International audienceWe report novel superlattice wave patterns at the interface of a fluid layer driven vertically. These patterns are described most naturally in terms of two interacting hexagonal sublattices. Two frequency forcing at very large aspect ratio is utilized in this work. A superlattice pattern (“superlattice-I”) consisting of two hexagonal lattices oriented at a relative angle of 22° is obtained with a 6:7 ratio of forcing frequencies. Several theoretical approaches that may be useful in understanding this pattern have been proposed. In another example, the waves are fully described by two superimposed hexagonal lattices with a wavelength ratio of \sqrt{3}, oriented at a relative angle of 30°. The time dependence of this “superlattice-II” wave pattern is unusual. The instantaneous patterns reveal a time-periodic stripe modulation that breaks the sixfold symmetry at any instant, but the stripes are absent in the time average. The instantaneous patterns are not simply amplitude modulations of the primary standing wave. A transition from the superlattice-II state to a 12-fold quasi-crystalline pattern is observed by changing the relative phase of the two forcing frequencies. Phase diagrams of the observed patterns (including superlattices, quasicrystalline patterns, ordinary hexagons, and squares) are obtained as a function of the amplitudes and relative phases of the driving accelerations

In this paper it is shown that the two-dimensional time-periodic vortex shedding régime observed in the cylinder wake at moderate Reynolds numbers may be interpreted as a nonlinear global structure and its naturally selected frequency obtained in the framework of hydrodynamic stability theory. The frequency selection criterion is based on the local absolute frequency curve derived from the unperturbed basic flow fields under the assumption of slow streamwise variations. Although the latter assumption is only approximately fulfilled in the vicinity of the obstacle, the theoretically predicted frequency is in good agreement with direct numerical simulations for Reynolds numbers Re > 100.

In the three-dimensional boundary layer produced by a rotating disk, the experimentally well-documented sharp transition from laminar to turbulent flow is shown to coincide with secondary absolute instability of the naturally selected primary nonlinear crossflow vortices. Fully saturated primary finite-amplitude waves and the associated nonlinear dispersion relation are first numerically computed using a local parallel flow approximation. Exploiting the slow radial development of the basic flow, the naturally selected primary self-sustained flow structure is then derived by asymptotic analysis. In this state, outward-spiralling nonlinear vortices are initiated at the critical radius where primary absolute instability first occurs. A subsequent secondary stability analysis reveals that as soon as the primary nonlinear waves come into existence they are absolutely unstable with respect to secondary perturbations. Secondary disturbances growing in time at fixed radial locations continuously perturb the primary vortices, thus triggering the direct route to turbulence prevailing in this configuration.

A family of slowly spatially developing wakes with variable pressure gradient is numerically demonstrated to sustain a synchronized finite-amplitude vortex street tuned at a well-defined frequency. This oscillating state is shown to be described by a steep global mode exhibiting a sharp Dee-Langer-type front at the streamwise station of marginal absolute instability. The front acts as a wavemaker which sends out nonlinear travelling waves in the downstream direction, the global frequency being imposed by the real absolute frequency prevailing at the front station. The nonlinear travelling waves are determined to be governed by the local nonlinear dispersion relation resulting from a temporal evolution problem on a local wake profile considered as parallel. Although the vortex street is fully nonlinear, its frequency is dictated by a purely linear marginal absolute instability criterion applied to the local linear dispersion relation.

International audienceA new frequency selection criterion valid in the fully nonlinear regime is presented for extended oscillating states in spatially developing media. The spatial structure and frequency of these modes are dominated by the existence of a sharp front connecting linear to nonlinear regions. A new type of fully nonlinear time harmonic solutions called steep global modes is identified in the context of the supercritical complex Ginzburg-Landau equation with slowly varying coefficients. A similar formulation is likely to be applicable to fully nonlinear synchronized global oscillations in spatially developing free shear flows

International audienceThe selection of fully nonlinear extended oscillating states is analysed in the context of one-dimensional nonlinear evolution equations with slowly spatially varying coefficients on a doubly-infinite domain. Two types of synchronized structures referred to as steep and soft global modes are shown to exist. Steep global modes are characterized by the presence of a sharp stationary front at the marginally absolutely unstable station and their frequency is determined by the corresponding linear absolute frequency, as in Dee--Langer propagating fronts. Soft global modes exhibit slowly varying amplitude and wavenumber over the entire domain and their frequency is determined by the application of a saddle point condition to the local nonlinear dispersion relation. The two selection criteria are compared and shown to be mutually exclusive. The onset of global instability first gives rise to a steep global mode via a saddle-node bifurcation as soon as local linear absolute instability is reached somewhere in the medium. As a result, such self-sustained structures may be observed while the medium is still linearly globally stable. Soft global modes only occur further above global onset and for sufficiently weak advection. The entire bifurcation scenario and state diagram are described in terms of three characteristic control parameters. The complete spatial structure of nonlinear global modes is analytically obtained in the framework of WKBJ approximations

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The dynamics of small-amplitude perturbations as well as the regime of fully developed nonlinear propagating waves is investigated for pulsatile channel flows. The timeperiodic base flows are known analytically and completely determined by the Reynolds number Re (based on the mean flowrate), the Womersley number Wo (a dimensionless expression of the frequency) and the flowrate waveform. This paper considers pulsatile flows with a single oscillating component and hence only three non-dimensional control parameters are present. Linear stability characteristics are obtained both by Floquet analyses and by linearized direct numerical simulations. In particular, the long-term growth or decay rates and the intracyclic modulation amplitudes are systematically computed. At large frequencies (mainly Wo 14), increasing the amplitude of the oscillating component is found to have a stabilizing effect, while it is destabilizing at lower frequencies; strongest destabilization is found for Wo ≃ 7. Whether stable or unstable, perturbations may undergo large-amplitude intracyclic modulations; these intracyclic modulation amplitudes reach huge values at low pulsation frequencies. For linearly unstable configurations, the resulting saturated fully developed finite-amplitude solutions are computed by direct numerical simulations of the complete Navier-Stokes equations. Essentially two types of nonlinear dynamics have been identified: "cruising" regimes for which nonlinearities are sustained throughout the entire pulsation cycle and which may be interpreted as modulated Tollmien-Schlichting waves, and "ballistic" regimes that are propelled into a nonlinear phase before subsiding again to small amplitudes within every pulsation cycle. Cruising regimes are found to prevail for weak baseflow pulsation amplitudes, while ballistic regimes are selected at larger pulsation amplitudes; at larger pulsation frequencies, however, the ballistic regime may be bypassed due to the stabilizing effect of the base-flow pulsating component. By investigating extended regions of a multi-dimensional parameter space and considering both two-dimensional and threedimensional perturbations, the linear and nonlinear dynamics are systematically explored and characterized.