Oliver T. Schmidt

We show that both temporal and spatial symmetry breaking in canonical K-type boundary layer transition arise as organised structures with quantifiable energetic pathways rather than unstructured noise. Before the skin-friction maximum, the flow is described by a periodic, spanwise-symmetric fundamental harmonic response (FHR) to the Tollmien–Schlichting wave. The FHR is spatially compact, produces hairpin packets and remains fully harmonic despite a turbulence-like appearance, thereby delimiting the deterministic regime. Past this point, a distinct regime change occurs: a hierarchy of quasi-periodic and aperiodic structures emerges, followed shortly by anti-symmetric structures that develop similarly despite no anti-symmetric inputs. We identify these structures as symmetry-decomposed spectral and space–time proper orthogonal modes that resolve the progression from deterministic harmonics to broadband dynamics. We introduce inter-modal and inter-symmetry energy budgets derived from symmetry-decomposed Navier–Stokes equations. They reveal a directed energy transfer from the FHR into the leading temporal and spatial symmetry breaking modes and, subsequently, into broadband residual fluctuations, showing that broadband dynamics grow only once inter-modal transfer is active, while inter-symmetry transfer also strongly amplifies broadband anti-symmetric fluctuations once asymmetry is present. These key insights support a view of laminar–turbulent transition as a sequence of symmetry breaking events, energetically driven by dominant space–time modes that route energy from harmonic flow to broadband turbulence.

Abstract We introduce a new statistical framework that operates directly in space and time to detect shoaling internal waves as seabed temperature structures observed by a fiber optic distributed temperature sensing array deployed in the coastal waters offshore of California. The nonlinear internal wave exhibited a mean shoaling velocity of approximately m , with the most commonly observed speed near 0.08 m , a mean period of minutes, and arrived at the fiber at an average angle of incidence of relative to shore‐normal. This angle is similar to the orientation of the nearby Scripps branch of the La Jolla Canyon, suggesting that the canyon is a generation site. Nonlinear wave events preferentially occurred during specific phases of the background internal tide, clustered around the ebb phase. To investigate the impact of the oblique angle of approach on the dynamics of shoaling, the observations were used as inspiration for a idealized simulation of a shoaling internal wave approaching a slope at an angle. In the simulation, an initial solitary‐like depression wave evolved into elevation boluses propagating obliquely to the isobaths, remaining closely attached to the seafloor as they transported cold water upslope. The shoaling bolus both developed an organized along‐crest trailing wake and entrained upslope waters in ways that were quantitatively consistent with the observations. The oblique incidence generated vertically sheared along‐crest flow, which may contribute to the coherent structures forming in the trailing wake.

Energy transfer across scales is fundamental in fluid dynamics, linking large-scale flow motions to small-scale turbulent structures in engineering and natural environments. Triadic interactions among three wave components form complex networks across scales, challenging understanding and model reduction. We introduce triadic orthogonal decomposition (TOD), a method that identifies coherent flow structures optimally capturing spectral momentum transfer, quantifies their coupling and energy exchange in an energy-budget bispectrum and reveals the regions where they interact. Triadic orthogonal decomposition distinguishes three components – a momentum recipient, donor and catalyst – and recovers laws governing pairwise, six-triad and global triad conservation. We apply TOD to three examples: the classical cylinder wake, experimental wind turbine wake data and a direct numerical simulation of isotropic turbulence. Energy transfer can be spatially distributed but vanish upon integration or spatially localised but facilitate net interscale exchange, so a complete characterisation of nonlinearity requires examination of both integral and local transfers. In the cylinder wake, we link backscatter of energy from high to low frequencies to a compact attenuation region downstream of the cylinder. In the turbine wake, we confirm the known association between energy amplification and decay and vortex tilting, but observe more complex secondary mechanisms in suboptimal modes. For isotropic turbulence, we derive and confirm inertial-range frequency scaling for convective–recipient covariances, then demonstrate self-similar energy transfer at each rank.

This study presents band-ensemble Spectral Proper Orthogonal Decomposition (bSPOD). The approach is inspired by frequency smoothing, a method used to reduce estimator variance in power spectral density estimates, and is here extended to SPOD. The algorithm estimates SPOD modes from consecutive Fourier coefficients obtained from a single Fourier transform of the full time record and thus avoids time segmentation. In this study, bSPOD is applied to artificial test data and to a PIV data set of a broadband-tonal cavity flow. Compared to the more commonly used Welch-based SPOD formulation, bSPOD reduces spectral leakage, permits increased frequency resolution, and retains frequency information of tonal components at comparable computational cost. These features enable reduced estimator variance while maintaining low bias for tonal components, making bSPOD particularly effective for broadband-tonal flows.

