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This study investigates the linear stability of a laminar premixed flame, anchored on a square cylinder and confined inside a channel. Many modern linear analysis concepts have been developed and validated around non-reacting bluff-body wake flows, and the objective of this paper is to explore whether those tools can be applied with the same success to the study of reacting flows in similar configurations. It is found that linear instability analysis of steady reacting flow states accurately predicts critical flow parameters for the onset of limit-cycle oscillations, when compared to direct numerical simulation performed with a simple one-step reaction scheme in the low Mach number limit. Furthermore, the linear analysis predicts a strong stabilising effect of flame ignition, consistent with documented experiments and numerical simulations. Instability in ignited wake flows is, however, found to set in at sufficiently high Reynolds number, and a linear wavemaker analysis characterises this instability as being driven by hydrodynamic mechanisms of a similar nature as in non-reacting wake flows. The frequency of nonlinear limit-cycle flame oscillations in this unstable regime is retrieved accurately by linear eigenmode analysis performed on the time-averaged mean flow, under the condition that the full set of the reacting flow equations is linearised. If, on the contrary, unsteadiness in the density and in the reaction rate are excluded from the linear model, then the congruence between linear and nonlinear dynamics is lost.

International audience; A framework for the computation of linear global modes, based on time stepping of a linearised Navier–Stokes solver with an Eulerian interface representation, is presented. The method is derived by linearising the nonlinear solver Basilisk, capable of computing immiscible two-phase flows, and offers several advantages over previous, matrix-based, multi-domain approaches to linear global stability analysis of interfacial flows. Using our linear solver, we revisit the study of Tammisola et al. (J. Fluid Mech., vol. 713, 2012, pp. 632–658), who found a counter-intuitive, destabilising effect of surface tension in planar wakes. Since their original study does not provide any validation, we further compute nonlinear results for the studied flows. We show that a surface-tension-induced destabilisation of plane wakes is observable which leads to periodic, quasiperiodic or chaotic oscillations depending on the Weber number of the flow. The predicted frequencies of the linear global modes, computed in the present study, are in good agreement with the nonlinear results, and the growth rates are comparable to the disturbance growth in the nonlinear flow before saturation. The bifurcation points of the nonlinear flow are captured accurately by the linear solver and the present results are as well in correspondence with the study of Tammisola et al. (J. Fluid Mech., vol. 713, 2012, pp. 632–658).

International audience; The presence of elongated streaks of high and low streamwise velocity in the shear layer of circular jets breaks the axisymmetry of their steady-state solution. If the streaks are considered to be part of the base flow, for the purpose of linear instability analysis, the instability eigenmodes are thus affected by their presence. The resulting changes of growth rate and spatial shapes of eigenmodes, related to the shear instability in jets, are investigated here for parallel base flows. Optimal streamwise vortices (rolls) with prescribed azimuthal periodicity are computed, such that the transient temporal growth of the streaks that they produce is maximal. The presence of nite-amplitude streaks requires the formulation of eigenvalue problems in a two-dimensional cross-plane. Sinuous rolls and streaks are found to have a stabilising effect on Kelvin-Helmholtz instability, whereas the varicose rolls and streaks have a destabilising effect. Absolute instability is not found to occur. This work shows that the effects of rolls and streaks need to be taken into account for more precise modelling of jet instability.

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. 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 optimizing 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., 11 pages, 3 figures, accepted manuscript

The effects of shear on double-diffusive fingering and on the settling-driven instability are assessed by means of a transient growth analysis. Employing Kelvin waves within a linearized framework allows for the consideration of time-dependent waveforms in uniform shear. In this way, the effects of boundaries and of shear-driven instability modes can be eliminated, so that the influence of shear on the double-diffusive and settling-driven instabilities can be analysed in isolation. Shear is seen to dampen both instabilities, which is consistent with previous findings by other authors. The shear damping is more pronounced for parameter values that produce larger unsheared growth. These trends can be explained in terms of instantaneous linear stability results for the unsheared case. For both double-diffusive and settling-driven instabilities, low Prandtl number ($Pr$) values result in less damping and an increased importance of the Orr mechanism, for which a quantitative scaling law is obtained.

The linear global stability of laminar buoyant jets and plumes is investigated under the low-Mach-number approximation. For Richardson numbers in the range $10^{-4}\leqslant Ri\leqslant 10^{3}$ and density ratios $S=\unicode[STIX]{x1D70C}_{\infty }/\unicode[STIX]{x1D70C}_{jet}$ between 1.05 and 7, only axisymmetric perturbations are found to exhibit global instability, consistent with experimental observations in helium jets. By varying the Richardson number over seven decades, the effects of buoyancy on the base flow and on the instability dynamics are characterised, and distinct behaviour is observed in the low-$Ri$ (jet) and in the high-$Ri$ (plume) regimes. A sensitivity analysis indicates that different physical mechanisms are responsible for the global instability dynamics in both regimes. In buoyant jets at low Richardson number, the baroclinic torque enhances the basic shear instability, whereas buoyancy provides the dominant instability mechanism in plumes at high Richardson number. The onset of axisymmetric global instability in both regimes is consistent with the presence of absolute instability. While absolute instability also occurs for helical perturbations, it appears to be too weak or too localised to give rise to a global instability.

