scholarly journals Bifurcations and instabilities in sliding Couette flow

2011 ◽  
Vol 678 ◽  
pp. 156-178 ◽  
Author(s):  
K. DEGUCHI ◽  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Stability Result ◽  
Axial Direction ◽  
Travelling Wave ◽  
Finite Amplitude ◽  
Axisymmetric Case ◽  
Sliding Motion ◽  
Axisymmetric Solutions ◽  
The Stability ◽  
Axisymmetric Perturbations

We carry out linear and nonlinear analyses on a flow between two infinitely long concentric cylinders with the radii a and b subject to a sliding motion of the inner cylinder in the axial direction. We confirm the linear stability result of Gittler (Acta Mechanica, vol. 101, 1993, p. 1) for the axisymmetric case, namely the flow is linearly stable against axisymmetric perturbations when the radius ratio η = a/b is greater than 0.1415. We extend his analysis to the non-axisymmetric case and find that the stability of the flow is still determined by axisymmetric perturbations. Our nonlinear analysis exhibits that (i) finite-amplitude axisymmetric solutions exist far below the linear critical Reynolds number for η < 0.1415 and (ii) non-axisymmetric travelling wave solutions appear abruptly at a finite Reynolds number even for η > 0.1415 where the linear critical state is absent.

A simple scenario of the laminar breakdown in liquid metal flows

2021 ◽  
Vol 57 (2) ◽  
pp. 191-210
Keyword(s):  
Reynolds Number ◽  
Electrical Potential ◽  
Finite Amplitude ◽  
Magnetic Reynolds Number ◽  
Small Length ◽  
Turbulent Transition ◽  
Local Perturbations ◽  
Magnetohydrodynamic Flows ◽  
The Stability ◽  
Velocity Pressure

In the article, authors present a numerical method for modelling a laminar-turbulent transition in magnetohydrodynamic flows. The small magnetic Reynolds number approach is considered. Velocity, pressure and electrical potential are decomposed to the sum of state values and finite amplitude perturbations. A solver based on the Nektar++ framework is described. The authors suggest using small-length local perturbations as a transition trigger. They can be imposed by blowing or by electrical enforcing. The stability of the Hartmann flow and the flow in the bend are considered as examples. Tables 4, Figs 19, Refs 28.

On the stability of pipe-Poiseuille flow to finite-amplitude axisymmetric and non-axisymmetric disturbances

1985 ◽  
Vol 158 ◽  
pp. 289-316 ◽  
Author(s):  
P. K. Sen ◽  
D. Venkateswarlu ◽  
S. Maji
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Higher Order ◽  
Equilibrium States ◽  
Minimum Threshold ◽  
Threshold Amplitude ◽  
The Stability ◽  
The One ◽  
Axisymmetric Disturbances

The stability of fully developed pipe-Poiseuille flow to finite-amplitude axisymmetric and non-axisymmetric disturbances has been studied using the equilibrium-amplitude method of Reynolds & Potter (1967). In both the cases the least-stable centre-modes were investigated. Also, for the non-axisymmetric case the mode investigated was the one with azimuthal wavenumber equal to one. Many higher-order Landau coefficients were calculated, and the Stuart-Landau series was analysed by the Shanks (1955) method and by using Padé approximants to look for the existence of possible equilibrium states. The results show in both cases that, for each value of the Reynolds number R, there is a preferred band of spatial wavenumbers α in which equilibrium states are likely to exist. Moreover, in both cases it was found that the magnitude of the minimum threshold amplitude for a given R decreases with increasing R. The scales of the various quantities obtained agree very well with those deduced by Davey & Nguyen (1971).

The effect of external forcing on the stability of plane Poiseuille flow

1978 ◽  
Vol 359 (1699) ◽  
pp. 453-478 ◽  
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Stable Solution ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Plane Poiseuille Flow ◽  
Critical Reynolds Numbers ◽  
Threshold Amplitude ◽  
The Stability

The stability of plane Poiseuille flow in a channel forced by a wavelike motion on one of the channel walls is investigated. The amplitude Є of this forcing is taken to be small. The most dangerous modes of forcing are identified and it is found in general the critical Reynolds number is changed by O (Є) 2 . However, we identify two particular modes of forcing which give rise to decrements of order Є 2/3 and Є in the critical Reynolds number. Some types of forcing are found to generate sub critical stable finite amplitude perturbations to plane Poiseuille flow. This contrasts with the unforced case where the only stable solution is the zero amplitude solution. The forcing also deforms the unstable subcritical limit cycle solution from its usual circular shape into a more complicated shape. This has an effect on the threshold amplitude ideas suggested by, for example, Meksyn & Stuart (1951). It is found that the phase of disturbances must also be considered when finding the amplitude dependent critical Reynolds numbers.

