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Mean flow generation by an intermittently unstable boundary layer over a sloping wall
- Abouzar Ghasemi, Marten Klein, Andreas Will, Uwe Harlander
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- Journal:
- Journal of Fluid Mechanics / Volume 853 / 25 October 2018
- Published online by Cambridge University Press:
- 22 August 2018, pp. 111-149
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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.
Inertial wave excitation and focusing in a liquid bounded by a frustum and a cylinder
- Marten Klein, Torsten Seelig, Michael V. Kurgansky, Abouzar Ghasemi V., Ion Dan Borcia, Andreas Will, Eberhard Schaller, Christoph Egbers, Uwe Harlander
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- Journal:
- Journal of Fluid Mechanics / Volume 751 / 25 July 2014
- Published online by Cambridge University Press:
- 18 June 2014, pp. 255-297
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The mechanism of localized inertial wave excitation and its efficiency is investigated for an annular cavity rotating with $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Omega _0$. Meridional symmetry is broken by replacing the inner cylinder with a truncated cone (frustum). Waves are excited by individual longitudinal libration of the walls. The geometry is non-separable and exhibits wave focusing and wave attractors. We investigated laboratory and numerical results for the Ekman number $E\approx 10^{-6}$, inclination $\alpha =5.71^\circ $ and libration amplitudes $\varepsilon \leq 0.2$ within the inertial wave band $0 < \omega < 2\Omega _0$. Under the assumption that the inertial waves do not essentially affect the boundary-layer structure, we use classical boundary-layer analysis to study oscillating Ekman layers over a librating wall that is at an angle $\alpha \neq 0$ to the axis of rotation. The Ekman layer erupts at frequency $\omega =f_{*}$, where $f_{*}\equiv 2 \Omega _0 \sin \alpha $ is the effective Coriolis parameter in a plane tangential to the wall. For the selected inclination this eruption occurs for the forcing frequency $\omega /\Omega _0=0.2$. For the librating lids eruption occurs at $\omega /\Omega _0=2$. The study reveals that the frequency dependence of the total kinetic energy $K_{\omega }$ of the excited wave field is strongly connected to the square of the Ekman pumping velocity $w_{{E}}(\omega )$ that, in the linear limit, becomes singular when the boundary layer erupts. This explains the frequency dependence of non-resonantly excited waves. By the localization of the forcing, the two configurations investigated, (i) frustum libration and (ii) lids together with outer cylinder in libration, can be clearly distinguished by their response spectra. Good agreement was found for the spatial structure of low-order wave attractors and periodic orbits (both characterized by a small number of reflections) in the frequency windows predicted by geometric ray tracing. For ‘resonant’ frequencies a significantly increased total bulk energy was found, while the energy in the boundary layer remained nearly constant. Inertial wave energy enters the bulk flow via corner beams, which are parallel to the characteristics of the underlying Poincaré problem. Numerical simulations revealed a mismatch between the wall-parallel mass fluxes near the corners. This leads to boundary-layer eruption and the generation of inertial waves in the corners.