3 results
Internal shear layers in librating spherical shells: the case of attractors
- Jiyang He, Benjamin Favier, Michel Rieutord, Stéphane Le Dizès
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- Journal:
- Journal of Fluid Mechanics / Volume 974 / 10 November 2023
- Published online by Cambridge University Press:
- 23 October 2023, A3
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Following our previous work on periodic ray paths (He et al., J. Fluid Mech., vol. 939, 2022, A3), we study asymptotically and numerically the structure of internal shear layers for very small Ekman numbers in a three-dimensional spherical shell and in a two-dimensional cylindrical annulus when the rays converge towards an attractor. We first show that the asymptotic solution obtained by propagating the self-similar solution generated at the critical latitude on the librating inner core describes the main features of the numerical solution. The internal shear layer structure and the scaling for its width and velocity amplitude in $E^{1/3}$ and $E^{1/12}$, respectively, are recovered. The amplitude of the asymptotic solution is shown to decrease to $E^{1/6}$ when it reaches the attractor, as is also observed numerically. However, some discrepancies are observed close to the particular attractors along which the phase of the wave beam remains constant. Another asymptotic solution close to those attractors is then constructed using the model of Ogilvie (J. Fluid Mech., vol. 543, 2005, pp. 19–44). The solution obtained for the velocity has an $O(E^{1/6})$ amplitude, but a self-similar structure different from the critical-latitude solution. It also depends on the Ekman pumping at the contact points of the attractor with the boundaries. We demonstrate that it reproduces correctly the numerical solution. Surprisingly, the numerical solution close to an attractor with phase shift (that is, an attractor touching the axis in three or two dimensions with a symmetric forcing) is also found to be $O(E^{1/6})$, but its amplitude is much weaker. However, its asymptotic structure remains a mystery.
Internal shear layers in librating spherical shells: the case of periodic characteristic paths
- Jiyang He, Benjamin Favier, Michel Rieutord, Stéphane Le Dizès
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- Journal:
- Journal of Fluid Mechanics / Volume 939 / 25 May 2022
- Published online by Cambridge University Press:
- 23 March 2022, A3
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Internal shear layers generated by the longitudinal libration of the inner core in a spherical shell rotating at a rate $\varOmega ^*$ are analysed asymptotically and numerically. The forcing frequency is chosen as $\sqrt {2}\varOmega ^*$ such that the layers issued from the inner core at the critical latitude in the form of concentrated conical beams draw a simple rectangular pattern in meridional cross-sections. The asymptotic structure of the internal shear layers is described by extending the self-similar solution known for open domains to closed domains where reflections on the boundaries occur. The periodic ray path ensures that the beams remain localised around it. Asymptotic solutions for both the main beam along the critical line and the weaker secondary beam perpendicular to it are obtained. The asymptotic predictions are compared with direct numerical results obtained for Ekman numbers as low as $E=10^{-10}$. The agreement between the asymptotic predictions and numerical results improves as the Ekman number decreases. The asymptotic scalings in $E^{1/12}$ and $E^{1/4}$ for the amplitudes of the main and secondary beams, respectively, are recovered numerically. Since the self-similar solution is singular on the axis, a new local asymptotic solution is derived close to the axis and is also validated numerically. This study demonstrates that, in the limit of vanishing Ekman numbers and for particular frequencies, the main features of the flow generated by a librating inner core are obtained by propagating through the spherical shell the self-similar solution generated by the singularity at the critical latitude on the inner core.
Effects of distributed roughness on crossflow instability through generalized resonance mechanisms
- Jiyang He, Adam Butler, Xuesong Wu
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- Journal:
- Journal of Fluid Mechanics / Volume 858 / 10 January 2019
- Published online by Cambridge University Press:
- 12 November 2018, pp. 787-831
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Experiments have shown that micron-sized distributed surface roughness can significantly promote transition in a three-dimensional boundary layer dominated by crossflow instability. This sensitive effect has not yet been fully explained physically and mathematically. Past studies focused on surface roughness exciting crossflow vortices and/or changing the local stability characteristics. The present paper seeks possible additional mechanisms by investigating the effects of distributed surface roughness on crossflow instability through resonant interactions with eigenmodes. A key observation is that the perturbation induced by roughness with specific wavenumbers can interact with two eigenmodes (travelling and stationary vortices) through triadic resonance, or interact with one eigenmode (stationary vortices) through Bragg scattering. Unlike the usual triadic resonance of neutral, or nearly neutral, eigenmodes, the present triadic resonance can take place among modes with $O(1)$ growth rates, provided that these are equal; unlike the usual Bragg scattering involving neutral waves, crossflow stationary vortices can also be unstable. For these amplifying waves, the generalized triadic resonance and Bragg scattering are put forward, and the resulting corrections to the growth rates are derived by a multiple-scale method. The analysis is extended to the case where up to four crossflow vortices interact with each other in the presence of suitable roughness components. The numerical results for Falkner–Skan–Cooke boundary layers show that roughness with a small height (a few percent of the local boundary-layer thickness) can change growth rates substantially (by a more-or-less $O(1)$ amount). This sensitive effect is attributed to two facts: (i) the resonant nature of the triadic interaction and Bragg scattering, which makes the correction to the growth rate proportional to the roughness height, and (ii) the wavenumbers of the roughness component required for the resonance are close to those of the neutral stationary crossflow modes, as a result of which a small roughness can generate a large response. Another important effect of roughness is that its presence renders the participating eigenmodes, which are otherwise independent, fully coupled. Our theoretical results suggest that micron-sized distributed surface roughness influences significantly both the amplification and spectral composition of crossflow vortices.