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Optimal triggering of jet bifurcation: an example of optimal forcing applied to a time-periodic base flow

Published online by Cambridge University Press:  06 January 2020

Léopold Shaabani-Ardali Open the ORCID record for Léopold Shaabani-Ardali [Opens in a new window] ,
Denis Sipp Open the ORCID record for Denis Sipp [Opens in a new window]  and
Lutz Lesshafft Open the ORCID record for Lutz Lesshafft [Opens in a new window]
Léopold Shaabani-Ardali*
Affiliation:
LadHyX, CNRS – Ecole Polytechnique – Institut Polytechnique de Paris, 91120 Palaiseau, France DAAA, ONERA, Université Paris-Saclay, 92190 Meudon, France
Denis Sipp
Affiliation:
DAAA, ONERA, Université Paris-Saclay, 92190 Meudon, France
Lutz Lesshafft
Affiliation:
LadHyX, CNRS – Ecole Polytechnique – Institut Polytechnique de Paris, 91120 Palaiseau, France
*
†Email address for correspondence: [email protected]
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Abstract

The present article aims at optimising the spread of a bifurcating jet: a jet that combines axisymmetric and helical forcing to achieve increased mixing in a preferential plane. Parekh et al. (Tech. Rep. TF-35, Stanford University, 1988) explained such a bifurcation as the result of nonlinear interaction between ring vortices (triggered by $m=0$ axisymmetric forcing), shifted off-axis in alternate directions (owing to $m=1$ helical forcing). Following this idea, we linearly optimise the periodic helical forcing to be applied at the inlet, in order to maximally displace the ring vortices of an axisymmetrically forced jet. Two norms are introduced for evaluating the effect of helical forcing onto the helical response: the standard ${\mathcal{L}}_{2}$-norm and a semi-norm reflecting the off-axis vortex displacement. The linear results show one dominant forcing mode over the entire Strouhal band studied ($0.35\leqslant St\leqslant 0.8$), with a large gain separation from suboptimals. The dominant forcing is mainly radial, independent of the chosen response norm, and provides a gain at least five times larger than what was achieved by previous ad hoc forcing strategies. Superposition of base flow and linear results show the alternate shifting and twisting provoked by the the small-amplitude helical forcing, which is an essential ingredient for triggering jet bifurcation. When tested in three-dimensional direct numerical simulations, low-amplitude helical forcing achieves efficient bifurcation at all Strouhal values studied. At high Strouhal numbers, an additional central branch emerges in the mean flow, leading to trifurcation. Across all frequencies, compared with ad hoc forcing strategies, the optimal forcing triggers a much stronger and robust spreading, by moving the bifurcation point upstream. As a result, bifurcating jets are observed over a much larger Strouhal band ($0.35\leqslant St\leqslant 0.8$) compared with the band where ad hoc forcing achieves bifurcation in our setting ($0.4\leqslant St\leqslant 0.5$).

JFM classification

Flow Control: Mixing enhancement Vortex Flows: Vortex interactions Wakes/Jets: Jets
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 885 , 25 February 2020 , A34
Copyright
© 2020 Cambridge University Press

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