The effects of non-normality and nonlinearity of the Navier–Stokes operator on the dynamics of a large laminar separation bubble

Abstract

. The effects of non-normality and nonlinearity of the Navier-Stokes operator on the dynamics of a large laminar separation bubble. The effects of non-normality and nonlinearity of the Navier-Stokes operator on the dynamics of a large laminar separation bubble, 2010, 22 (014102), pp.15. <10.1063/1.3276903>. <hal-00799782>The effects of non-normality and nonlinearity of the Navier-Stokes operator on the dynamics of a large laminar separation bubble The effects of non-normality and nonlinearity of the two-dimensional Navier-Stokes differential operator on the dynamics of a large laminar separation bubble over a flat plate have been studied in both subcritical and slightly supercritical conditions. The global eigenvalue analysis and direct numerical simulations have been employed in order to investigate the linear and nonlinear stability of the flow. The steady-state solutions of the Navier-Stokes equations at supercritical and slightly subcritical Reynolds numbers have been computed by means of a continuation procedure. Topological flow changes on the base flow have been found to occur close to transition, supporting the hypothesis of some authors that unsteadiness of separated flows could be due to structural changes within the bubble. The global eigenvalue analysis and numerical simulations initialized with small amplitude perturbations have shown that the non-normality of convective modes allows the bubble to act as a strong amplifier of small disturbances. For subcritical conditions, nonlinear effects have been found to induce saturation of such an amplification, originating a wave-packet cycle similar to the one established in supercritical conditions, but which is eventually damped. A transient amplification of finite amplitude perturbations has been observed even in the attached region due to the high sensitivity of the flow to external forcing, as assessed by a linear sensitivity analysis. For supercritical conditions, the non-normality of the modes has been found to generate low-frequency oscillations ͑flapping͒ at large times. The dependence of such frequencies on the Reynolds number has been investigated and a scaling law based on a physical interpretation of the phenomenon has been provided, which is able to explain the onset of a secondary flapping frequency close to transition.

A pressure-gradient mechanism for vortex shedding in constricted channels

“…The topological changes are found to trigger a rapid transition to an absolute instability. Topological changes in the base flow are also connected with the onset of unsteadiness in separation bubbles investigated by Cherubini et al 15 Vortices may shed via the classic inviscid Kelvin-Helmholtz instability in a free shear layer, where the shear layer rolls up into a sequence of vortices that subsequently detach from the layer. In addition, in axisymmetric jet flow into a quiescent medium, vortex rings are shed via a process termed "pinch-off."…”

“…10 Furthermore, transient growth studies suggest that the optimal perturbation may be in the form of a convective-type instability. For example, Alizard et al 10 and Cherubini et al 15 identify the optimal perturbation for the occurrence of unsteadiness in a separated boundary-layer flow over a flat plate as "convective waves" or a disturbance that is localized near the separation point of the recirculation region and amplified along the shear layer as it convects downstream through a pseudo-resonance of global modes. Blackburn, Sherwin, and Barkley 33 also find that optimal growth dynamics mimic a convective instability in a stenotic flow.…”

Instabilities in laminar separation bubbles

“…Several scenarios are possible. Cherubini, Robinet & De Palma (2010) have shown that when an LSB is large enough the modes associated with the shear layer can interact non-normally and nonlinearly to generate self-sustained low-frequency oscillations. Other scenarios are possible, for instance synchronization of flow through a hydrodynamic and/or pressure feedback mechanism (such as the Rossiter mechanism, observed in open cavities) can force the flow at low frequency and enhance a collective interaction mechanism.…”

Hairpin-like optimal perturbations in plane Poiseuille flow

“…Streaky structures are observed at different scales in transitional and turbulent shear flows (Brandt, Schlatter & Henningson 2004;Hwang & Cossu 2010) and their † Email address for correspondence: s.cherubini@gmail.com origin seems now to be well understood. As conjectured by Landahl (1980) some decades ago, elongated near-wall zones of low and high momentum are created by a mechanism of transient growth of the perturbations known as the lift-up effect, linked to the non-normality of the Navier-Stokes operator (Cherubini, Robinet & De Palma 2010b). Such a mechanism is based on the transport of the mean shear by rolls of streamwise vorticity.…”