scholarly journals A singular vorticity wave packet within a rapidly rotating vortex: spiralling versus oscillating motions

2019 ◽  
Vol 873 ◽  
pp. 688-741 ◽  
Author(s):  
Philippe Caillol
Keyword(s):  
Wave Packet ◽  
Radial Velocity ◽  
Three Dimensional ◽  
Critical Layer ◽  
Shear Mode ◽  
Vortex Interaction ◽  
Leading Order ◽  
Mean Flow ◽  
Radial Wave ◽  
Radar Images

This paper considers a free vorticity wave packet propagating within a rapidly rotating vortex in the quasi-steady regime, a long time after the wave packet strongly and unsteadily interacted with the vortex. We study a singular, nonlinear, helical and asymmetric shear mode inside a linearly stable, columnar and axisymmetric vortex on the $f$-plane. The amplitude-modulated mode enters resonance with the vortex at a certain radius $r_{c}$, where the phase angular speed is equal to the rotation frequency. The singularity in the modal equation at $r_{c}$ strongly modifies the flow in the three-dimensional helical critical layer, the region around $r_{c}$ where the wave/vortex interaction occurs. This interaction generates a vertically sheared three-dimensional mean flow of higher amplitude than the wave packet. The chosen envelope regime assumes the formation of a mean radial velocity of the same order as the wave packet amplitude, leading to the streamlines exhibiting a spiral motion in the neighbourhood of the critical layer. Radar images frequently show such spiral bands in tropical cyclones or tornadoes. Through matched asymptotic expansions, we find an analytical solution of the leading-order equations inside the critical layer. The generalized Batchelor integral condition applied to the quasi-steady, three-dimensional motion inside the separatrices yields a leading-order, non-uniform three-dimensional vorticity. The critical-layer pattern, strongly deformed by the mean radial velocity, loses its symmetries with respect to the azimuthal and radial directions, which makes the leading-order mean radial wave fluxes non-zero. Finally, a stronger wave/vortex interaction occurs with respect to previous studies where a steady neutral vortical mode or an envelope of larger extent was involved.

The development of three-dimensional wave-packets in unbounded parallel flows

1968 ◽  
Vol 32 (4) ◽  
pp. 801-808 ◽  
Author(s):  
M. Gaster ◽  
A. Davey
Keyword(s):  
Wave Packet ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Mean Flow ◽  
Physical Plane ◽  
Parallel Flows ◽  
Flow Velocity Profile ◽  
The Stability ◽  
Special Case ◽  
Dimensional Wave

In this paper we examine the stability of a two-dimensional wake profile of the form u(y) = U∞(1 – r e-sy2) with respect to a pulsed disturbance at a point in the fluid. The disturbed flow forms an expanding wave packet which is convected downstream. Far downstream, where asymptotic expansions are valid, the motion at any point in the wave packet is described by a particular three-dimensional wave having complex wave-numbers. In the special case of very unstable flows, where viscosity does not have a significant influence, it is possible to evaluate the three-dimensional eigenvalues in terms of two-dimensional ones using the inviscid form of Squire's transformation. In this way each point in the physical plane can be linked to a particular two-dimensional wave growing in both space and time by simple algebraic expressions which are independent of the mean flow velocity profile. Computed eigenvalues for the wake profile are used in these relations to find the behaviour of the wave packet in the physical plane.

On critical-layer and diffusion-layer nonlinearity in the three-dimensional stage of boundary-layer transition

1993 ◽  
Vol 443 (1917) ◽  
pp. 95-106 ◽  
Keyword(s):  
Boundary Layer ◽  
Parametric Resonance ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Boundary Layer Transition ◽  
Critical Layer ◽  
Transition To Turbulence ◽  
Maximum Magnitude ◽  
Nonlinear Interactions ◽  
Mean Flow

