scholarly journals Finite amplitude convection between stress-free boundaries; Ginzburg–Landau equations and modulation theory

1994 ◽  
Vol 5 (3) ◽  
pp. 267-282 ◽  
Author(s):  
Andrew J. Bernoff
Keyword(s):  
Stability Theory ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Mean Flow ◽  
Onset Of Convection ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equations ◽  
Modulation Theory ◽  
The Stability

The stability theory for rolls in stress-free convection at finite Prandtl number is affected by coupling with low wavenumber two-dimensional mean-flow modes. In this work, a set of modified Ginzburg–Landau equations describing the onset of convection is derived which accounts for these additional modes. These equations can be used to extend the modulation equations of Zippelius & Siggia describing the breakup of rolls, bringing their stability theory into agreement with the results of Busse & Bolton.

Stability of convection rolls in a layer with stress-free boundaries

1985 ◽  
Vol 150 ◽  
pp. 487-498 ◽  
Author(s):  
E. W. Bolton ◽  
F. H. Busse
Keyword(s):  
Three Dimensional ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Stability Boundaries ◽  
Prandtl Numbers ◽  
Steady Solutions ◽  
The Stability ◽  
Convection Rolls ◽  
Steady Convection

Steady finite-amplitude solutions for two-dimensional convection in a layer heated from below with stress-free boundaries are obtained numerically by a Galerkin method. The stability of the steady convection rolls with respect to arbitrary three-dimensional infinitesimal disturbances is investigated. Stability is found only in a small fraction of the Rayleigh-number-wavenumber space where steady solutions exist. The cross-roll instability and the oscillatory and monotonic skewed varicose instabilities are most important in limiting the stability of steady convection rolls. The Prandtlnumbers P = 0.71, 7, 104 areemphasized, but the stability boundaries are sufficiently smoothly dependent on the parameters of the problem to permit qualitative extrapolations to other Prandtl numbers.

On the stability of an inviscid shear layer which is periodic in space and time

1967 ◽  
Vol 27 (4) ◽  
pp. 657-689 ◽  
Author(s):  
R. E. Kelly
Keyword(s):  
Growth Rate ◽  
Shear Layer ◽  
Finite Amplitude ◽  
Periodic Component ◽  
Periodic Flow ◽  
Flow Profile ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean ◽  
The Stability

In experiments concerning the instability of free shear layers, oscillations have been observed in the downstream flow which have a frequency exactly half that of the dominant oscillation closer to the origin of the layer. The present analysis indicates that the phenomenon is due to a secondary instability associated with the nearly periodic flow which arises from the finite-amplitude growth of the fundamental disturbance.At first, however, the stability of inviscid shear flows, consisting of a non-zero mean component, together with a component periodic in the direction of flow and with time, is investigated fairly generally. It is found that the periodic component can serve as a means by which waves with twice the wavelength of the periodic component can be reinforced. The dependence of the growth rate of the subharmonic wave upon the amplitude of the periodic component is found for the case when the mean flow profile is of the hyperbolic-tangent type. In order that the subharmonic growth rate may exceed that of the most unstable disturbance associated with the mean flow, the amplitude of the streamwise component of the periodic flow is required to be about 12 % of the mean velocity difference across the shear layer. This represents order-of-magnitude agreement with experiment.Other possibilities of interaction between disturbances and the periodic flow are discussed, and the concluding section contains a discussion of the interactions on the basis of the energy equation.

Lubricated pipelining: stability of core–annular flow. Part 4. Ginzburg–Landau equations

1991 ◽  
Vol 227 ◽  
pp. 587-615 ◽  
Author(s):  
Kangping Chen ◽  
Daniel D. Joseph
Keyword(s):  
Annular Flow ◽  
Landau Equation ◽  
Finite Amplitude ◽  
Ginzburg Landau ◽  
Neutral Curves ◽  
Critical Reynolds Numbers ◽  
Ginzburg Landau Equations ◽  
Flow Part ◽  
Annular Flows ◽  
Monochromatic Waves

Nonlinear stability of core-annular flow near points of the neutral curves at which perfect core-annular flow loses stability is studied using Ginzburg-Landau equations. Most of the core-annular flows are always unstable. Therefore the set of core-annular flows having critical Reynolds numbers is small, so that the set of flows for which our analysis applies is small. An efficient and accurate algorithm for computing all the coefficients of the Ginzburg-Landau equation is implemented. The nonlinear flows seen in the experiments do not appear to be modulations of monochromatic waves, and we see no evidence for soliton-like structures. We explore the bifurcation structure of finite-amplitude monochromatic waves at criticality. The bifurcation theory is consistent with observations in some of the flow cases to which it applies and is not inconsistent in the other cases.

