scholarly journals Radiative instability of an anticyclonic vortex in a stratified rotating fluid

2012 ◽  
Vol 707 ◽  
pp. 381-392 ◽  
Author(s):  
Junho Park ◽  
Paul Billant
Keyword(s):  
Growth Rate ◽  
Radial Profile ◽  
Base Flow ◽  
Rossby Number ◽  
Stratified Fluids ◽  
Spontaneous Radiation ◽  
Columnar Vortex ◽  
Centrifugal Instability ◽  
Radiative Instability ◽  
Asymptotic Analyses

AbstractIn strongly stratified fluids, an axisymmetric vertical columnar vortex is unstable because of a spontaneous radiation of internal waves. The growth rate of this radiative instability is strongly reduced in the presence of a cyclonic background rotation $f/ 2$ and is smaller than the growth rate of the centrifugal instability for anticyclonic rotation, so it is generally expected to affect vortices in geophysical flows only if the Rossby number $Ro= 2\Omega / f$ is large (where $\Omega $ is the angular velocity of the vortex). However, we show here that an anticyclonic Rankine vortex with low Rossby number in the range $\ensuremath{-} 1\leq Ro\lt 0$, which is centrifugally stable, is unstable to the radiative instability when the azimuthal wavenumber $\vert m\vert $ is larger than 2. Its growth rate for $Ro= \ensuremath{-} 1$ is comparable to the values reported in non-rotating stratified fluids. In the case of continuous vortex profiles, this new radiative instability is shown to occur if the potential vorticity of the base flow has a sufficiently steep radial profile. The most unstable azimuthal wavenumber is inversely proportional to the steepness of the vorticity jump. The properties and mechanism of the instability are explained by asymptotic analyses for large wavenumbers.

scholarly journals Stability of an isolated pancake vortex in continuously stratified-rotating fluids

2016 ◽  
Vol 801 ◽  
pp. 508-553 ◽  
Author(s):  
Eunok Yim ◽  
Paul Billant ◽  
Claire Ménesguen
Keyword(s):  
Angular Velocity ◽  
Baroclinic Instability ◽  
Base Flow ◽  
Shear Instability ◽  
Vertical Shear ◽  
Rossby Number ◽  
Centrifugal Instability ◽  
Pancake Vortex ◽  
To Come ◽  
The Stability

This paper investigates the stability of an axisymmetric pancake vortex with Gaussian angular velocity in radial and vertical directions in a continuously stratified-rotating fluid. The different instabilities are determined as a function of the Rossby number $Ro$, Froude number $F_{h}$, Reynolds number $Re$ and aspect ratio ${\it\alpha}$. Centrifugal instability is not significantly different from the case of a columnar vortex due to its short-wavelength nature: it is dominant when the absolute Rossby number $|Ro|$ is large and is stabilized for small and moderate $|Ro|$ when the generalized Rayleigh discriminant is positive everywhere. The Gent–McWilliams instability, also known as internal instability, is then dominant for the azimuthal wavenumber $m=1$ when the Burger number $Bu={\it\alpha}^{2}Ro^{2}/(4F_{h}^{2})$ is larger than unity. When $Bu\lesssim 0.7Ro+0.1$, the Gent–McWilliams instability changes into a mixed baroclinic–Gent–McWilliams instability. Shear instability for $m=2$ exists when $F_{h}/{\it\alpha}$ is below a threshold depending on $Ro$. This condition is shown to come from confinement effects along the vertical. Shear instability transforms into a mixed baroclinic–shear instability for small $Bu$. The main energy source for both baroclinic–shear and baroclinic–Gent–McWilliams instabilities is the potential energy of the base flow instead of the kinetic energy for shear and Gent–McWilliams instabilities. The growth rates of these four instabilities depend mostly on $F_{h}/{\it\alpha}$ and $Ro$. Baroclinic instability develops when $F_{h}/{\it\alpha}|1+1/Ro|\gtrsim 1.46$ in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.

