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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, A2) showed that the resolvent-based estimator is optimal when the true space-time forcing statistics are utilized, 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 characterize 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 log-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., 40 pages, 31 figures

Made available in DSpace on 2022-04-28T19:45:18Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-11-25 We investigate spanwise-coherent structures in the turbulent flow around airfoils, motivated by their connection with trailing-edge noise. We analyse well-resolved large-eddy simulations (LES) of the flow around NACA 0012 and NACA 4412 airfoils, both at a Reynolds number of 400 000 based on the chord length. Spectral proper orthogonal decomposition performed on the data reveals that the most energetic coherent structures are hydrodynamic waves, extending over the turbulent boundary layers around the airfoils with significant amplitudes near the trailing edge. Resolvent analysis was used to model such structures, using the mean field as a base flow. We then focus on evaluating the dependence of such structures on the domain size, to ensure that they are not an artefact of periodic boundary conditions in small computational boxes. To this end, we performed incompressible LES of a zero-pressure-gradient turbulent boundary layer, for three different spanwise sizes, with the momentum-thickness Reynolds number matching those near the airfoils trailing edge. The same coherent hydrodynamic waves were observed for the three domains. Such waves are accurately modelled as the most amplified flow response from resolvent analysis. The signature of such wide structures is seen in non-premultiplied spanwise wavenumber spectra, which collapse for the three computational domains. These results suggest that the spanwise-elongated structures are not domain-size dependent for the studied simulations, indicating thus the presence of very wide structures in wall-bounded turbulent flows. São Paulo State University (UNESP) Campus of São João da Boa Vista, SP Divisão de Engenharia Aeronáutica Instituto Tecnológico de Aeronáutica, SP Flow Kth Engineering Mechanics Royal Institute of Technology São Paulo State University (UNESP) Campus of São João da Boa Vista, SP

We study a turbulent jet issuing from a cylindrical nozzle to characterise coherent structures evolving in the turbulent boundary layer. The analysis is performed using data from a large-eddy simulation of a Mach 0.4 jet. Azimuthal decomposition of the velocity field in the nozzle shows that turbulent kinetic energy predominantly resides in high azimuthal wavenumbers; the first three azimuthal wavenumbers, that are important for sound generation, contain much lower, but non-zero amplitudes. Using two-point statistics, low azimuthal modes in the nozzle boundary layer are shown to exhibit significant correlations with modes of same order in the free-jet region. Spectral Proper Orthogonal Decomposition (SPOD) is used to distill a low-rank approximation of the flow dynamics. This reveals the existence of tilted coherent structures within the nozzle boundary layer and shows that these are coupled with wavepackets in the jet. The educed nozzle boundary-layer structures are modelled using a local linear stability analysis of the nozzle mean flow. Projection of the leading SPOD modes on the stability eigenmodes shows that the organised boundary-layer structures can be modelled using a small number of stable eigenmodes of the boundary-layer branch of the eigenspectrum, indicating the prevalence of non-modal effects. Finally local and global resolvent analysis of the mean-flow are performed. It is shown that the most-energetic nozzle structures can be successfully described with optimal resolvent response modes, whose associated forcing modes are observed to tilt against the nozzle boundary-layer, suggesting that the Orr mechanism underpins these organised, turbulent, boundary-layer structures., 38 pages, 31 figures. Submitted to Journal of Fluid Mechanics

The linear stability of high-Reynolds-number corrugated jets is investigated by solving the compressible Rayleigh equation linearized about the time-averaged flow field. A Floquet ansatz is used to account for periodicity of this base flow in the azimuthal direction. The origin of multiple unstable solutions, which are known to appear in these non-circular configurations, is traced through gradual perturbations of a parametrized base-flow profile. It is shown that all unstable modes are corrugated jet continuations of the classical Kelvin–Helmholtz modes of circular jets, highlighting that the same instability mechanism, modified by corrugations, leads to the growth of disturbances in such flows. It is found that under certain conditions the eigenvalues may form saddles in the complex plane and display axis switching in their eigenfunctions. A parametric study is also conducted to understand how penetration and number of corrugations impact stability. The effect of these geometric properties on growth rates and phase speeds of the multiple unstable modes is explored, and the results provide guidelines for the development of nozzle configurations that more effectively modify the Kelvin–Helmholtz instability.