International audience; We extend the resolvent-based estimation approach recently introduced by Towne etal. (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.

International audience; The present article aims at optimising the spread of a bifurcating jet: a jet that combines axisymmetric and helical forcing to achieve increased mixing in a preferential plane. Parekh et al. (Tech. Rep. TF-35, Stanford University, 1988) explained such a bifurcation as the result of nonlinear interaction between ring vortices (triggered by m D 0 axisymmetric forcing), shifted off-axis in alternate directions (owing to mD1 helical forcing). Following this idea, we linearly optimise the periodic helical forcing to be applied at the inlet, in order to maximally displace the ring vortices of an axisymmetrically forced jet. Two norms are introduced for evaluating the effect of helical forcing onto the helical response: the standard L2-norm and a semi-norm reflecting the off-axis vortex displacement. The linear results show one dominant forcing mode over the entire Strouhal band studied (0:35 6 St 6 0:8), with a large gain separation from suboptimals. The dominant forcing is mainly radial, independent of the chosen response norm, and provides a gain at least five times larger than what was achieved by previous ad hoc forcing strategies. Superposition of base flow and linear results show the alternate shifting and twisting provoked by the the small-amplitude helical forcing, which is an essential ingredient for triggering jet bifurcation. When tested in three-dimensional direct numerical simulations, low-amplitude helical forcing achieves efficient bifurcation at all Strouhal values studied. At high Strouhal numbers, an additional central branch emerges in the mean flow, leading to trifurcation. Across all frequencies, compared with ad hoc forcing strategies, the optimal forcing triggers a much stronger and robust spreading, by moving the bifurcation point upstream. As a result, bifurcating jets are observed over a much larger Strouhal band (0:35 6 St 6 0:8) compared with the band where ad hoc forcing achieves bifurcation in our setting (0:46St 60:5).

Resolvent analysis is applied to a nonreacting and a reacting swirled jet flow. Time-averaged flows as input for the resolvent analysis and validation for the results of the resolvent analysis are obtained by experiments. We show that in the nonreacting (cold) flow case, the resolvent analysis is capable of predicting the hydrodynamic response to upstream harmonic acoustic forcing if the flow shows low-rank behavior. This is the case for low and moderate acoustic forcing amplitudes. Even for very strong acoustic velocity amplitudes that are of the same order of magnitude as the flow velocity, the resolvent analysis still provides reasonable results. The method also yields very good results for the reacting flow in terms of velocity fluctuation and heat release response to the acoustic forcing. This confirms the idea that the resolvent method could be applied to estimate the flame transfer function (FTF) based on the mean flow and flame.

International audience; In this work we investigate the use of multiple-input, multiple-output (MIMO) transfer functions obtained empirically from a large-eddy simulation of a turbulent jet to improve estimates of the space-time evolution of flow disturbances by single-input-single-output (SISO) transfer functions used in previous studies. The choice of sensor placement has been made based on results of linear stability analysis from the literature. The results have shown that MIMO transfer functions are capable of improving predictions of SISO transfer functions concerning both single-and two-point statistics. It has also been observed that MIMO estimates have good accuracy even for large separation distances between input and output and that the number of sensors necessary to converge the estimates depends strongly on Strouhal number. I. Nomenclature y = Input measurement z = Output measurement H 1 = Transfer function corresponding to H 1 estimator S zy = Cross-spectral density matrix between inputs and outputs S yy = Cross-spectral density matrix between input measurements h 1 = Time-domain transfer function γ 2 = Two-point coherence c = Speed of sound St = Strouhal number p = Fluctuating pressure u = Fluctuating streamwise velocity u r = Fluctuating radial velocity II. Introduction Input-output analysis has been increasingly used in control of fluid systems, because it provides a framework in which to associate control design and reduced-order models, notably the ones based on hydrodynamic stability theory[1]. In this framework the frequency response of the system to given excitations is described by means of transfer functions, and this information is used to feed actuators via a control law so as to achieve a certain objective. For a convectively unstable flow, the crucial step in this approach is the correct estimation of the evolution of flow disturbances as they are convected downstream. Therefore, a model that provides an accurate