The development of three-dimensional wave-packets in unbounded parallel flows

1968 ◽  
Vol 32 (4) ◽  
pp. 801-808 ◽  
Author(s):  
M. Gaster ◽  
A. Davey
Keyword(s):  
Wave Packet ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Mean Flow ◽  
Physical Plane ◽  
Parallel Flows ◽  
Flow Velocity Profile ◽  
The Stability ◽  
Special Case ◽  
Dimensional Wave

In this paper we examine the stability of a two-dimensional wake profile of the form u(y) = U∞(1 – r e-sy2) with respect to a pulsed disturbance at a point in the fluid. The disturbed flow forms an expanding wave packet which is convected downstream. Far downstream, where asymptotic expansions are valid, the motion at any point in the wave packet is described by a particular three-dimensional wave having complex wave-numbers. In the special case of very unstable flows, where viscosity does not have a significant influence, it is possible to evaluate the three-dimensional eigenvalues in terms of two-dimensional ones using the inviscid form of Squire's transformation. In this way each point in the physical plane can be linked to a particular two-dimensional wave growing in both space and time by simple algebraic expressions which are independent of the mean flow velocity profile. Computed eigenvalues for the wake profile are used in these relations to find the behaviour of the wave packet in the physical plane.

scholarly journals Finite disturbance effect on the stability of a laminar incompressible wake behind a flat plate

1970 ◽  
Vol 40 (2) ◽  
pp. 315-341 ◽  
Author(s):  
D. Ru-Sue Ko ◽  
T. Kubota ◽  
L. Lees
Keyword(s):  
Flat Plate ◽  
Finite Amplitude ◽  
Single Frequency ◽  
Shape Parameters ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Flow Parameters ◽  
Local Mean ◽  
The Mean ◽  
The Stability

An integral method is used to investigate the interaction between a two-dimensional, single frequency finite amplitude disturbance in a laminar, incompressible wake behind a flat plate at zero incidence. The mean flow is assumed to be a non-parallel flow characterized by a few shape parameters. Distribution of the fluctuation across the wake is obtained as functions of those mean flow parameters by solving the inviscid Rayleigh equation using the local mean flow. The variations of the fluctuation amplitude and of the shape parameters for the mean flow are then obtained by solving a set of ordinary differential equations derived from the momentum and energy integral equations. The interaction between the mean flow and the fluctuation through Reynolds stresses plays an important role in the present formulation, and the theoretical results show good agreement with the measurements of Sato & Kuriki (1961).

Stability of Bifurcating Solutions for the Ginzburg–Landau Equations

1998 ◽  
Vol 10 (05) ◽  
pp. 579-626 ◽  
Author(s):  
Catherine Bolley ◽  
Bernard Helffer
Keyword(s):  
Existence And Uniqueness ◽  
Dominant Role ◽  
Critical Value ◽  
Symmetric Problem ◽  
Classical Analysis ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equations ◽  
The Stability ◽  
Stability Results

This paper is concerned with superconducting solutions of the Ginzburg–Landau equations for a film. We study the structure and the stability of the bifurcating solutions starting from normal solutions as functions of the parameters (κ, d), where d is the thickness of the film and κ is the Ginzburg–Landau parameter characterizing the material. Although κ and d play independent roles in the determination of these properties, we will exhibit the dominant role taken up by the product κd in the existence and uniqueness of bifurcating solutions as much as in their stability. Using the semi-classical analysis developed in our previous papers for getting the existence of asymmetric solutions and asymptotics for the supercooling field, we prove in particular that the symmetric bifurcating solutions are stable for (κ, d) such that κd is small and [Formula: see text] (for any η>0) and unstable for (κ, d) such that κd is large. We also show the existence of an explicit critical value Σ0 such that, for κ≤Σ0-η and κd large, the asymmetric solutions are unstable, while, for κ≥Σ0+η and κd large, the asymmetric solutions are stable. Finally, we also analyze the symmetric problem which leads to other stability results.

Two-dimensional instabilities of steady double-diffusive interleaving

1992 ◽  
Vol 242 ◽  
pp. 99-116 ◽  
Author(s):  
Oliver S. Kerr
Keyword(s):  
Density Gradient ◽  
Linear Analysis ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Density Profiles ◽  
Vertical Density ◽  
Diffusive Effects ◽  
The Stability ◽  
Secondary Instabilities ◽  
Double Diffusive

The stability of finite-amplitude double–diffusive interleaving driven by linear gradients of salinity and temperature is considered. We show that as the sinusoidal interleaving predicted by linear analysis grows to finite amplitude it is subject to instabilities centred along the lines of minimum vertical density gradient and maximum shear. These secondary instabilities could lead to the step-like density profiles observed in experiments. We show that these instabilities can occur for large Richardson numbers and hence are not driven by shear, but are driven, by double-diffusive effects.

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