# zbMATH — the first resource for mathematics

Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion. (English) Zbl 1183.76721
Summary: Direct numerical simulations (DNS) of the wake of a circular disk placed normal to a uniform flow show that, as the Reynolds number is increased, the flow undergoes a sequence of successive bifurcations, each state being characterized by specific time and space symmetry breaking or recovering D. Fabre, F. Auguste, J. Magnaudet, Phys. Fluids 20, No. 5, Paper No. 051702, 4 p. (2008)]. To explain this bifurcation scenario, we investigate the stability of the axisymmetric steady wake in the framework of the global stability theory. Both the direct and adjoint eigenvalue problems are solved. The threshold Reynolds numbers Re and characteristics of the destabilizing modes agree with the study of R. Natarajan and A. Acrivos [J. Fluid Mech. 254, 323–344 (1993; Zbl 0780.76027)]: the first destabilization occurs for a stationary mode of azimuthal wavenumber $$m = 1$$ at $$Re_{c}^{A} = 116.9$$, and the second destabilization of the axisymmetric flow occurs for two oscillating modes of azimuthal wavenumbers $$m \pm 1$$ at $$Re_{c}^{B} = 125.3$$. Since these critical Reynolds numbers are close to one another, we use a multiple time scale expansion to compute analytically the leading-order equations that describe the nonlinear interaction of these three leading eigenmodes. This set of equations is given by imposing, at third order in the expansion, a Fredholm alternative to avoid any secular term. It turns out to be identical to the normal form predicted by symmetry arguments. Though, all coefficients of the normal form are here analytically computed as the scalar product of an adjoint global mode with a resonant third-order forcing term, arising from the second-order base flow modification and harmonics generation. We show that all nonlinear interactions between modes take place in the recirculation bubble, as the contribution to the scalar product of regions located outside the recirculation bubble is zero. The normal form accurately predicts the sequence of bifurcations, the associated thresholds and symmetry properties observed in the DNS calculations.

##### MSC:
 76E09 Stability and instability of nonparallel flows in hydrodynamic stability 76E30 Nonlinear effects in hydrodynamic stability 76D25 Wakes and jets
Full Text:
##### References:
 [1] DOI: 10.1103/PhysRevE.77.055308 [2] DOI: 10.1017/S0022112087002234 · Zbl 0639.76041 [3] DOI: 10.1007/BF00280698 · Zbl 0588.34030 [4] DOI: 10.1016/0167-2789(88)90063-2 · Zbl 0656.76088 [5] Friedrichs, Spectral Theory of Operators in Hilbert Space. (1973) · Zbl 0266.47001 [6] DOI: 10.1145/992200.992205 · Zbl 1072.65036 [7] DOI: 10.1063/1.2909609 · Zbl 1182.76238 [8] DOI: 10.1103/PhysRevLett.57.2935 [9] DOI: 10.1017/S0022112005005112 · Zbl 1073.76027 [10] Crawford, Annu. Rev. Fluid. Mech. 23 pp 341– (1991) [11] DOI: 10.1017/S0022112094000583 · Zbl 0813.76021 [12] Crawford, Dyn. Stab. Syst. 3 pp 159– (1988) · Zbl 0681.58029 [13] DOI: 10.1002/(SICI)1097-0363(19990930)31:23.0.CO;2-O [14] Chossat, Appl. Math. Sci. 102 (1994) [15] DOI: 10.1137/S0895479894246905 · Zbl 0884.65021 [16] Chomaz, Proceedings of the Conf. on New Trends in Nonlinear Dynamics and Pattern-Forming Phenomena: The Geometry of Nonequilibrium pp 259– (1990) [17] DOI: 10.1146/annurev.fluid.37.061903.175810 · Zbl 1117.76027 [18] DOI: 10.1017/S002211200200232X · Zbl 1026.76019 [19] DOI: 10.1209/epl/i2006-10168-7 [20] DOI: 10.1103/PhysRevLett.79.3893 [21] DOI: 10.1007/BF00127673 [22] DOI: 10.1006/jfls.2000.0362 [23] DOI: 10.1017/S0022112007008907 · Zbl 1172.76318 [24] Schmid, Stability and Transition in Shear Flows. (2001) · Zbl 0966.76003 [25] DOI: 10.1017/S0022112087002222 · Zbl 0641.76046 [26] DOI: 10.1017/S0022112008000736 · Zbl 1151.76473 [27] DOI: 10.1017/S0022112093002150 · Zbl 0780.76027 [28] Golubitsky, Singularities and Groups in Bifurcation Theory Vol. II – Applied Mathematical Sciences. (1988) · Zbl 0691.58003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.