International audience; The spontaneous pairing of rolled-up vortices in a laminar jet is investigated as a global secondary instability of a time-periodic spatially developing vortex street. The growth of subharmonic perturbations, associated with vortex pairing, is analysed both in terms of modal Floquet instability and in terms of transient growth dynamics. The article has the double objective to outline a toolset for global analysis of time-periodic flows, and to leverage such an analysis for a fresh view on the vortex pairing phenomenon. Axisymmetric direct numerical simulations (DNS) of jets with single-frequency inflow forcing are performed, in order to identify combinations of the Reynolds and Strouhal numbers for which vortex pairing is naturally observed. The same DNS calculations are then repeated with an added time-delay control term, which artificially suppresses pairing, so as to obtain time-periodic unpaired base flows for linear stability analysis. It is demonstrated that the natural occurrence of vortex pairing in nonlinear DNS coincides with a linear subharmonic Floquet instability of the underlying unpaired vortex street. However, DNS results suggest that the onset of pairing involves much stronger temporal growth of subharmonic perturbations than what is predicted by modal Floquet analysis, as well as a spatial distribution of these fast-growing perturbation structures that is inconsistent with the unstable Floquet mode. Singular value decomposition of the phase-shift operator (the operator that maps a given perturbation field to its state one flow period later) is performed for an analysis of optimal transient growth in the vortex street. Non-modal mechanisms near the jet inlet are thus found to provide a fast route towards the limit-cycle regime of established vortex pairing, in good agreement with DNS observations. It is concluded that modal Floquet analysis accurately predicts the parameter regime where sustained vortex pairing occurs, but that the bifurcation scenario under typical conditions is dominated by transient growth phenomena.

International audience; Organized structures in turbulent jets can be modeled as wavepackets. These are characterized by spatial amplification and decay, both of which are related to stability mechanisms, and they are coherent over several jet diameters, thereby constituting a noncompact acoustic source that produces a distinctive directivity in the acoustic field. In this review, we use simplified model problems to discuss the salient features of turbulent-jet wavepackets and their modeling frameworks. Two classes of model are considered. The first, that we refer to as kinematic, is based on Lighthill's acoustic analogy, and allows an evaluation of the radiation properties of sound-source functions postulated following observation of jets. The second, referred to as dynamic, is based on the linearized, inhomogeneous Ginzburg–Landau equation, which we use as a surrogate for the linearized, inhomogeneous Navier–Stokes system. Both models are elaborated in the framework of resolvent analysis, which allows the dynamics to be viewed in terms of an input–ouput system, the input being either sound-source or nonlinear forcing term, and the output, correspondingly, either farfield acoustic pressure fluctuations or nearfield flow fluctuations. Emphasis is placed on the extension of resolvent analysis to stochastic systems, which allows for the treatment of wavepacket jitter, a feature known to be relevant for subsonic jet noise. Despite the simplicity of the models, they are found to qualitatively reproduce many of the features of turbulent jets observed in experiment and simulation. Sample scripts are provided and allow calculation of most of the presented results.

This study aims to detect the unsteady features of the turbulent flow behind a wall-proximity rib using resolvent analysis, based on the mean flow field determined using an adjoint-based data assimilation (ABDA) model. The rib is located at gap ratios G/d = 0.25 and 0.50 with a flow Reynolds number Re = 7600 based on the rib size (d = 10 mm) and the free-stream velocity U0. The split fiber measurements at x/d = −0.25, 1.25, 4.25, and 9.25 are solely used as observational data, while the temperature sensitive paint and particle image velocimetry (PIV) results are used as the complement for the analysis and validation. First, the mean flows at both gap ratios are reproduced with the ABDA model using the streamwise velocity constraint of the observational data. It is shown that the global fields are accurately recovered, including the wall jet originating from the gap, which is absent from the PIV results. This finding indicates substantial heat transfer enhancement immediately behind the rib. Subsequently, the resolvent modes at Strouhal numbers St = 0.02, 0.05, 0.15, and 0.30 are obtained from the mean flows using a stochastic approach instead of performing the singular value decomposition directly on the resolvent operator, due to the large matrix size. With the help of the power spectral density of the split fiber measurement, the resolvent analysis identifies the large-scale flapping motion and the wall-jet fine scales that enhance the heat transfer in the case of G/d = 0.25, in addition to the Karman vortex shedding, which makes little contribution to the wall heat transfer in the case of G/d = 0.50. The flow dynamical features in both cases are reconstructed using the leading five resolvent modes at St = 0.15, showing good agreement with the proper orthogonal decomposition modes.

International audience; Perturbation wavepackets in a jet, induced by externally applied forcing, are computed from the linearized equations of motion. An experimentally measured mean flow serves as the base state for this linear analysis, and optimally amplified mono-frequency perturbations are identified, in the sense of maximal gain between the forcing energy input and the flow response energy. Sub-optimal orthogonal forcing/response structures are also discussed. Linear analysis results are then compared to measured perturbation wavepackets in the jet experiment. The study addresses the question to what extent the true dynamics of a turbulent jet can be represented by a model based on linear instability.

