Abstract

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.

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-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.

Abstract

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 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 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 𝑁𝜔 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.

Abstract

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.

Abstract

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.

Abstract

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.

Abstract

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.

Abstract

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.

Abstract

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.

Abstract

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(102) to O(103) 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.

Abstract

Screech tones in supersonic jets are underpinned by resonance between downstream-travelling Kelvin–Helmholtz waves and upstream-travelling acoustic waves. Specifically, recent works suggest that the relevant acoustic waves are guided within the jet and are described by a discrete mode of the linearised Euler equations. However, the reflection mechanism that converts downstream-travelling waves into upstream-travelling waves, and vice versa, has not been thoroughly addressed, leading to missing physics within most resonance models. In this work, we investigate the reflection and transmission of waves generated by the interaction between a Kelvin–Helmholtz wave and a normal shock in an under-expanded jet using a mode-matching approach. Both vortex-sheet and finite-thickness shear-layer models are explored, quantifying the impact of the shear layer in the reflection process. This approach could enable more quantitative predictions of resonance phenomena in jets and other fluid systems.

Abstract

Spatial marching methods, in which the flow state is spatially evolved in the downstream direction, can be used to produce low-cost models of flows containing a slowly varying direction, such as mixing layers, jets, and boundary layers. The parabolized stability equations (PSE) are popular due to their extremely low cost but can only capture a single instability mode; all other modes are damped or distorted by regularization methods required to stabilize the spatial march, precluding PSE from properly capturing non-modal behavior, acoustics, and interactions between multiple instability mechanisms. The one-way Navier-Stokes (OWNS) equations properly retain all downstream-traveling modes at a cost that is a fraction of that of global methods but still one to two orders of magnitude higher than PSE. In this paper, we introduce a new variant of OWNS whose cost, both in terms of CPU time and memory requirements, approaches that of PSE while still properly capturing the contributions of all downstream-traveling modes. The method is formulated in terms of a projection operator that eliminates upstream-traveling modes. Unlike previous OWNS variants, the action of this operator on a vector can be efficiently approximated using a series of equations that can be solved recursively, i.e., successively one after the next, rather than as a coupled set. In addition to highlighting the improved cost scaling of our method, we derive explicit error expressions and elucidate the relationship with previous OWNS variants. The properties, efficiency, and accuracy of our method are demonstrated for both free-shear and wall-bounded flows.

Abstract

Avoiding aliasing in time-resolved flow data obtained through high-fidelity simulations while keeping the computational and storage costs at acceptable levels is often a challenge. Well-established solutions such as increasing the sampling rate or low-pass filtering to reduce aliasing can be prohibitively expensive for large datasets. This paper provides a set of alternative strategies for identifying and mitigating aliasing that are applicable even to large datasets. We show how time-derivative data, which can be obtained directly from the governing equations, can be used to detect aliasing and to turn the ill-posed problem of removing aliasing from data into a well-posed problem, yielding a prediction of the true spectrum. Similarly, we show how spatial filtering can be used to remove aliasing for convective systems. We also propose strategies to prevent aliasing when generating a database, including a method tailored for computing nonlinear forcing terms that arise within the resolvent framework. These methods are demonstrated using a nonlinear Ginzburg–Landau model and large-eddy simulation data for a subsonic turbulent jet.

Abstract

Resolvent analysis is a powerful tool for modelling and analysing transitional and turbulent flows and, in particular, for approximating coherent flow structures. Despite recent algorithmic advances, computing resolvent modes for flows with more than one inhomogeneous spatial coordinate remains computationally expensive. In this paper we show how efficient and accurate approximations of resolvent modes can be obtained using a well-posed spatial marching method for flows that contain a slowly varying direction, i.e. one in which the mean flow changes gradually. First, we derive a well-posed and convergent one-way equation describing the downstream-travelling waves supported by the linearized Navier–Stokes equations. The method is based on a projection operator that isolates downstream-travelling waves. Integrating these one-way Navier–Stokes (OWNS) equations in the slowly varying direction, which requires significantly less CPU and memory resources than a direct solution of the linearized Navier–Stokes equations, approximates the action of the resolvent operator on a forcing vector. Second, this capability is leveraged to compute approximate resolvent modes using an adjoint-based optimization framework in which the forward and adjoint OWNS equations are marched in the downstream and upstream directions, respectively. This avoids the need to solve direct and adjoint globally discretized equations, therefore bypassing the main computational bottleneck of a typical global resolvent calculation. The method is demonstrated using the examples of a simple acoustics problem, a Mach 1.5 turbulent jet and a Mach 4.5 transitional zero-pressure-gradient flat-plate boundary layer. The optimal OWNS results are validated against corresponding global calculations, and the close agreement demonstrates the near-parabolic nature of these flows.

