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Mean flow generation by an intermittently unstable boundary layer over a sloping wall

Published online by Cambridge University Press:  22 August 2018

Abouzar Ghasemi Open the ORCID record for Abouzar Ghasemi [Opens in a new window] ,
Marten Klein ,
Andreas Will  and
Uwe Harlander
Abouzar Ghasemi*
Affiliation:
Department for Atmospheric and Environmental Sciences, Goethe University of Frankfurt am Main, Frankfurt/Main, Germany
Marten Klein
Affiliation:
Department of Numerical Fluid and Gas Dynamics, Brandenburg University of Technology (BTU) Cottbus-Senftenberg, Siemens-Halske-Ring 14, D-03046 Cottbus, Germany
Andreas Will
Affiliation:
Department of Environmental Meteorology, Brandenburg University of Technology (BTU) Cottbus-Senftenberg, Burger Chaussee 2, D-03044 Cottbus, Germany
Uwe Harlander
Affiliation:
Department of Aerodynamics and Fluid Mechanics, Brandenburg University of Technology (BTU) Cottbus-Senftenberg, Siemens-Halske-Ring 14, D-03046 Cottbus, Germany
*
†Email address for correspondence: [email protected]
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Abstract

Direct numerical simulations (DNS) of the flow in various rotating annular confinements have been conducted to investigate the effects of wall inclination on secondary fluid motions due to an unstable boundary layer. The inner wall resembles a truncated cone (frustum) whose apex half-angle is varied from $18^{\circ }$ to $0^{\circ }$ (straight cylinder). The large inner radius $r_{1}$, the mean rotation rate $\unicode[STIX]{x1D6FA}_{0}$ and the kinematic viscosity $\unicode[STIX]{x1D708}$ were kept constant resulting in the constant Ekman number $E=\unicode[STIX]{x1D708}/(\unicode[STIX]{x1D6FA}_{0}r_{1}^{2})=4\times 10^{-5}$. Flows were excited by time-harmonic modulation of the inner wall’s rotation rate (so-called longitudinal libration) by prescribing the amplitude $\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FA}_{0}$ and the forcing frequency $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D6FA}_{0}$. By steepening the inner wall and hence reducing the effect of the local Coriolis force in the boundary layer three different flow regimes can be realized: a rotation-dominated, a libration-dominated and an intermediate regime. In the present study we focus on the libration-dominated regime. For small libration amplitudes (here $\unicode[STIX]{x1D700}=0.2$), a laminar Ekman–Stokes boundary layer (ESBL) is realized at the librating wall. With the aid of laminar boundary layer theory and DNS we show that the ESBL exhibits an oscillatory mass flux along the librating wall (Ekman property) and an oscillatory azimuthal velocity, which resembles a radially damped wave (Stokes property). For large libration amplitudes (here $\unicode[STIX]{x1D700}=0.8$), the DNS results exhibit an intermittently unstable ESBL, which turns centrifugally unstable during the prograde (faster) part of a libration period. This instability is due to the Stokes property and gives rise to Görtler vortices, which are found to be tilted with respect to the azimuth when the librating wall is at a finite angle relative to the axis of rotation. We show that this tilt is related to the Ekman property of the ESBL. This suggests that linear and nonlinear dynamics are equally important for this intermittent instability. Our DNS results indicate further that the Görtler vortices propagate into the fluid bulk where they generate an azimuthal mean flow. This mean flow is notably different from the mean flow driven in the case of the stable ESBL. A diagnostic analysis of the Reynolds-averaged Navier–Stokes (RANS) equations in the unstable flow regime hints at a competition between the radial and axial turbulent transport terms which act as generating and destructing agents for the azimuthal mean flow, respectively. We show that the balance of both terms depends on the wall inclination, that is, on the wall-tangential component of the Coriolis force.

JFM classification

Instability: Instability Turbulent Flows: Intermittency Geophysical and Geological Flows: Rotating flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 853 , 25 October 2018 , pp. 111 - 149
Copyright
© 2018 Cambridge University Press 

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