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.

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 Dynamic analysis of submerged microscale plates: the effects of acoustic radiation and viscous dissipation

2016 ◽  
Vol 472 (2187) ◽  
pp. 20150728 ◽  
Author(s):  
Zhangming Wu ◽  
Xianghong Ma
Keyword(s):  
Viscous Fluid ◽  
Boundary Conditions ◽  
Viscous Dissipation ◽  
Stokes Equations ◽  
Acoustic Radiation ◽  
Navier Stokes ◽  
Rectangular Plates ◽  
Fluid Loading ◽  
Navier Stokes Equations ◽  
Loading Impedance

The aim of this paper is to study the dynamic characteristics of micromechanical rectangular plates used as sensing elements in a viscous compressible fluid. A novel modelling procedure for the plate–fluid interaction problem is developed on the basis of linearized Navier–Stokes equations and no-slip conditions. Analytical expression for the fluid-loading impedance is obtained using a double Fourier transform approach. This modelling work provides us an analytical means to study the effects of inertial loading, acoustic radiation and viscous dissipation of the fluid acting on the vibration of microplates. The numerical simulation is conducted on microplates with different boundary conditions and fluids with different viscosities. The simulation results reveal that the acoustic radiation dominates the damping mechanism of the submerged microplates. It is also proved that microplates offer better sensitivities (Q-factors) than the conventional beam type microcantilevers being mass sensing platforms in a viscous fluid environment. The frequency response features of microplates under highly viscous fluid loading are studied using the present model. The dynamics of the microplates with all edges clamped are less influenced by the highly viscous dissipation of the fluid than the microplates with other types of boundary conditions.

scholarly journals Thermal tides in rotating hot Jupiters

2019 ◽  
Vol 488 (2) ◽  
pp. 1960-1976 ◽  
Author(s):  
Umin Lee ◽  
Daiki Murakami
Keyword(s):  
Harmonic Functions ◽  
Series Expansions ◽  
Radiative Cooling ◽  
Tidal Torque ◽  
Hot Jupiters ◽  
The Core ◽  
Oscillation Modes ◽  
Convective Core ◽  
Thermal Tides ◽  
Spherical Harmonic Functions

ABSTRACT We calculate tidal torque due to semidiurnal thermal tides in rotating hot Jupiters, taking account of the effects of radiative cooling in the envelope and of the planets rotation on the tidal responses. We use a simple Jovian model composed of a nearly isentropic convective core and a thin radiative envelope. To represent the tidal responses of rotating planets, we employ series expansions in terms of spherical harmonic functions $Y_l^m$ with different ls for a given m. For low-forcing frequency, there occurs frequency resonance between the forcing and the g- and r-modes in the envelope and inertial modes in the core. We find that the resonance enhances the tidal torque, and that the resonance with the g- and r-modes produces broad peaks and that with the inertial modes very sharp peaks, depending on the magnitude of the non-adiabatic effects associated with the oscillation modes. We also find that the behaviour of the tidal torque as a function of the forcing frequency (or period) is different between prograde and retrograde forcing, particularly for long forcing periods because the r-modes, which have long periods, exist only on the retrograde side.

Similarity solutions for viscous vortex cores

1992 ◽  
Vol 238 ◽  
pp. 487-507 ◽  
Author(s):  
Ernst W. Mayer ◽  
Kenneth G. Powell
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Flow ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Axial Pressure Gradient ◽  
Self Similar ◽  
Similar Solutions

Results are presented for a class of self-similar solutions of the steady, axisymmetric Navier–Stokes equations, representing the flows in slender (quasi-cylindrical) vortices. Effects of vortex strength, axial gradients and compressibility are studied. The presence of viscosity is shown to couple the parameters describing the core growth rate and the external flow field, and numerical solutions show that the presence of an axial pressure gradient has a strong effect on the axial flow in the core. For the viscous compressible vortex, near-zero densities and pressures and low temperatures are seen on the vortex axis as the strength of the vortex increases. Compressibility is also shown to have a significant influence upon the distribution of vorticity in the vortex core.

Conical turbulent swirling vortex with variable eddy viscosity

1986 ◽  
Vol 403 (1825) ◽  
pp. 235-268 ◽  
Keyword(s):  
Eddy Viscosity ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Turbulent Vortex ◽  
The Core ◽  
Vortex Solutions ◽  
Variable Eddy Viscosity

In the one hundred years since Rankine suggested his well known two-dimensional vortex model with finite core, no one has ever found any exact vortex solutions of the Navier-Stokes equations that can satisfy a complete set of physical boundary conditions. In this paper a variable viscosity is introduced and the existence of conical turbulent vortex solutions of the Navier-Stokes equations is examined. It is found that for a class of deliberately chosen eddy viscosity function a steady turbulent vortex can, for the first time, satisfy both the regularity condition at the core and the adherence condition at the surface, except for a singularity at the origin inherent in all conical similarity solutions. In its asymptotic form, if the eddy viscosity only varies in a boundary layer near the surface or the core, outside the layer the solution given would approach one of the laminar solutions of Yih et al . ( Physics Fluids 25, 2147 (1982)) or that of Serrin ( Phil. Trans. R. Soc. Lond. A 271, 325 (1972)) respectively. These results reveal some remarkable relations between the behaviour, and even the existence, of a vortex and turbulence.

