Energy dissipation caused by boundary layer instability at vanishing viscosity

Natacha Nguyen van Yen (1) , Matthias Waidmann (1) , Rupert Klein (2) , Marie Farge (3) , Kai Schneider (4)

1 Department of Mathematics and Computer Science
2 Freie Universität Berlin
3 LMD - Laboratoire de Météorologie Dynamique (UMR 8539)
4 I2M - Institut de Mathématiques de Marseille

A qualitative explanation for the scaling of energy dissipation by high-Reynolds-number fluid flows in contact with solid obstacles is proposed in the light of recent mathematical and numerical results. Asymptotic analysis suggests that it is governed by a fast, small-scale Rayleigh-Tollmien-Schlichting instability with an unstable range whose lower and upper bounds scale as Re-3/8 and Re-1/2 , respectively. By linear superposition, the unstable modes induce a boundary vorticity flux of order Re-1, a key ingredient in detachment and drag generation according to a theorem of Kato. These predictions are confirmed by numerically solving the Navier-Stokes equations in a two-dimensional periodic channel discretized using compact finite differences in the wall-normal direction, and a spectral scheme in the wall-parallel direction.

Keywords: boundary layer stability; vortex shedding

Domains: Fluid Dynamics [physics.flu-dyn]; Mathematics [math]

Natacha Nguyen van Yen, Matthias Waidmann, Rupert Klein, Marie Farge, Kai Schneider. Energy dissipation caused by boundary layer instability at vanishing viscosity. Journal of Fluid Mechanics, 2018, 849, pp.676 - 717. ⟨10.1017/jfm.2018.396⟩. ⟨hal-01838153⟩