A vorticity wave packet breaking within a rapidly rotating vortex. Part I: The critical layer flow

2021 ◽  
Author(s):  
Philippe Caillol
Keyword(s):  
Wave Packet ◽  
Critical Layer ◽  
Layer Flow

Time-dependent critical layers in shear flows on the beta-plane

1984 ◽  
Vol 142 ◽  
pp. 431-449 ◽  
Author(s):  
Fred J. Hickernell
Keyword(s):  
Evolution Equation ◽  
Nonlinear Effects ◽  
Finite Amplitude ◽  
Critical Layer ◽  
Convolution Integral ◽  
Matched Asymptotics ◽  
Beta Plane ◽  
Neutral Mode ◽  
Critical Layers ◽  
Layer Flow

The problem of a finite-amplitude free disturbance of an inviscid shear flow on the beta-plane is studied. Perturbation theory and matched asymptotics are used to derive an evolution equation for the amplitude of a singular neutral mode of the Kuo equation. The effects of time-dependence, nonlinearity and viscosity are included in the analysis of the critical-layer flow. Nonlinear effects inside the critical layer rather than outside the critical layer determine the evolution of the disturbance. The nonlinear term in the evolution equation is some type of convolution integral rather than a simple polynomial. This makes the evolution equation significantly different from those commonly encountered in fluid wave and stability problems.

An experimental study of absolute instability of the rotating-disk boundary-layer flow

1996 ◽  
Vol 314 ◽  
pp. 373-405 ◽  
Author(s):  
R. J. Lingwood
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Wave Packet ◽  
Boundary Layer Flow ◽  
Critical Reynolds Number ◽  
Linear Stability Theory ◽  
Absolute Instability ◽  
Unstable Flow ◽  
Average Value ◽  
Layer Flow

In this paper, the results of experiments on unsteady disturbances in the boundary-layer flow over a disk rotating in otherwise still air are presented. The flow was perturbed impulsively at a point corresponding to a Reynolds numberRbelow the value at which transition from laminar to turbulent flow is observed. Among the frequencies excited are convectively unstable modes, which form a three-dimensional wave packet that initially convects away from the source. The wave packet consists of two families of travelling convectively unstable waves that propagate together as one packet. These two families are predicted by linear-stability theory: branch-2 modes dominate close to the source but, as the packet moves outwards into regions with higher Reynolds numbers, branch-1 modes grow preferentially and this behaviour was found in the experiment. However, the radial propagation of the trailing edge of the wave packet was observed to tend towards zero as it approaches the critical Reynolds number (about 510) for the onset of radial absolute instability. The wave packet remains convectively unstable in the circumferential direction up to this critical Reynolds number, but it is suggested that the accumulation of energy at a well-defined radius, due to the flow becoming radially absolutely unstable, causes the onset of laminar–turbulent transition. The onset of transition has been consistently observed by previous authors at an average value of 513, with only a small scatter around this value. Here, transition is also observed at about this average value, with and without artificial excitation of the boundary layer. This lack of sensitivity to the exact form of the disturbance environment is characteristic of an absolutely unstable flow, because absolute growth of disturbances can start from either noise or artificial sources to reach the same final state, which is determined by nonlinear effects.

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.

On the global instability of free disturbances with a time-dependent nonlinear viscous critical layer

1985 ◽  
Vol 157 ◽  
pp. 53-77 ◽  
Author(s):  
J. Gajjar ◽  
F. T. Smith
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution Equations ◽  
Time Dependent ◽  
Critical Layer ◽  
Mean Flow ◽  
Nonlinear Growth ◽  
Special Cases ◽  
Layer Flow ◽  
Scale Properties ◽  
High Reynolds

A theoretical study is made of the global nonlinear growth or decay, in space and time, of an unsteady non-neutral disturbance/wavepacket when a time-dependent nonlinear viscous critical layer is present. The basic flow considered is a steady quasiparallel channel flow, boundary layer or liquid-layer flow at high Reynolds number. The unsteadiness with regard to the critical layer shows itself less in the internal dynamics than in the relatively slow movement of the layer across the flow, the temporal and spatial rate of movement discussed being sufficient to affect the nonlinear viscous balance in the layer. This greatly reduces the mean-flow distortions produced. The disturbance amplitude, in contrast, responds nonlinearly on faster time- and space-scales, both inside and outside the critical layer. These slower- and faster-scale properties inside the critical layer and outside, i.e. globally, are coupled together in general. The work addresses first the structure and nonlinear evolution equations for the growing or decaying free disturbance and the critical layer. But preliminary analysis in special cases suggests, among other things, the significant result that previous nonlinear studies based on quasineutral assumptions give unstable subcritical threshold amplitudes, above which increasingly fast disturbance growth takes place globally.

Nonlinear critical-layer evolution of a forced gravity wave packet

2003 ◽  
Vol 493 ◽  
pp. 151-179 ◽  
Author(s):  
L. J. CAMPBELL ◽  
S. A. MASLOWE
Keyword(s):  
Wave Packet ◽  
Gravity Wave ◽  
Critical Layer
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