International audience; Coherent turbulent wavepacket structures in a jet at Reynolds number 460 000 and Mach number 0.4 are extracted from experimental measurements, and are modelled as linear fluctuations around the mean flow. The linear model is based on harmonic optimal forcing structures and their associated flow response at individual Strouhal numbers, obtained from analysis of the global linear resolvent operator. These forcing/response wavepackets ('re-solvent modes') are first discussed with regard to relevant physical mechanisms that provide energy gain of flow perturbations in the jet. Modal shear instability and the non-modal Orr mechanism are identified as dominant elements, cleanly separated between the optimal and sub-optimal forcing/response pairs. A theoretical development in the framework of spectral covariance dynamics then explicates the link between linear harmonic forcing/response structures and the cross-spectral density (CSD) of stochastic turbulent fluctuations. A low-rank model of the CSD at given Strouhal number is formulated from a truncated set of linear resolvent modes. Corresponding experimental CSD matrices are constructed from extensive two-point velocity measurements. Their eigenmodes (spectral proper orthogonal decomposition or SPOD modes) represent coherent wavepacket structures, and these are compared to their counterparts obtained from the linear model. Close agreement is demonstrated in the range of 'preferred mode' Strouhal numbers, around a value of 0.4, between the leading coherent wavepacket structures as educed from the experiment and from the linear resolvent-based model.

The temporal and spatiotemporal stability of thermal plumes is investigated for laminar velocity and temperature profiles, under the Boussinesq approximation, in the far self-similar region as well as in the region close to a finite-size inlet. In the self-similar case, Prandtl and Grashof numbers are systematically varied, and azimuthal wavenumbers $m=0$, 1 and 2 are considered. In the temporal analysis, helical modes of $m=1$ are found to be dominant throughout the unstable parameter space, with few exceptions. Axisymmetric modes typically present smaller growth rates, but they may dominate at very low Prandtl and Grashof numbers. Double-helical modes of $m=2$ are unstable over a very restricted range of parameters. Only the helical $m=1$ mode is found to ever become absolutely unstable, whereas $m=0$ and $m=2$ modes are at most convectively unstable. In a temporal setting, an analysis of the perturbation energy growth identifies buoyancy- and shear-related mechanisms as the two potentially destabilizing flow ingredients. Buoyancy is demonstrated to be important at low Grashof numbers and long wavelengths, whereas classical shear mechanisms are dominant at high Grashof numbers and shorter wavelengths. The physical mechanism of destabilization through the effect of buoyancy is investigated, and an interpretation is proposed. In the near-source region, both axisymmetric and helical modes may be unstable in a temporal sense over a significant range of wavenumbers. However, absolute instability is again only found for helical $m=1$ modes.

International audience; A numerical method is presented that allows to compute time-periodic flow states, even in the presence of hydrodynamic instabilities. The method is based on filtering non-harmonic components by way of delayed feedback control, as introduced by Pyragas [Phys. Lett. A 170, 421-428 (1992)]. Its use in flow problems is demonstrated here for the case of a periodically forced laminar jet, subject to a subharmonic instability that gives rise to vortex pairing. The optimal choice of the filter gain, which is a free parameter in the stabilization procedure, is investigated in the context of a low-dimensional model problem, and it is shown that this model predicts well the filter performance in the high-dimensional flow system. Vortex pairing in the jet is efficiently suppressed, so that the unstable periodic flow state in response to harmonic forcing is accurately retrieved. The procedure is straightforward to implement inside any standard flow solver. Memory requirements for the delayed feedback control can be significantly reduced by means of time interpolation between checkpoints. Finally, the method is extended for the treatment of periodic problems where the frequency is not known a priori. This procedure is demonstrated for a three-dimensional cubic lid-driven cavity in supercritical conditions.

The global stability of laminar axisymmetric low-density jets is investigated in the low Mach number approximation. The linear modal dynamics is found to he characterised by two features: a stable arc branch of eigenmodes and an isolated eigemnode. Both features are studied in detail, revealing that, whereas the former is highly sensitive to numerical domain size and is existence can he linked to spurious feedback from the outflow boundary, the latter is the physical eigenmode that is responsible for the appearance of self-sustained oscillations in low-density jets observed in experiments at low- Mach numbers. In contrast to previous local spatio-temporal stability analyses, the present global analysis permits, for the first time, the determination of the critical conditions for the onset of global instability, as well the frequency of the associated oscillations, without additional hypotheses, yielding predictions in fair agreement with previous experimental observations. It is shown that under the conditions of those experiments, viscosity variation with composition, as well as buoyancy, only have a small effect on the onset of instability. L.L. acknowledges financial support from ANR under the Cool Jazz project and from DGA under contract LADX2331. W.C. and A.S. were supported by the Spanish MICINN under project DPI2014-59292-C3-1-P and DPI2015-71901-REDT.