A numerical and experimental investigation of the stability of spiral Poiseuille flow

1981 ◽  
Vol 102 ◽  
pp. 101-126 ◽  
Author(s):  
Donald I. Takeuchi ◽  
Daniel F. Jankowski
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Concentric Cylinders ◽  
Axial Pressure Gradient ◽  
Secondary Motion ◽  
Wave Forms ◽  
Predicted Values ◽  
The Stability

The linear stability of the spiral motion induced between concentric cylinders by an axial pressure gradient and independent cylinder rotation is studied numerically and experimentally for a wide-gap geometry. A three-dimensional disturbance is considered. Linear stability limits in the form of Taylor numbers TaL are computed for the rotation ratios μ, = 0, 0·2, and -0·5 and for values of the axial Reynolds number Re up to 100. Depending on the values of μ and Re, the disturbance which corresponds to TaL can have a toroidal vortex structure or a spiral form. Aluminium-flake flow visualization is used to determine conditions for the onset of a secondary motion and its structure at finite amplitude. The experimental results agree with the predicted values of TaL for μ [ges ] 0, and low Reynolds number. For other cases in which agreement is only fair, apparatus length is shown to be a contributing influence. The comparison between experimental and predicted wave forms shows good agreement in overall trends.

scholarly journals Axisymmetric travelling waves in annular sliding Couette flow at finite and asymptotically large Reynolds number

2013 ◽  
Vol 720 ◽  
pp. 582-617 ◽  
Author(s):  
K. Deguchi ◽  
A. G. Walton
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Finite Amplitude ◽  
Vortical Flow ◽  
Navier Stokes ◽  
Large Reynolds Number ◽  
Equilibrium Solutions ◽  
Specific Flow

AbstractThe relationship between numerical finite-amplitude equilibrium solutions of the full Navier–Stokes equations and nonlinear solutions arising from a high-Reynolds-number asymptotic analysis is discussed for a Tollmien–Schlichting wave-type two-dimensional vortical flow structure. The specific flow chosen for this purpose is that which arises from the mutual axial sliding of co-axial cylinders for which nonlinear axisymmetric travelling-wave solutions have been discovered recently by Deguchi & Nagata (J. Fluid Mech., vol. 678, 2011, pp. 156–178). We continue this solution branch to a Reynolds number $R= 1{0}^{8} $ and confirm that the behaviour of its so-called lower branch solutions, which typically produce a smaller modification to the laminar state than the other solution branches, quantitatively agrees with the axisymmetric asymptotic theory developed in this paper. We further find that this asymptotic structure breaks down when the disturbance wavelength is comparable with $R$. The new structure which replaces it is investigated and the governing equations are derived and solved. The flow visualization of the resultant solutions reveals that they possess a streamwise localized structure, with the trend agreeing qualitatively with full Navier–Stokes solutions for relatively long-wavelength disturbances.

The Effect of Finite Amplitude Disturbance Magnitude on Departures From Laminar Conditions in Impulsively Started and Steady Pipe Entrance Flows

2001 ◽  
Vol 124 (1) ◽  
pp. 235-240 ◽  
Author(s):  
E. A. Moss ◽  
A. H. Abbot
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Shape Parameter ◽  
Critical Reynolds Number ◽  
Flow Instability ◽  
Finite Amplitude ◽  
Pipe Flows ◽  
Displacement Thickness ◽  
Layer Flow ◽  
The Stability

The aim of this study was to investigate first departures from laminar conditions in both impulsively started and steady pipe entrance flows. Wall shear stress measurements were conducted of transition in impulsively started pipe flows with large disturbances. These results were reconciled in a framework of displacement thickness Reynolds number and a velocity profile shape parameter, with existing measurements of pipe entrance flow instability, pipe-Poiseuille and boundary layer flow responses to large disturbances, and linear stability predictions. Limiting critical Reynolds number variations for each type of flow were thus inferred, corresponding to the small and gross disturbance limits respectively. Consequently, insights have been provided regarding the effect of disturbance levels on the stability of both steady and unsteady pipe flows.

