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Bifurcation scenario for a two-dimensional static airfoil exhibiting trailing edge stall

Published online by Cambridge University Press:  04 October 2021

Denis Busquet Open the ORCID record for Denis Busquet [Opens in a new window] ,
Olivier Marquet Open the ORCID record for Olivier Marquet [Opens in a new window] ,
François Richez Open the ORCID record for François Richez [Opens in a new window] ,
Matthew Juniper Open the ORCID record for Matthew Juniper [Opens in a new window]  and
Denis Sipp Open the ORCID record for Denis Sipp [Opens in a new window]
Denis Busquet*
Affiliation:
DAAA, ONERA, Université Paris Saclay, F-92190 Meudon, France Engineering Department, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
Olivier Marquet
Affiliation:
DAAA, ONERA, Université Paris Saclay, F-92190 Meudon, France
François Richez
Affiliation:
DAAA, ONERA, Université Paris Saclay, F-92190 Meudon, France
Matthew Juniper
Affiliation:
Engineering Department, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
Denis Sipp
Affiliation:
DAAA, ONERA, Université Paris Saclay, F-92190 Meudon, France
*
†Email address for correspondence: [email protected]
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Abstract

We numerically investigate stalling flow around a static airfoil at high Reynolds numbers using the Reynolds-averaged Navier–Stokes equations (RANS) closed with the Spalart–Allmaras turbulence model. An arclength continuation method allows to identify three branches of steady solutions, which form a characteristic inverted S-shaped curve as the angle of attack is varied. Global stability analyses of these steady solutions reveal the existence of two unstable modes: a low-frequency mode, which is unstable for angles of attack in the stall region, and a high-frequency vortex shedding mode, which is unstable at larger angles of attack. The low-frequency stall mode bifurcates several times along the three steady solutions: there are two Hopf bifurcations, two solutions with a two-fold degenerate eigenvalue and two saddle-node bifurcations. This low-frequency mode induces a cyclic flow separation and reattachment along the airfoil. Unsteady simulations of the RANS equations confirm the existence of large-amplitude low-frequency periodic solutions that oscillate around the three steady solutions in phase space. An analysis of the periodic solutions in the phase space shows that, when decreasing the angle of attack, the low-frequency periodic solution collides with the unstable steady middle-branch solution and thus disappears via a homoclinic bifurcation of periodic orbits. Finally, a one-equation nonlinear stall model is introduced to reveal that the disappearance of the limit cycle, when increasing the angle of attack, is due to a saddle-node bifurcation of periodic orbits.

JFM classification

Instability: Nonlinear instability Nonlinear Dynamical Systems: Bifurcation Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 928 , 10 December 2021 , A3
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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