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.

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.

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.

Stability of modulated finite-gap cylindrical Couette flow: linear theory

1981 ◽  
Vol 108 ◽  
pp. 19-42 ◽  
Author(s):  
S. Carmi ◽  
J. I. Tustaniwskyj
Keyword(s):  
Unsteady Flow ◽  
Couette Flow ◽  
Floquet Theory ◽  
Analytic Solution ◽  
Outer Cylinder ◽  
Three Dimensional ◽  
Linear Perturbation ◽  
Cylindrical Couette Flow ◽  
The Stability ◽  
The Galerkin Method

The linear stability of an extensively modulated cylindrical Couette flow is investigated in the finite-gap range. A closed form analytic solution is obtained for the basic unsteady flow after modulation is introduced through the boundary conditions. The general linear perturbation equations for three-dimensional disturbances are then derived and subsequently solved using the Galerkin method with the stability analysed by the Floquet theory. Modulation is found to destabilize the flow in most cases and results compare very favourably with the ones obtained experimentally. Stabilization is possible only for some cases of outer cylinder modulation.

The effect of viscous heating on the stability of Taylor–Couette flow

2002 ◽  
Vol 462 ◽  
pp. 111-132 ◽  
Author(s):  
U. A. AL-MUBAIYEDH ◽  
R. SURESHKUMAR ◽  
B. KHOMAMI
Keyword(s):  
Stability Analysis ◽  
Secondary Flow ◽  
Linear Stability ◽  
Couette Flow ◽  
Temperature Difference ◽  
Time Dependent ◽  
Viscous Heating ◽  
Taylor Couette ◽  
The Stability ◽  
Taylor Couette Flow

The influence of viscous heating on the stability of Taylor–Couette flow is investigated theoretically. Based on a linear stability analysis it is shown that viscous heating leads to significant destabilization of the Taylor–Couette flow. Specifically, it is shown that in the presence of viscous dissipation the most dangerous disturbances are axisymmetric and that the temporal characteristic of the secondary flow is very sensitive to the thermal boundary conditions. If the temperature difference between the two cylinders is small, the secondary flow is stationary as in the case of isothermal Taylor–Couette flow. However, when the temperature difference between the two cylinders is large, time-dependent secondary states are predicted. These linear stability predictions are in agreement with the experimental observations of White & Muller (2000) in terms of onset conditions as well as the spatiotemporal characteristics of the secondary flow. Nonlinear stability analysis has revealed that over a broad range of operating conditions, the bifurcation to the time-dependent secondary state is subcritical, while stationary states result as a consequence of supercritical bifurcation. Moreover, the supercritically bifurcated stationary state undergoes a secondary bifurcation to a time-dependent flow. Overall, the structure of the time-dependent state predicted by the analysis compares very well with the experimental observations of White & Muller (2000) that correspond to slowly moving vortices parallel to the cylinder axis. The significant destabilization observed in the presence of viscous heating arises as the result of the coupling of the perturbation velocity and the base-state temperature gradient that gives rise to fluctuations in the radial temperature distribution. Due to the thermal sensitivity of the fluid these fluctuations greatly modify the fluid viscosity and reduce the dissipation of disturbances provided by the viscous stress terms in the momentum equation.

Tertiary solutions and their stability in rotating plane Couette flow

1998 ◽  
Vol 358 ◽  
pp. 357-378 ◽  
Author(s):  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Stability Boundary ◽  
Streamwise Vortex ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Plane Couette Flow ◽  
Streamwise Direction ◽  
The Stability ◽  
Oscillatory Instabilities

The stability of nonlinear tertiary solutions in rotating plane Couette flow is examined numerically. It is found that the tertiary flows, which bifurcate from two-dimensional streamwise vortex flows, are stable within a certain range of the rotation rate when the Reynolds number is relatively small. The stability boundary is determined by perturbations which are subharmonic in the streamwise direction. As the Reynolds number is increased, the rotation range for the stable tertiary motions is destroyed gradually by oscillatory instabilities. We expect that the tertiary flow is overtaken by time-dependent motions for large Reynolds numbers. The results are compared with the recent experimental observation by Tillmark & Alfredsson (1996).

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.

Nonlinear dynamics of viscoelastic Taylor–Couette flow: effect of elasticity on pattern selection, molecular conformation and drag

2009 ◽  
Vol 620 ◽  
pp. 353-382 ◽  
Author(s):  
D. G. THOMAS ◽  
B. KHOMAMI ◽  
R. SURESHKUMAR
Keyword(s):  
Shear Rate ◽  
Linear Stability ◽  
Couette Flow ◽  
Drag Force ◽  
Molecular Conformation ◽  
Vortex Structures ◽  
Time Dependent ◽  
Pattern Selection ◽  
Stability Threshold ◽  
Flow Transitions

Three-dimensional and time-dependent simulations of viscoelastic Taylor–Couette flow of dilute polymer solutions are performed using a fully implicit parallel spectral time-splitting algorithm to discover flow patterns with various spatio-temporal symmetries, namely rotating standing waves (RSWs), disordered oscillations (DOs) and solitary vortex structures referred to as oscillatory strips (OSs) and diwhirls (DWs). A detailed account of the impact of flow transitions on molecular conformation and viscoelastic stress, velocity profiles, hydrodynamic drag force and energy spectra of time-dependent flow states is presented. Overall, predicted pattern selection and flow features compare very favourably with experimental observations. For elasticity number E, that signifies the ratio of elastic to viscous forces, >0.1, and when the shear rate (cylinder rotation speed) is increased above the linear stability threshold, the circular Couette flow (CCF) becomes unstable to RSWs which are characterized by a checkerboard-like pattern in the space–time plot of radial velocity, implying symmetry between inflow/outflow (I/O) regions. As the shear rate is further increased, perturbations that break the I/O symmetry are amplified leading to DOs and/or flame-like patterns with spectral mechanical energy transfer reminiscent of elastically induced low-Reynolds-number turbulence. However, when the shear rate is decreased from those at which such chaotic states are observed, the radially inward acting polymer body force created by flow-induced molecular stretching causes the development of narrow inflow regions surrounded by much broader weak outflow domains. This promotes the formation of solitary vortex structures, which can be stationary and axisymmetric (DWs) or time-dependent (OSs). The dynamics of the formation of these structures by merging and coalescence of vortex pairs and the implication of such events on instantaneous hydrodynamic force are studied. For O(1) values of E, OSs and DWs appear approximately at constant values of the We, defined as the ratio of polymer relaxation time to the inverse shear rate in the gap. As shear rate is decreased further, DWs decay to CCF although at We values less than the linear stability threshold. The flow transitions are hysteretic with respect to We, as evidenced by a plot of drag force versus We.

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