Effect of electron inertia on radiative instability of rotating two-component gaseous plasma

2012 ◽  
Vol 90 (12) ◽  
pp. 1209-1221 ◽  
Author(s):  
A.K. Patidar ◽  
R.K. Pensia ◽  
V. Shrivastava
Keyword(s):  
Magnetic Field ◽  
Thermal Conductivity ◽  
Growth Rate ◽  
Electrical Resistivity ◽  
Heat Loss ◽  
Loss Function ◽  
Mode Analysis ◽  
Electron Inertia ◽  
Radiative Instability ◽  
Jeans Criterion

The problem of radiative instability of homogeneous rotating partially ionized plasma incorporating viscosity, porosity, and electron inertia in the presence of a magnetic field is investigated. A general dispersion relation is obtained using normal mode analysis with the help of relevant linearized perturbation equations of the problem. The modified Jeans criterion of instability is obtained. The conditions of Jeans instabilities are discussed in the different cases of interest. It is found that the simultaneous effect of viscosity, rotation, finite conductivity, and porosity of the medium does not essentially change the Jeans criterion of instability. It is also found that the presence of arbitrary radiative heat-loss function and thermal conductivity modified the conditions of Jeans instability for longitudinal propagation. It is found that, for longitudinal propagation, the conditions of radiative instability are independent of magnetic field, viscosity, rotation, finite electrical resistivity, and electron inertia, but for the transverse mode of propagation it depends upon finite electrical resistivity and strength of magnetic field and is independent of viscosity, electron inertia, and rotation. From the curves we find that viscosity has a stabilizing effect on the growth rate of instability but the thermal conductivity and density-dependent heat-loss function has a destabilizing effect on the instability growth rate.

scholarly journals Numerical stability analysis of a vortex ring with swirl

2019 ◽  
Vol 878 ◽  
pp. 5-36 ◽  
Author(s):  
Yuji Hattori ◽  
Francisco J. Blanco-Rodríguez ◽  
Stéphane Le Dizès
Keyword(s):  
Numerical Simulation ◽  
Growth Rate ◽  
Direct Numerical Simulation ◽  
Vortex Ring ◽  
Stokes Equations ◽  
Base Flow ◽  
Critical Layer ◽  
Navier Stokes ◽  
Instability Mode ◽  
Navier Stokes Equations

The linear instability of a vortex ring with swirl with Gaussian distributions of azimuthal vorticity and velocity in its core is studied by direct numerical simulation. The numerical study is carried out in two steps: first, an axisymmetric simulation of the Navier–Stokes equations is performed to obtain the quasi-steady state that forms a base flow; then, the equations are linearized around this base flow and integrated for a sufficiently long time to obtain the characteristics of the most unstable mode. It is shown that the vortex rings are subjected to curvature instability as predicted analytically by Blanco-Rodríguez & Le Dizès (J. Fluid Mech., vol. 814, 2017, pp. 397–415). Both the structure and the growth rate of the unstable modes obtained numerically are in good agreement with the analytical results. However, a small overestimation (e.g. 22 % for a curvature instability mode) by the theory of the numerical growth rate is found for some instability modes. This is most likely due to evaluation of the critical layer damping which is performed for the waves on axisymmetric line vortices in the analysis. The actual position of the critical layer is affected by deformation of the core due to the curvature effect; as a result, the damping rate changes since it is sensitive to the position of the critical layer. Competition between the curvature and elliptic instabilities is also investigated. Without swirl, only the elliptic instability is observed in agreement with previous numerical and experimental results. In the presence of swirl, sharp bands of both curvature and elliptic instabilities are obtained for $\unicode[STIX]{x1D700}=a/R=0.1$, where $a$ is the vortex core radius and $R$ the ring radius, while the elliptic instability dominates for $\unicode[STIX]{x1D700}=0.18$. New types of instability mode are also obtained: a special curvature mode composed of three waves is observed and spiral modes that do not seem to be related to any wave resonance. The curvature instability is also confirmed by direct numerical simulation of the full Navier–Stokes equations. Weakly nonlinear saturation and subsequent decay of the curvature instability are also observed.

scholarly journals Inertia–gravity-wave radiation by a sheared vortex

2008 ◽  
Vol 596 ◽  
pp. 169-189 ◽  
Author(s):  
E. I. ÓLAFSDÓTTIR ◽  
A. B. OLDE DAALHUIS ◽  
J. VANNESTE
Keyword(s):  
Gravity Wave ◽  
Couette Flow ◽  
Gravity Waves ◽  
Dimensional Structure ◽  
Analytic Description ◽  
Wave Radiation ◽  
Rossby Number ◽  
Spontaneous Radiation ◽  
Initial State ◽  
Inertia Gravity Waves