This paper considers nonlinear interactions in the three-dimensional stage of transition to turbulence, taking an accelerating boundary layer as a prototype flow. Attention is focused on transition via subharmonic resonance in the upper-branch scaling régime. It is shown that the (weakly) nonlinear instability of the flow is described by a seven-zoned structure, cf. the five-zoned structure for the linear problem. The dominant nonlinear interactions take place both in a critical layer and in ‘diffusion layers’. The nonlinearly generated mean flow in turn interacts with the wall to attain a maximum magnitude near the wall. It is emphasized that both the nonlinear mechanism and the flow structure are generic for three-dimensional disturbances. And there is some similarity with the work in the context of wave/vortex interaction. Numerical solutions of the amplitude equations indicate that if the oblique modes initially have a small amplitude, they first experience a rapid growth caused by parametric resonance. Following this the cubic interactions of the oblique modes inhibit the growth and lead to a wavelength shortening. However, if the initial amplitudes of the oblique modes are sufficiently large, the parametric resonance can be completely bypassed. Numerical solutions also suggest that oblique modes with unequal initial amplitudes evolve to an equal-amplitude state.

scholarly journals Isopycnal Mixing Suppression by the Antarctic Circumpolar Current and the Southern Ocean Meridional Overturning Circulation

2017 ◽  
Vol 47 (8) ◽  
pp. 2023-2045 ◽  
Author(s):  
Christopher Chapman ◽  
Jean-Baptiste Sallée
Keyword(s):  
Southern Ocean ◽  
Three Dimensional ◽  
Eddy Diffusivity ◽  
Meridional Overturning Circulation ◽  
Critical Layer ◽  
Ekman Transport ◽  
Volume Flux ◽  
Mean Flow ◽  
Overturning Circulation ◽  
Meridional Overturning

AbstractThe meridional overturning circulation (MOC) in the Southern Ocean is investigated using hydrographic observations combined with satellite measurements of sea surface height. A three-dimensional (spatial and vertical) estimate of the isopycnal eddy diffusivity in the Southern Ocean is obtained using the theory of Ferrari and Nikurashin that includes the influence of suppression of the diffusivity by the strong, time-mean flows. It is found that the eddy diffusivity is enhanced at depth, reaching a maximum at the critical layer near 1000 m. The estimate of diffusivity is used with a simple diffusive parameterization to estimate the meridional eddy volume flux. This estimate of eddy volume flux is combined with an estimate of the Ekman transport to reconstruct the time-mean overturning circulation. By comparing the reconstruction with, and without, suppression of the eddy diffusivity by the mean flow, the influence of the suppression on the overturning is illuminated. It is shown that the suppression of the eddy diffusivity results in a large reduction of interior eddy transports and a more realistic eddy-induced overturning circulation. Finally, a simple conceptual model is used to show that the MOC is influenced not only by the existence of enhanced diffusivity at depth but also by the details of the vertical structure of the eddy diffusivity, such as the depth of the critical layer.

Membrane theory

2017 ◽  
Author(s):  
David J. Steigmann
Keyword(s):  
Three Dimensional ◽  
Thin Sheets ◽  
Two Dimensional ◽  
Leading Order ◽  
Membrane Theory ◽  
Dimensional Theory ◽  
Small Thickness

This chapter develops two-dimensional membrane theory as a leading order small-thickness approximation to the three-dimensional theory for thin sheets. Applications to axisymmetric equilibria are developed in detail, and applied to describe the phenomenon of bulge propagation in cylinders.

