost of computing resolvent modes. In particular, the CPU and memory requirements of state-of-the-art algorithms scale poorly with the problem dimension, \ie the number of discrete degrees of freedom. In this paper, we present RSVD-$Δt$, a novel approach that overcomes these limitations by combining randomized singular value decomposition with an optimized time-stepping method for computing the action of the resolvent operator. Critically, the CPU cost and memory requirements of the algorithm scale linearly with the problem dimension. We develop additional strategies to minimize these costs and control errors. We validate the algorithm using a Ginzburg-Landau test problem and demonstrate RSVD-$Δt$'s low cost and improved scaling using a three-dimensional discretization of a turbulent jet. Lastly, we use it to study the impact of low-speed streaks on the development of Kelvin-Helmholtz wavepackets in the jet via secondary stability analysis, a problem that would have been intractable using previous algorithms.

Time-delay embedding is an increasingly popular starting point for data-driven reduced-order modeling efforts. In particular, the singular value decomposition (SVD) of a block Hankel matrix formed from successive delay embeddings of the state of a dynamical system lies at the heart of several popular reduced-order modeling methods. In this paper, we show that the left singular vectors of this Hankel matrix are a discrete approximation of space-time proper orthogonal decomposition (POD) modes, and the singular values are square roots of the POD energies. Analogous to the connection between the SVD of a data matrix of snapshots and space-only POD, this connection establishes a clear interpretation of the Hankel modes grounded in classical theory, and we gain insights into the Hankel modes by instead analyzing the equivalent discrete space-time POD modes in terms of the correlation matrix formed by multiplying the Hankel matrix by its conjugate transpose. These insights include the distinct meaning of rows and columns, the implied norm in which the modes are optimal, the impact of the time step between snapshots on the modes, and an interpretation of the embedding dimension/height of the Hankel matrix in terms of the time window on which the modes are optimal. Moreover, the connections we establish offer opportunities to improve the convergence and computation time in certain practical cases, and to improve the accuracy of the modes with the same data. Finally, popular variants of POD, namely the standard space-only POD and spectral POD, are recovered in the limits that snapshots used to form each column of the Hankel matrix represent flow evolution over short and long times, respectively.

Modelling the noise emitted by turbulent jets is made difficult by their acoustic inefficiency: only a tiny fraction of the near-field turbulent kinetic energy is propagated to the far field as acoustic waves. As a result, jet-noise models must accurately capture this small, acoustically efficient component hidden among comparatively inefficient fluctuations. In this paper, we identify this acoustically efficient near-field source from large-eddy simulation data and use it to inform a predictive model. Our approach uses the resolvent framework, in which the source takes the form of nonlinear fluctuation terms that act as a forcing on the linearised Navier–Stokes equations. First, we identify the forcing that, when acted on by the resolvent operator, produces the leading spectral proper orthogonal decomposition modes in the acoustic field for a Mach 0.4 jet. Second, the radiating components of this forcing are isolated by retaining only portions with a supersonic phase speed. This component makes up less than 0.05 % of the total forcing energy but generates most of the acoustic response, especially at peak (downstream) radiation angles. Finally, we propose an empirical model for the identified acoustically efficient forcing components. The model is tested at other Mach numbers and flight-stream conditions and predicts noise within 2 dB accuracy for a range of frequencies, downstream angles and flight conditions.

We investigate the intermittency of the coupling behaviour in screeching twin round supersonic jets at low Mach numbers across a range of nozzle spacings. Application of proper orthogonal decomposition combined with time-frequency wavelet analysis and spectral proper orthogonal decomposition shows that intermittency can manifest in twin jets as either a competition between the two symmetries, or the jets uncoupling and recoupling. The time scales on which symmetry switching occurs can vary strongly, ranging from $O(10^2)$ to $O(10^3)$ screech cycles. A transition from one symmetry to another is accompanied by a slight change in the screech frequency ranging from 0.30 % to 0.63 %. It was observed that complete uncoupling occurred only at the largest nozzle spacing of $s/D=6$ and at Mach numbers close to modal staging. When the jets are uncoupled they screech at slightly different frequencies, with a disparity of approximately 0.6 %. The coupling is particularly intermittent in the transition from the A1 to A2 branch, where the A2 mode is first observed, and tends toward steady coupling with increasing Mach number.

