Continuous spectra of the Batchelor vortex

2011 ◽  
Vol 681 ◽  
pp. 1-23 ◽  
Author(s):  
XUERUI MAO ◽  
SPENCER SHERWIN
Keyword(s):  
Vortex Core ◽  
Evolution Operator ◽  
Free Stream ◽  
Finite Amplitude ◽  
Far Field ◽  
Potential Region ◽  
Convergence Study ◽  
Spectra And Pseudospectra ◽  
Discrete Spectra ◽  
Stream Mode

The spectra of the Batchelor vortex are obtained by discretizing its linearized evolution operator using a modified Chebyshev polynomial approximation at a Reynolds number of 1000 and zero azimuthal wavenumber. Three types of eigenmodes are identified from the spectra: discrete modes, potential modes and free-stream modes. The discrete modes have been extensively documented but the last two modes have received little attention. A convergence study of the spectra and pseudospectra supports the classification that discrete modes correspond to discrete spectra while the other two modes correspond to continuous spectra. Free-stream modes have finite amplitude in the far field whilst potential modes decay to zero in the far field. The free-stream modes are therefore a limiting form of the potential modes when the radial decay rate of velocity components reduces to zero. The radial form of the free-stream modes with axial and radial wavenumbers is investigated and the penetration of the free-stream mode into the vortex core highlights the possibility of interaction between the potential region and the vortex core. A wavepacket pseudomode study confirms the existence of continuous spectra and predicts the locations and radial wavenumbers of the eigenmodes. The pseudomodes corresponding to the potential modes are observed to be in the form of one or two wavepackets while the free-stream modes are not observed to be in the form of wavepackets.

On the dynamics of finite-amplitude baroclinic waves as a function of supercriticality

1976 ◽  
Vol 78 (3) ◽  
pp. 621-637 ◽  
Author(s):  
Joseph Pedlosky
Keyword(s):  
Steady State ◽  
Baroclinic Instability ◽  
Finite Amplitude ◽  
Steady State Mode ◽  
Final State ◽  
Nonlinear Modes ◽  
Stream Structure ◽  
The Stability ◽  
Amplitude Mode ◽  
Stream Mode

A finite-amplitude model of baroclinic instability is studied in the case where the cross-stream scale is large compared with the Rossby deformation radius and the dissipative and advective time scales are of the same order. A theory is developed that describes the nature of the wave field as the shear supercriticality increases beyond the stability threshold of the most unstable cross-stream mode and penetrates regions of higher supercriticality. The set of possible steady nonlinear modes is found analytically. It is shown that the steady cross-stream structure of each finite-amplitude mode is a function of the supercriticality.Integrations of initial-value problems show, in each case, that the final state realized is the state characterized by the finite-amplitude mode with the largest equilibrium amplitude. The approach to this steady state is oscillatory (nonmonotonic). Further, each steady-state mode is a well-defined mixture of linear cross-stream modes.

scholarly journals Unsteady flow in trailing vortices

1991 ◽  
Vol 227 ◽  
pp. 107-134 ◽  
Author(s):  
S. I. Green ◽  
A. J. Acosta
Keyword(s):  
Vortex Core ◽  
Axial Velocity ◽  
Free Stream ◽  
Flow Direction ◽  
Free Stream Velocity ◽  
Stream Velocity ◽  
Centripetal Acceleration ◽  
California Institute Of Technology ◽  
Aspect Ratios ◽  
Trailing Vortices

The instantaneous velocity distribution in trailing vortices generated by lifting hydrofoils has been measured in the Low Turbulence Water Tunnel at the California Institute of Technology. Two different rectangular planform hydrofoils with small aspect ratios were tested. Double-pulsed holography of injected microbubbles, which act much as Lagrangian flow tracers, was used to determine instantaneous axial and tangential velocities. Measurements were made at various free-stream velocities, angles of attack, and downstream distances. The vortex core mean axial velocity is consistently greater than the free-stream velocity near the hydrofoil trailing edge, and decreases with downstream distance. The mean axial velocity is strongly Reynolds-number dependent.Axial flow in the trailing vortex is highly unsteady for all the flow conditions studied; peak-to-peak fluctuations on the centreline as large as the free-stream velocity have been observed. The amplitude of these fluctuations falls rapidly with increasing distance from the centreline. For an angle of attack of 10° the fluctuations consist of both ‘fast’ and ‘slow’ components, whereas for α = 5° only ‘fast’ fluctuations have been observed. Peak decelerations of the centreline fluid occur with amplitude comparable to the maximum centripetal acceleration around the centreline. Certain unusual structures of the vortex core - regions in which the flow direction quickly diverges from the free-stream direction, and then equally quickly recovers - have been labelled ‘vortex kinks.’