Three methods are evaluated to estimate the streamwise velocity fluctuations of a zero-pressure-gradient turbulent boundary layer of momentum-thickness-based Reynolds number up to $Re_{\unicode[STIX]{x1D703}}\simeq 8200$, using as input velocity fluctuations at different wall-normal positions. A system identification approach is considered where large-eddy simulation data are used to build single and multiple-input linear and nonlinear transfer functions. Such transfer functions are then treated as convolution kernels and may be used as models for the prediction of the fluctuations. Good agreement between predicted and reference data is observed when the streamwise velocity in the near-wall region is estimated from fluctuations in the outer region. Both the unsteady behaviour of the fluctuations and the spectral content of the data are properly predicted. It is shown that approximately 45 % of the energy in the near-wall peak is linearly correlated with the outer-layer structures, for the reference case $Re_{\unicode[STIX]{x1D703}}=4430$. These identified transfer functions allow insight into the causality between the different wall-normal locations in a turbulent boundary layer along with an estimation of the tilting angle of the large-scale structures. Differences in accuracy of the methods (single- and multiple-input linear and nonlinear) are assessed by evaluating the coherence of the structures between wall-normally separated positions. It is shown that the large-scale fluctuations are coherent between the outer and inner layers, by means of an interactions which strengthens with increasing Reynolds number, whereas the finer-scale fluctuations are only coherent within the near-wall region. This enables the possibility of considering the wall-shear stress as an input measurement, which would more easily allow the implementation of these methods in experimental applications. A parametric study was also performed by evaluating the effect of the Reynolds number, wall-normal positions and input quantities considered in the model. Since the methods vary in terms of their complexity for implementation, computational expense and accuracy, the technique of choice will depend on the application under consideration. We also assessed the possibility of designing and testing the models at different Reynolds numbers, where it is shown that the prediction of the near-wall peak from wall-shear-stress measurements is practically unaffected even for a one order of magnitude change in the corresponding Reynolds number of the design and test, indicating that the interaction between the near-wall peak fluctuations and the wall is approximately Reynolds-number independent. Furthermore, given the performance of such methods in the prediction of flow features in turbulent boundary layers, they have a good potential for implementation in experiments and realistic flow control applications, where the prediction of the near-wall peak led to correlations above 0.80 when wall-shear stress was used in a multiple-input or nonlinear scheme. Errors of the order of 20 % were also observed in the determination of the near-wall spectral peak, depending on the employed method.

A model problem for analysing the interaction between coherent structures in shear flows with the presence of a convective instability is proposed in this work. Starting from Couette flow, a permanent forcing in the shape of a hyperbolic tangent is introduced in the laminar equations, leading to a wall-bounded flow with an inflection point, which triggers a hydrodynamic instability. Temporal linear stability analysis applied to this new flow model shows that this flow is unstable at low Reynolds numbers, giving rise to Kelvin–Helmholtz-like vortices. Due to the presence of shear, streaks and rolls (streamwise vortices), predicted by resolvent analysis, are also present in the flow, and these structures will interact with vortices via oblique waves. Results of locally parallel analysis inspired the design of a computational box for a direct numerical simulation of such flow and the numerical results exhibit a limit cycle involving streaks, vortices, rolls, oblique waves and the mean flow, so that the flow becomes periodically unstable for the present case. The flow dynamics is shown to reproduce some of the features of jets and mixing layers, such as jitter and translational instability, showing that the present model can potentially clarify some of the phenomena involved in the turbulent dynamics of such flows.

The cross-spectral density (CSD) of the non-linear forcing in resolvent analyses is here quantified for the first time for turbulent channel flows. Direct numerical simulations (DNS) at $Re_{\tau} =179$ and $Re_{\tau} =543$ are performed. The CSDs are computed for highly energetic structures typical of buffer-layer and large-scale motions, for different temporal frequencies. The CSD of the non-linear forcing is shown not to be uncorrelated (white) in space, which implies the forcing is structured. Since the non-linear forcing is non-solenoidal by construction and the velocity of an incompressible flow is affected only by the solenoidal part of the forcing, this solenoidal part is evaluated. It is shown that the solenoidal part of the non-linear forcing is the combination of oblique streamwise vortices and a streamwise component which counteract each other, as in a destructive interference. It is shown that a rank-2 approximation of the forcing, with only the most energetic SPOD (spectral proper orthogonal decomposition) modes, leads to the bulk of the response. The projections of the non-linear forcing onto the right-singular vectors of the resolvent are evaluated. The left-singular vectors of the resolvent associated with very low-magnitude singular values are non-negligible since the non-linear forcing term has a non-negligible projection onto the linear sub-optimals of resolvent analysis. The same projections are computed when the forcing is modelled with an eddy-viscosity approach. It is clarified that this modelling improves the accuracy of the prediction since the projections are closer to those associated with the non-linear forcing from DNS data.

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.