Third-order resonance effects and the nonlinear stability of drop oscillations

1987 ◽  
Vol 183 ◽  
pp. 95-121 ◽  
Author(s):  
Ramesh Natarajan ◽  
Robert A. Brown
Keyword(s):  
Nonlinear Oscillations ◽  
Three Dimensional ◽  
Normal Modes ◽  
Travelling Wave ◽  
Finite Amplitude ◽  
Amplitude Equations ◽  
Third Order ◽  
Resonant Coupling ◽  
Resonance Effects ◽  
Axisymmetric Perturbations

The three-dimensional nonlinear oscillations of an isolated, inviscid drop with surface tension are studied by a multiple timescale analysis and pre-averaging applied to the variational principle for the appropriate Lagrangian. Amplitude equations are derived which describe the generic cubic resonance caused by the spatial degeneracy of the eigenfrequencies of the linear normal modes. This resonant coupling leads to the instability of the finite amplitude axisymmetric oscillations to small non-axisymmetric perturbations, as is demonstrated here for the three-and four-lobed normal modes. Solutions to the interaction equations that describe finite amplitude, non-axisymmetric travelling-wave solutions are also obtained and their stability is investigated. A non-generic cubic resonance between the two-lobed and four-lobed oscillatory modes leads to quasi-periodic motions.

scholarly journals On the stability of the boundary layer at the bottom of propagating surface waves

2021 ◽  
Vol 928 ◽  
Author(s):  
Paolo Blondeaux ◽  
Jan Oscar Pralits ◽  
Giovanna Vittori
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Second Harmonic ◽  
Harmonic Component ◽  
Threshold Value ◽  
Finite Amplitude ◽  
Turbulent Regime ◽  
Laminar Regime ◽  
Steady Streaming ◽  
The Stability

This study contributes to an improved understanding of the stability of the boundary layer generated at the bottom of a propagating surface wave of small but finite amplitude such that both a second harmonic component and a steady streaming component, which are superimposed on the main oscillatory flow, assume significant values. A linear stability analysis of the laminar flow is made to determine the conditions leading to transition and turbulence appearance. The Reynolds number of the phenomenon is assumed to be large and a ‘momentary’ criterion of stability is used. The results show that, at a given location, the laminar regime becomes unstable when the flow close to the bottom reverses its direction from the onshore to the offshore direction and the Reynolds number exceeds a first critical value $R_{\delta ,c1}$ . However, close to the critical condition, the flow is expected to relaminarize during the other phases of the cycle. Only when the Reynolds number is increased does turbulence tend to appear also after the passage of the wave trough when the flow close to the bottom reverses from the offshore to the onshore direction. When the Reynolds number is further increased and becomes larger than a second ‘threshold’ value, the growth rate of the perturbations becomes positive over the entire wave period. The obtained results suggest the existence of four different flow regimes: the laminar regime, the disturbed laminar regime, the intermittently turbulent regime and the fully developed turbulent regime.

Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation

2019 ◽  
Vol 14 (1) ◽  
pp. 52-58 ◽  
Author(s):  
A.D. Nizamova ◽  
V.N. Kireev ◽  
S.F. Urmancheev
Keyword(s):  
Reynolds Number ◽  
Spectral Characteristics ◽  
Wave Number ◽  
Critical Parameters ◽  
Fluid Viscosity ◽  
Flat Channel ◽  
Stability Equation ◽  
The Stability ◽  
Thermoviscous Fluid ◽  
Sommerfeld Equation

The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

Global stability result for parabolic Cauchy problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mourad Choulli ◽  
Masahiro Yamamoto
Keyword(s):  
New York ◽  
Partial Differential Equations ◽  
Global Stability ◽  
Classical Problem ◽  
Stability Result ◽  
Cauchy Problems ◽  
Smooth Solutions ◽  
Linear Partial Differential Equations ◽  
Ill Posed ◽  
The Stability

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

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