scholarly journals Linear stability of inviscid vortex rings to axisymmetric perturbations

2019 ◽  
Vol 874 ◽  
pp. 1115-1146
Author(s):  
Bartosz Protas
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Equilibrium Configuration ◽  
Three Dimensional ◽  
Vortex Rings ◽  
Meridional Plane ◽  
The Family ◽  
A Free Boundary Problem ◽  
Shape Differentiation ◽  
Axisymmetric Perturbations

We consider the linear stability to axisymmetric perturbations of the family of inviscid vortex rings discovered by Norbury (J. Fluid Mech., vol. 57, 1973, pp. 417–431). Since these vortex rings are obtained as solutions to a free-boundary problem, their stability analysis is performed using recently developed methods of shape differentiation applied to the contour-dynamics formulation of the problem in the three-dimensional axisymmetric geometry. This approach allows us to systematically account for the effects of boundary deformations on the linearized evolution of the vortex ring. We investigate the instantaneous amplification of perturbations assumed to have the same the circulation as the vortex rings in their equilibrium configuration. These stability properties are then determined by the spectrum of a singular integro-differential operator defined on the vortex boundary in the meridional plane. The resulting generalized eigenvalue problem is solved numerically with a spectrally accurate discretization. Our results reveal that while thin vortex rings remain neutrally stable to axisymmetric perturbations, they become linearly unstable to such perturbations when they are sufficiently ‘fat’. Analysis of the structure of the eigenmodes demonstrates that they approach the corresponding eigenmodes of Rankine’s vortex and Hill’s vortex in the thin-vortex and fat-vortex limit, respectively. This study is a stepping stone on the way towards a complete stability analysis of inviscid vortex rings with respect to general perturbations.

scholarly journals Linear stability of Hill’s vortex to axisymmetric perturbations

2016 ◽  
Vol 799 ◽  
pp. 579-602 ◽  
Author(s):  
Bartosz Protas ◽  
Alan Elcrat
Keyword(s):  
Linear Stability ◽  
Eigenvalue Problems ◽  
Three Dimensional ◽  
Spectral Approach ◽  
Classical Analysis ◽  
Contour Dynamics ◽  
Stagnation Points ◽  
A Free Boundary Problem ◽  
Shape Differentiation ◽  
Axisymmetric Perturbations

We consider the linear stability of Hill’s vortex with respect to axisymmetric perturbations. Given that Hill’s vortex is a solution of a free-boundary problem, this stability analysis is performed by applying methods of shape differentiation to the contour dynamics formulation of the problem in a three-dimensional axisymmetric geometry. This approach allows us to systematically account for the effect of boundary deformations on the linearized evolution of the vortex under the constraint of constant circulation. The resulting singular integro-differential operator defined on the vortex boundary is discretized with a highly accurate spectral approach. This operator has two unstable and two stable eigenvalues complemented by a continuous spectrum of neutrally stable eigenvalues. By considering a family of suitably regularized (smoothed) eigenvalue problems solved with a range of numerical resolutions, we demonstrate that the corresponding eigenfunctions are in fact singular objects in the form of infinitely sharp peaks localized at the front and rear stagnation points. These findings thus refine the results of the classical analysis by Moffatt & Moore (J. Fluid Mech., vol. 87, 1978, pp. 749–760).

Secondary motion in convection layers generated by lateral heating of a solute gradient

2002 ◽  
Vol 455 ◽  
pp. 1-19 ◽  
Author(s):  
CHO LIK CHAN ◽  
WEN-YAU CHEN ◽  
C. F. CHEN
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Convection Cell ◽  
Experimental Conditions ◽  
Longitudinal Plane ◽  
Secondary Motion ◽  
Solute Gradient

The three-dimensional motion observed by Chen & Chen (1997) in the convection cells generated by sideways heating of a solute gradient is further examined by experiments and linear stability analysis. In the experiments, we obtained visualizations and PIV measurements of the velocity of the fluid motion in the longitudinal plane perpendicular to the imposed temperature gradient. The flow consists of a horizontal row of counter-rotating vortices within each convection cell. The magnitude of this secondary motion is approximately one-half that of the primary convection cell. Results of a linear stability analysis of a parallel double-diffusive flow model of the actual ow show that the instability is in the salt-finger mode under the experimental conditions. The perturbation streamlines in the longitudinal plane at onset consist of a horizontal row of counter-rotating vortices similar to those observed in the experiments.

scholarly journals Linear stability of confined flow around a 180-degree sharp bend

2017 ◽  
Vol 822 ◽  
pp. 813-847 ◽  
Author(s):  
Azan M. Sapardi ◽  
Wisam K. Hussam ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Stability

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

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