Abstract

Global COVID-19 pandemic is caused by infection with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Continuous emergence of new variants and their rapid spread are jeopardizing vaccine countermeasures to a significant extent. While currently available vaccines are effective at preventing illness associated with SARS-CoV-2 infection, these have been shown to be less effective at preventing breakthrough infection and transmission from a vaccinated individual to others. Here we demonstrate broad antiviral activity of cysteamine HCl in vitro against major emergent infectious variants of SARS-CoV-2 in a highly permissible Vero cell line. Cysteamine HCl inhibited infection of wild type, alpha, beta, gamma, delta, lambda, and omicron variants effectively. Cysteamine is a very well-tolerated US FDA-approved drug used chronically as a topical ophthalmic solution to treat ocular cystinosis in patients who receive it hourly or QID lifelong at concentrations 6 times higher than that required to inhibit SARS CoV-2 in tissue culture. Application of cysteamine as a topical nasal treatment can potentially 1) mitigate existing infection 2) prevent infection in exposed individuals, and 3) limit the contagion in vulnerable populations.

Abstract

We propose an approach to predict the modulation of wave packets in shock-containing jets. With a modeled ideally expanded mean flow as input, an approximation of the shock-cell structure is obtained from the parabolized stability equations (PSE) at zero frequency. This solution is then used to define a new shock-containing mean flow, which is a function of the shock-cell wave number at each streamwise station. Linearization of the Navier-Stokes equations around this quasiperiodic mean flow allows us to postulate a solution based on the Floquet ansatz, and further manipulation of the equations leads to a system called the parabolized Floquet equations [PFE; Ran et al., Phys. Rev. Fluids 4, 023901 (2019)] that bears several similarities to PSE. The modulation wave numbers are marched spatially together with the central Kelvin-Helmholtz wave number, leading to a modulated wave packet as the final solution. The limitations of PFE are highlighted, and the method is applied to two sample cases: a canonical slowly diverging jet at low supersonic Mach number and a heated overexpanded jet, for which large-eddy simulation (LES) data are available. Good agreement is observed between the wave packets predicted by PFE and the leading spectral proper orthogonal decomposition (SPOD) modes from the LES, suggesting that the method is able to capture the underlying physical mechanism associated with wave-packet modulation: the extraction of energy from the mean flow by the Kelvin-Helmholtz mode and a redistribution of energy to modulation wave numbers due to the interaction of this mode with the shock-cell structure.

Abstract

The application of control tools to complex flows frequently requires approximations, such as reduced-order models and/or simplified forcing assumptions, where these may be considered low rank or defined in terms of simplified statistics (e.g. white noise). In this work, we propose a resolvent-based control methodology with causality imposed via a Wiener–Hopf formalism. Linear optimal causal estimation and control laws are obtained directly from full-rank, globally stable systems with arbitrary disturbance statistics, circumventing many drawbacks of alternative methods. We use efficient, matrix-free methods to construct the matrix Wiener–Hopf problem, and we implement a tailored method to solve the problem numerically. The approach naturally handles forcing terms with space-time colour; it allows inexpensive parametric investigation of sensor/actuator placement in scenarios where disturbances/targets are low rank; it is directly applicable to complex flows disturbed by high-rank forcing; it has lower cost in comparison to standard methods; it can be used in scenarios where an adjoint solver is not available; or it can be based exclusively on experimental data. The method is particularly well suited for the control of amplifier flows, for which optimal control approaches are typically robust. Validation of the approach is performed using the linearized Ginzburg–Landau equation. Flow over a backward-facing step perturbed by high-rank forcing is then considered. Sensor and actuator placement are investigated for this case, and we show that while the flow response downstream of the step is dominated by the Kelvin–Helmholtz mechanism, it has a complex, high-rank receptivity to incoming upstream perturbations, requiring multiple sensors for control.