External turbulence-induced axial flow and instability in a vortex

2016 ◽  
Vol 793 ◽  
pp. 353-379 ◽  
Author(s):  
Eric Stout ◽  
Fazle Hussain
Keyword(s):  
Stokes Equations ◽  
Axial Flow ◽  
Underlying Mechanism ◽  
Outer Edge ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Flow Generation ◽  
Axial Pressure Gradient ◽  
External Turbulence

External turbulence-induced axial flow in an incompressible, normal-mode stable Lamb–Oseen (two-dimensional) vortex column is studied via direct numerical simulations of the Navier–Stokes equations. Azimuthally oriented vorticity filaments, formed from external turbulence, advect radially towards or away from the vortex axis (depending on the filament’s swirl direction), resulting in a net induced axial flow in the vortex core; axial flow increases with increasing vortex Reynolds number ($Re=$ vortex circulation/viscosity). This contrasts the viscous mechanism for axial flow generation downstream of a lifting body, wherein an axial pressure gradient is produced by viscous diffusion of the swirl (Batchelor, J. Fluid Mech., vol. 20, 1964, pp. 645–658). Analysis of the self-induced motion of an arbitrarily curved external filament shows that any non-axisymmetric filament undergoes radial advection. We then studied the evolution of a vortex column starting with an imposed optimal transient growth perturbation. For a range of Re values, axial flow develops and initially grows as (time)$^{5/2}$ before decreasing after two turnover times; for $Re=10\,000$ – the highest computationally achievable – axial flow at late times becomes sufficiently strong to induce vortex instability. Contrary to a prior claim of a parent–offspring mechanism at the outer edge of the core, vorticity tilting within the core by axial flow is the underlying mechanism producing energy growth. Thus, external perturbations in practical flows (at $Re\sim 10^{7}$) produce destabilizing axial flow, possibly leading to the sought-after vortex breakup.

Analysis of the dripping–jetting transition in compound capillary jets

2010 ◽  
Vol 649 ◽  
pp. 523-536 ◽  
Author(s):  
M. A. HERRADA ◽  
J. M. MONTANERO ◽  
C. FERRERA ◽  
A. M. GAÑÁN-CALVO
Keyword(s):  
Stability Analysis ◽  
Convective Instability ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Outer Interface ◽  
Spatio Temporal ◽  
Capillary Jets ◽  
The Stability

We examine the behaviour of a compound capillary jet from the spatio-temporal linear stability analysis of the Navier–Stokes equations. We map the jetting–dripping transition in the parameter space by calculating the Weber numbers for which the convective/absolute instability transition occurs. If the remaining dimensionless parameters are set, there are two critical Weber numbers that verify Brigg's pinch criterion. The region of absolute (convective) instability corresponds to Weber numbers smaller (larger) than the highest value of those two Weber numbers. The stability map is affected significantly by the presence of the outer interface, especially for compound jets with highly viscous cores, in which the outer interface may play an important role even though it is located very far from the core. Full numerical simulations of the Navier–Stokes equations confirm the predictions of the stability analysis.

Re-entrant corner flows of Oldroyd-B fluids

2005 ◽  
Vol 461 (2060) ◽  
pp. 2573-2603 ◽  
Author(s):  
J.D Evans
Keyword(s):  
Boundary Layers ◽  
Newtonian Fluid ◽  
Similarity Solution ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Outer Solution ◽  
The Core ◽  
Retardation Parameter

