On the stability of circular Couette flow with radial heating

1990 ◽  
Vol 220 ◽  
pp. 53-84 ◽  
Author(s):  
Mohamed Ali ◽  
P. D. Weidman
Keyword(s):  
Linear Stability ◽  
Couette Flow ◽  
Outer Cylinder ◽  
Boussinesq Approximation ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Critical Stability ◽  
Circular Couette Flow ◽  
Prandtl Numbers ◽  
The Stability

The stability of circular Couette flow with radial heating across a vertically oriented annulus with inner cylinder rotating and outer cylinder stationary is investigated using linear stability theory. Infinite aspect ratio and constant fluid properties are assumed and critical stability boundaries are calculated for a conduction-regime base flow. Buoyancy is included through the Boussinesq approximation and stability is tested with respect to both toroidal and helical disturbances of uniform wavenumber. Symmetries of the linearized disturbance equations based on the sense of radial heating and the sense of cylinder rotation and their effect on the kinematics and morphology of instability waveforms are presented. The numerical investigation is primarily restricted to radius ratios 0.6 and 0.959 at Prandtl numbers 4.35, 15 and 100. The results follow the development of critical stability from Taylor cells at zero heating through a number of asymmetric modes to axisymmetric cellular convection at zero rotation. Increasing the Prandtl number profoundly destabilizes the flow in both wide and narrow gaps and the number of contending critical modes increases with increasing radius ratio. Specific calculations made to compare with the stability measurements of Snyder & Karlsson (1964) and Sorour & Coney (1979) exhibit good agreement considering the idealizations built into the linear stability analysis.

Linear stability of modulated circular Couette flow

1976 ◽  
Vol 75 (4) ◽  
pp. 625-646 ◽  
Author(s):  
P. J. Riley ◽  
R. L. Laurence
Keyword(s):  
Linear Stability ◽  
Couette Flow ◽  
Floquet Theory ◽  
Outer Cylinder ◽  
Time Dependent ◽  
Narrow Gap ◽  
Circular Couette Flow ◽  
Steady Rotation ◽  
The Stability ◽  
Axisymmetric Disturbances

The linear stability of modulated circular Couette flow to axisymmetric disturbances is examined in the narrow-gap limit. The outer cylinder is assumed stationary, while the inner is modulated both with and without a mean rotation. The equations governing the disturbance motion are solved by a Galerkin expansion with time-dependent coefficients, and the stability of the motion determined by Floquet theory. Modulation is found, in general, to destabilize the flow due to steady rotation, although weak stabilization is found for some modulation amplitudes at intermediate frequencies.

Stability of the Prandtl model for katabatic slope flows

2019 ◽  
Vol 865 ◽  
Author(s):  
Cheng-Nian Xiao ◽  
Inanc Senocak
Keyword(s):  
Linear Stability ◽  
Stability Theory ◽  
Slope Angle ◽  
Transverse Mode ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Vortical Flow ◽  
Longitudinal Modes ◽  
Prandtl Model ◽  
The Stability

We investigate the stability of the Prandtl model for katabatic slope flows using both linear stability theory and direct numerical simulations. Starting from Prandtl’s analytical solution for uniformly cooled laminar slope flows, we use linear stability theory to identify the onset of instability and features of the most unstable modes. Our results show that the Prandtl model for parallel katabatic slope flows is prone to transverse and longitudinal modes of instability. The transverse mode of instability manifests itself as stationary vortical flow structures aligned in the along-slope direction, whereas the longitudinal mode of instability emerges as waves propagating in the base-flow direction. Beyond the stability limits, these two modes of instability coexist and form a complex flow structure crisscrossing the plane of flow. The emergence of a particular form of these instabilities depends strongly on three dimensionless parameters, which are the slope angle, the Prandtl number and a newly introduced stratification perturbation parameter, which is proportional to the relative importance of the disturbance to the background stratification due to the imposed surface buoyancy flux. We demonstrate that when this parameter is sufficiently large, then the stabilising effect of the background stratification can be overcome. For shallow slopes, the transverse mode of instability emerges despite meeting the Miles–Howard stability criterion of $Ri>0.25$. At steep slope angles, slope flow can remain linearly stable despite attaining Richardson numbers as low as $3\times 10^{-3}$.