Abstract

We present an analysis of the linear stability characteristics of shock-containing jets. The flow is linearised around a spatially periodic mean, which acts as a surrogate for a mean flow with a shock-cell structure, leading to a set of partial differential equations with periodic coefficients in space. Disturbances are written using the Floquet ansatz and Fourier modes in the streamwise direction, leading to an eigenvalue problem for the Floquet exponent. The characteristics of the solution are directly compared with the locally parallel case, and some of the features are similar. The inclusion of periodicity induces minor changes in the growth rate and phase velocity of the relevant modes for small shock amplitudes. On the other hand, the eigenfunctions are now subject to modulation related to the periodicity of the flow. Analysis of the spatio-temporal growth rates led to the identification of a saddle point between the Kelvin–Helmholtz mode and the guided jet mode, characterising an absolute instability mechanism. Frequencies and mode shapes related to the saddle points for two conditions (associated with axisymmetric and helical modes) are compared with screech frequencies and the most energetic coherent structures of screeching jets, resulting in a good agreement for both. The analysis shows that a periodic shock-cell structure has an impulse response that grows upstream, leading to oscillator behaviour. The results suggest that screech can occur in the absence of a nozzle, and that the upstream reflection condition is not essential for screech frequency selection. Connections to previous models are also discussed.

Abstract

Resolvent analysis has demonstrated encouraging results for modeling coherent structures in jets when compared against their data-educed counterparts from high-fidelity large-eddy simulations (LES). We formulate resolvent analysis as an acoustic analogy that relates the near-field resolvent forcing to the near- and far-field pressure. We use an LES database of round, isothermal, Mach 0.9 and 1.5 jets to produce an ensemble of realizations for the acoustic field that we project onto a limited set of resolvent modes. In the near-field, we perform projections on a restricted acoustic output domain, 𝑟/𝐷=[5,6], while the far-field projections are performed on a Kirchhoff surface comprising a 100-diameter arc centered at the nozzle. This allows the LES realizations to be expressed in the resolvent basis via a data-deduced, low-rank, cross-spectral density matrix. We find that a single resolvent mode reconstructs the most energetic regions of the acoustic field across Strouhal numbers, 𝑆𝑡=[0−1], and azimuthal wavenumbers, 𝑚=[0,2]. Finally, we present a simple function that results in a rank-1 resolvent model agreeing within 2 dB of the peak noise for both jets.

Abstract

We consider the resonance mechanism underpinning generation of A1 and A2 screech tones in an under-expanded supersonic jet. Starting from the resonance model recently proposed by Mancinelli et al. (2019), where the upstream-travelling wave is a neutrally-stable, guided jet mode, we here present a more complete linear-stability-based model for screech prediction. We study temperature and shear-layer thickness effects and show that, in order to accurately describe the experimental data, the effect of the finite thickness of the shear layer has to be incorporated in the jet-dynamics model. We then present an improved resonance model for screech-frequency predictions in which both downstream- and upstream-travelling waves may have complex wavenumber and frequency. This resonance model requires knowledge of the reflection coefficients at the upstream and downstream locations of the resonance loop. We explore the effect of the reflection coefficients on the resonance model and propose an approach for their identification. The complex-mode model allows to identify frequency-flow regions of positive values of the frequency imaginary part for which the resonance loop is amplified in time and resonance is sustained and is finally found to provide the most complete description of the measured data.

Abstract

We employ a resolvent-based methodology to estimate velocity and pressure fluctuations within turbulent channel flows at friction Reynolds numbers of approximately 180, 550 and 1000 using measurements of shear stress and pressure at the walls, taken from direct numerical simulation (DNS) databases. Martini et al. (J. Fluid Mech., vol. 900, 2021, p. A2) showed that the resolvent-based estimator is optimal when the true space–time forcing statistics are utilised, thus providing an upper bound for the accuracy of any linear estimator. We use this framework to determine the flow structures that can be linearly estimated from wall measurements, and we characterise these structures and the estimation errors in both physical and wavenumber space. We also compare these results to those obtained using approximate forcing models – an eddy-viscosity model and white-noise forcing – and demonstrate the significant benefit of using true forcing statistics. All models lead to accurate results up to the buffer layer, but only using the true forcing statistics allows accurate estimation of large-scale logarithmic-layer structures, with significant correlation between the estimates and DNS results throughout the channel. The eddy-viscosity model displays an intermediate behaviour, which may be related to its ability to partially capture the forcing colour. Our results show that structures that leave a footprint on the channel walls can be accurately estimated using the linear resolvent-based methodology, and the presence of large-scale wall-attached structures enables accurate estimations through the logarithmic layer.

Abstract

The acoustic field of high-speed turbulent jets is dominated by a small number of low-wavenumber azimuthal Fourier modes. Accordingly, it is of interest to directly obtain individual azimuthal modes of the acoustic field from simulation data or models of the jet near-field. To this end, we manipulate the Ffowcs Williams and Hawkings (FW–H) equation to obtain a new formulation that lives entirely in the azimuthal Fourier domain—it delivers individual azimuthal modes of the acoustic field as a function of the same azimuthal modes of the near-field FW–H source term or, upon linearization of the source terms, as a function of the same azimuthal modes of the near-field flow variables. As an added benefit, all surface integrals are converted into line integrals in the streamwise–radial plane. After verifying and validating our formulation using a monopole problem with an exact solution and large-eddy simulation data, respectively, we show how our method can be used to efficiently and naturally compute the acoustic field associated with resolvent modes of a Mach 1.5 jet, thus avoiding the need to compute the modes on a large computational domain to capture their acoustic radiation.

