On flow in weakly precessing cylinders: the general asymptotic solution

2012 ◽  
Vol 709 ◽  
pp. 610-621 ◽  
Author(s):  
Xinhao Liao ◽  
Keke Zhang
Keyword(s):  
Numerical Analysis ◽  
Asymptotic Solution ◽  
Symmetry Axis ◽  
Slip Boundary Condition ◽  
Frame Of Reference ◽  
Rotating Frame ◽  
Homogeneous Fluid ◽  
Ekman Number ◽  
Slip Boundary ◽  
The Galerkin Method

AbstractWe investigate, through both asymptotic and numerical analysis, precessionally driven flow of a homogeneous fluid confined in a fluid-filled circular cylinder that rotates rapidly about its symmetry axis and precesses slowly about a different axis that is fixed in space. After demonstrating that the inviscid approximation is always divergent even far away from resonance, we derive a general asymptotic solution for an asymptotically small Ekman number in the rotating frame of reference describing the weakly precessing flow that satisfies the no-slip boundary condition and that is valid at or near or away from resonance. Numerical analysis of the same problem using the Galerkin method in terms of a Chebyshev polynomial expansion is also carried out, showing satisfactory agreement between the general asymptotic solution and the corresponding numerical solution at or near or away from resonance.

scholarly journals The non-resonant response of fluid in a rapidly rotating sphere undergoing longitudinal libration

2013 ◽  
Vol 720 ◽  
pp. 212-235 ◽  
Author(s):  
Keke Zhang ◽  
Kit H. Chan ◽  
Xinhao Liao ◽  
Jonathan M. Aurnou
Keyword(s):  
Pressure Difference ◽  
Oscillatory Flow ◽  
Slip Boundary Condition ◽  
Dimensionless Frequency ◽  
Homogeneous Fluid ◽  
Ekman Number ◽  
Slip Boundary ◽  
Characteristic Value ◽  
Fixed Axis ◽  
Asymptotic Scaling

AbstractWe investigate the problem of oscillatory flow of a homogeneous fluid with viscosity $\nu $ in a fluid-filled sphere of radius $a$ that rotates rapidly about a fixed axis with angular velocity ${\Omega }_{0} $ and that undergoes weak longitudinal libration with amplitude $\epsilon {\Omega }_{0} $ and frequency $\hat {\omega } {\Omega }_{0} $, where $\epsilon $ is the Poincaré number with $\epsilon \ll 1$ and $\hat {\omega } $ is dimensionless frequency with $0\lt \hat {\omega } \lt 2$. Three different methods are employed in this investigation: (i) asymptotic analysis at small Ekman numbers $E$ defined as $E= \nu / ({a}^{2} {\Omega }_{0} )$; (ii) linear numerical analysis using a spectral method; and (iii) nonlinear direct numerical simulation using a finite-element method. A satisfactory agreement among the three different sets of solutions is achieved when $E\leq 1{0}^{- 4} $. It is shown that the flow amplitude $\vert \boldsymbol{u}\vert $ is nearly independent of both the Ekman number $E$ and the libration frequency $\hat {\omega } $, always obeying the asymptotic scaling $\vert \boldsymbol{u}\vert = O(\epsilon )$ even though various spherical inertial modes are excited by longitudinal libration at different libration frequencies $\hat {\omega } $. Consequently, resonances do not occur in this system even when $\hat {\omega } $ is at the characteristic value of an inertial mode. It is also shown that the pressure difference along the axis of rotation is anomalous: this quantity reaches a sharp peak when $\hat {\omega } $ approaches a characteristic value. In contrast, the pressure difference measured at other places in the sphere, such as in the equatorial plane, and the volume-integrated kinetic energy are nearly independent of both the Ekman number $E$ and the libration frequency $\hat {\omega } $. Absence of resonances in a fluid-filled sphere forced by longitudinal libration is explained through the special properties of the analytical solution that satisfies the no-slip boundary condition and is valid for $E\ll 1$ and $\epsilon \ll 1$.

scholarly journals Asymptotic theory of resonant flow in a spheroidal cavity driven by latitudinal libration

