Stability and optimal control of time-periodic flows : application to a pulsed jet

Abstract : This thesis describes the linear instability analysis and the design of linear optimal control of time-periodic flows. The numerical framework developed is applied to the study of pulsed jets.When a laminar round jet is forced axisymmetrically and time-periodically at the inlet, a regular street of vortex rings is formed. Two instability phenomena of such arrays are investigated. Firstly, intrinsic mechanisms may trigger vortex pairing. Secondly, if an additional subharmonic helical component is superposed onto the fundamental axisymmetric forcing, jet bifurcation is induced. Both phenomena result in strongly increased spreading and mixing in the mean flow.In a first step, a numerical stabilisation technique is devised, allowing the computation of exact periodic flow solutions, even when they are subject to instrinsic instabilities. This method, based on a time-delayed feedback, is then applied in order to recover unpaired periodic flow states of pulsed jets, in parameter regimes where vortex pairing naturally occurs. These unpaired flow states form the basis for the following instability and optimal control calculations.In a second step, the instrinsic perturbation dynamics in pulsed jets is investigated. Modal instability properties, governing the long-time flow behaviour, are examined in the framework of Floquet theory. Numerically, a Krylov basis is constructed from linear time-stepping using a block-Arnoldi algorithm to maximise efficiency. Transient dynamics, governing the short-time growth of initial perturbations, are characterised by an optimal perturbation analysis. While the modal Floquet analysis accurately predicts the critical Reynolds and Strouhal numbers of the long-time occurrence of vortex pairing, transient growth dynamics dominates the bifurcation.Finally, the optimal way to trigger jet bifurcation through subharmonic inlet forcing is computed. Inlet helical forcing is identified that maximises the jet spreading and mixing in one privileged meridional plane. This optimal forcing is implemented in direct numerical simulations, and its efficiency in the nonlinear regime is compared to that of ad hoc forcing used in previous studies. The optimal forcing results in bifurcation further upstream, at higher spreading angles, and over a much wider range of Strouhal numbers than found previously.
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Léopold Shaabani Ardali. Stability and optimal control of time-periodic flows : application to a pulsed jet. Mechanics of the fluids [physics.class-ph]. Université Paris-Saclay, 2018. English. ⟨NNT : 2018SACLX090⟩. ⟨tel-02275156v2⟩

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