The method of matched asymptotic expansions is used to construct solutions for the planar steady flow of Oldroyd-B fluids around re-entrant corners of angles π / α (1/2≤ α <1). Two types of similarity solutions are described for the core flow away from the walls. These correspond to the two main dominant balances of the constitutive equation, where the upper convected derivative of stress either dominates or is balanced by the upper convected derivative of the rate of strain. The former balance gives the incompressible Euler or inviscid flow equations and the latter balance the incompressible Navier–Stokes equations. The inviscid flow similarity solution for the core is that first derived by Hinch (Hinch 1993 J. Non-Newtonian Fluid Mech. 50 , 161–171) with a core stress singularity that depends upon the corner angle and radial distance as O ( r −2(1− α ) ) and a velocity behaviour that vanishes as O ( r α (3− α )−1 ). Extending the analysis of Renardy (Renardy 1995 J. Non-Newtonian Fluid Mech. 58 , 83–39), this outer solution is matched to viscometric wall behaviour for both upstream and downstream boundary layers. This structure is shown to hold for the majority of the retardation parameter range. In contrast, the similarity solution associated with the Navier–Stokes equations has a velocity behaviour O ( r λ ) where λ ∈(0,1) satisfies a nonlinear eigenvalue problem, dependent upon the corner angle and an associated Reynolds number defined in terms of the ratio of the retardation and relaxation times. This similarity solution is shown to hold as an outer solution and is matched into stress boundary layers at the walls which recover viscometric behaviour. However, the matching is restricted to values of the retardation parameter close to the relaxation parameter. In this case the leading order core stress is Newtonian with behaviour O ( r −(1− λ ) ).

Thermal Aspects of a Novel Viscous Pump

1998 ◽  
Vol 120 (1) ◽  
pp. 99-107 ◽  
Author(s):  
M. C. Sharatchandra ◽  
M. Sen ◽  
M. Gad-el-Hak
Keyword(s):  
Viscous Dissipation ◽  
Temperature Rise ◽  
Channel Wall ◽  
Stokes Equations ◽  
Coupled System ◽  
Operating Conditions ◽  
Navier Stokes ◽  
Bulk Temperature ◽  
Temperature Dependent Viscosity ◽  
Pump Operation

We have previously introduced a novel method for pumping fluids via a viscous mechanism. The device essentially consists of a cylindrical rotor eccentrically placed in a channel, and it is suited for hauling highly viscous polymers in macroducts, or more common fluids in microducts. Under certain operating conditions, viscous dissipation can be important, and a significant attendant temperature rise can have adverse effects on the pump operation. For this reason, we have conducted a numerical experiment to characterize the associated phenomena. The coupled system of the two-dimensional Navier-Stokes equations, with temperature-dependent viscosity, and the energy equation, with viscous dissipation terms retained, are solved using a finite-volume method. Different types of thermal boundary conditions at the rotor-fluid interface are explored in the numerical scheme. An approximate theoretical model is also developed to analyze flow in the region between the rotor and the nearest plate (for small gaps). The results indicate that although the bulk temperature rise is minimal for typical microscale situations, significantly steep temperature gradients are observed in the region between the rotor and the nearest channel wall where the most intense shear stress occurs. For certain combinations of Re, Ec, and Pr, temperature rises along the channel wall of the order of 30 K were calculated. Moreover, for very small values of this gap, large errors in the computed flowrates and pumping power estimates can arise for large Brinkman numbers, if the effects of viscous dissipation are ignored. Furthermore, the existence of an optimum value of rotor position, such that the bulk velocity is a maximum, is demonstrated. These findings are significant, as they are indicative of trends associated with the flow of highly viscous polymeric liquids, where much larger temperature rises and their attendant degradation in performance are likely to occur.

Self-induced flow in a rotating tube

1991 ◽  
Vol 230 ◽  
pp. 505-524 ◽  
Author(s):  
S. Gilham ◽  
P. C. Ivey ◽  
J. M. Owen ◽  
J. R. Pincombe
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Laser Doppler Anemometry ◽  
Glass Tube ◽  
The Self ◽  
Navier Stokes ◽  
Ekman Number ◽  
Induced Flow ◽  
Short Tube

When a tube, sealed at one end and open to a quiescent environment at the other, is rotated about its axis, fluid flows from the open end along the axis towards the sealed end and returns in an annular boundary layer on the cylindrical wall. This paper describes the first known study to be made of this self-induced flow. Numerical solutions of the Navier–Stokes equations are shown to be in mainly good agreement with experimental results obtained using flow visualization and laser–Doppler anemometry in a rotating glass tube.The self-induced flow in the tube can be described in terms of the length-to-radius ratio, G, and the Ekman number, E. However, for large values of G (G [ges ] 20), the flow outside the boundary layer on the endwall of the tube can be characterized by a single, modified, Ekman number, E*, where E* = GE. Although most of the fluid entering the open end of the tube is entrained into the annular (Stewartson-type) boundary layer, for small values of E* (E* < 0.2) some flow reaches the sealed end. For this so-called 'short-tube case’, the flow in the boundary layer on the endwall is shown to be similar to that associated with a disk rotating in a quiescent environment: the free disk. The self-induced flow for the short-tube case is believed to be responsible for the ’ hot-poker effect’ used, on some jet engines, to provide ice protection for the nose bullet.

Sign in / Sign up

Export Citation Format

Share Document