scholarly journals Hairpin-like optimal perturbations in plane Poiseuille flow

2015 ◽  
Vol 775 ◽  
Author(s):  
Mirko Farano ◽  
Stefania Cherubini ◽  
Jean-Christophe Robinet ◽  
Pietro De Palma
Keyword(s):  
Poiseuille Flow ◽  
Nonlinear Effects ◽  
Optimal Growth ◽  
Turbulent Shear ◽  
Hairpin Vortex ◽  
Plane Poiseuille Flow ◽  
Hairpin Vortices ◽  
Optimal Perturbation ◽  
Target Time ◽  
Lagrange Multiplier Technique

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.

scholarly journals Optimal perturbation growth in axisymmetric intrusions

2016 ◽  
Vol 811 ◽  
Author(s):  
Bruce R. Sutherland ◽  
C. P. Caulfield
Keyword(s):  
Gravity Current ◽  
Optimal Growth ◽  
Base Flow ◽  
Optimal Perturbation ◽  
Target Time ◽  
Experimental Parameters ◽  
Radial Spokes ◽  
Perturbation Energy ◽  
The One ◽  
Perturbation Growth

The cylindrical lock-release laboratory experiments of Sutherland & Nault (J. Fluid Mech., vol. 586, 2007, pp. 109–118) showed that a radially advancing symmetric intrusive gravity current spreads not as an expanding annulus (as is the case for bottom-propagating gravity currents), but rather predominantly along azimuthally periodic radial ‘spokes’. Here, we investigate whether the spokes are associated with azimuthal perturbations that undergo ‘optimal’ growth. We use a nonlinear axisymmetric numerical simulation initialised with the experimental parameters to compute the time-evolving axisymmetric base state of the collapsing lock fluid. Using fields from this rapidly evolving base state together with the linearised perturbation equations and their adjoint, the ‘direct–adjoint looping’ method is employed to identify, as a function of the azimuthal wavenumber $m$, the vertical–radial structure of the set of initial perturbations that exhibit the largest total perturbation energy gain over a target time $T$. Of this set of perturbations, the one that extracts energy fastest, and so is expected to be observed to emerge first from the base flow, has azimuthal wavenumber comparable to the number of spokes observed in the experiment.

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