The influence of initial condition on the linear stability of time-dependent circular Couette flow

1985 ◽  
Vol 28 (2) ◽  
pp. 749 ◽  
Author(s):  
J.-C. Chen ◽  
G. P. Neitzel ◽  
D. F. Jankowski
Keyword(s):  
Linear Stability ◽  
Couette Flow ◽  
Time Dependent ◽  
Initial Condition ◽  
Circular Couette Flow

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.

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.

Linear stability of a circular Couette flow under a radial thermoelectric body force

2015 ◽  
Vol 91 (3) ◽  
Author(s):  
H. N. Yoshikawa ◽  
A. Meyer ◽  
O. Crumeyrolle ◽  
I. Mutabazi
Keyword(s):  
Linear Stability ◽  
Couette Flow ◽  
Body Force ◽  
Circular Couette Flow

Stratified circular Couette flow: instability and flow regimes

1995 ◽  
Vol 292 ◽  
pp. 333-358 ◽  
Author(s):  
B. M. Boubnov ◽  
E. B. Gledzer ◽  
E. J. Hopfinger
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Couette Flow ◽  
Flow Instability ◽  
Buoyancy Frequency ◽  
Vertical Wavelength ◽  
Taylor Vortices ◽  
Circular Couette Flow ◽  
Vortex Size ◽  
Gap Width

The stability conditions of the flow between two concentric cylinders with the inner one rotating (circular Couette flow) have been investigated experimentally and theoretically for a fluid with axial, stable linear density stratification. The behaviour of the flow, therefore, depends on the Froude number Fr = Ω/N (where Ω is the angular velocity of the inner cylinder and N is the buoyancy frequency of the fluid) in addition to the Reynolds number and the non-dimensional gap width ε, here equal to 0.275.Experiments show that stratification has a stabilizing effect on the flow with the critical Reynolds number depending on N, in agreement with linear stability theory. The selected, most amplified, vertical wavelength at onset of instability is reduced by the stratification effect and is for the geometry considered only about half the gap width. Furthermore, the observed instability is non-axisymmetric. The resulting vortex motion causes some mixing and this leads to layer formation, clearly visible on shadowgraph images, with the height of the layer being determined by the vertical vortex size. This regime of vertically reduced vortex size is referred to as the S-regime.For larger Reynolds and Froude numbers the role of stratification decreases and the most amplified vertical wavelength is determined by the gap width, giving rise to the usual Taylor vortices (we call this the T-regime). The layers which emerge are determined by these vortices. For relatively small Reynolds number when Fr ≈ 1 the Taylor vortices are stable and the layers have a height h equal to the gap width; for larger Reynolds number or Fr ≈ 2 the Taylor vortices interact in pairs (compacted Taylor vortices, regime CT) and layers of twice the gap width are predominant. Stratification inhibits the azimuthal wavy vortex flow observed in homogeneous fluid. By further increasing the Reynolds number, turbulent motions appear with Taylor vortices superimposed like in non-stratified fluid.The theoretical analysis is based on a linear stability consideration of the axisymmetric problem. This gives bounds of instability in the parameter space (Ω, N) for given vertical and radial wavenumbers. These bounds are in qualitative agreement with experiments. The possibility of oscillatory-type instability (overstability) observed experimentally is also discussed.

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.

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