• Citer

Global modes, absolute instabilities and acoustic feedback: a model study

par Gopalakrishnan, Shyam Sunder
Promoteur Prof. Patrick Huerre, ;Prof. Peter Schmid,
Publication Non publié, 2010-09-07

Mémoire

• Détails
• Contenu

Résumé :

Global modes in shear flows typically arise when a finite region of absolute instabilityprevails within the physical domain of interest. Examples include the wake behinda circular cylinder, hot jets, counterflow mixing layers, etc. There are, however, othersituations where global modes are triggered by acoustic feedback from downstream.The prime example in this case is the flow along a cavity of finite streamwise extent.The vortices produced as a result of the instability of the shear layer in the cavityinteract with the downstream boundary to produce acoustic waves which may in turngenerate new vortices at the upstream edge of the cavity. Such a feedback interactionleads to a synchronization of the unsteady flow, in the form of a global mode of specificfrequency.In the present work this phenomenon was studied using the one dimensional complexGinzburg-Landau (CGL) equation on a finite domain and an infinite domain. Theacoustic feedback was implemented explicitly through the boundary conditions as wellas by an acoustic plane wave which is coupled with the linear Ginzburg-Landau equation.Using this approach a finite amount of feedback from the downstream boundaryto the upstream boundary along with a finite time delay was enforced and the roleof these parameters were qualitatively and quantitatively investigated. The linearstability analysis was carried out on a spatially homogeneous and a spatially varyingdomain and the influence of the acoustic feedback on the globally stable (locallyconvectively unstable) and globally unstable (locally absolutely unstable) nature ofinstabilities present in the flow has been analysed. The regimes where the unstableglobal mode is triggered by the absolute instability present in the flow (from theGinzburg-Landau equation) or by the acoustic feedback were studied. The role ofthe acoustic feedback in the transient growth phenomenon of the non-normal systemwas analysed. Significant energy amplification was observed of around three ordersof magnitude higher than the system without any feedback due to the synchronizationmechanism. Input-output analysis was carried out to generalize the observationsmade from the hydrodynamic stability theory by studying the energy amplificationand decay with respect to specific input disturbances. Tools like stochastic forcing,impulse response and transfer functions were used in the investigation. The role ofthe acoustic feedback in the nonlinear regime was studied using the nonlinear complexGinzburg-Landau equation and the differences with the linear regime were noted.A closed loop Linear Quadratic Regulator (LQR) based control was applied to thesystem of linear CGL with acoustic plane wave feedback and the decay in the energygrowth due to the control has been shown.