2012 ◽  
Vol 692 ◽  
pp. 420-445 ◽  
Author(s):  
Keke Zhang ◽  
Kit H. Chan ◽  
Xinhao Liao
Keyword(s):  
Numerical Simulation ◽  
Boundary Layer ◽  
Asymptotic Solution ◽  
Slip Boundary Condition ◽  
Viscous Boundary Layer ◽  
Slip Boundary ◽  
Oblate Spheroidal ◽  
Spheroidal Cavity ◽  
Viscous Boundary

AbstractWe consider a homogeneous fluid of viscosity $\nu $ confined within an oblate spheroidal cavity, ${x}^{2} / {a}^{2} + {y}^{2} / {a}^{2} + {z}^{2} / ({a}^{2} (1\ensuremath{-} {\mathscr{E}}^{2} ))= 1$, with eccentricity $0\lt \mathscr{E}\lt 1$. The spheroidal container rotates rapidly with an angular velocity ${\mbit{\Omega} }_{0} $, which is fixed in an inertial frame and defines a small Ekman number $E= \nu / ({a}^{2} \vert {\mbit{\Omega} }_{0} \vert )$, and undergoes weak latitudinal libration with frequency $\hat {\omega } \vert {\mbit{\Omega} }_{0} \vert $ and amplitude $\mathit{Po}\vert {\mbit{\Omega} }_{0} \vert $, where $\mathit{Po}$ is the Poincaré number quantifying the strength of Poincaré force resulting from latitudinal libration. We investigate, via both asymptotic and numerical analysis, fluid motion in the spheroidal cavity driven by latitudinal libration. When $\vert \hat {\omega } \ensuremath{-} 2/ (2\ensuremath{-} {\mathscr{E}}^{2} )\vert \gg O({E}^{1/ 2} )$, an asymptotic solution for $E\ll 1$ and $\mathit{Po}\ll 1$ in oblate spheroidal coordinates satisfying the no-slip boundary condition is derived for a spheroidal cavity of arbitrary eccentricity without making any prior assumptions about the spatial–temporal structure of the librating flow. In this case, the librationally driven flow is non-axisymmetric with amplitude $O(\mathit{Po})$, and the role of the viscous boundary layer is primarily passive such that the flow satisfies the no-slip boundary condition. When $\vert \hat {\omega } \ensuremath{-} 2/ (2\ensuremath{-} {\mathscr{E}}^{2} )\vert \ll O({E}^{1/ 2} )$, the librationally driven flow is also non-axisymmetric but latitudinal libration resonates with a spheroidal inertial mode that is in the form of an azimuthally travelling wave in the retrograde direction. The amplitude of the flow becomes $O(\mathit{Po}/ {E}^{1/ 2} )$ at $E\ll 1$ and the role of the viscous boundary layer becomes active in determining the key property of the flow. An asymptotic solution for $E\ll 1$ describing the librationally resonant flow is also derived for an oblate spheroidal cavity of arbitrary eccentricity. Three-dimensional direct numerical simulation in an oblate spheroidal cavity is performed to demonstrate that, in both the non-resonant and resonant cases, a satisfactory agreement is achieved between the asymptotic solution and numerical simulation at $E\ll 1$.

scholarly journals On precessing flow in an oblate spheroid of arbitrary eccentricity

2014 ◽  
Vol 743 ◽  
pp. 358-384 ◽  
Author(s):  
Keke Zhang ◽  
Kit H. Chan ◽  
Xinhao Liao
Keyword(s):  
Numerical Simulation ◽  
Angular Velocity ◽  
Direct Numerical Simulation ◽  
Asymptotic Solution ◽  
Temporal Structure ◽  
Oblate Spheroid ◽  
Frame Of Reference ◽  
Viscous Effects ◽  
Homogeneous Fluid ◽  
Spheroidal Cavity

