ON THE SECONDARY INSTABILITY IN PLANAR SHEAR FLOW PAST A CIRCULAR CYLINDER

초록

The eects of planar shear on the secondary instability in the ow past a circular cylinder have been studied by using Floquet stability ananlysis. An immersed boundary method [1] was used in order to implement a circular cylinder in a Cartesian grid system. The key parameters are the Reynolds number (Re) based on the inlet center velocity (Uc ) and the cylinder diameter (D) and the shear parameter (K) dened as K = GD/Uc. Here, G is the velocity gradient in the vertical direction. The ranges of Re and K considered in the present investigation are and , respectively. Increasing shear renders the ow more unstable to the primary instability, known as Hopf bifurcation in which a steady ow bifurcates into a time-periodic ow. The critical Reynolds number of the primary instability (Rec1) becomes lower with increasing K. Above another higher critical Reynolds number (Rec2), a secondary instability of three-dimensional (3D) nature sets in. In the case of uniform ow (K = 0), the secondary instability manifests itself in the form of 3D modes, namely A, B, and QP in the ascending order of Rec2 [2]. We found, however, that presence of planar shear breaks the symmetry of the time-periodic base ow, signicantly aecting the characteristics of the secondary instability. Mode QP is turned into mode C by the asymmetry induced by the planar shear. The planar shear stabilizes modes A and B, but destabilizes mode C. As a result, crossover of criticality between C and A, B occurs with increasing K. In the case of K = 0.2, the maximum planar shear considered in the present study, mode C is even dominant over modes A and B, and Rec2 of mode C is even lower than that of mode A without shear. The eects of planar shear on the spatial and temporal characteristics of the 3D modes are discussed. Neutral stability curves of the 3D modes for each K are also presented. Direct Numerical Simulation was performed in order to verify our Floquet stability analysis, yielding good agreement with o