We present a publicly accessible database specifically designed to aid in the conception, training, demonstration, evaluation, and comparison of reduced-complexity models for fluid mechanics. Availability of high-quality flow data is essential for all of these aspects of model development for both data-driven and physics-based methods. The current database is unique in that it has been curated with this need in mind. The database contains time-resolved data for six distinct datasets: a large eddy simulation of a turbulent jet, direct numerical simulations of a zero-pressure-gradient turbulent boundary layer, particle-image-velocimetry measurements for the same boundary layer at several Reynolds numbers, direct numerical simulations of laminar stationary and pitching flat-plate airfoils, particle-image-velocimetry and force measurements of an airfoil encountering a gust, and a large eddy simulation of the separated, turbulent flow over an airfoil. These six cases span several key flow categories: laminar and turbulent, statistically stationary and transient, tonal and broadband spectral content, canonical and application-oriented, wall-bounded and free-shear flow, and simulation and experimental measurements. For each dataset, we describe the flow setup and computational/experimental methods, catalog the data available in the database, and provide examples of how these data can be used for reduced-complexity modeling. All data can be downloaded using a browser interface or Globus. Our vision is that the common testbed provided by this database will aid the fluid mechanics community in clarifying the distinct capabilities of new and existing methods.

This work aims to provide a more complete understanding of the resonance mechanisms that occur in turbulent jets at high subsonic Mach number, as shown by Towne et al. (J. Fluid Mech., vol. 825, 2017, pp. 1113-1152). Resonance was suggested by that study to exist between upstream- and downstream-travelling guided waves. Five possible resonance mechanisms were postulated, each involving different families of guided waves that reflect in the nozzle exit plane and at a number of downstream turning points. However, that study did not identify which of the five resonance mechanisms underpin the observed spectral peaks. In this work, the waves underpinning resonance are identified via a biorthogonal projection of Large Eddy Simulation data on eigenbases provided by a locally parallel linear stability analysis. Two of the five scenarios postulated by Towne et al. are thus confirmed to exist in the turbulent jet. The reflection-coefficients in the nozzle exit and turning-point planes are, furthermore, identified. Such information is required as input for simplified resonance-modelling strategies such as developed in Jordan et al. (J. Fluid Mech., vol. 853, 2018, pp. 333-358) for jet-edge resonance, and in Mancinelli et al. (Exp. Fluids, vol. 60, 2019, pp. 1-9) for supersonic screech.

The theory of transient growth describes how linear mechanisms can cause temporary amplification of disturbances even when the linearized system is asymptotically stable as defined by its eigenvalues. This growth is traditionally quantified by finding the initial disturbance that generates the maximum response, in terms of energy gain, at the peak time of its evolution. While this bounds the growth, it can vastly overstate the growth of a real disturbance. In this paper, we introduce a statistical perspective on transient growth that models statistics of the energy amplification of the disturbances. We derive a formula for the mean energy amplification in terms of the two-point spatial correlation of the initial disturbance. We also derive an accurate approximation of the probability density function of the energy of the growing disturbance, from which confidence bounds on the growth can be obtained. Applying our analysis to Poisseuille flow yields a number of observations. First, the mean gain can be drastically smaller than the maximum, especially when the disturbances are broadband in wavenumber content. In these cases, it is exceedingly unlikely to achieve near-optimal growth due to the exponential behavior which we observe in the probability density function. Second, the characteristic length scale of the initial disturbances has a significant impact on the expected growth; specifically, large-scale initial disturbances produce orders-of-magnitude-larger expected growth than smaller scales, indicating that the length scale of incoming disturbances may be key in determining whether transient growth leads to transition for a particular flow. Finally, while the optimal growth scales quadratically with Reynolds number, we observe that the mean energy amplification scales only linearly for certain reasonable choices of the initial correlations.

View Video Presentation: https://doi.org/10.2514/6.2023-0077.vid We use a resolvent-based approach, recently developed by Martini et al. (J. Fluid Mech., vol. 937, 2022, A19), to estimate unsteady fluctuations in the near wake of a NACA 0012 airfoil at Ma = 0.3, Re = 5000, and AoA = 6.5. The flow is simulated using direct numerical simulation, and global stability and resolvent analyses about the mean flow are performed to verify the accuracy of the linearization and elucidate the dominant flow physics. The resolvent-based estimators are obtained using two approaches: 1) an operator-based approach, resulting in low computational cost without the need for a priori model reduction, and 2) a data-driven approach that avoids building the linearized Navier-Stokes operator and statistically accounts for the non-linearity of the flow. In both cases, a Wiener-Hopf formalism is used to optimally enforce causality. The resolvent-based estimators are then used to estimate unsteady fluctuations in the wake for the linear and nonlinear systems, which are forced by random upstream perturbations to break the periodic limit cycle produced by the vortex shedding and trigger chaotic fluctuations. The results demonstrate good accuracy in the near wake.