Energy stability of modulated circular Couette flow

1977 ◽  
Vol 79 (3) ◽  
pp. 535-552 ◽  
Author(s):  
Peter J. Riley ◽  
Robert L. Laurence
Keyword(s):  
Couette Flow ◽  
Nonlinear Stability ◽  
Energy Analysis ◽  
Outer Cylinder ◽  
Reynolds Numbers ◽  
Initial Energy ◽  
Critical Parameters ◽  
Narrow Gap ◽  
Circular Couette Flow ◽  
The Stability

The stability of circular Couette flow when the outer cylinder is at rest and the inner is modulated both with and without a mean shear is examined in the narrow-gap limit. Disturbances are assumed to be axisymmetric. Two criteria are used to determine conditions for stability; the first requires that the motion be strongly stable, the second only that disturbances of arbitrary initial energy decay from cycle to cycle. The behaviour of critical parameters as a function of frequency is similar for the linear and the energy analysis. The range of Reynolds numbers bounded above by certain instability and below by conditional nonlinear stability is enlarged by modulation.

scholarly journals Complex eigenvalues for the stability of Couette flow

1984 ◽  
Vol 396 (1810) ◽  
pp. 75-94 ◽  
Keyword(s):  
Eigenvalue Problem ◽  
Linear Stability ◽  
Couette Flow ◽  
Perturbation Methods ◽  
Stability Boundary ◽  
Outer Cylinder ◽  
Concentric Cylinders ◽  
Complex Eigenvalues ◽  
The Stability ◽  
Axisymmetric Disturbances

The eigenvalue problem for the linear stability of Couette flow between rotating concentric cylinders to axisymmetric disturbances is considered. It is shown by numerical calculations and by formal perturbation methods that when the outer cylinder is at rest there exist complex eigenvalues corresponding to oscillatory damped disturbances. The structure of the first few eigenvalues in the spectrum is discussed. The results do not contradict the ‘principle of exchange of stabilities’; namely, for a fixed axial wavenumber the first mode to become unstable as the speed of the inner cylinder is increased is non-oscillatory as the stability boundary is crossed.

Optimal Disturbances in Three-Dimensional Natural Convection Flows

2012 ◽  
Author(s):  
Elia Merzari ◽  
Paul Fischer ◽  
W. David Pointer
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Natural Circulation ◽  
Three Dimensional ◽  
Boussinesq Approximation ◽  
Spectral Element ◽  
Base Flow ◽  
The Stability ◽  
Optimal Disturbances

Buoyancy-driven systems are subject to several types of flow instabilities. To evaluate the performance of such systems it is becoming increasingly crucial to be able to predict the stability of a given base flow configuration. Traditional Modal Linear stability Analysis requires the solution of very large eigenvalue systems for three-dimensional flows, which make this problem difficult to tackle. An alternative to modal Linear stability Analysis is the use of adjoint solvers [1] in combination with a power iteration [2]. Such methodology allows for the identification of an optimal disturbance or forcing and has been recently used to evaluate the stability of several isothermal flow systems [2]. In this paper we examine the extension of the methodology to non-isothermal flows driven by buoyancy. The contribution of buoyancy in the momentum equation is modeled through the Boussinesq approximation. The method is implemented in the spectral element code Nek5000. The test case is the flow is a two-dimensional cavity with differential heating and conductive walls and the natural circulation flow in a toroidal thermosiphon.

