Unsteadiness in the wake of disks and spheres: Instability, receptivity and control using direct and adjoint global stability analyses

Abstract : We consider the stability of the steady, axisymmetric wake of a disk and a sphere as a function of the Reynolds number. Both the direct and adjoint eigenvalue problems are solved. The threshold Reynolds numbers and characteristics of the destabilizing modes agree with that documented in previous studies: for both configurations, the first destabilization occurs for a stationary mode of azimuthal wavenumber m = 1, and the second destabilization for an oscillating mode of same azimuthal wavenumber. For both geometries, the adjoint mode computation allows us to determine the receptivity of each mode to particular initial conditions or forcing and to define control strategies. We show that the adjoint global mode reaches a maximum amplitude within the recirculating bubble and downstream of the separation point for both the disk and the sphere. In the case of the sphere, the optimal forcing corresponds to a displacement of the separation point along the sphere surface with no tilt of the separation line. However, in the case of the disk, its blunt shape does not allow such displacement and the optimal forcing corresponds to a tilt of the separation line with no displacement of the separation point. As a result, the magnitudes of the adjoint global modes are larger for the sphere than for the disk, showing that the wake of the sphere is more receptive to forcing than the disk. In the case of active control at the boundary through blowing and suction at the body wall, the actuator should be placed close to the separation point, where the magnitude of the adjoint pressure reaches its maximum in the four cases. In the case of passive control, we show that the region of the wake that is most sensitive to local modifications of the linearized Navier-Stokes operator, including base flow alterations, is limited to the recirculating bubble for both geometries and both instability modes. This region may therefore be identified as the intrinsical wavemaker. © 2009 Elsevier Ltd. All rights reserved.
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Philippe Meliga, Jean-Marc Chomaz, D. Sipp. Unsteadiness in the wake of disks and spheres: Instability, receptivity and control using direct and adjoint global stability analyses. Journal of Fluids and Structures, Elsevier, 2009, 25 (4), pp.601-616. 〈10.1016/j.jfluidstructs.2009.04.004〉. 〈hal-01002606〉



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