Optimal wave packets in a boundary layer and initial phases of a turbulent spot

Abstract

is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. The three-dimensional global optimal dynamics of a flat-plate boundary layer is studied by means of an adjoint-based optimization in a spatial domain of longbut finite -streamwise dimension. The localized optimal initial perturbation is characterized by a pair of streamwise-modulated counter-rotating vortices, tilted upstream, yielding at the optimal time elongated streaks of alternating sign in the streamwise direction. This indicates that perturbations with non-zero streamwise wavenumber have a role in the transient dynamics of a boundary layer. A scaling law is provided, describing the variation of the streamwise modulation of the optimal initial perturbation with respect to the streamwise domain length and to the Reynolds number. For spanwise-extended domains, a near-optimal three-dimensional perturbation is extracted during the optimization process; it is localized also in the spanwise direction, resulting in a wave packet of elongated disturbances modulated in the spanwise and streamwise directions. The nonlinear evolution of the optimal and near-optimal perturbations is investigated by means of direct numerical simulations. Both perturbations are found to induce transition at lower levels of the initial energy than local optimal and suboptimal perturbations. Moreover, it is observed that transition occurs in a well-defined region of the convected wave packet, close to its centre, via a mechanism including at the same time oscillations of the streaks of both quasi-sinuous and quasi-varicose nature. Hairpin vortices are observed before transition; they have an active role in the breakdown of the streaks and result in a turbulent spot which spreads out in the boundary layer.

Nonlinear control of unsteady finite-amplitude perturbations in the Blasius boundary-layer flow

“…Periodicity of the perturbation is imposed in the spanwise direction. The zero perturbation condition at inflow and outflow points is enforced by means of a fringe region (Cherubini et al (2010b)), which allows the perturbation at the exit boundary to vanish smoothly.…”

“…The direct and adjoint equations are parabolic in the forward and backward time direction, respectively, so that they can be solved by a coupled iterative approach, similar to the one used by Cherubini et al (2010b). By solving the direct and adjoint equations at each step of the iterative procedure, the first variation of the augmented functional with respect to q and q † is set to zero.…”

“…Two types of initial perturbations have been considered, namely, a linear and a nonlinear optimal perturbation, defined as the perturbations of the base flow which induce the largest energy growth at a given target time, in a linear and a nonlinear framework, respectively. These perturbations have been computed by means of an optimization of the perturbation energy gain using the linearized and the nonlinear Navier-Stokes equations, respectively (see Cherubini et al (2010bCherubini et al ( , 2011b). In both cases, the initial energy of the optimal perturbations is chosen in order to achieve transition.…”

“…Both perturbations evolve in time initially amplifying themselves via the Orr and the lift-up mechanisms. In particular, the LOP induces an array of streaks alternated in the streamwise and spanwise directions, shown in the left frame of figure 4, which then experience secondary instability of both varicose and sinuous type (see Cherubini et al (2010b)). On the other hand, the evolution of the NLOP is more complex, since the dislocations of the initial localized patches of finite amplitude streamwise disturbance generate Λ-shaped structures of slow and fast fluid (see the green surfaces on the right frame of figure 4), while the vortices are further inclined, sustained by nonlinear coupling, and stretched by the mean flow, creating Λ and hairpin vortices of finite amplitude (see Cherubini et al (2011b) for further details).…”

The minimal seed of turbulent transition in the boundary layer

“…Following these ideas, Cherubini et al (2010b) attempted to identify, in a linear framework, initial localized disturbances capable of provoking breakdown to turbulence most effectively in a Blasius boundary layer. The procedure to find the optimal wave packet was that of the linear global optimization theory, the optimization of the perturbation energy being performed for a laminar non-parallel boundary-layer flow without any assumption on the shape or on the frequency spectrum of the perturbation.…”