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.

Centrifugal instability in non-axisymmetric vortices

2015 ◽  
Vol 769 ◽  
pp. 26-45 ◽  
Author(s):  
David Nagarathinam ◽  
A. Sameen ◽  
Manikandan Mathur
Keyword(s):  
Aspect Ratio ◽  
Growth Rates ◽  
Axial Velocity ◽  
Local Stability ◽  
Axial Flow ◽  
Sufficient Criterion ◽  
Concentration Parameter ◽  
Centrifugal Instability ◽  
Axisymmetric Vortex ◽  
New Criterion

We study the centrifugal instability of non-axisymmetric vortices in the presence of an axial flow ($w$) and a background rotation (${\it\Omega}_{z}$) using the local stability approach. Analytically solving the local stability equations for an axisymmetric vortex with $w$ and ${\it\Omega}_{z}$, growth rates for wave vectors that are periodic upon evolution around a closed streamline are calculated. The resulting sufficient criterion for centrifugal instability in an axisymmetric vortex is then heuristically extended to non-axisymmetric vortices and written in terms of integral quantities and their derivatives with respect to the streamfunction on a streamline. The new criterion for non-axisymmetric vortices, which converges to the exact criterion of Bayly (Phys. Fluids, vol. 31, 1988, pp. 56–64) in the absence of background rotation and axial flow, is validated by comparisons with numerically calculated growth rates for two different anticyclonic vortices: the Stuart vortex (specified by the concentration parameter ${\it\rho},~0<{\it\rho}\leqslant 1$) and the Taylor–Green vortex (specified by the aspect ratio $E,~0<E\leqslant 1$). With no axial velocity and finite background rotation, the criterion predicts a lower and an upper threshold of $|{\it\Omega}_{z}|$ between which centrifugal instability is present. We further demonstrate that the criterion represents an improvement over the criterion of Sipp & Jacquin (Phys. Fluids, vol. 12, 2000, pp. 1740–1748). Finally, in the presence of both axial velocity and background rotation, the criterion is shown to be accurate for large enough ${\it\rho}$ and $E$.

The instability nature of the Vogel–Escudier flow

2015 ◽  
Vol 766 ◽  
pp. 590-610 ◽  
Author(s):  
Miguel A. Herrada ◽  
Vladimir N. Shtern ◽  
M. M. Torregrosa
Keyword(s):  
Shear Layer ◽  
Axial Velocity ◽  
Inflection Point ◽  
Critical Reynolds Number ◽  
Axisymmetric Flow ◽  
Rotating Disk ◽  
Base Flow ◽  
Cylinder Length ◽  
Centrifugal Instability ◽  
Circulatory Flow

AbstractThe instability of the steady axisymmetric flow in a sealed elongated cylinder, driven by a rotating end disk, is studied with the help of numerical simulations. It is argued that this instability is of the shear-layer type, being caused by the presence of an inflection point in the radial distribution of axial velocity of the base circulatory flow. The disturbance kinetic energy is localized in both the radial and axial directions, reaching its peak near the rotating disk, where the magnitude of base-flow axial velocity is close to its maximum. The critical Reynolds number, $\mathit{Re}_{cr}$, is found to be nearly $h$-independent for $h>5$; $h$ is the cylinder length-to-radius ratio. It is shown that the sidewall co-rotation suppresses the instability. As the co-rotation increases, the centrifugal instability becomes the most dangerous, i.e. determines $\mathit{Re}_{cr}$. Physical explanations are given for the stabilizing effect of the co-rotation, which is stronger (weaker) for the shear-layer (centrifugal) instability.

scholarly journals Diffusive effects in local instabilities of a baroclinic axisymmetric vortex

2021 ◽  
Vol 928 ◽  
Author(s):  
Suraj Singh ◽  
Manikandan Mathur
Keyword(s):  
Local Stability ◽  
Base Flow ◽  
Mode Analysis ◽  
Neutral Stability ◽  
Curvature Effects ◽  
Axisymmetric Vortex ◽  
Out Of Plane ◽  
Double Diffusive ◽  
Symmetric Instability ◽  
Oscillatory Instabilities

