Librational forcing of a rapidly rotating fluid-filled cube

2018 ◽  
Vol 842 ◽  
pp. 469-494 ◽  
Author(s):  
Ke Wu ◽  
Bruno D. Welfert ◽  
Juan M. Lopez
Keyword(s):  
Stokes Equations ◽  
Rotation Axis ◽  
Rotation Period ◽  
Rotation Frequency ◽  
Shear Layers ◽  
Navier Stokes ◽  
Rotating Fluid ◽  
Ekman Number ◽  
Slip Boundary ◽  
The Mean

The flow response of a rapidly rotating fluid-filled cube to low-amplitude librational forcing is investigated numerically. Librational forcing is the harmonic modulation of the mean rotation rate. The rotating cube supports inertial waves which may be excited by libration frequencies less than twice the rotation frequency. The response is comprised of two main components: resonant excitation of the inviscid inertial eigenmodes of the cube, and internal shear layers whose orientation is governed by the inviscid dispersion relation. The internal shear layers are driven by the fluxes in the forced boundary layers on walls orthogonal to the rotation axis and originate at the edges where these walls meet the walls parallel to the rotation axis, and are hence called edge beams. The relative contributions to the response from these components is obscured if the mean rotation period is not small enough compared to the viscous dissipation time, i.e. if the Ekman number is too large. We conduct simulations of the Navier–Stokes equations with no-slip boundary conditions using parameter values corresponding to a recent set of laboratory experiments, and reproduce the experimental observations and measurements. Then, we reduce the Ekman number by one and a half orders of magnitude, allowing for a better identification and quantification of the contributions to the response from the eigenmodes and the edge beams.

scholarly journals Tidal oscillations of rotating hot Jupiters

2020 ◽  
Vol 494 (3) ◽  
pp. 3141-3155 ◽  
Author(s):  
Umin Lee
Keyword(s):  
Viscous Dissipation ◽  
Stokes Equations ◽  
Rotation Frequency ◽  
Inertial Range ◽  
Navier Stokes ◽  
Ekman Number ◽  
Hot Jupiters ◽  
The Core ◽  
Convective Core ◽  
Tidal Torques

ABSTRACT We calculate small amplitude gravitational and thermal tides of uniformly rotating hot Jupiters composed of a nearly isentropic convective core and a geometrically thin radiative envelope. We treat the fluid in the convective core as a viscous fluid and solve linearized Navier–Stokes equations to obtain tidal responses of the core, assuming that the Ekman number, Ek, is a constant parameter. In the radiative envelope, we take account of the effects of radiative dissipations on the responses. The properties of tidal responses depend on thermal time-scales τ* in the envelope and Ekman number, Ek, in the core and on whether the forcing frequency ω is in the inertial range or not, where the inertial range is defined by |ω| ≤ 2Ω for the rotation frequency Ω. If Ek ≳ 10−7, the viscous dissipation in the core is dominating the thermal contributions in the envelope for τ* ≳ 1 d. If Ek ≲ 10−7, however, the viscous dissipation is comparable to or smaller than the thermal contributions and the envelope plays an important role to determine the tidal torques. If the forcing is in the inertial range, frequency resonance of the tidal forcing with core inertial modes significantly affects the tidal torques, producing numerous resonance peaks of the torque. Depending on the sign of the torque in the peaks, we suggest that there exist cases in which the resonance with core inertial modes hampers the process of synchronization between the spin and orbital motion of the planets.

Dynamics of a vortex ring in a rotating fluid

1996 ◽  
Vol 317 ◽  
pp. 215-239 ◽  
Author(s):  
R. Verzicco ◽  
P. Orlandi ◽  
A. H. M. Eisenga ◽  
G. J. F. Van Heijst ◽  
G. F. Carnevale
Keyword(s):  
Vortex Ring ◽  
Stokes Equations ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Swirl Flow ◽  
Degree Of Polarization ◽  
Navier Stokes ◽  
Vortex Rings ◽  
Rotating Fluid ◽  
Translation Velocity

The formation and the evolution of axisymmetric vortex rings in a uniformly rotating fluid, with the rotation axis orthogonal to the ring vorticity, have been investigated by numerical and laboratory experiments. The flow dynamics turned out to be strongly affected by the presence of the rotation. In particular, as the background rotation increases, the translation velocity of the ring decreases, a structure with opposite circulation forms ahead of the ring and an intense axial vortex is generated on the axis of symmetry in the tail of the ring. The occurrence of these structures has been explained by the presence of a self-induced swirl flow and by inspection of the extra terms in the Navier–Stokes equations due to rotation. Although in the present case the swirl was generated by the vortex ring itself, these results are in agreement with those of Virk et al. (1994) for polarized vortex rings, in which the swirl flow was initially assigned as a ‘degree of polarization’.If the rotation rate is further increased beyond a certain value, the flow starts to be dominated by Coriolis forces. In this flow regime, the impulse imparted to the fluid no longer generates a vortex ring, but rather it excites inertial waves allowing the flow to radiate energy. Evidence of this phenomenon is shown.Finally, some three-dimensional numerical results are discussed in order to justify some asymmetries observed in flow visualizations.