Linear stability analysis of cylindrical Rayleigh–Bénard convection

2012 ◽  
Vol 711 ◽  
pp. 27-39 ◽  
Author(s):  
Bo-Fu Wang ◽  
Dong-Jun Ma ◽  
Cheng Chen ◽  
De-Jun Sun
Keyword(s):  
Stability Analysis ◽  
Prandtl Number ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Axisymmetric Flow ◽  
Base Flow ◽  
Prandtl Numbers ◽  
The Stability ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractThe instabilities and transitions of flow in a vertical cylindrical cavity with heated bottom, cooled top and insulated sidewall are investigated by linear stability analysis. The stability boundaries for the axisymmetric flow are derived for Prandtl numbers from 0.02 to 1, for aspect ratio $A$ ($A= H/ R= \mathrm{height} / \mathrm{radius} $) equal to 1, 0.9, 0.8, 0.7, respectively. We found that there still exists stable non-trivial axisymmetric flow beyond the second bifurcation in certain ranges of Prandtl number for $A= 1$, $0. 9$ and 0.8, excluding the $A= 0. 7$ case. The finding for $A= 0. 7$ is that very frequent changes of critical mode (azimuthal Fourier mode) of the second bifurcation occur when the Prandtl number is changed, where five kinds of steady modes $m= 1, 2, 8, 9, 10$ and three kinds of oscillatory modes $m= 3, 4, 6$ are presented. These multiple modes indicate different flow structures triggered at the transitions. The instability mechanism of the flow is explained by kinetic energy transfer analysis, which shows that the radial or axial shear of base flow combined with buoyancy mechanism leads to the instability results.

scholarly journals On the linear stability of compressible plane Couette flow

1994 ◽  
Vol 258 ◽  
pp. 131-165 ◽  
Author(s):  
Peter W. Duck ◽  
Gordon Erlebacher ◽  
M. Yousuff Hussaini
Keyword(s):  
Linear Stability ◽  
Couette Flow ◽  
Small Amplitude ◽  
Reynolds Numbers ◽  
Inviscid Limit ◽  
Plane Couette Flow ◽  
Moving Wall ◽  
Type Boundary ◽  
The Stability ◽  
Radiation Type

The linear stability of compressible plane Couette flow is investigated. The appropriate basic velocity and temperature distributions are perturbed by a small-amplitude normal-mode disturbance. The full small-amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in some detail. It is found that instabilities can occur, although the corresponding growth rates are often quite small; the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wave speed of the disturbances approaches the velocity of either of the walls, and these regimes are also analysed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.

scholarly journals Decomposition of the temporal growth rate in linear instability of compressible gas flows

2015 ◽  
Vol 778 ◽  
pp. 120-132 ◽  
Author(s):  
Mario Weder ◽  
Michael Gloor ◽  
Leonhard Kleiser
Keyword(s):  
Growth Rate ◽  
Energy Balance ◽  
Linear Stability ◽  
Couette Flow ◽  
Stability Theory ◽  
Compressible Flows ◽  
Linear Instability ◽  
Linear Stability Theory ◽  
Gas Flows ◽  
Temporal Growth Rate

We present a decomposition of the temporal growth rate ${\it\omega}_{i}$ which characterises the evolution of wave-like disturbances in linear stability theory for compressible flows. The decomposition is based on the disturbance energy balance by Chu (Acta Mech., vol. 1 (3), 1965, pp. 215–234) and provides terms for production, dissipation and flux of energy as components of ${\it\omega}_{i}$. The inclusion of flux terms makes our formulation applicable to unconfined flows and flows with permeable or vibrating boundaries. The decomposition sheds light on the fundamental mechanisms determining temporal growth or decay of disturbances. The additional insights gained by the proposed approach are demonstrated by an investigation of two model flows, namely compressible Couette flow and a plane compressible jet.

Three-dimensional instability of axisymmetric buoyant convection in cylinders heated from below

1996 ◽  
Vol 326 ◽  
pp. 399-415 ◽  
Author(s):  
M. Wanschura ◽  
H. C. Kuhlmann ◽  
H. J. Rath
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Onset Of Convection ◽  
Buoyant Convection ◽  
Thermal Boundary Conditions ◽  
Aspect Ratios ◽  
Prandtl Numbers

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

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