We present a local stability analysis of an idealized model of the stratified vortices that appear in geophysical settings. The base flow comprises an axisymmetric vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in the thermal wind balance with the out-of-plane momentum gradient. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on the Schmidt number $Sc$ (ratio between momentum and mass diffusivities) is studied, in the presence of curvature effects. In the diffusion-free limit, the well-known symmetric instability is recovered. In the viscous, double-diffusive regime, instability characteristics are shown to depend on three non-dimensional parameters (including $Sc$ ), and two different instabilities are identified: (i) a monotonic instability (same as symmetric instability at $Sc = 1$ ), and (ii) an oscillatory instability (absent at $Sc = 1$ ). Separating the base flow and perturbation characteristics, two each of base flow and perturbation parameters (apart from $Sc$ ) are identified, and the entire parameter space is explored for the aforementioned instabilities. In comparison with $Sc = 1$ , monotonic and oscillatory instabilities are shown to significantly expand the instability region in the space of base flow parameters as $Sc$ moves away from unity. Neutral stability boundaries on the plane of $Sc$ and a modified gradient Richardson number are then identified for both these instabilities. In the absence of curvature effects, our results are shown to be consistent with previous studies based on normal mode analysis, thus establishing that the local stability approach is well suited to capturing symmetric and double-diffusive instabilities. The paper concludes with a discussion of curvature effects, and the likelihood of monotonic and oscillatory instabilities in typical oceanic settings.

Decaying, Swirling Flow in a Pipe: Axial Velocity Field Development With Reversed Axial Flow

2003 ◽  
Author(s):  
Heather L. McClusky ◽  
Donald E. Beasley
Keyword(s):  
Velocity Field ◽  
Axial Velocity ◽  
Vortex Breakdown ◽  
Swirling Flow ◽  
Axial Flow ◽  
Axial Direction ◽  
Velocity Profiles ◽  
Field Development ◽  
Constant Diameter ◽  
Spatial Oscillations

Streamwise development of the axial velocity field in a confined, decaying swirling flow is explored in the present study. A tangential injection mechanism produces swirling flow at the inlet of a constant diameter pipe. Particle image velocimetry is employed for axial velocity measurements. Representative axial velocity profiles are presented for axial locations of 3 to 67 pipe diameters. The axial velocity profiles are asymmetric relative to the pipe centerline and the asymmetries persist as the flow develops in the axial direction. The eccentricity of the swirling flow spatially oscillates as the flow develops in the axial direction. The spatial oscillations of the axial velocity suggest that a vortex breakdown may be located near the inlet of the pipe and the entire pipe is the wake region of the vortex breakdown. The centerline axial velocity is also used to document the axial development of the flow. A comprehensive view of the flow field is provided by considering theoretical explanations presented in the literature for decaying, swirling pipe flows and for vortex breakdown.

Spin-up from rest of a compressible fluid in a rapidly rotating cylinder

1992 ◽  
Vol 237 ◽  
pp. 413-434 ◽  
Author(s):  
Jae Min Hyun ◽  
Jun Sang Park
Keyword(s):  
Mach Number ◽  
Compressible Fluid ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Direction ◽  
Ekman Layer ◽  
Infinite Cylinder ◽  
Full Time ◽  
Viscous Effects ◽  
Initial State

Spin-up flows of a compressible gas in a finite, closed cylinder from an initial state of rest are studied, The flow is characterized by small reference Ekman numbers, and the peripheral Mach number is O(1). Comprehensive numerical solutions have been obtained for the full, time-dependent compressible Navier-Stokes equations. The details of the flow, temperature, and density evolution are described. In the early phase of spin-up, owing to the thermoacoustic disturbances caused by the compressible Rayleigh effect, the flows are oscillatory, and this oscillatory behaviour is pronounced at higher Mach numbers. The principal dynamical role of the Ekman layer is dominant over moderate times of orders of the homogeneous spin-up timescales. Owing to the density stratification in the radial direction, the Ekman layer is thicker in the central region of the interior. The interior azimuthal flows are mainly uniform in the axial direction. As the Mach number increases, the rate of spin-up in the interior becomes slower, and the propagating shear front is more diffusive. Explicit comparisons with the results for an infinite cylinder are made to ascertain the contributions of the endwall disks. In contrast to the usual incompressible spin-up from rest, the viscous effects are relatively more important for the case of a compressible fluid.

The Elimination of Wall Effects in Axial-Flow Compressor Stages

1953 ◽  
Vol 57 (508) ◽  
pp. 241-243
Author(s):  
J. M. Stephenson
Keyword(s):  
Velocity Profile ◽  
Axial Velocity ◽  
Radial Profile ◽  
Axial Flow ◽  
Gas Velocity ◽  
Wall Effects ◽  
Axial Velocity Component ◽  
Axial Velocity Profile ◽  
Flow Compressor ◽  
Work Done

Compressor stages are usually designed on the assumption that the gas velocity is nowhere affected by the friction at the walls. The only way in which viscosity is taken into account is in the assumed efficiency, and in a guessed “work-done factor,” which ensures that by aiming high the required work is actually attained.It is known that the radial profile of the axial velocity component becomes more and more peaked through successive stages of a compressor, so that the assumptions just quoted become very inaccurate. It is possible that the efficiency of a stage could be raised considerably if the axial velocity profile were controlled; moreover up to 20 per cent. more work could be done if a “ work-done factor ” did not have to be applied.