Three methods are considered for estimating the downstream evolution of wavepackets in turbulent jets based on upstream measurements. The parabolised stability equations are used to compute a transfer function between axially and radially separated points in the flow, and the performance of this theoretical model is compared with that of two empirical approaches, direct transfer function calculation and autoregressive moving-average exogenous system identification, both of which require unsteady experimental data. The three approaches, which perform equally well, prove suitable for estimation of the downstream evolution of wavepackets using pressure data measured in the near-nozzle region. Over distances of the order of a couple of jet diameters, correlations of up to 80 % are observed between estimation and measurement. The performance deteriorates as axial separation between input and output is increased. While the two empirical approaches are limited in terms of both the number of input–output pairs and the number of flow variables that can be reasonably considered, the parabolised stability equations-based approach has no such limitation and can be used to perform full-field estimates comprising all of the dependent variables; in this it constitutes a potentially formidable means by which to perform single-input–multiple-output estimation. It has the further advantage of not requiring unsteady data for its construction, the only necessary ingredients being the mean flow and the linearised equations of motion.

Linear instability waves, or wavepackets, are key building blocks for the jet-noise problem. It has been shown in previous work that linear models correctly predict the evolution of axisymmetric wavepackets up to the end of the potential core of subsonic turbulent jets. Beyond this station, linear models fail, and nonlinearity is the likely missing piece. The essential underlying nonlinear mechanisms are unknown, and it remains unclear how these should be incorporated in a reduced-order model. The nonlinear interactions are considered in this work as an ‘external’ harmonic forcing added to the standard linear model. This modelling framework is explored using a locally parallel resolvent analysis to determine optimal forcing and associated responses, and a global approach based on 4D-Var data assimilation aimed at finding the optimal forcing of the parabolised stability equations that would minimise errors in the predictions of wavepackets. In all of the problems considered, the critical layer is found to be relevant: it is the position where sensitivity of wavepackets to nonlinearity is greatest. It is seen that disturbances are forced around the critical layer, and tilted by shear as they are advected, in a manner suggestive of an Orr-like mechanism. The ensemble of results suggests that critical-layer effects play a central role in the dynamics of wavepackets in subsonic turbulent jets, and that inclusion of such effects may remedy the shortcomings of linear reduced-order models.

This work deals with the closed-loop control of streaky structures induced by free-stream turbulence in a zero-pressure gradient, transitional boundary layer, by means of localized sensors and actuators. A linear quadratic gaussian regulator is considered along with a system identification technique to build reduced-order models for control. Three actuators are developed with different spatial supports, corresponding to a baseline shape with only vertical forcing, and to two other shapes obtained by different optimization procedures. A computationally efficient method is derived to obtain an actuator which aims to induce the exact structures which are inside the boundary layer, given in terms of their first spectral proper orthogonal decomposition mode, and an actuator that maximizes the energy of induced downstream structures. Two free-stream turbulence levels were evaluated, corresponding to 3.0% and 3.5%, and closed-loop control is applied in large-eddy simulations of transitional boundary layers. All three actuators lead to significant delays in the transition to turbulence and were shown to be robust to mild variations in the free-stream turbulence levels. Differences are understood in terms of the SPOD of actuation and FST-induced fields along with the causality of the control scheme. The actuator optimized to generate the leading downstream SPOD mode, representing the streaks in the open-loop flow, leads to the highest transition delay, which can be understood due to its capability of closely cancelling structures in the boundary layer. However, it is shown that even with the actuator located downstream of the input measurement it may become impossible to cancel incoming disturbances in a causal way, depending on the wall-normal position of the output and on the actuator considered, which limits sensor and actuator placement capable of good closed-loop performance., 33 pages, 19 figures

We present a study of the turbulent and acoustic fields of subsonic jets, controlled by means of a novel actuator that introduces perturbations via steady-fluidic actuation from a rotating centrebody. The actuation can produce louder or quieter jets, and these are analysed using time-resolved stereoscopic particle image velocimetry and a hot-wire anemometer. We place the analysis in the framework of wavepackets and linear stability theory, whence we show, using solutions of the linear parabolised stability equations, that the quieter flows can be understood to result from a mean-flow deformation that modifies wavepacket dynamics, and in particular their phase velocities, which are significantly reduced. The mean-flow deformation is shown, by a triple decomposition, to be due to the generation of Reynolds stresses associated with incoherent turbulence (rather than coherent structures) which arises when the actuation energises the flow with a frequency–azimuthal wavenumber (${\it\omega}$–$m$) combination to which the mean flow is stable. When the actuation excites the flow with an ${\it\omega}$–$m$ combination to which the mean flow is unstable, the response is dominated by coherent structures, whose rapid growth takes them beyond the linear limit, where they undergo quadratic wave interactions and lead, consequently, to a louder flow.