Whenever linear eigenmodes of open flows are computed on a numerical domain that is truncated in the streamwise direction, artificial boundary conditions may give rise to spurious pressure signals that are capable of providing unwanted perturbation feedback to upstream locations. The manifestation of such feedback in the eigenmode spectrum is analysed here for two simple configurations. First, explicitly prescribed feedback in a Ginzburg-Landau model is shown to produce a spurious eigenmode branch, named the 'arc branch', that strongly resembles a characteristic family of eigenmodes typically present in open shear flow calculations. Second, corresponding mode branches in the global spectrum of an incompressible parallel jet in a truncated domain are examined. It is demonstrated that these eigenmodes of the numerical model depend on the presence of spurious forcing of a local k+ instability wave at the inflow, caused by pressure signals that appear to be generated at the outflow. Multiple local k+ branches result in multiple global eigenmode branches. For the particular boundary treatment chosen here, the strength of the pressure feedback from the outflow towards the inflow boundary is found to decay with the cube of the numerical domain length. It is concluded that arc-branch eigenmodes are artifacts of domain truncation, with limited value for physical analysis. It is demonstrated, for the example of a non-parallel jet, how spurious feedback may be reduced by an absorbing layer near the outflow boundary., Comment: to appear in Theoretical and Computational Fluid Dynamics (2017 or 2018)

This article provides theoretical conditions for the use and meaning of a stability analysis around a mean flow. As such, it may be considered as an extension of the works by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to non-parallel flows and by Turton et al. (Phys. Rev. E, vol. 91 (4), 2015, 043009) to broadband flows. Considering a Reynolds decomposition of the flow field, the spectral (or temporal Fourier) mode of the fluctuation field is found to be equal to the action on a turbulent forcing term by the resolvent operator arising from linearisation about the mean flow. The main result of the article states that if, at a particular frequency, the dominant singular value of the resolvent is much larger than all others and if the turbulent forcing at this frequency does not display any preferential direction toward one of the suboptimal forcings, then the spectral mode is directly proportional to the dominant optimal response mode of the resolvent at this frequency. Such conditions are generally met in the case of weakly non-parallel open flows exhibiting a convectively unstable mean flow. The spatial structure of the singular mode may in these cases be approximated by a local spatial stability analysis based on parabolised stability equations (PSE). We have also shown that the frequency spectrum of the flow field at any arbitrary location of the domain may be predicted from the frequency evolution of the dominant optimal response mode and the knowledge of the frequency spectrum at one or more points. Results are illustrated in the case of a high Reynolds number turbulent backward facing step flow.

Linear stability analysis of fluid flows usually involves the numerical solution of large eigenvalue problems. We present a spectral transformation allowing the computation of the least stable eigenmodes in a prescribed frequency range, based on the filtering of the linearized equations of motion. This "shift-relax" method has the advantage of low memory requirements and is therefore suitable for large two- or three-dimensional problems. For demonstration purposes, this new method is applied to compute eigenmodes of a compressible jet.

According to a recent hypothesis corroborated by experimental investigations, large-scale coherent structures in turbulent jet flows can be modelled as wavepackets. This conjecture suggests that these structures can be responsible for low-frequency acoustic radiation. In this contribution, we compute optimally forced stability wavepackets, based on meanflow data of a turbulent jet. The dataset is obtained from fully compressible DNS. The final aim is to assess the validity of the hypothesis by verifying to what extent the jet noise can be predicted by means of these linear structures.

The linear global stability of light jets and plumes is analyzed to examine the role of buoyancy on the dynamics of the flow by varying the Froude number under low Mach number approximation. In this study, buoyancy is generated due to the heating of fluid injected at the inlet (thermal plume). The results also closely describe the dynamics of an isothermal light jet at the corresponding density ratio, \(\rho _{\infty }/\rho _{j}\), where \(\rho _{j}\) and \(\rho _{\infty }\) are the jet and ambient density respectively. The analysis is limited to a thermal plume which has a density ratio \(\rho _{\infty }/\rho _{j} = 7\) corresponding to the value for pure helium injected into air. At large Froude number (or small Richardson number), the dynamics of the flow are shown to be determined by the baroclinic torque resulting in the Kelvin Helmholtz instability as in the case of hot jets. However, as the Froude number is lowered, the dominant growth rate and Strouhal number increase monotonically and the dynamics become strongly buoyancy dependent. The variation of Strouhal number with Richardson number (or Froude number) resulting from the global instability is found to match closely with the experimental and DNS investigations on helium jets injected into air.

The linear amplification of axisymmetric external forcing in incompressible jet flows is investigated within a fully non-parallel framework. Experimental and numerical studies have shown that isothermal jets preferably amplify external perturbations for Strouhal numbers in the range $0. 25\leq {\mathit{St}}_{D} \leq 0. 5$, depending on the operating conditions. In the present study, the optimal forcing of an incompressible jet is computed as a function of the excitation frequency. This analysis characterizes the preferred amplification as a pseudo-resonance with a dominant Strouhal number of around $0. 45$. The flow response at this frequency takes the form of a vortical wavepacket that peaks inside the potential core. Its global structure is characterized by the cooperation of local shear-layer and jet-column modes.

International audience; The linear stability dynamics of incompressible and compressible isothermal jets are investigated by means of their optimal initial perturbations and of their temporal eigenmodes. The transient growth analysis of optimal perturbations is robust and allows physical interpretation of the salient instability mechanisms. In contrast, the modal representation appears to be inadequate, as neither the computed eigenvalue spectrum nor the eigenmode shapes allow a characterization of the flow dynamics in these settings. More surprisingly, numerical issues also prevent the reconstruction of the dynamics from a basis of computed eigenmodes. An investigation of simple model problems reveals inherent problems of this modal approach in the context of a stable convection-dominated configuration. In particular, eigenmodes may exhibit an exponential growth in the streamwise direction even in regions where the flow is locally stable.

