is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible.This is an author-deposited version published in: http://sam.ensam.eu Handle ID: .http://hdl.handle.net/10985/6862 To cite this version :Frédéric ALIZARD, Stefania CHERUBINI, Jean-Christophe ROBINET -Sensitivity and optimal forcing response in separated boundary layer flows -Physics of Fluids -Vol. 21, n°064108, p.3 -2009 Any correspondence concerning this service should be sent to the repository The optimal asymptotic response to time harmonic forcing of a convectively unstable two-dimensional separated boundary layer on a flat plate is numerically revisited from a global point of view. By expanding the flow disturbance variables and the forcing term as a summation of temporal modes, the linear convective instability mechanism associated with the response leading to the maximum gain in energy is theoretically investigated. Such a response is driven by a pseudoresonance of temporal modes due to the non-normality of the underlying linearized evolution operator. In particular, the considered expansion on a limited number of modes is found able to accurately simulate the linear instability mechanism, as suggested by a comparison between the global linear stability analysis and a linearized direct numerical simulation. Furthermore, the dependence of such a mechanism on the Reynolds number and the adverse pressure gradient is investigated, outlining a physical description of the destabilization of the flow induced by the rolling up of the shear layer. Therefore, the convective character of the problem suggests that the considered flat plate separated flows may act as a selective noise amplifier. In order to verify such a possibility, the responses of the flow to the optimal forcing and to a small level of noise are compared, and their connection to the onset of self-excited vortices observed in literature is investigated. For that purpose, a nonlinear direct numerical simulation is performed, which is initialized by a random noise superposed to the base flow at the inflow boundary points. The band of excited frequencies as well as the associated peak match with the ones computed by the asymptotic global analysis. Finally, the connection between the onset of unsteadiness and the optimal response is further supported by a comparison between the optimal circular frequency and a typical Strouhal number predicted by numerical simulations of previous authors in similar cases.

The linear stability of a weakly nonparallel flow, the case of a flat plate boundary layer, is revisited by a linear global stability approach where the two spatial directions are taken as inhomogeneous, leading to a fully nonparallel stability method. The resulting discrete eigenvalues obtained by the fully nonparallel approach seem to be in agreement with classical Tollmien–Schlichting waves. Then the different modes are compared with classical linear stability approach and weakly nonparallel method based on linear parabolized stability equations (PSEs). It is illustrated that the nonparallel correction provided by the linear global stability approach is well matched by linear PSE. Furthermore, physical interpretation of these spatio-temporal global modes is given where a real pulsation, which has more physical interest, is considered. In particular the use of a Gaster transformation and the pseudospectrum illustrate the local and global properties of these Tollmien–Schlichting modes. Finally, the contribution of different components of global modes (normal and streamwise) in the transient amplifying behavior associated with the convectively unstable boundary layer is analyzed and compared with a classical steepest descent method. Then, a discussion of an equivalent of the continuous branch is given.

A fully three-dimensional linear stability analysis is carried out to investigate the unstable bifurcations of a compressible viscous fluid past a sphere. A time-stepper technique is used to compute both equilibrium states and leading eigenmodes. In agreement with previous studies, the numerical results reveal a regular bifurcation under the action of a steady mode and a supercritical Hopf bifurcation that causes the onset of unsteadiness but also illustrate the limitations of previous linear approaches, based on parallel and axisymmetric base flow assumptions, or weakly nonlinear theories. The evolution of the unstable bifurcations is investigated up to low-supersonic speeds. For increasing Mach numbers, the thresholds move towards higher Reynolds numbers. The unsteady fluctuations are weakened and an axisymmetrization of the base flow occurs. For a sufficiently high Reynolds number, the regular bifurcation disappears and the flow directly passes from an unsteady planar-symmetric solution to a stationary axisymmetric stable one when the Mach number is increased. A stability map is drawn by tracking the bifurcation boundaries for different Reynolds and Mach numbers. When supersonic conditions are reached, the flow becomes globally stable and switches to a noise-amplifier system. A continuous Gaussian white noise forcing is applied in front of the shock to examine the convective nature of the flow. A Fourier analysis and a dynamic mode decomposition show a modal response that recalls that of the incompressible unsteady cases. Although transition in the wake does not occur for the chosen Reynolds number and forcing amplitude, this suggests a link between subsonic and supersonic dynamics.