AbstractWe consider a homogeneous fluid of viscosity $\nu $ confined within an oblate spheroidal cavity of arbitrary eccentricity $\mathcal{E}$ marked by the equatorial radius $d$ and the polar radius $d \sqrt{1-\mathcal{E}^2}$ with $0<\mathcal{E}<1$. The spheroidal container rotates rapidly with an angular velocity ${\boldsymbol{\Omega}}_0 $ about its symmetry axis and precesses slowly with an angular velocity ${\boldsymbol{\Omega}}_p$ about an axis that is fixed in space. It is through both topographical and viscous effects that the spheroidal container and the viscous fluid are coupled together, driving precessing flow against viscous dissipation. The precessionally driven flow is characterized by three dimensionless parameters: the shape parameter $\mathcal{E}$, the Ekman number ${\mathit{Ek}}=\nu /(d^2 \delimiter "026A30C {\boldsymbol{\Omega}}_0\delimiter "026A30C )$ and the Poincaré number ${\mathit{Po}}=\pm \delimiter "026A30C {\boldsymbol{\Omega}}_p\delimiter "026A30C / \delimiter "026A30C \boldsymbol{\Omega}_0\delimiter "026A30C $. We derive a time-dependent asymptotic solution for the weakly precessing flow in the mantle frame of reference satisfying the no-slip boundary condition and valid for a spheroidal cavity of arbitrary eccentricity at ${\mathit{Ek}}\ll 1$. No prior assumptions about the spatial–temporal structure of the precessing flow are made in the asymptotic analysis. We also carry out direct numerical simulation for both the weakly and the strongly precessing flow in the same frame of reference using a finite-element method that is particularly suitable for non-spherical geometry. A satisfactory agreement between the asymptotic solution and direct numerical simulation is achieved for sufficiently small Ekman and Poincaré numbers. When the nonlinear effect is weak with $\delimiter "026A30C {\mathit{Po}}\delimiter "026A30C \ll 1$, the precessing flow in an oblate spheroid is characterized by an azimuthally travelling wave without having a mean azimuthal flow. Stronger nonlinear effects with increasing $\delimiter "026A30C {\mathit{Po}}\delimiter "026A30C $ produce a large-amplitude, time-independent mean azimuthal flow that is always westward in the mantle frame of reference. Implications of the precessionally driven flow for the westward motion observed in the Earth’s fluid core are also discussed.

scholarly journals Asymptotic theory for torsional convection in rotating fluid spheres

2017 ◽  
Vol 813 ◽  
Author(s):  
Keke Zhang ◽  
Kameng Lam ◽  
Dali Kong
Keyword(s):  
Boundary Condition ◽  
Asymptotic Solution ◽  
Torsional Oscillation ◽  
Slip Boundary Condition ◽  
Quantitative Agreement ◽  
Rotating Fluid ◽  
Axially Symmetric ◽  
Slip Boundary ◽  
Boussinesq Fluid ◽  
Fluid Spheres

This paper is concerned with the classical, well-studied problem of convective instabilities in rapidly rotating, self-gravitating, internally heated Boussinesq fluid spheres. Sanchez et al. (J. Fluid Mech., vol. 791, 2016, R1) recently found, unexpectedly via careful numerical simulation, that non-magnetic convection in the form of axially symmetric, equatorially antisymmetric torsional oscillation is physically preferred in a special range of small Prandtl number for rapidly rotating fluid spheres with the stress-free boundary condition. We derive an asymptotic solution describing convection-driven torsional oscillation – whose flow velocity and pressure are fully analytical and in closed form – that confirms the result of the numerical analysis and is in quantitative agreement with the numerical solution. We also demonstrate, through the derivation of a different asymptotic solution, that convection-driven torsional oscillation cannot occur in rapidly rotating fluid spheres with the no-slip boundary condition.

The Effect of Wall Conduction on the Stability of a Fluid in a Right Circular Cylinder Heated From Below

1983 ◽  
Vol 105 (2) ◽  
pp. 255-260 ◽  
Author(s):  
J. C. Buell ◽  
I. Catton
Keyword(s):  
Critical Rayleigh Number ◽  
Slip Boundary Condition ◽  
Dimensionless Temperature ◽  
Axisymmetric Mode ◽  
Slip Boundary ◽  
Wall Conductivity ◽  
Stream Functions ◽  
Cylindrical Volume ◽  
The Stability ◽  
The Galerkin Method

The onset of natural convection in a cylindrical volume of fluid bounded above and below by rigid, perfectly conducting surfaces and laterally by a wall of arbitrary thermal conductivity is examined. The critical Rayleigh number (dimensionless temperature difference) is determined as a function of aspect (radius to height) ratio and wall conductivity. The first three asymmetric modes as well as the axisymmetric mode are considered. Two sets of stream functions are employed to represent a velocity field that satisfies the no-slip boundary condition on all surfaces and conservation of mass everywhere. The Galerkin method is then used to reduce the linearized perturbation equations to an eigenvalue problem. The results for perfectly insulating and conducting walls are compared with the work of Charlson and Sani[9].

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