This study presents band-ensemble Spectral Proper Orthogonal Decomposition (bSPOD). The approach is inspired by frequency smoothing, a method used to reduce estimator variance in power spectral density estimates, and is here extended to SPOD. The algorithm estimates SPOD modes from consecutive Fourier coefficients obtained from a single Fourier transform of the full time record and thus avoids time segmentation. In this study, bSPOD is applied to artificial test data and to a PIV data set of a broadband-tonal cavity flow. Compared to the more commonly used Welch-based SPOD formulation, bSPOD reduces spectral leakage, permits increased frequency resolution, and retains frequency information of tonal components at comparable computational cost. These features enable reduced estimator variance while maintaining low bias for tonal components, making bSPOD particularly effective for broadband-tonal flows.

We investigate the feasibility of applying reinforcement learning (RL) to the active flow control of a supersonic twin-rectangular turbulent jet for noise reduction. Large-eddy simulations (LES) of the Mach 1.5 twin jet, with a Reynolds number of one million, constitute the learning environment. Control of the jet flow is realized through plasma actuators placed within the nozzles. The environment is observed via a three-dimensional grid of near-field pressure sensors. The reward signal is provided by an additional far-field pressure sensor, which measures the overall sound pressure level (OASPL) at a location relative to the nozzles that is representative of the final checker on an aircraft carrier. To learn a control policy that seeks to minimize the OASPL, we employ proximal policy optimization. For computational tractability, we train RL agents independently on low- and intermediate-resolution meshes. The agent trained on the low-resolution mesh is able to increase its reward over 1000 episodes of training, but its learned policy does not successfully generalize to a fully-resolved LES. The agent trained on the intermediate-resolution mesh is unable to outperform a random control policy after 300 episodes of training. In the context of numerical simulations, our results demonstrate that significant hurdles remain in the application of RL to turbulent jet noise reduction, and to the control of high-dimensional, highly stochastic systems more broadly.

This study investigates the nonlinear dynamics of an open cavity flow, specifically addressing the origin of side peaks observed around the Rossiter modes. A spanwise-periodic direct numerical simulation (DNS) at Mach 0.5 and a length-to-depth ratio of two is utilized to examine the interactions between centrifugal instabilities and Rossiter modes. By employing spanwise-decomposed multitaper spectral proper orthogonal decomposition (SPOD), we isolate multiple low-frequency, three-dimensional centrifugal modes and associate each peak observed in the spectrum with its spanwise wavenumber. The triadic resonance condition is precisely satisfied between the side peaks, centrifugal modes, and the Rossiter modes within each spanwise wavenumber. We utilize triadic orthogonal decomposition (TOD) to analyze the energy transfer of inter-wavenumber triad interactions. The modal energy budget confirms that the centrifugal instabilities energize the Rossiter modes, forming inter-wavenumber frequency triads.

A simple method is presented for calculating the structure of entropy layers generated by bow shocks over blunt-nosed wedges in hypersonic flow. The model employs a streamfunction-based framework to track entropy jumps introduced at the bow shock. It is shown that entropy conservation along streamlines, combined with the condition of uniform total enthalpy, determines the transverse profiles of velocity and thermodynamic properties in the downstream parallel-flow region at distances much larger than the wedge radius. A key finding is that the variation of entropy with streamfunction, computed numerically for ideal gas at freestream Mach numbers ranging from 3 to 10 and wedge half-angles ranging from 0 to 10 degrees, identifies a specific nondimensional streamfunction scaled by freestream parameters that collapses the entropy data onto a single quasi-universal curve. The proposed model provides a robust and efficient means of reconstructing the density and velocity fields within the entropy layer for prescribed freestream Mach number and wedge angle, eliminating the need for direct CFD simulation and enabling rapid generation of base states for stability analysis and preliminary design.

The accuracy and robustness of reduced-order models (ROMs) often suffer from parameter variations, such as Mach number or angle of attack for fluid mechanics problems. Specifically, the reduced order basis obtained via spectral proper orthogonal decomposition (SPOD) suffers from reduced robustness due to frequency drifting of the dominant modes. Although the Grassmannian manifold interpolation method has proven successful for POD-based ROMs, the SPOD-based ROMs introduce an additional spectral component that challenges this interpolation method. This paper presents an interpolation method for the dominant SPOD modes on the Grassmannian manifold based on frequency alignment, reducing the SPOD mode interpolation problem back to the standard subspace interpolation problem. This method preserves mode orthogonality and ensures frequency consistency during interpolation via resampling the time series data. This technique is demonstrated with a vortex shedding problem behind a NACA 0012 airfoil and benchmarked against the direct interpolation method, showing improved interpolation results for the SPOD basis when the angle of attack is varied.

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.