Nonlinear Kelvin–Helmholtz instability of a finite vortex layer

1985 ◽  
Vol 157 ◽  
pp. 225-263 ◽  
Author(s):  
C. Pozrikidis ◽  
J. J. L. Higdon
Keyword(s):  
Growth Rate ◽  
Layer Thickness ◽  
Vortex Core ◽  
Numerical Procedure ◽  
Interfacial Area ◽  
Finite Amplitude ◽  
Computational Effort ◽  
Thin Layers ◽  
Nonlinear Growth ◽  
Vortex Layer

The nonlinear growth of periodic disturbances on a finite vortex layer is examined. Under the assumption of constant vorticity, the evolution of the layer may be analysed by following the contour of the vortex region. A numerical procedure is introduced which leads to higher-order accuracy than previous methods with negligible increase in computational effort. The response of the vortex layer is studied as a function of layer thickness and the amplitude and form of the initial disturbance. For small initial disturbances, all unstable layers form a large rotating vortex core of nearly elliptical shape. The growth rate of the disturbances is strongly affected by the layer thickness; however, the final amplitude of the disturbance is relatively insensitive to the thickness and reaches a maximum value of approximately 20% of the wavelength. In the fully developed layers, the amplitude shows a small oscillation owing to the rotation of the vortex core. For finite-amplitude initial disturbances, the evolution of the layer is a function of the initial amplitude. For thin layers with thickness less than 3% of the wavelength, three different patterns were observed in the vortex-core region: a compact elliptic core, an elongated S-shaped core and a bifurcation into two orbiting cores. For thicker layers, stationary elliptic cores may develop if the thickness exceeds 15% of the wavelength. The spacing and eccentricity of these cores is in good agreement with previously discovered steady-state solutions. The growth rate of interfacial area (or length of the vortex contour) is calculated and is found to approach a constant value in well-developed vortex layers.

Vortex core behaviour in confined and unconfined geometries: a quasi-one-dimensional model

2001 ◽  
Vol 449 ◽  
pp. 61-84 ◽  
Author(s):  
D. L. DARMOFAL ◽  
R. KHAN ◽  
E. M. GREITZER ◽  
C. S. TAN
Keyword(s):  
Vortex Core ◽  
Core Area ◽  
Navier Stokes ◽  
Dominant Factor ◽  
Far Field ◽  
Confined Geometries ◽  
Outer Flow ◽  
One Dimensional ◽  
The Core ◽  
Confined Flows

Axisymmetric vortex core flows, in unconfined and confined geometries, are examined using a quasi-one-dimensional analysis. The goal is to provide a simple unified view of the topic which gives insight into the key physical features, and the overall parametric dependence, of the core area evolution due to boundary geometry or far-field pressure variation. The analysis yields conditions under which waves on vortex cores propagate only downstream (supercritical flow) or both upstream and downstream (subcritical flow), delineates the conditions for a Kelvin–Helmholtz instability arising from the difference in core and outer flow axial velocities, and illustrates the basic mechanism for suppression of this instability due to the presence of swirl. Analytic solutions are derived for steady smoothly, varying vortex cores in unconfined geometries with specified far-field pressure and in confined flows with specified bounding area variation. For unconfined vortex cores, a maximum far-field pressure rise exists above which the vortex cannot remain smoothly varying; this coincides with locally critical conditions (axial velocity equal to wave speed) in terms of wave propagation. Comparison with axisymmetric Navier–Stokes simulations and experimental results indicate that this maximum correlates with the appearance of vortex breakdown and marked core area increase in the simulations and experiments. For confined flows, the core stagnation pressure defect relative to the outer flow is found to be the dominant factor in determining conditions for large increases in core size. Comparisons with axisymmetric Navier–Stokes computations show that the analysis captures qualitatively, and in many instances, quantitatively, the evolution of vortex cores in confined geometries. Finally, a strong analogy with quasi-one-dimensional compressible flow is demonstrated by construction of continuous and discontinuous flows as a function of imposed downstream core edge pressure.

Transient growth associated with continuous spectra of the Batchelor vortex

2012 ◽  
Vol 697 ◽  
pp. 35-59 ◽  
Author(s):  
X. Mao ◽  
S. J. Sherwin
Keyword(s):  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Vortex Core ◽  
Azimuthal Angle ◽  
Original Position ◽  
Local Analysis ◽  
Transient Growth ◽  
Transient Effects ◽  
Potential Region ◽  
Optimal Perturbation