Abstract

Resolvent analysis of the linearized Navier–Stokes equations provides useful insight into the dynamics of transitional and turbulent flows and can provide a model for the dominant coherent structures within the flow, particularly for flows where the linear operator selectively amplifies one particular force component, known as the optimal force mode. Force and response modes are typically obtained from a singular-value decomposition of the resolvent operator. Despite recent progress, the cost of resolvent analysis for complex flows remains considerable, and explicit construction of the resolvent operator is feasible only for simplified problems with a small number of degrees of freedom. In this paper we propose two new matrix-free methods for computing resolvent modes based on the integration of the linearized equations and the corresponding adjoint system in the time domain. Our approach achieves an order of magnitude speedup when compared with previous matrix-free time-stepping methods by enabling all frequencies of interest to be computed simultaneously. Two different methods are presented: one based on analysis of
the transient response, providing leading modes with fine frequency discretization; and another based on the steady-state response to periodic forcing, providing optimal and suboptimal modes for a discrete set of frequencies. The methods are validated using a linearized Ginzburg–Landau equation and applied to the three-dimensional flow around a parabolic body.

Abstract

The interaction between various wave-like structures in screeching jets is considered via both experimental measurements and linear stability theory. Velocity snapshots of screeching jets are used to produce a reduced-order model of the screech cycle via proper orthogonal decomposition. Streamwise Fourier filtering is then applied to isolate the negative and positive wavenumber components, which for the waves of interest in this jet correspond to upstream- and downstream-travelling waves. A global stability analysis on an experimentally derived base flow is conducted, demonstrating a close match to the results obtained via experiment, indicating that the mechanisms considered here are well represented in a linear framework. In both the global stability analysis
and the experimental decomposition, three distinct wave-like structures are evident; these waves are also solutions to the cylindrical vortex-sheet dispersion relation. One of the waves is the well-known downstream-travelling Kelvin–Helmholtz mode. Another is the upstream-travelling guided jet mode that has been a topic of recent discussion by a number of authors. The third component, with positive phase velocity, has not previously been identified in screeching jets. Via a local stability analysis, we provide evidence that this downstream-travelling wave is a duct-like mode similar to that recently identified in high-subsonic jets. We further demonstrate that both of the latter two waves are generated by the interaction between the Kelvin–Helmholtz wavepacket and the shock cells in the flow. Finally, we consider the periodic spatial modulation of the coherent velocity fluctuation evident in screeching jets, and show that this modulation can be at least partially explained by the superposition of the three wave-like structures, in addition to any possible modulation of the Kelvin–Helmholtz wavepacket by the shocks themselves.

Abstract

This paper studies the amplitude of large-scale coherent wave-packet structures in jets, modeled by the parabolized stability equations (PSEs). Linear PSEs can retrieve the shape of the wave packets, but linearity leads to solutions with a free amplitude, which has traditionally been obtained in an ad hoc manner using limited data. We systematically determine the free amplitude as a function of frequency and azimuthal wave number by comparing the fluctuation fields retrieved from PSEs with coherent structures educed from large-eddy simulation data using spectral proper orthogonal decomposition. The wave-packet amplitude is shown to decay exponentially with the Strouhal number for axisymmetric and helical modes at both Mach numbers considered in the study: 0.4 and 0.9. Analytical fit functions are proposed, and the scaled wave packets provide reasonable reconstructions of pressure and velocity spectra on the jet centerline and lip line over a range of streamwise positions.