Abstract Space–time projection (STP) is introduced as a data-driven forecasting approach for high-dimensional, time-resolved data. The method computes extended space–time proper orthogonal modes from training data spanning a prediction horizon comprising both hindcast and forecast intervals. Forecasts are generated by projecting the hindcast portion of these modes onto new data, leveraging their orthogonality and optimal correlation with the forecast extension. Rooted in proper orthogonal decomposition (POD) theory, dimensionality reduction and time-delay embedding are intrinsic to the approach. The only tunable parameters are the truncation rank and the hindcast length; no additional hyperparameters are required. Hindcast accuracy serves as a reliable indicator for short-term forecast accuracy. The method’s efficacy is demonstrated using two datasets: transient, highly anisotropic simulations of supernova explosions in a turbulent interstellar medium and experimental velocity fields of a turbulent high-subsonic engineering flow. In a comparative study with standard dynamic mode decomposition (DMD) and long short-term memory (LSTM) networks (acknowledging that alternative architectures or training strategies may yield different outcomes) STP achieved the lowest errors at short and long lead times and was comparable at intermediate horizons. Considering its simplicity and robust performance, STP offers an interpretable and competitive baseline for forecasting high-dimensional transient chaotic processes, relying purely on spatio-temporal correlation.

SPOD identifies coherent structures in statistically stationary flows. The method estimates the cross-spectral density matrix from an ensemble of temporal Fourier modes obtained by segmenting time series data into blocks. This statistical approach enables the extraction of coherent structures even in broadband turbulence. However, SPOD faces a tradeoff: better statistical convergence requires more blocks, which reduces frequency resolution. This variance–bias tradeoff necessitates long time series for accurate mode extraction. In this talk we present a modified version of SPOD. Instead of block-segmenting the time series, the Fourier transform is computed using the full time history. SPOD modes are then obtained by averaging neighboring Fourier modes via Proper Orthogonal Decomposition. Importantly, information about the contribution of individual Fourier modes to each SPOD mode are stored in the expansion coefficients. Compared to standard SPOD, the new approach offers two key advantages: (1) the number of Fourier modes used to estimate SPOD modes can be varied with frequency, enabling adaptive tradeoff between spectral variance and spectral bias; and (2) the contribution of individual frequencies to each SPOD mode can be directly quantified, allowing for more accurate extraction of tonal features without sacrificing statistical convergence. This makes the new algorithm particularly effective for analyzing flows that exhibit both broadband and tonal components.

We present a framework for parametric proper orthogonal decomposition (POD)-Galerkin reduced-order modelling (ROM) of fluid flows that accommodates variations in flow parameters and control inputs. As an initial step, to explore how the locally optimal POD modes vary with parameter changes, we demonstrate a sensitivity analysis of POD modes and their spanned subspace, respectively rooted in Stiefel and Grassmann manifolds. The sensitivity analysis, by defining distance between POD modes for different parameters, is applied to the flow around a rotating cylinder with varying Reynolds numbers and rotation rates. The sensitivity of the subspace spanned by POD modes to parameter changes is represented by a tangent vector on the Grassmann manifold. For the cylinder case, the inverse of the subspace sensitivity on the Grassmann manifold is proportional to the Roshko number, highlighting the connection between geometric properties and flow physics. Furthermore, the Reynolds number at which the subspace sensitivity approaches infinity corresponds to the lower bound at which the characteristic frequency of the Kármán vortex street exists (Noack & Eckelmann, J. Fluid Mech. , 1994, vol. 270, pp. 297–330). From the Stiefel manifold perspective, sensitivity modes are derived to represent the flow field sensitivity, comprising the sensitivities of the POD modes and expansion coefficients. The temporal evolution of the flow field sensitivity is represented by superposing the sensitivity modes. Lastly, we devise a parametric POD-Galerkin ROM based on subspace interpolation on the Grassmann manifold. The reconstruction error of the ROM is intimately linked to the subspace-estimation error, which is in turn closely related to subspace sensitivity.

A stochastic data-driven reduced-order model applicable to a wide range of turbulent natural and engineering flows is presented. Combining ideas from Koopman theory and spectral model order reduction, the stochastic low-dimensional inflated convolutional Koopman model accurately forecasts short-time transient dynamics while preserving long-term statistical properties. A discrete Koopman operator is used to evolve convolutional coordinates that govern the temporal dynamics of spectral orthogonal modes, which, in turn, represent the energetically most salient large-scale coherent flow structures. Turbulence closure is achieved in two steps: first, by inflating the convolutional coordinates to incorporate nonlinear interactions between different scales, and second, by modelling the residual error as a stochastic source. An empirical dewhitening filter informed by the data is used to maintain the second-order flow statistics. The model uncertainty is quantified through either Monte–Carlo simulation or by directly propagating the model covariance matrix. The model is demonstrated on the Ginzburg–Landau equations, large-eddy simulation data of a turbulent jet, and particle image velocimetry data of the flow over an open cavity. In all cases, the model is predictive over time horizons indicated by a detailed error analysis and integrates stably over arbitrary time horizons, generating realistic surrogate data.

