A minimal flow-elements model for the generation of packets of hairpin vortices in shear flows

Abstract

Packets of hairpin-shaped vortices and streamwise counter-rotating vortex pairs (CVPs) appear to be key structures during the late stages of the transition process as well as in low-Reynolds-number turbulence in wall-bounded flows. In this work we propose a robust model consisting of minimal flow elements that can produce packets of hairpins. Its three components are: simple shear, a CVP having finite streamwise vorticity magnitude and a two-dimensional (2D) wavy (in the streamwise direction) spanwise vortex sheet. This combination is inherently unstable: the CVP modifies the base flow due to the induced velocity forming an inflection point in the base-flow velocity profile. Consequently, the 2D wavy vortex sheet is amplified, causing undulation of the CVP. The undulation is further enhanced as the wave continues to be amplified and eventually the CVP breaks down into several segments. The induced velocity generates highly localized patches of spanwise vorticity above the regions connecting two consecutive streamwise elements of the CVP. These patches widen with time and join with the streamwise vortical elements situated beneath them forming a packet of hairpins. The results of the unbounded (having no walls) model are compared with pipe and channel flow experiments and with a direct numerical simulation of a transition process in Couette flow. The good agreement in all cases demonstrates the universality and robustness of the model.

Hairpin-like optimal perturbations in plane Poiseuille flow

“…Moreover, in figure 8(b) one can observe that the hairpin head is placed right in the zone of interaction of the stronger ejection with the stronger sweep, indicating that the formation of the spanwise vortex connecting the initial quasi-streamwise vortices might be a consequence of an inflectional instability taking place in this interaction zone. This observation is in agreement with the minimal flow-element model proposed by Cohen, Karp & Mehta (2014), in which a wavy spanwise vortex sheet was necessary to provide the inflection points for creating hairpin vortices from streamwise counter-rotating vortex pairs. Figure 9(a,b) provides the instantaneous velocity and vorticity profiles at t = 10, computed solving the nonlinear and the linearized NS equations, respectively, extracted along a vertical axis passing through the hairpin head obtained in the nonlinear case.…”

Tracking stages of transition in Couette flow analytically

“…The vortex dynamics during the stages of transition is very similar to the model proposed by Cohen et al (2009aCohen et al ( , 2014 for the formation of packets of hairpins. In both cases, the inflectional instability leads to the undulation of the CVPs and consequently to the formation of a packet of hairpins.…”

“…Nevertheless, other features of the nonlinear optimization procedure such as streamwise dependence and localization of the disturbance (which allow the vortices to 'unpack') are required for triggering transition more effectively. Finally, the evolution of the vortical structures associated with the nonlinear transient growth scenario (figure 13) is very similar to the mechanism of generation of a packet of hairpins proposed in the model by Cohen, Karp & Shukhman (2009a) and Cohen, Karp & Mehta (2014). Accordingly, three basic flow elements are required: pure shear, CVP and a wavy spanwise vortex disturbance.…”

Boundary layer transition experiments with embedded streamwise vortices

“…Whenever transition was observed the origin could be traced to streak instability. In a recent study, a hairpin packet evolved from a counterrotating vortex pair in a shear layer without any wall [33]. Nevertheless, it was surprising that even after local separation was induced, transition was not triggered by the separating shear layer.…”