We investigate source mechanisms for subsonic jet noise using experimentally obtained datasets of high-Reynolds-number Mach 0.4 and 0.6 turbulent jets. The focus is on the axisymmetric mode which dominates downstream sound radiation for low polar angles and the frequency range at which peak noise occurs. A linearized Euler equation (LEE) solver with an inflow boundary condition is used to generate single-frequency hydrodynamic instability waves, and the resulting near-field fluctuations and far-field acoustics are compared with those from experiments and linear parabolized stability equation (LPSE) computations. It is found that the near-field velocity fluctuations closely agree with experiments and LPSE computations up to the end of the potential core, downstream of which deviations occur, but the LEE results match experiments better than the LPSE results. Both the near-field wavepackets and the sound field are observed directly from LEE computations, but the far-field sound pressure levels (SPLs) obtained are more than an order of magnitude lower than experimental values despite close statistical agreement of the near hydrodynamic field up to the potential core region. We explore the possibility that this discrepancy is due to the mismatch between the decay of two-point coherence with increasing distance in experimental flow fluctuations and the perfect coherence in linear models. To match the near-field coherence, experimentally obtained coherence profiles are imposed on the two-point cross-spectral density (CSD) at cylindrical and conical surfaces that enclose near-field structures generated with LEEs. The surface pressure is propagated to the far field using boundary value formulations based on the linear wave equation. Coherence matching yields far-field SPLs which show improved agreement with experimental results, indicating that coherence decay is the main missing component in linear models. The CSD on the enclosing surfaces reveals that the application of a decaying coherence profile spreads the hydrodynamic component of the linear wavepacket source on to acoustic wavenumbers, resulting in a more efficient acoustic source.

Wavepackets obtained by a linear stability analysis of the turbulent mean flow were shown in recent works to agree closely with some relevant statistics of turbulent jets, such as power spectral densities and averaged phases of flow fluctuations. However, when such wavepacket models were used to calculate the far-field sound, satisfactory agreement was only obtained for flows that were supersonic relative to the ambient speed of sound; attempts with subsonic flows led to errors of more than an order of magnitude. We investigate here the reasons for such discrepancies by developing the integral solution of the Helmholtz equation in terms of the cross-spectral densities of turbulent quantities. It is shown that agreement of a statistical source, such as would be obtained by the above-mentioned wavepacket models, in averaged amplitudes and phases in the near field is not a sufficient condition for exact agreement of the far-field sound. The sufficient condition is that, in addition to the amplitudes and phases, the statistical source should also match the coherence function of the flow fluctuations. This is exemplified in a model problem, where we show that the effect of coherence decay on sound radiation is more prominent for subsonic convection velocities, and its neglect leads to discrepancies of more than an order of magnitude in the far-field sound. For supersonic flows errors are reduced for the peak noise direction, but for other angles the coherence decay is also seen to have a significant effect. Coherence decay in the model source is seen to lead to similar decays in the coherence of two points in the far acoustic field, these decays being significantly faster for higher Mach numbers. The limitations of linear wavepacket models are illustrated with another simplified problem, showing that superposition of time-periodic solutions can lead to a correlation decay between two points. However, the coherence between any pair of points in such models remains unity, and cannot thus represent the behaviour observed in turbulent flows.

International audience; This study is focused on the prediction of coherent structures, propagating within a turbulent channel flow. We propose a derivation of the linearised problem based on a stochastic formulation of the Navier-Stokes equations. It consists in considering the transport of quantities by a resolved velocity (i.e. solution of the model) perturbed by a Brownian motion which models the unresolved turbulent fluctuations over the timeaveraged field, here thought of as the underlying background turbulence. The associated linearised model, considering the mean velocity profile as given, predicts linear solutions evolving within a corrected mean velocity field and perturbed by modelled background turbulence. Two ways to define the statistics of the Brownian motion are proposed and compared: one based on full simulation data, and the second, data-free, based on preliminary predictions from resolvent analysis. The technique is applied on turbulent channel flows at friction Reynolds number Re τ = 180 and Re τ = 550, and predictions are compared to direct numerical simulation results. We show that the principal components of an ensemble of solutions of this stochastic linearised system are able to represent the leading proper orthogonal decomposition (SPOD) modes with a similar accuracy than optimal responses coming form resolvent analysis with eddy viscosity model at scales where strong production occurs. For the other scales, receiving energy by non-linear redistribution, the present strategy improves the prediction. Moreover, the second mode is systematically well predicted over all scales. This behaviour is understood by the ability of the stochastic modelling to model positive and negative inter-scale energy transfers through stochastic diffusion and random stochastic transport, while the eddy viscosity term in resolvent analysis is purely diffusive.