Two-dimensional global low-frequency oscillations in a separating boundary-layer flow

Abstract

A separated boundary-layer flow at the rear of a bump is considered. Two-dimensional equilibrium stationary states of the Navier-Stokes equations are determined using a nonlinear continuation procedure varying the bump height as well as the Reynolds number. A global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The onset of instability is shown to be characterized by a family of modes with localized structures around the reattachment point becoming almost simultaneously unstable. The optimal perturbation analysis, by projecting the initial disturbance on the set of temporal eigenmodes, reveals that the non-normal modes are able to describe localized initial perturbations associated with the large transient energy growth. At larger time a global low-frequency oscillation is found, accompanied by a periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier-Stokes time integration and is shown to trigger the nonlinear 'flapping' typical of separation bubbles. It is possible to follow the stationary equilibrium state on increasing the Reynolds number far beyond instability, ruling out for the present flow case the hypothesis of some authors that topological flow changes are responsible for the 'flapping'.

Open-loop control of a separated boundary layer

“…These large gain values suggest that an incoming noise of low amplitude might be significantly amplified through linear mechanisms, enough to reach order one and possibly modify the base flow, or even completely destabilize the overall flow behavior. The largest values of optimal gain are obtained for frequencies corresponding to the most unstable global eigenvalues near critical conditions (0.15 ω 0.30 in [10]). Figure 3 shows the spatial structure of the optimal forcing and optimal response at Re = 580.…”

“…The bump geometry y = y b (x) is shown in figure 1 and is the same as in Marquillie and Ehrenstein [12] and following studies [10,11,13]. The incoming flow has a Blasius boundary layer profile, of displacement thickness δ * at the reference position x = 0.…”

“…The Reynolds number is defined as Re = U ∞ δ * /ν, with ν the fluid kinematic viscosity. Previous studies using direct numerical simulations [12] and linear global stability analysis [10] reported a 2D critical Reynolds number Re c between 590 and 610. (See [14] for details about the 3D flow.)…”

“…The optimal response has a wave packet-like structure in the x direction, whose wavelength decreases with ω. The location of the optimal response moves upstream with ω; at most amplified frequencies the optimal response is located just around the reattachment point, and its spatial structure is reminiscent of that of the most unstable global eigenmodes [10].…”

“…In this study, the flow past a wall-mounted bump is considered (section 2). This separated flow is characterized by a long recirculation region, high shear, strong backflow, and exhibits large transient growth [10,11]. Optimal gains are computed at different frequencies (sec.…”