Vortex pairing in jets as a global Floquet instability: modal and transient dynamics

International audienceThe 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

Stabilité et contrôle optimal d'écoulements périodiques en temps - application au jet pulsé

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

Time-delayed feedback technique for suppressing instabilities in time-periodic flow

International audienceA 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

Mixing enhancement through optimally controlled jet bifurcation

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Optimal triggering of jet bifurcation: an example of optimal forcing applied to a time-periodic base flow

International audienceThe 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)

Stability of vicinal surfaces: beyond the quasistatic approximation

International audienceWe revisit the step bunching instability without recourse to the quasistatic approximation and show thatthe stability diagrams are significantly altered, even in the low-deposition regime where it was thoughtsufficient. In particular, steps are unstable against bunching for attachment-detachment limited growth. Byaccounting for the dynamics and chemical effects, we can explain the onset of step bunching inSið111Þ-ð7 × 7Þ and GaAs(001) without resort to the inverse Schwoebel barrier or step-edge diffusion.Further, the size-scaling analysis of step-bunch growth, as induced by these two combined effects, agreeswith the bunching regime observed at 750 °C in Si(111)-(7x7)

Recovering the transfer function of a turbulent jet and a laminar flame from linear mean flow analysis

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