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 study the stationary, intermittent and nonlinear dynamics of nominally ideally expanded, natural and forced supersonic twin-rectangular turbulent jets using spectral modal decomposition. We decompose large-eddy simulation data into four reflectional symmetry components about the major and minor axes. In the natural jet, spectral proper orthogonal decomposition (SPOD) uncovers two resonant instabilities antisymmetric about the major axis. Known as screech tones, the more energetic of the two is a steady flapping instability, while the other is an intermittent double-flapping instability. We test the hypothesis that symmetry breaking can be leveraged for control design. Time-periodic forcing symmetric about the major and minor axes is implemented using a plasma actuation model, and succeeds in removing screech from a different symmetry component. We investigate the spectral peaks of the forced jet using an extension of bispectral mode decomposition (BMD), where the bispectrum is bounded by unity and which conditionally recovers the SPOD. We explain the appearance of harmonic peaks as three sets of triadic interactions between reflectional symmetries, forming an interconnected triad network. BMD modes of active triads distil coherent structures comprising multiple coupled instabilities, including Kelvin–Helmholtz, core and guided-jet modes (G-JM). Downstream-propagating core modes can be symmetric or antisymmetric about the major axis, whereas upstream-propagating G-JM responsible for screech closure (Edgington-Mitchell et al. J. Fluid Mech. 945, 2022, p. A8) are antisymmetric only. The dependence of G-JM on symmetry hence translates from the azimuthal symmetry of the round jet to the dihedral group symmetry of the twin-rectangular jet, and explains why the twin jet exhibits antisymmetric but not symmetric screech modes.

This study investigates the onset of linear instabilities and their later nonlinear interactions in the shear layer of an initially laminar jet using high-fidelity simulations. We present a quantitative analysis of the vortex-pairing phenomenon by computing the spatial growth rates and energy budget of the dominant frequencies. Compared with a turbulent jet, the hydrodynamic instabilities and vortex pairing are enhanced in an initially laminar jet. Using local linear theory, we identify the fundamental as the frequency with the largest spatial growth rate, and its exponential growth causes the shear layer to roll up into vortices. Visualisations and conditional $x$ – $t$ plots reveal that fundamental vortices pair to form subharmonic vortices, which then merge to produce second subharmonic vortices. The energy transfer during this process is evaluated using the spectral turbulent kinetic energy equation, focusing on dominant coherent structures identified through spectral proper orthogonal decomposition. Spectral production and nonlinear transfer terms show that the fundamental frequency gains energy solely from the mean flow, while subharmonics gain energy both linearly from the mean flow and nonlinearly through backscatter from the fundamental frequency. Our results confirm Monkewitz’s theoretical model of a resonance mechanism between the fundamental and subharmonic, which supplies energy to the subharmonic. We highlight the energetic versus dynamical importance of tonal frequencies. The second subharmonic corresponds to the largest spectral peak, while the fundamental, though the fourth largest spectral peak, is dynamically dominant, as it determines all other spectral peaks and supplies energy to the subharmonics through a reverse energy cascade.

This study investigates boundary layer transition over an open cavity under subsonic flow conditions, motivated by the economic and environmental impact of reducing aerodynamic drag. Using Direct Numerical Simulation (DNS), Linear Stability Theory (LST), and Spectral Proper Orthogonal Decomposition (SPOD), the research examines how cavity-induced instabilities, such as Rossiter (acoustic feedback-driven) and centrifugal modes, interact to drive transition, building on prior work linking surface imperfections to bypass transition mechanisms. Results reveal nonlinear couplings between these modes, with Rossiter modes persisting into turbulent regions and centrifugal instabilities introducing three-dimensional flow distortions, highlighting their combined role in destabilizing the boundary layer.

Space-Time Projection (STP) is introduced as a data-driven forecasting approach for high-dimensional and time-resolved data. The method computes extended space-time proper orthogonal modes from training data spanning a prediction horizon comprising both hindcast and forecast intervals. Forecasts are then generated by projecting the hindcast portion of these modes onto new data, simultaneously leveraging their orthogonality and optimal correlation with the forecast extension. Rooted in Proper Orthogonal Decomposition (POD) theory, dimensionality reduction and time-delay embedding are intrinsic to the approach. For a given ensemble and fixed prediction horizon, the only tunable parameter is the truncation rank--no additional hyperparameters are required. The hindcast accuracy serves as a reliable indicator for short-term forecast accuracy and establishes a lower bound on forecast errors. The efficacy of the method is demonstrated using two datasets: transient, highly anisotropic simulations of supernova explosions in a turbulent interstellar medium, and experimental velocity fields of a turbulent high-subsonic engineering flow. In a comparative study with standard Long Short-Term Memory (LSTM) neural networks--acknowledging that alternative architectures or training strategies may yield different outcomes--the method consistently provided more accurate forecasts. Considering its simplicity and robust performance, STP offers an interpretable and competitive benchmark for forecasting high-dimensional transient and chaotic processes, relying purely on spatiotemporal correlation information.

