Hairpin-like optimal perturbations in plane Poiseuille flow

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. In this work it is shown that hairpin vortex structures can be the outcome of a nonlinear optimal growth process, in a similar way as streaky structures can be the result of a linear optimal growth mechanism. With this purpose, nonlinear optimizations based on a Lagrange multiplier technique coupled with a direct-adjoint iterative procedure are performed in a plane Poiseuille flow at subcritical values of the Reynolds number, aiming at quickly triggering nonlinear effects. Choosing a suitable time scale for such an optimization process, it is found that the initial optimal perturbation is composed of sweeps and ejections resulting in a hairpin vortex structure at the target time. These alternating sweeps and ejections create an inflectional instability occurring in a localized region away from the wall, generating the head of the primary and secondary hairpin structures, quickly inducing transition to turbulent flow. This result could explain why transitional and turbulent shear flows are characterized by a high density of hairpin vortices.

Nonlinear Nonmodal Stability Theory

“…There is also further indirect evidence for this robustness to the choice T in that similar optimals are reported across a variety of flows: in plane Couette flow (Cherubini & De Palma 2013, 2014a; the Blasius boundary layer (Cherubini et al 2010(Cherubini et al , 2011(Cherubini et al , 2012; the asymptotic suction boundary layer (Cherubini et al 2015) and plane Poiseuille flow (Farano et al 2015(Farano et al , 2016). If T is too small, transients can obscure the situation (Rabin et al 2012) or new optimals become preferred (Pringle et al 2015, Farano 2015. Another potential problem is studying a flow where the turbulent state is not separated from the laminar-turbulent boundary or edge as far as the objective functional is concerned.…”

“…see Figure 22 of Cherubini et al 2011). Hairpin vortices are also found to be generated by optimals computed in plane Poiseuille flow using a very short T on the timescale of the Orr mechanism (Farano et al 2015).…”

“…As a result, the looping procedure becomes ill-conditioned due to the extreme sensitivity of the final state energy to changes in the initial condition (Pringle et al 2012). Reducing T allows NLOPs to be converged at higher initial energies (Cherubini et al 2010, Cherubini & De Palma 2013, Farano et al 2015 whereas increasing T reduces Ec(T ; Re). This implies the existence of a minimal finite (since the base flow is linearly stable) value Ec(∞; Re) which represents the infimum on the disturbance energy which can trigger transition or, equivalently, equals the minimum energy of any state on the laminar-turbulent boundary or edge.…”

Subcritical transition scenarios via linear and nonlinear localized optimal perturbations in plane Poiseuille flow

“…These solutions are localized in the three spatial directions and do not present any symmetry. Whereas, for large values of E(0) and very small target times (T opt < 18), we obtain hairpin-like nonlinear optimal solutions (see also Farano et al (2015)), shown in the fourth subframe and represented by diamonds in the top frame of figure 2. The latter type of nonlinear optimal perturbation is associated with very large values of the initial energies, much larger than the transition threshold energy characterizing the other types of perturbations.…”

Boundary layer transition experiments with embedded streamwise vortices

“…Further downstream, these vortices connect above interspersed high-speed streaks, forming lifted hairpins, which in turn break down, spawning spots. Quasi-streamwise vortices are not just natural features of streak instability; they have also been found to be the optimal, finite-amplitude perturbation that develops into hairpins in a short target time [14].…”