AbstractThe spectrum of the Batchelor vortex can be broadly split into a discrete spectrum, a potential spectrum and a free-stream spectrum where, since the last two spectra are both continuous, they can also be considered as one continuous spectrum. The discrete spectrum has been extensively studied but the continuous spectrum has received limited attention in the context of vortex flow. A local transient growth study is conducted and the contribution of the discrete spectrum and the continuous spectrum to the transient growth is separated by constructing optimal perturbations on the discrete or continuous sub-eigenspaces separately. It is found that the significant transient growth is mainly due to the non-normality of the continuous eigenmodes/spectrum whilst the discrete eigenmodes/spectrum have little contribution to the transient energy growth. A matrix-free method, which reduces to the local analysis when appropriate periodic boundary conditions are imposed, is also applied to investigate the transient growth in both a plane of constant azimuthal angle and a plane constant axial location. Previously studies by other authors have demonstrated that at zero azimuthal wavenumber the transient growth reaches infinitely large values over infinite time intervals while the optimal perturbations are located far from the vortex core. Therefore we limited our scope to small values of the time horizon so as to obtain reasonably strong transient effects stemming from physically relevant optimal perturbations. Two mechanisms of transient growth are observed: namely a redistribution of the azimuthal velocity to the azimuthal vorticity and interaction between out-of-vortex-core structures with those within the vortex core. A direct numerical simulation (DNS) of the vortex perturbed by optimal perturbations is conducted to investigate the nonlinear development of the optimal perturbations. In the azimuthally constant decomposed case, it is found that the optimal perturbation induces a string of bubble structures to be generated as a consequence of the non-orthogonality of continuous eigenmodes and the breakdown bubble is induced by viscous diffusion, while in the axially constant decomposition transient growth analysis, it is observed that the optimal perturbations associated with the continuous eigenmodes drive the vortex to vibrate around the initial vortex centre before eventually returning to its original position at larger times. This transient effect provides a mechanism for the ‘vortex meandering’ observed in previous experimental and numerical studies. These optimal perturbations associated with the continuous spectrum with out-of-vortex-core structures are observed to be activated by anisotropic inflow perturbations in the potential region.

scholarly journals Instability of Surface Quasigeostrophic Vortices

2009 ◽  
Vol 66 (4) ◽  
pp. 1051-1062 ◽  
Author(s):  
Xavier Carton
Keyword(s):  
Vortex Core ◽  
Temperature Anomaly ◽  
Finite Amplitude ◽  
Temperature Gradients ◽  
Barotropic Instability ◽  
Vortex Instability ◽  
Considerable Similarity ◽  
Mean Temperature ◽  
Model Evidence ◽  
2D Model

Abstract The instability of circular vortices is studied numerically in the surface quasigeostrophic (SQG) model, and their evolutions are compared with those of barotropically unstable 2D vortices. The growth rates in the SQG model evidence similarity with their barotropic counterparts for moderate radial gradients of temperature (or of vorticity in the 2D model). For stronger gradients, SQG vortices are more unstable than 2D vortices. The nonlinear, finite-amplitude evolutions of perturbed vortices provide evidence that moderately unstable, elliptically perturbed vortices form tripoles. When they are more unstable, they break into two dipoles. Weakly unstable vortices with triangular perturbations form transient quadrupoles that break; they stabilize only for large gradients of mean temperature. Finally, with square perturbations, pentapoles degenerate into dipoles, at least for the range of mean temperature gradients explored here. The analysis of nonlinear stabilizations reveals that the deformation of the vortex core and the leak of its temperature anomaly to the periphery are essential ingredients to stabilize the perturbation at finite amplitude. In conclusion, SQG vortex instability exhibits considerable similarity to the barotropic instability of 2D vortices.

Possible applications of fraunhofer holography in high resolution electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 222-223 ◽  
Author(s):  
N. Bonnet ◽  
M. Troyon ◽  
P. Gallion
Keyword(s):  
Electron Microscopy ◽  
High Resolution ◽  
Transfer Function ◽  
Radiation Damage ◽  
High Resolution Electron Microscopy ◽  
Far Field ◽  
Field Region ◽  
Crystalline Materials ◽  
Resolution Electron ◽  
The Ideal

Two main problems in high resolution electron microscopy are first, the existence of gaps in the transfer function, and then the difficulty to find complex amplitude of the diffracted wawe from registered intensity. The solution of this second problem is in most cases only intended by the realization of several micrographs in different conditions (defocusing distance, illuminating angle, complementary objective apertures…) which can lead to severe problems of contamination or radiation damage for certain specimens.Fraunhofer holography can in principle solve both problems stated above (1,2). The microscope objective is strongly defocused (far-field region) so that the two diffracted beams do not interfere. The ideal transfer function after reconstruction is then unity and the twin image do not overlap on the reconstructed one.We show some applications of the method and results of preliminary tests.Possible application to the study of cavitiesSmall voids (or gas-filled bubbles) created by irradiation in crystalline materials can be observed near the Scherzer focus, but it is then difficult to extract other informations than the approximated size.

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