Tangential oscillations of a circular disk in a viscous stratified fluid

Davis, A. M. J.; Smith, Stefan G. Llewellyn

doi:10.1017/s0022112010001205

Cited by 13 publications

(19 citation statements)

References 25 publications

(28 reference statements)

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“…Axisymmetric wave fields have traditionally been experimentally excited using an oscillating sphere and exploring the shape of the wave beams [16][17][18][19][20][21]. While the form of the wave field close to the oscillating body is nontrivial, modeling studies have explored the limit states of the wave beams in terms of plane waves with a spherical amplitude decreasing as r −1/2 , r being the radial distance from the sphere, computed from the Green function of the moving source [22], or as infinite sums of Bessel functions with complex coefficients [7,23]. The amplitude decrease and the viscous decay of the conical wave beam emitted by an oscillating sphere has been explored in laboratory experiments by Flynn et al [17] showing good agreement with theoretical predictions.…”

Section: Introductionmentioning

confidence: 99%

Excitation and resonant enhancement of axisymmetric internal wave modes

2019

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To date, axisymmetric internal wave fields, which have relevance to atmospheric internal wave fields generated by storm cells and oceanic near-inertial wave fields generated by surface storms, have been experimentally realized using an oscillating sphere or torus as the source. Here, we use a novel wave generator configuration capable of exciting axisymmetric internal wave fields of arbitrary radial form to generate axisymmetric internal wave modes. After establishing the theoretical background for axisymmetric mode propagation, taking into account lateral and vertical confinement, and also accounting for the effects of weak viscosity, we experimentally generate and study modes of different order. We characterize the efficiency of the wave generator through careful measurement of the wave amplitude based upon group velocity arguments. This established, we investigate the ability of vertical confinement to induce resonance, identifying a series of experimental resonant peaks that agree well with theoretical predictions. In the vicinity of resonance, the wave fields undergo a transition to non-linear behaviour that is initiated on the central axis of the domain and proceeds to erode the wave field throughout the domain.

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Section: Introductionmentioning

confidence: 99%

Excitation and resonant enhancement of axisymmetric internal wave modes

2019

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“…Like Hendershott (1969), we include rotation with stratification, with the latter dominant in the main text. Viscous effects facilitate the enforcement of only outwardly propagating waves at infinity without consideration of group velocity (Davis & Llewellyn Smith 2010). Extensive reviews of such calculations are given in the two cited papers by Voisin.…”

Section: Introductionmentioning

confidence: 99%

Generation of internal waves from rest: extended use of complex coordinates, for a sphere but not a disk

Davis

2012

J. Fluid Mech.

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The anisotropy created by stratification and or rotation places restrictions, severe if viscosity is present, on the construction of analytic solutions to wave generation and scattering problems. Consequently, much literature is devoted to frequency space and so careful consideration of oscillatory motion generated from rest is advisable. Moreover, the use of complex coordinates in the inviscid case has been incompletely presented. The detailed inversion of the Fourier time transform for a breathing or heaving sphere demonstrates an expanded, perhaps more crucial, role for the complex coordinates and shows that the known phase changes in the energy propagation regions are present throughout the St Andrew's cross that circumscribes the sphere. However, the different solution structure for the heaving disk requires and allows a more direct calculation. Though the inclusion of rotation does not affect the dynamics, it enables the significance of their relative magnitude to be identified and reference to rotation only results achieved.

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“…Using (3.3a-d), these equations form a system of integral equations for the coefficients V j , j = 1, 2, 3, 4. Analogous equations were obtained by Tanzosh & Stone (1995) and Davis & Llewellyn Smith (2010) for instance. Here, using the following property of the Bessel functions (Watson 1952, P406):…”

Section: Harmonic Responsementioning

confidence: 88%

“…A comprehensive list of references, especially for stratified fluids, can be found in Voisin (2003) and Voisin, Ermanyuk & Flór (2011). The case of the disk has been considered in numerous works, for steady displacements in a rotating fluid (Stewartson 1957;Moore & Saffman 1969; Vedensky & Ungarish 1994; Tanzosh & Stone 1995), for oscillating displacements in a stratified fluid (Il'inyhk & Chashechkin 2004;Bardakov, Vasil'ev & Chashechkin 2007;Davis & Llewellyn Smith 2010) or for more complicated surface fluctuations (Walton 1975;Kerswell 1995). The method of resolution is based on the use of the Hankel transform, which leads to a system of dual integral equations.…”

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confidence: 99%

“…The method of resolution is based on the use of the Hankel transform, which leads to a system of dual integral equations. These equations can then be converted into a system of algebraic equations using expansion in terms of Bessel functions (see, for instance, Davis & Llewellyn Smith 2010). Here, we use the same method but the integral equations will turn out to have a simple explicit solution.…”

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confidence: 99%

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Wave field and zonal flow of a librating disk

Dizès

2015

J. Fluid Mech.

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In this work, we provide a viscous solution of the wave field generated by librating a disk (harmonic oscillation of the rotation rate) in a stably stratified rotating fluid. The zonal flow (mean flow correction) generated by the nonlinear interaction of the wave field is also calculated in the weakly nonlinear framework. We focus on the low dissipative limit relevant for geophysical applications and for which the wave field and the zonal flow exhibit generic features (Ekman scaling, universal structures, etc.). General expressions are obtained which depend on the disk radius a * , the libration frequency ω * , the rotation rate Ω * of the frame, the buoyancy frequency N * of the fluid, its kinematic diffusion ν * and its thermal diffusivity κ * . When the libration frequency is in the inertia-gravity frequency interval (min(Ω * , N * ) < ω * < max(Ω * , N * )), the presence of conical internal shear layers is observed in which the spatial structures of the harmonic response and of the mean flow correction are provided. At the point of focus of these internal shear layers on the rotation axis, the largest amplitudes are obtained: the angular velocity of the harmonic response and the mean flow correction are found to be O(εE −1/3 ) and (ε 2 E −2/3 ) respectively, where ε is the libration amplitude and E = ν * /(Ω * a * 2 ) is the Ekman number. We show that the solution in the internal shear layers and in the focus region is at leading order the same as that generated by an oscillating source of axial flow localized at the edge of the disk (oscillating Dirac ring source).

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Tangential oscillations of a circular disk in a viscous stratified fluid

Cited by 13 publications

References 25 publications

Excitation and resonant enhancement of axisymmetric internal wave modes

Excitation and resonant enhancement of axisymmetric internal wave modes

Generation of internal waves from rest: extended use of complex coordinates, for a sphere but not a disk

Wave field and zonal flow of a librating disk

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