The global stability of a subsonic jet is investigated using a model base ow designed to t experimental results for turbulent mean ows. Eigenmodes are computed for axisymmetric perturbations in order to investigate the nature of typically observed large-scale coherent oscillations (\preferred mode"). We do not nd evidence that this preferred mode corresponds to the least damped global mode. Non-modal stability is also considered through the computation of optimal perturbations. Although non-axisymmetric perturbations (in particular for azimuthal wavenumber m = 1) are subject to larger transient growth, these reach their peak amplitude far downstream of the potential core, and therefore they are less likely to be observed.

International audience; A new approach is introduced for turbidite modeling, leveraging the potential of computational fluid dynamics methods to simulate the flow processes that led to turbidite formation. The practical use of numerical flow simulation for the purpose of turbidite modeling so far is hindered by the need to specify parameters and initial flow conditions that are a priori unknown. The present study proposes a method to determine optimal simulation parameters via an automated optimization process. An iterative procedure matches deposit predictions from successive flow simulations against available localized reference data, as in practice may be obtained from well logs, and aims at convergence towards the best-fit scenario. The final result is a prediction of the entire deposit thickness and local grain size distribution. The optimization strategy is based on a derivative-free, surrogate-based technique. Direct numerical simulations are performed to compute the flow dynamics. A proof of concept is successfully conducted for the simple test case of a two-dimensional lock-exchange turbidity current. The optimization approach is demonstrated to accurately retrieve the initial conditions used in a reference calculation.

International audience; A linear stability analysis is carried out for the interaction of an erodible sediment bed with a sediment-laden, stratified flow above the bed, such as a turbidity or bottom current. The fluid motion is described by the full, two-dimensional Navier-Stokes equations in the Boussinesq approximation, while erosion is modelled as a diffusive flux of particles from the bed into the fluid. The stability analysis shows the existence of both Tollmien-Schlichting and internal wave modes in the stratified boundary layer. For the internal wave mode, the stratified boundary layer acts as a wave duct, whose height can be determined analytically from the Brunt-Val frequency criterion. Consistent with this criterion, distinct unstable perturbation wavenumber regimes exist for the internal wave mode, which are associated with different numbers of pressure extrema in the wall-normal direction. For representative turbidity current parameters, the analysis predicts unstable wavelengths that are consistent with field observations. As a key condition for instability to occur, the base flow velocity boundary layer needs to be thinner than the corresponding concentration boundary layer. For most of the unstable wavenumber ranges, the phase relations between the sediment bed deformation and the associated wall shear stress and concentration perturbations are such that the sediment waves migrate in the upstream direction, which again is consistent with field observations. © 2011 Cambridge University Press.

The absolute/convective character of the linear instability of axisymmetric jets is investigated for a wide range of parallel velocity and density profiles. An adjoint-based sensitivity analysis is carried out in order to maximize the absolute growth rate of jet profiles with and without density variations. It is demonstrated that jets without counterflow may display absolute instability at density ratios well above the previously assumed threshold ρjet/ρ∞ = 0.72, and even in homogeneous settings. Absolute instability is promoted by a strong velocity gradient in the low-velocity region of the shear layer, as well as by a step-like density variation near the location of maximum shear. A new efficient algorithm for the computation of the absolute instability mode is presented.

International audience; The acoustic field generated by the synchronized vortex street in self-excited hot subsonic jets is investigated via direct numerical simulation of the compressible equations of motion in an axisymmetric geometry. The simulation simultaneously resolves both the aerodynamic near field and the acoustic far field. Self-sustained near-field oscillations in the present flow configurations have been described as nonlinear global modes in an earlier study. The associated acoustic far field is found to be that of a compact dipole, emanating from the location of vortex roll-up. A far-field solution of the axisymmetric Lighthill equation is derived, on the basis of the source term formulation of Lilley (AGARD-CP, vol. 131, 1974, pp. 13.1-13.12). With the near-field source distributions obtained from the direct numerical simulations, the Lighthill solution is in good agreement with the far-field simulation results. Fluctuations of the enthalpy flux within the jet are identified as the dominant aeroacoustic source. Superdirective effects are found to be negligible. © 2010 Cambridge University Press.

The convective/absolute nature of the instability of miscible core-annular flow with variable viscosity is investigated via linear stability analysis and nonlinear simulations. From linear analysis, it is found that miscible core-annular flows with the more viscous fluid in the core are at most convectively unstable. On the other hand, flows with the less viscous fluid in the core exhibit absolute instability at high viscosity ratios, over a limited range of core radii. Nonlinear direct numerical simulations in a semi-infinite domain display self-excited intrinsic oscillations if and only if the underlying base flow exhibits absolute instability. This oscillator-type flow behaviour is demonstrated to be associated with the presence of a nonlinear global mode. Both the parameter range of global instability and the intrinsically selected frequency of nonlinear oscillations, as observed in the simulation, are accurately predicted from linear criteria. In convectively unstable situations, the flow is shown to respond to external forcing over an unstable range of frequencies, in quantitative agreement with linear theory. As discussed in part 1 of this study (d'Olce, Martin, Rakotomalala, Salin and Talon,J. Fluid Mech., vol. 618, 2008, pp. 305–322), self-excited synchronized oscillations were also observed experimentally. An interpretation of these experiments is attempted on the basis of the numerical results presented here.