A stochastic data-driven reduced-order model applicable to a wide range of turbulent natural and engineering flows is presented. Combining ideas from Koopman theory and spectral model order reduction, the stochastic low-dimensional inflated convolutional Koopman model (SLICK) accurately forecasts short-time transient dynamics while preserving long-term statistical properties. A discrete Koopman operator is used to evolve convolutional coordinates that govern the temporal dynamics of spectral orthogonal modes, which in turn represent the energetically most salient large-scale coherent flow structures. Turbulence closure is achieved in two steps: first, by inflating the convolutional coordinates to incorporate nonlinear interactions between different scales, and second, by modeling the residual error as a stochastic source. An empirical dewhitening filter informed by the data is used to maintain the second-order flow statistics within the long-time limit. The model uncertainty is quantified through either Monte Carlo simulation or by directly propagating the model covariance matrix. The model is demonstrated on the Ginzburg-Landau equations, large-eddy simulation (LES) data of a turbulent jet, and particle image velocimetry (PIV) data of the flow over an open cavity. In all cases, the model is predictive over time horizons indicated by a detailed error analysis and integrates stably over arbitrary time horizons, generating realistic surrogate data.

We investigate the forcing-induced transient between statistically stationary and cyclostationary states. The transient dynamics of a turbulent supersonic twin-rectangular jet flow, forced symmetrically at a Strouhal number of 0.9, are studied using synchronized large-eddy simulations (LES) and space-time proper orthogonal decomposition (space-time POD). Under plasma-actuated control, the statistically stationary jet evolves towards a cyclostationary state over a transient phase. Forcing-induced perturbations of the natural jet are extracted using synchronized simulations of the natural and forced jets. A database is collected that captures an ensemble of realizations of the perturbations within the initial transient. The spatiotemporal dynamics and statistics of the transient are analyzed using space-time POD for each symmetry component. The eigenvalue spectra unveil low-rank dynamics in the symmetric component. The spatial and temporal structures of the leading modes indicate that the initial pulse of the actuators produces large, impulsive perturbations to the flow field. The symmetric mode reveals the contraction of the shock cells due to the forcing, and shows the evolution of the mean flow deformation transient.

We present a framework for parametric proper orthogonal decomposition (POD)-Galerkin reduced-order modeling (ROM) of fluid flows that accommodates variations in flow parameters and control inputs. As an initial step, to explore how the locally optimal POD modes vary with parameter changes, we demonstrate a sensitivity analysis of POD modes and their spanned subspace, respectively rooted in Stiefel and Grassmann manifolds. The sensitivity analysis, by defining distance between POD modes for different parameters, is applied to the flow around a rotating cylinder with varying Reynolds numbers and rotation rates. The sensitivity of the subspace spanned by POD modes to parameter changes is represented by a tangent vector on the Grassmann manifold. For the cylinder case, the inverse of the subspace sensitivity on the Grassmann manifold is proportional to the Roshko number, highlighting the connection between geometric properties and flow physics. Furthermore, the Reynolds number at which the subspace sensitivity approaches infinity corresponds to the lower bound at which the characteristic frequency of the Kármán vortex street exists (Noack & Eckelmann, JFM, 1994). From the Stiefel manifold perspective, sensitivity modes are derived to represent the flow field sensitivity, comprising the sensitivities of the POD modes and expansion coefficients. The temporal evolution of the flow field sensitivity is represented by superposing the sensitivity modes. Lastly, we devise a parametric POD-Galerkin ROM based on subspace interpolation on the Grassmann manifold. The reconstruction error of the ROM is intimately linked to the subspace-estimation error, which is in turn closely related to subspace sensitivity.