Abstract

We extend the resolvent-based estimation approach recently introduced by Towne et al. (J. Fluid Mech., vol. 883, 2020, A17) to obtain optimal, non-causal estimates of time-varying flow quantities from low-rank measurements. We derive optimal transfer functions between the measurements and certain nonlinear terms that act as a forcing on the linearised Navier–Stokes equations, and show that the resulting transfer function to the flow state is equivalent to a multiple-input, multiple-output Wiener filter if the colour of the forcing statistics is known. A matrix-free implementation is developed based on integration of the direct and adjoint linearised Navier–Stokes operators, enabling application to the large systems encountered for transitional and turbulent flows without the need for a priori model reduction. Using a linearised Ginzburg–Landau problem, we show that the non-casual resolvent-based method outperforms a casual Kalman filter for general sensor configurations and recovers the Kalman filter transfer function in specific cases, leading to causal estimates at a significantly reduced computational cost. Additionally, our method is shown to be more accurate and robust than popular approaches based on truncation of the resolvent operator to its leading modes. The applicability of the method to transitional and turbulent flows is demonstrated via application to a (linearised) transitional boundary layer and a (nonlinear) turbulent channel flow. Errors on the order of 2 % are achieved for the boundary layer, and the channel flow case highlights the need to account for the forcing colour to achieve accurate flow estimates. In practice, our method can be used as a post-processing tool to reconstruct unmeasured quantities from limited experimental data, and, in cases where the transfer function can be accurately truncated to its causal components, as a low-cost estimator for flow control.

Abstract

Linearisation of the Navier–Stokes equations about the mean of a turbulent flow forms the foundation of popular models for energy amplification and coherent structures, including resolvent analysis. While the Navier–Stokes equations can be equivalently written using many different sets of dependent variables, we show that the properties of the linear operator obtained via linearisation about the mean depend on the variables in which the equations are written prior to linearisation, and can be modified under nonlinear transformation of variables. For example, we show that using primitive and conservative variables leads to differences in the singular values and modes of the resolvent operator for turbulent jets, and that the differences become more severe as variable-density effects increase. This lack of uniqueness of mean-flow-based linear analysis provides new opportunities for optimising models by specific choice of variables while also highlighting the importance of carefully accounting for the nonlinear terms that act as a forcing on the resolvent operator.

Abstract

We study the behaviors of pressure fluctuations in high Reynolds number wall-bounded flows. Pressure fluctuations are small-scale quantities compared to velocity fluctuations in a wall-bounded flow (Tsuji, Marusic, & Johansson, Int. J. Heat Fluid Flow, vol. 61, 2016, pp. 2–11.): at a given wall-normal distance y, the premultiplied velocity spectrum peaks at a streamwise wavelength on the order of the boundary layer thickness (λ_x = O(δ)), whereas the premultiplied pressure spectrum peaks at λ_x<O(y). The differing scales of pressure and velocity pose a challenge to modeling, and the scaling of the pressure spectrum in wall-bounded flows remains an unsolved issue from both a theoretical and measurement standpoint. To address this unresolved issue, we incorporate Kolmogorov’s theory (K41) within the framework of Townsend’s attached eddy hypothesis to account for the small scale nature of pressure fluctuations, leading to the first derivation that is consistent with both theories. Our main result is that at a wall-normal distance in the logarithmic layer the premultiplied pressure power spectrum scales as [k_x E_pp] ~  λ_x^(n-1) y^(-(3+n)/4),  for λ_x < y/tan (θ), and as [k_x E_pp] ~ λ_x^((3n-7)/4) for λ_x > y/tan (θ).  Here, θ is the attached-eddy inclination angle, k_x is the streamwise wavenumber, the velocity spectrum follows a k^(-1) scaling for 1/k_x > y/tan(θ) and a k^(-5/3) scaling for 1/k_x < y/tan (θ), and n is a Reynolds-number-dependent constant. This result conforms to Kolmogorov’s theory of small scale turbulence, i.e., it yields a −7/3 scaling for the small scales at high Reynolds numbers, and also yields the anticipated −1 scaling for the logarithmic layer scales. Detailed analysis shows that pressure and spanwise velocity have differently statistical properties: while an outer peak emerges in the premultiplied spanwise velocity spectrum at high Reynolds numbers, no outer peak is expected in the premultiplied pressure spectrum. The derived scalings are confirmed using data from a direct numerical simulation of a channel flow at friction Reynolds number Re = 5200.

Abstract

A new methodology to construct three-dimensional, temporally stationary but spatially inhomogeneous, incompressible turbulence is presented. The method combines use of the data-driven spectral proper orthogonal decomposition (SPOD) to identify and isolate large-scale coherent motions of the flow, together with a physics-based enrichment algorithm using spatiotemporally localized Gabor modes that capture the inertial subrange turbulence. This fusion of data-driven and physics-based methods enables a statistically correct reconstruction of broadband turbulent flows using fewer modes than would be required using SPOD alone. To demonstrate the approach, we consider the problem of reconstructing wake turbulence on a plane downstream of a dragging actuator disk impinged by homogeneous isotropic turbulence. The reconstructed flow has single- and two-point correlations that are consistent with the reference high-resolution simulation data and could be used to generate statistically consistent inflow boundary conditions for subsequent simulations.