International audience; The self-sustained formation of synchronized ring vortices in hot subsonic jets is investigated by direct numerical simulation of the axisymmetric equations of motion. The onset of global instability and the global frequency of synchronized oscillations are examined as functions of the ambient-to-jet temperature ratio and the initial jet shear layer thickness. The numerical results are found to follow the predictions from nonlinear global instability theory; global instability sets in as the unperturbed flow is absolutely unstable over a region of finite streamwise extent at the inlet, and the global frequency near the global instability threshold corresponds to the absolute frequency of the inlet profile. In strongly supercritical thin shear layer jets, however, the simulations display global frequencies well above the absolute frequency, in agreement with experimental results. The inner structure of rolled-up vortices in hot jets displays fine layers of positive and negative vorticity that are produced and maintained by the action of the baroclinic torque. © 2007 American Institute of Physics.

International audience; Since the experiments of Monkewitz et al. (J. Fluid Mech. vol. 213, 1990, p. 611), sufficiently hot circular jets have been known to give rise to self-sustained synchronized oscillations induced by a locally absolutely unstable region. In the present investigation, numerical simulations are carried out in order to determine if such synchronized states correspond to a nonlinear global mode of the underlying base flow, as predicted in the framework of Ginzburg - Landau model equations. Two configurations of slowly developing base flows are considered. In the presence of a pocket of absolute instability embedded within a convectively unstable jet, global oscillations are shown to be generated by a steep nonlinear front located at the upstream station of marginal absolute instability. The global frequency is given, within 10% accuracy, by the absolute frequency at the front location and, as expected on theoretical grounds, the front displays the same slope as a k--wave. For jet flows displaying absolutely unstable inlet conditions, global instability is observed to arise if the streamwise extent of the absolutely unstable region is sufficiently large: While local absolute instability sets in for ambient-to-jet temperature ratios S = 0.453, global modes only appear for S = 0.3125. In agreement with theoretical predictions, the selected frequency near the onset of global instability coincides with the absolute frequency at the inlet. For lower S, it gradually departs from this value. © 2006 Cambridge University Press.

The occurrence of self‐sustained synchronized oscillations in a family of hot axisymmetric jets is investigated in direct numerical simulations. With parameter settings close to those used in the experiments of Monkewitz, Bechert, Barsikow & Lehmann, 1 such synchronized states are found numerically to arise when the ambient‐to‐jet temperature ratio S falls below a critical value. At a Reynolds number Re = 3750, based on the jet radius, global oscillations of Strouhal number St 0.45 occur for S 0.6, which corresponds well to the ‘mode II’ state of Monkewitz et al. 1 The distinct ‘mode I’ oscillations of St 0.3 reported in the experiments 1 are assumed to correspond to the subharmonic of a synchronized state that is numerically observed only to arise at a higher Reynolds number Re = 7500. Both modes are compared to theoretical predictions pertaining to nonlinear global modes which may exist due to the presence of absolute instability, as well as to the subharmonic acoustic resonance model of Laufer & Monkewitz. 2 It is found that the synchronization of the ‘mode I’ state sets in at temperature ratios for which the entire flow is convectively unstable. A nonlinear global mode therefore cannot exist in these cases. The onset of the ‘mode II’ state occurs in the presence of an absolutely unstable flow region of finite streamwise extent; however, even near the global instability threshold, the ‘mode II’ frequency does not match the absolute frequency at the inlet, as should be expected from global mode theory. The subharmonic acoustic resonance model, 2 which assumes a feedback loop between the inlet and the station of vortex pairing, is found not to apply in any of the observed cases of self‐sustained synchronized oscillations.

The linear impulse response of axisymmetric jets is examined for a family of variable-temperature profiles typical of the potential core. The influence of jet heating, shear layer thickness, and Reynolds and Mach numbers on the spatiotemporal stability of both axisymmetric and helical modes is investigated. The linear impulse response is retrieved from a numerical solution of the spatial eigenvalue problem, which is derived from the fully compressible equations of motion. Changes in the spatiotemporal stability of heated versus isothermal jets are shown to arise solely from the effect of the baroclinic torque. By considering the full linear impulse response, the competition between jet column modes and shear layer modes is characterized. Jet column modes are only found to occur for axisymmetric disturbances. In thin shear layer jets, the jet column mode is shown to prevail at low group velocities, whereas axisymmetric and helical shear layer modes dominate at high group velocities. The absolute mode of zero group velocity is found to always be of the jet column type. Although only convectively unstable, the maximum growth rates of the shear layer modes greatly exceed those of the jet column modes in thin shear layer jets. In thick shear layer jets, axisymmetric modes of mixed jet column/shear layer type arise. The weakened maximum growth rate of mixed modes accounts for the dominance of helical modes in temporal stability studies of thick shear layer jets.