Tilting at wave beams: a new perspective on the St. Andrew’s Cross

2017 ◽  
Vol 830 ◽  
pp. 660-680 ◽  
Author(s):  
T. Kataoka ◽  
S. J. Ghaemsaidi ◽  
N. Holzenberger ◽  
T. Peacock ◽  
T. R. Akylas
Keyword(s):  
Wave Field ◽  
Finite Length ◽  
Line Source ◽  
Cutoff Frequency ◽  
Three Dimensional ◽  
Resonance Phenomenon ◽  
Buoyancy Frequency ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Viscous Effects

The generation of internal gravity waves by a vertically oscillating cylinder that is tilted to the horizontal in a stratified Boussinesq fluid of constant buoyancy frequency, $N$, is investigated. This variant of the widely studied horizontal configuration – where a cylinder aligned with a plane of constant gravitational potential induces four wave beams that emanate from the cylinder, forming a cross pattern known as the ‘St. Andrew’s Cross’ – brings out certain unique features of radiated internal waves from a line source tilted to the horizontal. Specifically, simple kinematic considerations reveal that for a cylinder inclined by a given angle $\unicode[STIX]{x1D719}$ to the horizontal, there is a cutoff frequency, $N\sin \unicode[STIX]{x1D719}$, below which there is no longer a radiated wave field. Furthermore, three-dimensional effects due to the finite length of the cylinder, which are minor in the horizontal configuration, become a significant factor and eventually dominate the wave field as the cutoff frequency is approached; these results are confirmed by supporting laboratory experiments. The kinematic analysis, moreover, suggests a resonance phenomenon near the cutoff frequency as the group-velocity component perpendicular to the cylinder direction vanishes at cutoff; as a result, energy cannot be easily radiated away from the source, and nonlinear and viscous effects are likely to come into play. This scenario is examined by adapting the model for three-dimensional wave beams developed in Kataoka & Akylas (J. Fluid Mech., vol. 769, 2015, pp. 621–634) to the near-resonant wave field due to a tilted line source of large but finite length. According to this model, the combination of three-dimensional, nonlinear and viscous effects near cutoff triggers transfer of energy, through the action of Reynolds stresses, to a circulating horizontal mean flow. Experimental evidence of such an induced mean flow near cutoff is also presented.

scholarly journals The relationship between photometric and spectroscopic oscillation amplitudes from 3D stellar atmosphere simulations

2021 ◽  
Author(s):  
Yixiao Zhou ◽  
Thomas Nordlander ◽  
Luca Casagrande ◽  
Meridith Joyce ◽  
Yaguang Li ◽  
...  
Keyword(s):  
Radial Velocity ◽  
Three Dimensional ◽  
Quantitative Relationship ◽  
Stellar Atmospheres ◽  
Spectral Lines ◽  
Wavelength Shift ◽  
The Sun ◽  
Theoretical Derivation ◽  
Velocity Amplitude ◽  
Good Agreement

Abstract We establish a quantitative relationship between photometric and spectroscopic detections of solar-like oscillations using ab initio, three-dimensional (3D), hydrodynamical numerical simulations of stellar atmospheres. We present a theoretical derivation as proof of concept for our method. We perform realistic spectral line formation calculations to quantify the ratio between luminosity and radial velocity amplitude for two case studies: the Sun and the red giant ε Tau. Luminosity amplitudes are computed based on the bolometric flux predicted by 3D simulations with granulation background modelled the same way as asteroseismic observations. Radial velocity amplitudes are determined from the wavelength shift of synthesized spectral lines with methods closely resembling those used in BiSON and SONG observations. Consequently, the theoretical luminosity to radial velocity amplitude ratios are directly comparable with corresponding observations. For the Sun, we predict theoretical ratios of 21.0 and 23.7 ppm/[m s−1] from BiSON and SONG respectively, in good agreement with observations 19.1 and 21.6 ppm/[m s−1]. For ε Tau, we predict K2 and SONG ratios of 48.4 ppm/[m s−1], again in good agreement with observations 42.2 ppm/[m s−1], and much improved over the result from conventional empirical scaling relations which gives 23.2 ppm/[m s−1]. This study thus opens the path towards a quantitative understanding of solar-like oscillations, via detailed modelling of 3D stellar atmospheres.