Abstract

We develop a method to estimate space–time flow statistics from a limited set of known data. While previous work has focused on modelling spatial or temporal statistics independently, space–time statistics carry fundamental information about the physics and coherent motions of the flow and provide a starting point for low-order modelling and flow control efforts. The method is derived using a statistical interpretation of resolvent analysis. The central idea of our approach is to use known data to infer the statistics of the nonlinear terms that constitute a forcing on the linearized Navier–Stokes equations, which in turn imply values for the remaining unknown flow statistics through application of the resolvent operator. Rather than making an a priori assumption that the flow is dominated by the leading singular mode of the resolvent operator, as in some previous approaches, our method allows the known input data to select the most relevant portions of the resolvent operator for describing the data, making it well suited for high-rank turbulent flows. We demonstrate the predictive capabilities of the method, which we call resolvent-based estimation, using two examples: the Ginzburg–Landau equation, which serves as a convenient model for a convectively unstable flow, and a turbulent channel flow at low Reynolds number.

Abstract

The parabolized stability equations (PSE) are a ubiquitous tool for studying the stability and evolution of disturbances in weakly nonparallel, convectively unstable flows. The PSE method was introduced as an alternative to asymptotic approaches to these problems. More recently, PSE has been applied with mixed results to a more diverse set of problems, often involving flows with multiple relevant instability modes. This paper investigates the limits of validity of PSE via a spectral analysis of the PSE operator. We show that PSE is capable of accurately capturing only disturbances with a single wavelength at each frequency and that other disturbances are not necessarily damped away or properly evolved, as often assumed. This limitation is the result of regularization techniques that are required to suppress instabilities arising from the ill-posedness of treating a boundary value problem as an initial value problem. These findings are valid for both incompressible and compressible formulations of PSE and are particularly relevant for applications involving multiple modes with different wavelengths and growth rates, such as problems involving multiple instability mechanisms, transient growth, and acoustics. Our theoretical results are illustrated using a generic problem from acoustics and a dual-stream jet, and the PSE solutions are compared to both global solutions of the linearized Navier–Stokes equations and a recently developed alternative parabolization.

Abstract

The purpose of this paper is to characterise and model the A1 and A2 screech modes in supersonic jets operating at off-design conditions. The usual screech-modelling scenario involves a feedback loop between a downstream-travelling Kelvin–Helmholtz instability wave and an upstream-travelling acoustic wave. We review state-of-the-art screech-frequency prediction models and associated limitations. Following the work of Edgington-Mitchell et al. (J Fluid Mech 855, 2018), a new prediction approach is proposed where the feedback loop is closed by the upstream-travelling jet modes first discussed in Tam and Hu (J Fluid Mech 201:447–483, 1989) in lieu of the free-stream sound waves. The Kelvin–Helmholtz and upstream-travelling jet modes are obtained using a cylindrical vortex-sheet model. The predictions provide a better agreement with experimental observations than does the classical screech-prediction approach. Screech dynamics associated with the staging process is explored through a wavelet analysis, highlighting that staging involves mutually exclusive switching that is underpinned by non-linear interactions.

Abstract

A streaming algorithm to compute the spectral proper orthogonal decomposition (SPOD) of stationary random processes is presented. As new data becomes available, an incremental update of the truncated eigenbasis of the estimated cross-spectral density (CSD) matrix is performed. The algorithm requires access to only one temporal snapshot of the data at a time and converges orthogonal sets of SPOD modes at discrete frequencies that are optimally ranked in terms of energy. We define measures of error and convergence, and demonstrate the algorithm’s performance on two datasets. The first example considers a high-fidelity numerical simulation of a turbulent jet, and the second example uses optical flow data obtained from high-speed camera recordings of a stepped spillway experiment. For both cases, the most energetic SPOD modes are reliably converged. The algorithm’s low memory requirement enables realtime deployment and allows for the convergence of second-order statistics from arbitrarily long streams of data. A MATLAB implementation of the algorithm along with a test database for the jet example, can be found in the Supplementary material.

Abstract

A direct numerical simulation of the flow field around a controlled-diffusion airfoil within an anechoic wind-tunnel at 5° incidence and a high Reynolds number of 1.5×105 is performed for the first time using a lattice Boltzmann method. The simulation compares favorably with experimental measurements of wall-pressure, wake statistics, and far-field sound. The simulation noticeably captures experimentally observed high-amplitude acoustic tones that rise above a broadband hump. Both noise components are related to a breathing of a recirculation bubble formed around 65–70% of the chord, and to Kelvin-Helmholtz instabilities in the separated shear layer that yield rollers that break down into turbulent vortices whose diffraction at the trailing edge produces a strong dipole acoustic field. A wavelet analysis of the wall-pressure signals combined with some flow visualization has shown that the flow statistics are dominated by intense events caused by intermittent, large and intense bursting rollers. Several modal analyses of these events are performed on both the wall-pressure fluctuations and the span averaged flow field in order to analyse the boundary layer instability which triggers the typical sharp tones over a broadband hump in airfoil noise. A suction side Kelvin-Helmholtz instability is observed to be coupled with a pressure side vortex shedding induced by the sudden transition to turbulence and the blunt trailing edge.