We consider the linear evolution of a localized vortex with Gaussian potential vorticity that is superposed on a horizontal Couette flow in a rapidly rotating strongly stratified fluid. The Rossby number, defined as the ratio of the shear of the Couette flow to the Coriolis frequency, is assumed small. Our focus is on the inertia–gravity waves that are generated spontaneously during the evolution of the vortex. These are exponentially small in the Rossby number and hence are neglected in balanced models such as the quasi-geostrophic model and its higher-order generalizations. We develop an exponential-asymptotic approach, based on an expansion in sheared modes, to give an analytic description of the three-dimensional structure of the inertia–gravity waves emitted by the vortex. This provides an explicit example of the spontaneous radiation of inertia–gravity waves by localized balanced motion in the small-Rossby-number regime.The inertia–gravity waves are emitted as a burst of four wavepackets propagating downstream of the vortex. The approach employed reduces the computation of inertia–gravity-wave fields to a single quadrature, carried out numerically, for each spatial location and each time. This makes it possible to unambiguously define an initial state that is entirely free of inertia–gravity waves, and circumvents the difficulties generally associated with the separation between balanced motion and inertia–gravity waves.

Linear three-dimensional global and asymptotic stability analysis of incompressible open cavity flow

2015 ◽  
Vol 768 ◽  
pp. 113-140 ◽  
Author(s):  
Vincenzo Citro ◽  
Flavio Giannetti ◽  
Luca Brandt ◽  
Paolo Luchini
Keyword(s):  
Growth Rate ◽  
Reynolds Number ◽  
Stability Analysis ◽  
Closed Orbit ◽  
Three Dimensional ◽  
Base Flow ◽  
Open Cavity ◽  
Critical Conditions ◽  
Global Mode ◽  
Structural Sensitivity

The viscous and inviscid linear stability of the incompressible flow past a square open cavity is studied numerically. The analysis shows that the flow first undergoes a steady three-dimensional bifurcation at a critical Reynolds number of 1370. The critical mode is localized inside the cavity and has a flat roll structure with a spanwise wavelength of about 0.47 cavity depths. The adjoint global mode reveals that the instability is most efficiently triggered in the thin region close to the upstream tip of the cavity. The structural sensitivity analysis identifies the wavemaker as the region located inside the cavity and spatially concentrated around a closed orbit. As the flow outside the cavity plays no role in the generation mechanisms leading to the bifurcation, we confirm that an appropriate parameter to describe the critical conditions in open cavity flows is the Reynolds number based on the average velocity between the two upper edges. Stabilization is achieved by a decrease of the total momentum inside the shear layer that drives the core vortex within the cavity. The mechanism of instability is then studied by means of a short-wavelength approximation considering pressureless inviscid modes. The closed streamline related to the maximum inviscid growth rate is found to be the same as that around which the global wavemaker is concentrated. The structural sensitivity field based on direct and adjoint eigenmodes, computed at a Reynolds number far higher than that of the base flow, can predict the critical orbit on which the main instabilities inside the cavity arise. Further, we show that the sub-leading unstable time-dependent modes emerging at supercritical conditions are characterized by a period that is a multiple of the revolution time of Lagrangian particles along the orbit of maximum growth rate. The eigenfrequencies of these modes, computed by global stability analysis, are in very good agreement with the asymptotic results.

scholarly journals Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices

2014 ◽  
Vol 758 ◽  
pp. 565-585 ◽  
Author(s):  
Manikandan Mathur ◽  
Sabine Ortiz ◽  
Thomas Dubos ◽  
Jean-Marc Chomaz
Keyword(s):  
Axial Velocity ◽  
Local Stability ◽  
Numerical Solutions ◽  
Axial Flow ◽  
Axial Direction ◽  
Threshold Value ◽  
Base Flow ◽  
Centrifugal Instability ◽  
Axisymmetric Vortex ◽  
Stuart Vortices

AbstractLinear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz & Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich & Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.

scholarly journals Effect of Pressure Gradient on the Development of Görtler Vortices

2019 ◽  
Author(s):  
Leandro Marochio Fernandes ◽  
Marcio Teixeira de Mendonça
Keyword(s):  
Pressure Gradient ◽  
Linear Stability ◽  
Boundary Layers ◽  
Adverse Pressure Gradient ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Wave Number ◽  
Local Mode ◽  
Effect Of Pressure ◽  
Centrifugal Instability

