An EBE finite element method for simulating nonlinear flows in rotating spheroidal cavities

Chan, Kit H.; Zhang, Keke; Liao, Xinhao

doi:10.1002/fld.2088

Cited by 19 publications

(7 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A sketch of the non-uniform finite element mesh used in computing the nonlinear precessing flow is illustrated in figure 1. In our finite element method, a mixed finite element of the Hood-Taylor type is adopted: in each tetrahedral element, a piecewise quadratic polynomial is employed to approximate the velocity u while a piecewise linear polynomial is used to approximate the pressure p. The EBE (element-by-element) method for parallelizing code used in the present study is largely similar to that for spheroidal geometry (Chan et al 2010).…”

Section: Numerical Simulation Using a 3-d Finite Element Methodsmentioning

confidence: 99%

On the transition from the laminar to disordered flow in a precessing spherical-like cylinder

Kong

Cui

Liao

et al. 2014

Geophysical & Astrophysical Fluid Dynamics

Self Cite

View full text Add to dashboard Cite

We investigate, through both asymptotic analysis and direct numerical simulation, precessionally driven flow of a homogeneous fluid confined in a fluid-filled circular cylinder that rotates rapidly about its symmetry axis and precesses about a different axis that is fixed in space. A particular emphasis is placed on a spherical-like cylinder whose diameter is nearly the same as its length. At this special aspect ratio, the strongest direct resonance occurs between the spatially simplest inertial mode and the precessional Poincaré forcing. An asymptotic analytical solution in closed form describing weakly precessing flow is derived in the mantle frame of reference for asymptotically small Ekman numbers. We also construct a nonlinear three-dimensional finite element model -which is validated against both the asymptotic solution and a constructed exact solution -for elucidating the nonlinear transition leading to disordered flow in the precessing spherical-like cylinder. Properties of both weakly and strongly precessing flows are investigated with the aid of a complete inertial-mode decomposition of the fully nonlinear solution. Despite a large effort being made, the well-known triadic resonance is not found in the precessing spherical-like cylinder. The energy contained in the precessionally forced inertial mode is primarily transferred, through nonlinear effects in the viscous boundary layers, to the geostrophic flow that becomes predominant when the precessional Poincaré force is sufficiently large. It is found that the nonlinear flow evolutes gradually and progressively from the laminar to disordered as the precessional force increases.

show abstract

Section: Numerical Simulation Using a 3-d Finite Element Methodsmentioning

confidence: 99%

On the transition from the laminar to disordered flow in a precessing spherical-like cylinder

Kong

Cui

Liao

et al. 2014

Geophysical & Astrophysical Fluid Dynamics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, the three-dimensional mesh is constructed in a way such that more nodes placed in the vicinity of the bounding surface for the purpose of resolving the spherical viscous boundary layer. The detail of the numerical method and its validation can be found in Chan, Zhang & Liao (2010). An advantage of using a finite-element method in this particular problem is that the pressure p, in addition to the velocity u, are obtained directly from numerical simulation.…”

Section: Nonlinear Numerical Simulationmentioning

confidence: 99%

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

et al. 2013

Self Cite

View full text Add to dashboard Cite

We investigate the problem of oscillatory flow of a homogeneous fluid with viscosity ν in a fluid-filled sphere of radius a that rotates rapidly about a fixed axis with angular velocity Ω 0 and that undergoes weak longitudinal libration with amplitude Ω 0 and frequencyωΩ 0 , where is the Poincaré number with 1 andω is dimensionless frequency with 0 <ω < 2. Three different methods are employed in this investigation: (i) asymptotic analysis at small Ekman numbers E defined as E = ν/(a 2 Ω 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 10 −4 . It is shown that the flow amplitude |u| is nearly independent of both the Ekman number E and the libration frequencyω, always obeying the asymptotic scaling |u| = O( ) even though various spherical inertial modes are excited by longitudinal libration at different libration frequenciesω. Consequently, resonances do not occur in this system even whenω 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 whenω 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 frequencyω. 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 1 and 1.

show abstract

“…These equations are solved efficiently on modern parallel computers, starting from an arbitrary initial condition to find u n+1 and p n+1 from given u n and u n−1 . A discussion regarding the accuracy and convergence of our spheroidal finite element code can be found in Chan, Zhang & Liao (2010).…”

Section: Numerical Simulationmentioning

confidence: 99%

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

2012

Self Cite

View full text Add to dashboard Cite

We consider a homogeneous fluid of viscosity ν confined within an oblate spheroidal cavity, x 2 /a 2 + y 2 /a 2 + z 2 /(a 2 (1 − E 2 )) = 1, with eccentricity 0 < E < 1. The spheroidal container rotates rapidly with an angular velocity Ω 0 , which is fixed in an inertial frame and defines a small Ekman number E = ν/(a 2 |Ω 0 |), and undergoes weak latitudinal libration with frequencyω|Ω 0 | and amplitude Po|Ω 0 |, where 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 |ω − 2/(2 − E 2 )| O(E 1/2 ), an asymptotic solution for E 1 and Po 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 nonaxisymmetric with amplitude O(Po), and the role of the viscous boundary layer is primarily passive such that the flow satisfies the no-slip boundary condition. When |ω − 2/(2 − E 2 )| 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(Po/E 1/2 ) at E 1 and the role of the viscous boundary layer becomes active in determining the key property of the flow. An asymptotic solution for E 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 1.

show abstract

An EBE finite element method for simulating nonlinear flows in rotating spheroidal cavities

Cited by 19 publications

References 32 publications

On the transition from the laminar to disordered flow in a precessing spherical-like cylinder

On the transition from the laminar to disordered flow in a precessing spherical-like cylinder

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

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

Contact Info

Product

Resources

About