Compressor Design

1953 ◽  
Vol 57 (511) ◽  
pp. 463-463
Author(s):  
R. G. Taylor
Keyword(s):  
Velocity Profile ◽  
Axial Velocity ◽  
Axial Flow ◽  
Technical Note ◽  
Basic Design ◽  
Wall Effects ◽  
Axial Flow Compressor ◽  
Axial Velocity Profile ◽  
Compressor Design ◽  
Flow Compressor

In Mr. J. M. Stephenson's Technical Note, “ The Elimination of Wall Effects in Axial-Flow Compressor Stages,” in the April 1953 issue of the Journal, the author suggests that the blade rows of an axial flow compressor are so closely spaced as to ensure that the axial velocity profile is unchanged across the rows. Whether this statement is correct or not such an assumption regarding the axial velocity profile is a basic design condition and when made it will not leave any flexibility in the choice of the function f(r).

Similarity solutions for viscous vortex cores

1992 ◽  
Vol 238 ◽  
pp. 487-507 ◽  
Author(s):  
Ernst W. Mayer ◽  
Kenneth G. Powell
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Flow ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Axial Pressure Gradient ◽  
Self Similar ◽  
Similar Solutions

Results are presented for a class of self-similar solutions of the steady, axisymmetric Navier–Stokes equations, representing the flows in slender (quasi-cylindrical) vortices. Effects of vortex strength, axial gradients and compressibility are studied. The presence of viscosity is shown to couple the parameters describing the core growth rate and the external flow field, and numerical solutions show that the presence of an axial pressure gradient has a strong effect on the axial flow in the core. For the viscous compressible vortex, near-zero densities and pressures and low temperatures are seen on the vortex axis as the strength of the vortex increases. Compressibility is also shown to have a significant influence upon the distribution of vorticity in the vortex core.

Forced Flow of Vapor Condensing Over a Horizontal Plate (Problem of Cess and Koh): Steady and Unsteady Solutions of the Full 2D Problem

2010 ◽  
Vol 132 (10) ◽  
Author(s):  
S. Kulkarni ◽  
A. Narain ◽  
S. Mitra
Keyword(s):  
Numerical Solutions ◽  
Saturated Vapor ◽  
Threshold Value ◽  
Steady Solution ◽  
Forced Flow ◽  
Film Condensation ◽  
Horizontal Plate ◽  
Governing Equations ◽  
Long Time ◽  
Steady Solutions

Accurate steady and unsteady numerical solutions of the full 2D governing equations—which model the forced film condensation flow of saturated vapor over a semi-infinite horizontal plate (the problem of Cess and Koh)—are obtained over a range of flow parameters. The results presented here are used to better understand the limitations of the well-known similarity solutions given by Koh. It is found that steady/quasisteady filmwise solution exists only if the inlet speed is above a certain threshold value. Above this threshold speed, steady/quasisteady film condensation solutions exist and their film thickness variations are approximately the same as the similarity solution given by Koh. However, these steady solutions differ from the Koh solution regarding pressure variations and associated effects in the leading part of the plate. Besides results based on the solutions of the full steady governing equations, this paper also presents unsteady solutions that characterize the steady solutions’ attainability, stability (response to initial disturbances), and their response to ever-present minuscule noise on the condensing-surface. For this shear-driven flow, the paper finds that if the uniform vapor speed is above a threshold value, an unsteady solution that begins with any reasonable initial-guess is attracted in time to a steady solution. This long time limiting solution is the same—within computational errors—as the solution of the steady problem. The reported unsteady solutions that yield the steady solution in the long time limit also yield “attraction rates” for nonlinear stability analysis of the steady solutions. The attraction rates are found to diminish gradually with increasing distance from the leading edge and with decreasing inlet vapor speed. These steady solutions are generally found to be stable to initial disturbances on the interface as well as in any flow variable in the interior of the flow domain. The results for low vapor speeds below the threshold value indicate that the unsteady solutions exhibit nonexistence of any steady limit of filmwise flow in the aft portion of the solution. Even when a steady solution exists, the flow attainability is also shown to be difficult (because of waviness and other sensitivities) at large downstream distances.

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.

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