Boundary layers over concave surfaces may become unstable due to centrifugal instability that manifests itself as stationary streamwise counter rotating vortices. The centrifugal instability mechanism in boundary layers has been extensively studied and there is a large number of publications addressing different aspects of this problem. The results on the effect of pressure gradient show that favorable pressure gradients are stabilizing and adverse pressure gradient enhances the instability. The objective of the present investigation is to complement those works, looking particularly at the effect of pressure gradient on the stability diagram and on the determination of the spanwise wave number corresponding to the fastest growth. This study is based on the classic linear stability theory, where the parallel boundary layer approximation is assumed. Therefore, results are valid for Görtler numbers above 7, the lower limit where local mode linear stability analysis was identified in the literature as valid. For the base flow given by the Falkner-Skan solution, the linear stability equations are solved by a shooting method where the eigenvalues are the Görtler number, the spanwise wavenumber and the growth rate. The results show stabilization due to favorable pressure gradient as the constant amplification rate curves are displaced to higher Görtler numbers, with the opposite effect for adverse pressure gradient. Results previously unavailable in the literature identifying the fastest growing mode spanwise wavelength for a range of Falkner-Skan acceleration parameters are presented.

scholarly journals Formation–breakdown cycle of turbulent jets in a rotating fluid

2019 ◽  
Vol 868 ◽  
pp. 666-697 ◽  
Author(s):  
I. U. Atthanayake ◽  
P. Denissenko ◽  
Y. M. Chung ◽  
P. J. Thomas
Keyword(s):  
Turbulent Jets ◽  
Rossby Number ◽  
Rotating Fluid ◽  
Flow Conditions ◽  
Centrifugal Instability ◽  
Image Velocimetry ◽  
The Reformation ◽  
Internal Velocity ◽  
Time Periodic ◽  
Potential Origin

Results of comprehensive particle image velocimetry measurements investigating the dynamics of turbulent jets in a rotating fluid are presented. It is observed that background system rotation induces a time-periodic formation–breakdown cycle of the jets. The flow dynamics associated with this process is studied in detail. It is found that the frequency of the cycle increases linearly with the background rotation rate. The data show that the onset of the breakdown phase and of the reformation phase of the cycle can be characterized in terms of a local Rossby number employing an internal velocity and a length scale of the jet. The critical values for this local Rossby number, for onset of breakdown and reformation, scale linearly with a global Rossby number based on the flow conditions at the source. The analysis of the experimental data suggests centrifugal instability as the potential origin of the formation–breakdown cycle.

Linear stability analysis of two-way coupled particle-laden density current

2017 ◽  
Vol 95 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Pouriya Amini ◽  
Ehsan Khavasi ◽  
Navid Asadizanjani
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Base Flow ◽  
Interfacial Instability ◽  
Linear Stability Theory ◽  
Flow Density ◽  
Density Current ◽  
Density Currents

Stability of two-way coupled particle-laden density current is studied with the aim of linear stability analysis. Interfacial instability can be found in density currents, which effects entrainment and the rate of effective mixing. In this paper, we investigate the density current interfacial instability using linear stability theory, considering the particles attendance. The ultimate goal is to extract the governing equation for current with particles and study the effect of different parameters on stability of such currents. Base flow has velocity and density profiles of tangent hyperbolic type. Main current and particles are studied in two separate phases. It is found that current will be more stable as M0 (M0 = S∗N∗/ρ∗ where ρ∗ is the non-dimensional flow density, S∗ is the Stokes’ drag coefficient, and N∗ is the particles’ number density) grows, this is a result of number of particles and their radius, and also viscosity effects. The current is more stable as the growth rate increases. As the Richardson number in M0 rises, the growth rate value decreases. As the slope of the river bed increases, the current is less stable.

On the stability of slowly varying flow between concentric cylinders

1977 ◽  
Vol 355 (1681) ◽  
pp. 209-224 ◽  
Keyword(s):  
Growth Rate ◽  
Outer Cylinder ◽  
Low Frequency ◽  
Base Flow ◽  
Taylor Number ◽  
Sinusoidal Modulation ◽  
Concentric Cylinders ◽  
The Stability

We have applied the W.K.B. type of approximation to the stability of flow between concentric cylinders when the speed of rotation is varying slowly. For the case of a fixed outer cylinder we have shown that if the speed of the inner cylinder is increasing slowly then the growth rate for an axisymmetric disturbance is reduced considerably compared with that calculated by using a steady base flow. This leads to an increase of the Taylor number at which growth would first occur of the order of 20 % in cases where the theory seems applicable. We have also obtained results for a small sinusoidal modulation of low frequency to the inner cylinder velocity. These confirm Hall’s (1975) results of slight destabilization, almost independent of the frequency.

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