Flow Field Unsteadiness in the Tip Region of a Transonic Compressor Rotor

1997 ◽  
Vol 119 (1) ◽  
pp. 122-128 ◽  
Author(s):  
S. L. Puterbaugh ◽  
W. W. Copenhaver
Keyword(s):  
Flow Field ◽  
High Performance ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Vortex Interaction ◽  
Tip Leakage Vortex ◽  
Tip Leakage ◽  
Transonic Compressor ◽  
Compressor Rotor ◽  
Operating Points

An experimental investigation concerning tip flow field unsteadiness was performed for a high-performance, state-of-the-art transonic compressor rotor. Casing-mounted high frequency response pressure transducers were used to indicate both the ensemble averaged and time varying flow structure present in the tip region of the rotor at four different operating points at design speed. The ensemble averaged information revealed the shock structure as it evolved from a dual shock system at open throttle to an attached shock at peak efficiency to a detached orientation at near stall. Steady three-dimensional Navier Stokes analysis reveals the dominant flow structures in the tip region in support of the ensemble averaged measurements. A tip leakage vortex is evident at all operating points as regions of low static pressure and appears in the same location as the vortex found in the numerical solution. An unsteadiness parameter was calculated to quantify the unsteadiness in the tip cascade plane. In general, regions of peak unsteadiness appear near shocks and in the area interpreted as the shock-tip leakage vortex interaction. Local peaks of unsteadiness appear in mid-passage downstream of the shock-vortex interaction. Flow field features not evident in the ensemble averaged data are examined via a Navier-Stokes solution obtained at the near stall operating point.

Turbulent flow between counter-rotating concentric cylinders: a direct numerical simulation study

2008 ◽  
Vol 615 ◽  
pp. 371-399 ◽  
Author(s):  
S. DONG
Keyword(s):  
Turbulent Flow ◽  
Large Scale ◽  
Outer Cylinder ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Taylor Vortex ◽  
Small Scale ◽  
Mean Flow ◽  
Concentric Cylinders ◽  
High Reynolds

We report three-dimensional direct numerical simulations of the turbulent flow between counter-rotating concentric cylinders with a radius ratio 0.5. The inner- and outer-cylinder Reynolds numbers have the same magnitude, which ranges from 500 to 4000 in the simulations. We show that with the increase of Reynolds number, the prevailing structures in the flow are azimuthal vortices with scales much smaller than the cylinder gap. At high Reynolds numbers, while the instantaneous small-scale vortices permeate the entire domain, the large-scale Taylor vortex motions manifested by the time-averaged field do not penetrate a layer of fluid near the outer cylinder. Comparisons between the standard Taylor–Couette system (rotating inner cylinder, fixed outer cylinder) and the counter-rotating system demonstrate the profound effects of the Coriolis force on the mean flow and other statistical quantities. The dynamical and statistical features of the flow have been investigated in detail.

scholarly journals Spin-Orbit Entanglement in Time Evolution of Radial Wave Packets in Hydrogenic Systems

2004 ◽  
Vol 11 (04) ◽  
pp. 401-409
Author(s):  
Marcin Turek ◽  
Piotr Rozmej
Keyword(s):  
Wave Packet ◽  
Time Evolution ◽  
Time Scale ◽  
Degrees Of Freedom ◽  
Wave Packets ◽  
Characteristic Time Scale ◽  
Radial Wave ◽  
Mean Values ◽  
Breathing Motion ◽  
Wave Packet Revival

Time evolution of radial wave packets built from the eigenstates of Dirac equation for a hydrogenic system is considered. Radial wave packets are constructed from the states of different n quantum numbers and the same lowest angular momentum. In general they exhibit a kind of breathing motion with dispersion and (partial) revivals. Calculations show that for some particular preparations of the wave packet one can observe interesting effects in spin motion, coming from inherent entanglement of spin and orbital degrees of freedom. These effects manifest themselves through some oscillations in the mean values of spin operators and through changes of spatial probability density carried by upper and lower components of the wave function. It is also shown that the characteristic time scale of predicted effects (called T ls ) is much smaller for radial wave packets than in other cases, reaching values comparable to (or even less than) the time scale for the wave packet revival.

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