Bifurcation to fully nonlinear synchronized structures in slowly varying media

Abstract

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

Boundary-Layer Transition on Broad Cones rotating in an Imposed Axial Flow

“…Note however that this does not imply that absolute instability is not involved in transition. In order to see this one must include nonlinearity: Pier et al 33,34 have shown that a self-excited, nonlinear global mode will always exist in the presence of a region of local absolute instability. This is in contrast to linear theory, since publications 22,32 have demonstrated that the region of local absolute instability on the disk is not sufficient to support an unstable linear global mode.…”

Nonlinear self-sustained structures and fronts in spatially developing wake flows

“…The forcing frequency ω f and the nonlinear response wavenumber k n + satisfy the nonlinear dispersion relation (4.4). For a more complete discussion of the relationship between linear and nonlinear spatial branches, see Pier (1999) and Pier et al (2001). Figure 8 illustrates the spatial response of the parallel CU wake velocity profile (a) prevailing at X = 1, at a Reynolds number Re = 100.…”

“…The reader is referred to Soward (2001) for a review of related wkbj asymptotic studies in the astrophysical context. The transition scenarii towards fully nonlinear global modes have been analysed by Pier (1999) and Pier, Huerre & Chomaz (2001) for the CGL evolution model in an infinite domain. The results of interest to the present investigation are as follows.…”

On the frequency selection of finite-amplitude vortex shedding in the cylinder wake

“…Provansal, Mathis & Boyer (1987), Williamson (1988), Norberg (1994), Leweke & Provansal (1995); for a review see Williamson (1996). On the theoretical side, understanding of the spatiotemporal dynamics of oscillatory flows has proceeded by successively considering linear model equations (Chomaz, Huerre & Redekopp 1991;Le, Dizès et al 1996), the linearized version of the Navier-Stokes equations (Monkewitz, Huerre & Chomaz 1993), and nonlinear model equations on semi-infinite (Couairon & Chomaz 1996, 1997a, b, 1999a and infinite (Pier & Huerre 1996;Pier et al 1998;Pier, Huerre & Chomaz 2001) domains. In the framework of the fully nonlinear Navier-Stokes equations, the frequency selection criterion has been obtained (Pier & Huerre 2001a) for a particular 'synthetic' wake: a wake with no solid obstacle and no reverse flow region.…”

“…In infinite systems, self-sustained time-periodic finite-amplitude structures have been found as soft ('hat') modes (Pier & Huerre 1996) or steep ('elephant' †) modes (Pier et al 1998), and the respective frequency selection criteria have been established. The analysis of the relevant transition scenarios (Pier et al 2001) has shown that the unperturbed basic state always first bifurcates to an elephant structure. Moreover, hat modes may only exist in situations of weak mean flow advection, so that they are ruled out in wake flows.…”