Linear global modes are computed for round helium jets in air at low Reynolds number. The governing equations are formulated in the limit of zero Mach number, and all parameters are chosen such as to model the experiments of Hallberg & Strykowski1. A single axisymmetric mode is found to become unstable beyond a critical Reynolds number value that depends on the initial shear layer thickness. The neutral curve for linear global instability is traced as a function of these two parameters, and this result compares favorably with the experimental reference measurements.

This thesis focuses on the stability and the optimisation of time-periodic flows. The resulting framework has been applied to the pulsed jet study.In a laminar round jet, due to its large noise amplification properties, ring vortices are generated spontaneously. This generation can be controlled by imposing an axisymmetric forcing at the inlet. In this configuration, two kinds of instabilities arise. On the one hand, in an intrinsic way, these vortices, can undergo vortex pairing: rings merge two by two, increasing the jet thickness and mixing. On the other hand, if an additional subharmonic helical term is superimposed to the axisymmetric forcing, ring vortices can be shifted off-axis in an alternate fashion, leading to a bifurcating jet. The mutual induction between rings leads to a large spreading in a preferential bifurcating plane.First, a stabilisation technique for unstable periodic orbits, based on a time-delayed feedback, has been developed. This method has been applied to the axisymmetric pulsed jet to recover unpaired periodic states. This chapter provides the base flow used for stability and optimisation studies in the following.Second, the modal and non-modal stability of the un-paired jet has been investigated using Floquet theory. While modal stability accurately predicts vortex pairing occurrence, transient growth provides a vigorous transition mechanism from unpaired to paired state.Then, an optimisation of the helical forcing applied to bifurcating jets has been carried out to improve their spreading and mixing. The optimal forcing has been plugged in direct numerical simulations and results have been compared with those obtained with simpler forcing used in the literature. Optimal forcing moves strongly upstream the bifurcation point, triggering a larger flaring of the jet. In addition, optimal forcing dramatically increases the Strouhal number domain in which jet bifurcation can be encountered.Finally, as an Appendix, results from a collaboration with the Laboratoire de Mécanique des Solides (LMS,École Polytechnique) on the stability of the growth of vicinal surfaces are presented.; Cette thèse étudie la stabilité et l’optimisation d'écoulements périodiques en temps. Le cadre développé a été appliqué au cas du jet rond pulsé. Dans la couche de mélange d’un jet, du fait de sa grande réceptivité aux perturbations, des anneaux de vorticité se forment spontanément. Ces derniers sont contrôles en imposant une pulsation axi-symétrique au niveau de la buse. Dans cette configuration, deux types d’instabilités surgissent. D’une part, de fac¸on intrinsèque, un appariement tourbillonnaire peut parfois survenir : les anneaux fusionnent deux par deux, conduisant à un épaississement du jet et à un mélange accru. D’autre part, si un terme hélicoïdal sous-harmonique est ajouté au forçage axi-symétrique, les anneaux formes sont alternativement excentrés de part et d’autre de l’axe, conduisant à un jet bifurqué. L’induction mutuelle des anneaux permet un large évasement du jet dans un plan préférentiel de bifurcation.Dans un premier chapitre, nous avons développé une méthode de stabilisation d’orbites périodiques instables basée sur un contrôle volumique avec un terme de retard. Cette méthode a été appliquée au cas du jet axisymétrique pulsé pour obtenir un écoulement non-apparié. Cette partie fournit l'écoulement de base qui sera utilisé pour les études de stabilité et d’optimisation des parties suivantes.Dans un deuxième chapitre, nous avons étudié la stabilité modale et non-modale de ce jet non-apparié à l’aide de la théorie de Floquet. La stabilité modale prédit avec précision la présence ou non d’un appariement tourbillonnaire tandis que la croissance non-modale fournit un mécanisme de transition très intense entre l'état non-apparié et l'état apparié.Dans un troisième chapitre, le forçage hélicoïdal appliqué au jet bifurqué a ´été optimisé pour améliorer ses propriétés de mélange et d'évasement. Nous avons ensuite comparé par simulation numérique directe les jets bifurqués issus de forçages optimaux avec ceux issus de forçages plus simples précédemment utilisés dans la littérature. Le forçage optimal déclenche la bifurcation beaucoup plus tôt, ce qui permet un évasement bien plus marqué du jet qu’avec un forçage simple. En outre, le forçage optimal augmente considérablement la gamme de nombres de Strouhal dans laquelle cette bifurcation a lieu.Enfin, en annexe, des travaux effectués en collabo-ration avec le Laboratoire de Mécanique des Solides (LMS, École Polytechnique) sur la stabilité et la croissance des surfaces vicinales sont présentés.