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

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.

Evidence for wind-induced Ekman layer resonance based on rotating tank experiments

“…This velocity yields a bulk Reynolds number (inverse Ekman number) Re = U 2 0 /(Ω 0 ν) ≈ 13, 000. The corresponding steady [30] and librating [29] Ekman flows are already turbulent in this regime, as should be the flow in the experiment.…”

“…The governing equations are the simplified Navier-Stokes equations that includes rotation (f -plane equations) [2]. For an oscillating horizontal wall velocity U 0 cos(Ωt) the asymptotic leading-order solution reads [29]…”

A stochastic modeling strategy for intermittently unstable Ekman–Stokes boundary layers

“…The turbulent ODT results reveal that the vertical extent of nonzero velocity has notably increased indicating that the turbulent boundary layer is thicker than the laminar one. This meets the expectation and is in qualitative agreement with available reference data [5,6]. Note that the two turbulent flow profiles shown in each panel are not identical.…”

“…This is not a statistical fluctuation but a dynamical effect that hints at the temporal variability of the turbulent ESBL. The ensemble average used preserves some temporal information about the transient turbulence properties that are subject to periodic generation and destruction [5]. The stochastic modeling approach is able to capture these effects to leading order.…”

“…The laminar solution takes the form of two nested rotating Ekman layers with boundary-layer thicknesses D ± = D/|1 ± σ| [1,[4][5][6], in which σ = ω/f denotes the dimensionless forcing frequency and D = 2ν/f the conventional Ekman-layer thickness that serves as reference length scale for σ O(1). The laminar ESBL exhibits a boundary-layer resonance at σ = 1 where D − → ∞.…”