We study the stationary, intermittent, and nonlinear dynamics of natural and forced supersonic twin-rectangular turbulent jets using spectral modal decomposition. We decompose large-eddy simulation data into four reflectional symmetry components about the major and minor axes. In the natural jet, spectral proper orthogonal decomposition (SPOD) uncovers two resonant instabilities antisymmetric about the major axis. Known as screech tones, the more energetic of the two is symmetric about the minor axis and steady, while the other is intermittent. We test the hypothesis that flow symmetry can be leveraged for control design. Time-periodic forcing symmetric about the major and minor axes is implemented using a plasma actuation model, and succeeds in removing screech from a different symmetry component. We investigate the spectral peaks of the forced jet using an extension of bispectral mode decomposition (BMD), where the bispectrum is bounded by unity and which conditionally recovers the SPOD. We explain the appearance of harmonic peaks as three sets of triadic interactions between reflectional symmetries, forming an interconnected triad network. BMD modes of active triads distil coherent structures comprising multiple coupled instabilities, including Kelvin-Helmholtz, core, and guided-jet modes (G-JM). Downstream-propagating core modes can be symmetric or antisymmetric about the major axis, whereas upstream-propagating G-JM responsible for screech closure (Edgington-Mitchell et al., 2022, JFM) are antisymmetric only. The dependence of G-JM on symmetry hence translates from the azimuthal symmetry of the round jet to the dihedral group symmetry of the twin-rectangular jet, and explains why the twin jet exhibits antisymmetric but not symmetric screech modes.

A framework for fusing numerical and experimental databases using heterogeneous forms of the gappy-POD is presented in order to reconstruct real observations from reduced sensor sets. The problem is demonstrated in the context of the jet noise problem where the observation space comprises wavefronts corresponding to the unsteady pressure field within the hydrodynamic periphery of a Mach 3 jet flow. The data fusion approach leverages a numerical model of the unsteady density field (generated by way of large eddy simulation) for training, followed by entries from experiments (high speed schlieren) for real-time re- construction. The compatibility of these databases is evaluated to ensure that observations are generated using the smallest number of synthesized basis functions. The input to the reconstruction is a sensor set from experimentally captured schlieren images of the same jet flow. A further simplification utilizes heterogeneous forms of the gappy-POD so that only a reduced sensor set is required for the reconstruction. This comprises the XX-topos and YX forms where the linear algebraic system of equations are formed using the eigenvectors and expansion coefficients, respectively while the gappy sensor set is determined using a random number generator. An error analysis reinforces the general requirement that 2��+1 sensors are needed to properly resolve the ��th POD mode. Sample reconstructions of the density field are performed in real time using only 10% of available POD modes and 20% of the sensors and are shown to generate qualitatively satisfactory images of wavefronts evolving across space and time.

Introduction It is now known that capillary waves driven upon a fluid interface by high frequency ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> MHz) ultrasound exhibit capillary wave turbulence: the appearance of waves with phase and wavelength far removed from the excitation signal that drives them. These waves are responsible in significant part for atomization, a useful application for ultrasound, though the physics responsible for their appearance is poorly understood. Methods We use high-speed digital holographic microscopy to observe these capillary waves, an important step towards understanding their generation and atomization phenomena. Results We observe Zakharov-Kolmogorov weak wave turbulence for a limited range of input power, and find broader turbulence phenomena outside this range. We see discrete thresholds as the input power is increased, where higher and higher frequency responses are driven in the capillary waves with sudden onset between regimes. Discussion We employed spatial analysis to find extensions of the capillary wave response to higher frequencies, suggesting there is additional information in the spatial distribution of the capillary wave that is rarely if ever measured. We verified via frequency modulation that nonlinear resonance broadening is present, which undermines the use of Faraday wave or parametric wave theories to characterize these waves, important in the context of atomization which is now, definitively, not a Faraday wave process.

This work investigates the spatio-temporal evolution of coherent structures in the wake of a generic high-speed train, based on a three-dimensional database from large eddy simulation. Spectral proper orthogonal decomposition (SPOD) is used to extract energy spectra and energy ranked empirical modes for both symmetric and antisymmetric components of the fluctuating flow field. The spectrum of the symmetric component shows overall higher energy and more pronounced low-rank behaviour compared with the antisymmetric one. The most dominant symmetric mode features periodic vortex shedding in the near wake, and wave-like structures with constant streamwise wavenumber in the far wake. The mode bispectrum further reveals the dominant role of self-interaction of the symmetric component, leading to first harmonic and subharmonic triads of the fundamental frequency, with remarkable deformation of the mean field. Then, the stability of the three-dimensional wake flow is analysed based on two-dimensional local linear stability analysis combined with a non-parallelism approximation approach. Temporal stability analysis is first performed for both the near-wake and the far-wake regions, showing a more unstable condition in the near-wake region. The absolute frequency of the near-wake eigenmode is determined based on spatio-temporal analysis, then tracked along the streamwise direction to find out the global mode growth rate and frequency, which indicate a marginally stable global mode oscillating at a frequency very close to the most dominant SPOD mode. The global mode wavemaker is then located, and the structural sensitivity is calculated based on the direct and adjoint modes derived from a local spatial analysis, with the maximum value localized within the recirculation region close to the train tail. Finally, the global mode shape is computed by tracking the most spatially unstable eigenmode in the far wake, and the alignment with the SPOD mode is computed as a function of streamwise location. By combining data-driven and theoretical approaches, the mechanisms of coherent structures in complex wake flows are well identified and isolated.

