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Linear stability and weakly nonlinear analysis of the flow past rotating spheres

Citro, Vincenzo and Tchoufag, Joël and Fabre, David and Giannetti, Flavio and Luchini, Paolo Linear stability and weakly nonlinear analysis of the flow past rotating spheres. (2016) Journal of Fluid Mechanics, 807. 62-86. ISSN 0022-1120

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Official URL: https://doi.org/10.1017/jfm.2016.596

Abstract

We study the flow past a sphere rotating in the transverse direction with respect to the incoming uniform flow, and particularly consider the stability features of the wake as a function of the Reynolds number Re and the sphere dimensionless rotation rate Ω . Direct numerical simulations and three-dimensional global stability analyses are performed in the ranges 150⩽Re⩽300 and 0⩽Ω⩽1.2 . We first describe the base flow, computed as the steady solution of the Navier–Stokes equation, with special attention to the structure of the recirculating region and to the lift force exerted on the sphere. The stability analysis of this base flow shows the existence of two different unstable modes, which occur in different regions of the Re/Ω parameter plane. Mode I, which exists for weak rotations ( Ω<0.4 ), is similar to the unsteady mode existing for a non-rotating sphere. Mode II, which exists for larger rotations ( Ω>0.7 ), is characterized by a larger frequency. Both modes preserve the planar symmetry of the base flow. We detail the structure of these eigenmodes, as well as their structural sensitivity, using adjoint methods. Considering small rotations, we then compare the numerical results with those obtained using weakly nonlinear approaches. We show that the steady bifurcation occurring for Re>212 for a non-rotating sphere is changed into an imperfect bifurcation, unveiling the existence of two other base-flow solutions which are always unstable.

Item Type:Article
HAL Id:hal-03240794
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE)
Université de Toulouse > Université Toulouse III - Paul Sabatier - UT3 (FRANCE)
Other partners > Università degli Studi di Salerno - UniSa (ITALY)
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Deposited On:18 May 2021 11:51

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