Abstract

Experimental evidence is provided to demonstrate that the upstream-travelling waves in two jets screeching in the A1 and A2 modes are not free-stream acoustic waves, but rather waves with support within the jet. Proper orthogonal decomposition is used to educe the coherent fluctuations associated with jet screech from a set of randomly sampled velocity fields. A streamwise Fourier transform is then used to isolate components with positive and negative phase speeds. The component with negative phase speed is shown, by comparison with a vortex-sheet model, to resemble the upstream-travelling jet wave first studied by Tam & Hu (J. Fluid Mech., vol. 201, 1989, pp. 447–483). It is further demonstrated that screech tones are only observed over the frequency range where this upstream-travelling wave is propagative.

Abstract

Motivated by the problem of jet–flap interaction noise, we study the tonal dynamics that occurs when an isothermal turbulent jet grazes a sharp edge. We perform hydrodynamic and acoustic pressure measurements to characterise the tones as a function of Mach number and streamwise edge position. The observed distribution of spectral peaks cannot be explained using the usual edge-tone model, in which resonance is underpinned by coupling between downstream-travelling Kelvin–Helmholtz wavepackets and upstream-travelling sound waves. We show, rather, that the strongest tones are due to coupling between Kelvin–Helmholtz wavepackets and a family of trapped, upstream-travelling acoustic modes in the potential core, recently studied by Towne et al. (J. Fluid Mech. vol. 825, 2017) and Schmidt et al. (J. Fluid Mech. vol. 825, 2017). We also study the band-limited nature of the resonance, showing the high-frequency cutoff to be due to the frequency dependence of the upstream-travelling waves. Specifically, at high Mach number, these modes become evanescent above a certain frequency, whereas at low Mach number they become progressively trapped with increasing frequency, which inhibits their reflection in the nozzle plane.

Abstract

Abstract

We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space–time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic Tools in Turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time, while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg–Landau equation and a turbulent jet.

Abstract

We present diagnostic experiments aimed at understanding and mitigating supersonic jet noise from coherent wavepacket structures that are the source of peak aft-angle mixing noise. Both isothermal and heated, nearly perfectly-expanded Mach 1.5 jets were forced in the near-nozzle region with air injection generated by a spinning-valve device designed to excite the jet at frequencies approaching those of the dominant turbulent structures. Substantial reductions in the peak aft-angle radiation were achieved with steady blowing at amplitudes corresponding to 2-6% of the mass flow rate of the primary jet. The noise benefit saturates at mass flow rates above 4%, with as much
as 6 dB reduction in OASPL at aft angles. Increasing mass flow rates yield a monotonically increasing high-frequency noise penalty at sideline, where noise levels in the natural jet are already 15 dB lower than the aft-angle peak, so that the penalty due to actuation is minor. Although both steady and periodic unsteady mass injection are produced by the spinning valve when it rotates, it was calibrated to hold the steady mass flow rate constant as the frequency of unsteady blowing was changed. In this way, the effect of steady and unsteady blowing on the acoustic field could be decoupled.  We show that the noise benefit is uniquely associated with the steady component of blowing, whereas the unsteady component resulted in additive tones in the spectra.  This implied linearity is consistent with theory and experiments showing that the wavepacket structures, which give rise to the dominant aft-angle radiation, evolve in the turbulent mean flow field in a nearly linear fashion from their origin in the near nozzle region. The interpretation of noise reduction is that the steady component of blowing spreads the mean flow more rapidly resulting in weaker wavepackets. Periodic unsteady blowing forces coherent wavepackets that are largely uncorrelated from the random natural ones, which then leads to the observed additive tones.