This study presents an approach to investigate the role of eddy viscosity in linearized mean-field analysis of broadband turbulent flows. The procedure is based on spectral proper orthogonal decomposition (SPOD), resolvent analysis and the energy budget of coherent structures and is demonstrated using the example of a turbulent jet. The focus is on the coherent component of the Reynolds stresses, the nonlinear interaction term of the fluctuating velocity component in frequency space, which appears as an unknown in the derivation of the linearized Navier–Stokes equations and which is the quantity modelled by the Boussinesq approach. For the considered jet the coherent Reynolds stresses are found to have a mostly dissipative effect on the energy budget of the dominant coherent structures. Comparison of the energy budgets of SPOD and resolvent modes demonstrates that dissipation caused by nonlinear energy transfer must be explicitly considered within the linear operator to achieve satisfactory results with resolvent analysis. Non-modelled dissipation distorts the energy balance of the resolvent modes and is not, as often assumed, compensated for by the resolvent forcing vector. A comprehensive analysis, considering different predictive and data-driven eddy viscosities, demonstrates that the Boussinesq model is highly suitable for modelling the dissipation caused by nonlinear energy transfer for the considered flow. Suitable eddy viscosities are analysed with regard to their frequency, azimuthal wavenumber and spatial dependence. In conclusion, the energetic considerations reveal that the role of eddy viscosity is to ensure that the energy the structures receive from the mean-field is dissipated.

Energy transfer across scales is fundamental in fluid dynamics, linking large-scale flow motions to small-scale turbulent structures in engineering and natural environments. Triadic interactions among three wave components form complex networks across scales, challenging understanding and model reduction. We introduce Triadic Orthogonal Decomposition (TOD), a method that identifies coherent flow structures optimally capturing spectral momentum transfer, quantifies their coupling and energy exchange in an energy budget bispectrum, and reveals the regions where they interact. TOD distinguishes three components--a momentum recipient, donor, and catalyst--and recovers laws governing pairwise, six-triad, and global triad conservation. Applied to unsteady cylinder wake and wind turbine wake data, TOD reveals networks of triadic interactions with forward and backward energy transfer across frequencies and scales.

Experimental measurements often present corrupted data and outliers that can strongly affect the main coherent structures extracted with the classical modal analysis techniques. This effect is amplified at high frequencies, whose corresponding modes are more susceptible to contamination from measurement noise and uncertainties. Such limitations are overcome by a novel approach proposed here, the robust spectral proper orthogonal decomposition (robust SPOD), which implements the robust principal component analysis within the SPOD technique. The new technique is firstly presented with details on its algorithm, and its effectiveness is tested on two different fluid dynamics problems: the subsonic jet flow field numerically simulated, and the flow within an open cavity experimentally analyzed in [48] . The analysis of the turbulent jet data, corrupted both with salt and pepper and Gaussian noise, shows how the robust SPOD produces more converged and physically interpretable modes than the classical SPOD; moreover, the use of the robust SPOD as a tool for de-noising data, based on the signal reconstruction from de-noised modes, is also presented. Applying robust SPOD to the open cavity flow has revealed that it yields smoother spatial distributions of modes, particularly at high frequencies and when considering higher-order modes, compared to standard SPOD.

Different transition to turbulence routes for the flow around blunt bodies are possible. Non-modal amplification of perturbations via the lift-up effect has recently been explored to explain transition near the stagnation point in axisymmetric bodies. However, only perturbations already present in the boundary layer can be amplified, and the mechanisms by which free-stream perturbations enter the boundary layer have not yet been fully explored. In this study, we present an investigation of how disturbances enter the boundary layer via the stagnation point. This linear mechanism is expected to dominate over non-linear mechanisms previously identified on the formation of boundary layer perturbations at low turbulence intensity levels. A parametric investigation is presented, revealing trends with Reynolds and Mach numbers.

We present LinStab2D, an easy-to-use linear stability analysis MATLAB tool capable of handling complex domains, performing temporal and spatial linear stability, and resolvent analysis. We present the theoretical foundations of the code, including the linear stability and resolvent analysis frameworks, finite differences discretization schemes, and the Floquet ansatz. These concepts are explored in five different examples, highlighting and illustrating the different code capabilities, including mesh masking, mapping, imposition of boundary constraints, and the analysis of periodic flows using Cartesian or axisymmetric coordinates. These examples were constructed to be a departure point for studying other flows.