The motion generated by a rising particle in a rotating fluid – numerical solutions. Part 1. A short container

2000 ◽  
Vol 413 ◽  
pp. 111-148 ◽  
Author(s):  
E. MINKOV ◽  
M. UNGARISH ◽  
M. ISRAELI
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Time Development ◽  
Navier Stokes ◽  
Quasi Steady State ◽  
Rotating Fluid ◽  
Axial Motion ◽  
Good Qualitative Agreement ◽  
Navier Stokes Equations ◽  
Ekman Number

Numerical finite-difference results of the full axisymmetric incompressible Navier–Stokes equations are presented for the problem of the slow axial motion of a disk particle in an incompressible, rotating fluid in a cylindrical container. The governing parameters are the Ekman number, E, the Rossby number, Ro, and the dimensionless height of the container, H (with respect to the diameter of the particle). The study concerns small values of E, Ro, and HE−1/2 and compares the numerical results with predictions of previous analytical (mostly approximate) studies. Special attention is focused on the drag force. First, developed (quasi-steady state) cases are considered. Excellent agreement with the exact linear (Ro = 0) solution of Ungarish & Vedensky (1995) is obtained when the computational Ro = 10−4. The effects of the nonlinear momentum advection terms are analysed and shown to be proportional to RoE−1/2. Next, the time-development for both (a) impulsive start and (b) start under a constant axial force are considered, and good qualitative agreement with previous analytical results (including the appearance of oscillations in case (b)) is indicated.

Parallel iterative finite-element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kangrui Zhou ◽  
Yueqiang Shang
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Parallel Algorithms ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Parallel Iterative ◽  
Nonlinear Slip Boundary Conditions

AbstractBased on full domain partition, three parallel iterative finite-element algorithms are proposed and analyzed for the Navier–Stokes equations with nonlinear slip boundary conditions. Since the nonlinear slip boundary conditions include the subdifferential property, the variational formulation of these equations is variational inequalities of the second kind. In these parallel algorithms, each subproblem is defined on a global composite mesh that is fine with size h on its subdomain and coarse with size H (H ≫ h) far away from the subdomain, and then we can solve it in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. Compared with the corresponding serial iterative finite-element algorithms, the parallel algorithms proposed in this paper can yield an approximate solution with a comparable accuracy and a substantial decrease in computational time. Contributions of this paper are as follows: (1) new parallel algorithms based on full domain partition are proposed for the Navier–Stokes equations with nonlinear slip boundary conditions; (2) nonlinear iterative methods are studied in the parallel algorithms; (3) new theoretical results about the stability, convergence and error estimates of the developed algorithms are obtained; (4) some numerical results are given to illustrate the promise of the developed algorithms.

scholarly journals Involving the Navier–Stokes equations in the derivation of boundary conditions for the lattice Boltzmann method

2011 ◽  
Vol 369 (1944) ◽  
pp. 2184-2192 ◽  
Author(s):  
Joris C. G. Verschaeve
Keyword(s):  
Boundary Condition ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Grid Spacing ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Boltzmann Method

By means of the continuity equation of the incompressible Navier–Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.

scholarly journals Determining the Tangential Velocity profile of a Rotating Fluid in a Static Container using Navier–Stokes equations

2020 ◽  
Author(s):  
RAJDEEP TAH ◽  
SARBAJIT MAZUMDAR ◽  
Krishna Kant Parida
Keyword(s):  
Angular Velocity ◽  
Velocity Profile ◽  
Velocity Gradient ◽  
Stokes Equations ◽  
Tangential Velocity ◽  
Navier Stokes ◽  
Rotating Fluid ◽  
Navier Stokes Equations ◽  
Similar Principle ◽  
Tangential Velocity Profile

The shape of the liquid surface for a fluid present in a uniformly rotating cylinder is generally determined by making a Tangential velocity gradient along the radius of the rotating cylindrical container. A very similar principle can be applied if the direction of the produced velocity gradient is reversed, for which the source of rotation will be present at the central axis of the cylindrical vessel in which the liquid is present. Now if the described system is completely closed, the angular velocity will decrease as a function of time. But when the surface of the rotating fluid is kept free, then the Tangential velocity profile would be similar to that of the Taylor-Couette Flow, with a modification that; due to formation of a curvature at the surface, the Navier-Stokes law is to be modified. Now the final equation may not seem to have a proper general solution, but can be approximated to certain solvable expressions for specific cases of angular velocity.

Sign in / Sign up

Export Citation Format

Share Document