Abstract

The purpose of this paper is to characterize and model waves that are observed within the potential core of subsonic jets and relate them to previously observed tones in the near-nozzle region. The waves are detected in data from a large-eddy simulation of a Mach 0.9 isothermal jet and modelled using parallel and weakly non-parallel linear modal analysis of the Euler equations linearized about the turbulent mean flow, as well as simplified models based on a cylindrical vortex sheet and the acoustic modes of a cylindrical soft duct. In addition to the Kelvin–Helmholtz instability waves, three types of waves with negative phase velocities are identified in the potential core: upstream- and downstream-propagating duct-like acoustic modes that experience the shear layer as a pressure-release surface and are therefore radially confined to the potential core, and upstream-propagating acoustic modes that represent a weak coupling between the jet core and the free stream. The slow streamwise contraction of the potential core imposes a frequency-dependent end condition on the waves that is modelled as the turning points of a weakly non-parallel approximation of the waves. These turning points provide a mechanism by which the upstream- and downstream-travelling waves can interact and exchange energy through reflection and transmission processes. Paired with a second end condition provided by the nozzle, this leads to the possibility of resonance in limited frequency bands that are bound by two saddle points in the complex wavenumber plane. The predicted frequencies closely match the observed tones detected outside of the jet. The vortex-sheet model is then used to systematically explore the Mach number and temperature ratio dependence of the phenomenon. For isothermal jets, the model suggests that resonance is likely to occur in a narrow range of Mach number, 0.82<M<1.

Abstract

Coherent features of a turbulent Mach 0.9, Reynolds number jet are educed from a high-fidelity large eddy simulation. Besides the well-known Kelvin–Helmholtz instabilities of the shear layer, a new class of trapped acoustic waves is identified in the potential core. A global linear stability analysis based on the turbulent mean flow is conducted. The trapped acoustic waves form branches of discrete eigenvalues in the global spectrum, and the corresponding global modes accurately match the educed structures. Discrete trapped acoustic modes occur in a hierarchy determined by their radial and axial order. A local dispersion relation is constructed from the global modes and found to agree favourably with an empirical dispersion relation educed from the simulation data. The product between direct and adjoint modes is then used to isolate the trapped waves. Under certain conditions, resonance in the form of a beating occurs between trapped acoustic waves of positive and negative group velocities. This resonance explains why the trapped modes are prominently observed in the simulation and as tones in previous experimental studies. In the past, these tones were attributed to external factors. Here, we show that they are an intrinsic feature of high-subsonic jets that can be unambiguously identified by a global linear stability analysis.

Abstract

In this paper, we develop and demonstrate a method for constructing well-posed one-way approximations
of linear hyperbolic systems. We use a semi-discrete approach that allows the method to be applied
to a wider class of problems than existing methods based on analytical factorization of idealized dispersion
relations. After establishing the existence of an exact one-way equation for systems whose coefficients
do not vary along the axis of integration, efficient approximations of the one-way operator are constructed
by generalizing techniques previously used to create nonreflecting boundary conditions. When physically
justified, the method can also be applied to systems with slowly varying coefficients in the direction of
integration. To demonstrate the accuracy and computational efficiency of the approach, the method is applied
to model problems in acoustics and fluid dynamics via the linearized Euler equations; in particular we
consider the scattering of sound waves from a vortex and the evolution of hydrodynamic wavepackets in a
spatially evolving jet. The latter problem shows the potential of the method to offer a systematic, convergent
alternative to ad hoc regularizations such as the parabolized stability equations.

Abstract

The increasing importance of improving efficiency and reducing capital costs has led to significant work studying advanced Brayton cycles for high temperature energy conversion. Using compact, highly efficient, diffusion-bonded heat exchangers for the recuperators has been a noteworthy improvement in the design of advanced carbon dioxide Brayton cycles. These heat exchangers will operate near the pseudocritical point of carbon dioxide, making use of the drastic variation of the thermophysical properties. This paper focuses on the experimental measurements of heat transfer under cooling conditions, as well as pressure drop characteristics within a prototypic printed circuit heat exchanger. Studies utilize type-316 stainless steel, nine channel, semi-circular test section, and supercritical carbon dioxide serves as the working fluid throughout all experiments. The test section channels have a hydraulic diameter of 1.16 mm and a length of 0.5 m. The mini-channels are fabricated using current chemical etching technology, emulating techniques used in current diffusion-bonded printed circuit heat exchanger manufacturing. Local heat transfer values were determined using measured wall temperatures and heat fluxes over a large set of experimental parameters that varied system pressure, inlet temperature, and mass flux. Experimentally determined heat transfer coefficients and pressure drop data are compared to correlations and earlier data available in literature. Modeling predictions using the computational fluid dynamics (CFD) package FLUENT are included to supplement experimental data. All nine channels were modeled using known inlet conditions and measured wall temperatures as boundary conditions. The CFD results show excellent agreement in total heat removal for the near pseudocritical region, as well as regions where carbon dioxide is a high or low density fluid.