The transient dynamics of a turbulent supersonic twin-rectangular jet flow, forced symmetrically at a Strouhal number of 0.9, are investigated using large-eddy simulations (LES). The forcing is provided by localized arc filament plasma actuators (LAFPA), modeled as source terms in the energy equation. Under plasma-actuated control, the statistically stationary jet evolves towards a cyclostationary state over a transient phase. Forcing-induced perturbations of the natural jet are extracted using synchronized simulations of the natural and forced jets. A database is collected that captures an ensemble of realizations of the perturbations within the initial transient. The spatiotemporal dynamics and statistics of the transient are investigated using space-time proper orthogonal decomposition (space-time POD) for each D2 symmetry component. The eigenvalue spectra unveil low-rank dynamics in the symmetric component. The spatial and temporal structures of the leading modes indicate that the initial pulse of the actuators produces large, impulsive perturbations to the flow field. The symmetric mode reveals the contraction of the shock cells due to the forcing, and shows the evolution of the mean flow deformation transient.

An orthogonal modal decomposition for identifying triadic interactions in fluid flows is presented. The decomposition is based on spectral momentum transfer and extracts coherent structures that partake in three-wave interactions and are optimal in terms of the third-order space-time flow statistics. The method distinguishes between two quadratically interacting components, one acting as a catalyst and the other as a donor of momentum, that collectively contribute to a tertiary component, the recipient. The resulting modes maximize the covariance between the donor and recipient for each triad. The method can be understood as an extension of bispectral mode decomposition (BMD) by considering the exact form of the quadratic nonlinearity of the Navier-Stokes equations. Unlike BMD, and more similar to classical proper orthogonal decomposition (POD), it provides ranked bases for the donor and recipient that are jointly optimal and orthonormal in their respective inner products. Two applications are considered: numerical data of a canonical unsteady cylinder wake and experimental data of a turbulent wind turbine wake by Biswas and Buxton (JFM, 2024).

An adaptive algorithm for spectral proper orthogonal decomposition (SPOD) of mixed broadband-tonal turbulent flows is developed. Sharp peak resolution at tonal frequencies is achieved by locally minimizing bias of the spectrum. Smooth spectrum estimates of broadband regions are achieved by locally reducing variance of the spectrum. The method utilizes multitaper estimation with sine tapers. An iterative criterion based on modal convergence is introduced to enable the SPOD to adapt to spectral features. For tonal flows, the adaptivity is controlled by a single user input; for broadband flows, a constant number of sine tapers is recommended without adaptivity. The discrete version of Parseval’s theorem for SPOD is stated. Proper normalization of the tapers ensures that Parseval’s theorem is satisfied in expectation. Drastic savings in computational complexity and memory usage are facilitated by two aspects: (i) sine tapers, which permit post hoc windowing of a single Fourier transform; and (ii) time-domain lossless compression using a QR or eigenvalue decomposition. Sine-taper SPOD is demonstrated on time-resolved particle image velocimetry (TR-PIV) data from an open cavity flow (Zhang et al. in Exp Fluids 61(226):1–12, https://doi.org/10.1007/s00348-020-03057-8, 2020) and high-fidelity large-eddy simulation (LES) data from a round jet (Brès et al. in J. Fluid Mech. 851:83–124, https://doi.org/10.1017/jfm.2018.476, 2018), with and without adaptivity. For the tonal cavity flow, the adaptive algorithm outperforms Slepian-based multitaper SPOD in terms of variance and local bias of the spectrum, mode convergence, and memory usage. The tonal frequencies associated with the Rossiter instability are accurately identified. For both the tonal cavity and the broadband jet flows, results comparable to or better than those from standard SPOD based on Welch’s overlapped segment averaging are obtained with up to 75% fewer snapshots, including similar convergence of the Rossiter modes and Kelvin-Helmholtz wavepacket structures for the cavity and jet examples, respectively. Drawing from these examples, we establish best practices.

Bispectral mode decomposition is used to investigate triadic interactions within a Mach 0.4 turbulent jet. We explore its potential to identify dominant triadic interactions and their associated spatial structures in an unforced turbulent jet. The bispectral measure is broadband in frequency for each azimuthal wavenumber triad. The [1,1,2] and [0,0,0] azimuthal wavenumber triads are dominant, emphasizing the importance of the self-interactions of the helical and axisymmetric components. Bispectral modes reveal that streaky structures are fed by the interaction of a Kelvin-Helmholtz wavepacket with its conjugate. Streaks are also observed in other frequency interactions, occurring in regions where the structures of these frequencies are spatially active. Furthermore, integral interaction maps and nonlinear transfer terms are computed to determine the direction of energy transfer and to pinpoint the spatial regions where nonlinearity is most active. As the shear layer develops, small scales interact nonlinearly, transferring energy to larger scales. Moving downstream, near the potential core closure, larger scales become more active, resulting in a forward energy cascade.