He’s Multiple-Scale Solution for the Three-dimensional Nonlinear KH Instability of Rotating Magnetic Fluids

2019 ◽  
Vol 9 (1) ◽  
pp. 52-69 ◽  
Author(s):  
Yusry Osman El-Dib ◽  
Amal A Mady
Keyword(s):  
Multiple Scales ◽  
Velocity Ratio ◽  
Three Dimensional ◽  
Multiple Scale ◽  
Stability Criteria ◽  
Helmholtz Instability ◽  
Vertical Field ◽  
Horizontal Flux ◽  
The Stability ◽  
Transition Curves

This paper elucidates a trend in solving nonlinear oscillators of the rotating Kelvin-Helmholtz instability. The system is constituted by the vertical flux or the horizontal flux. He’s multiple-scales perturbation methodology has been applied and therefore the system is represented by a generalized homotopy equation. This approach ends up in a periodic answer to a nonlinear oscillator with high nonlinearity. The cubic-quintic nonlinear Duffing equation is obligatory as a condition to uniformly answer. This equation is employed to derive the stability criteria. The transition curves are plotted to investigate the stability image. It's shown that the angular velocity suppresses the instability. The tangential flux plays a helpful role, whereas the vertical field encompasses a destabilizing influence. Within the existence of the rotation, the velocity ratio reduces stability configuration.

Equilibrium and Stability of a Three-Dimensional Toroidal MHD Configuration Near its Magnetic Axis

1976 ◽  
Vol 31 (11) ◽  
pp. 1277-1288 ◽  
Author(s):  
D. Lortz ◽  
J. Nührenberg
Keyword(s):  
Three Dimensional ◽  
Magnetic Axis ◽  
Stability Criteria ◽  
Sufficient Criterion ◽  
Third Order ◽  
Stagnation Points ◽  
The Third ◽  
Longitudinal Current ◽  
The Stability

Abstract The expansion of a three-dimensional toroidal magnetohydrostatic equilibrium around its magnetic axis is reconsidered. Equilibrium and stability plasma-β estimates are obtained in connection with a discussion of stagnation points occurring in the third-order flux surfaces. The stability criteria entering the β-estimates are: (i) a necessary criterion for localized disturbances, (ii) a new sufficient criterion for configurations without longitudinal current. Hamada coordinates are used to evaluate these criteria.

A Study of Spike and Modal Stall Phenomena in a Low-Speed Axial Compressor

1997 ◽  
Author(s):  
T. R. Camp ◽  
I. J. Day
Keyword(s):  
High Speed ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Operating Conditions ◽  
Two Dimensional ◽  
Stability Criteria ◽  
Stall Inception ◽  
Low Speed ◽  
The Stability ◽  
Dimensional Instability

This paper presents a study of stall inception mechanisms a in low-speed axial compressor. Previous work has identified two common flow breakdown sequences, the first associated with a short lengthscale disturbance known as a ‘spike’, and the second with a longer lengthscale disturbance known as a ‘modal oscillation’. In this paper the physical differences between these two mechanisms are illustrated with detailed measurements. Experimental results are also presented which relate the occurrence of the two stalling mechanisms to the operating conditions of the compressor. It is shown that the stability criteria for the two disturbances are different: long lengthscale disturbances are related to a two-dimensional instability of the whole compression system, while short lengthscale disturbances indicate a three-dimensional breakdown of the flow-field associated with high rotor incidence angles. Based on the experimental measurements, a simple model is proposed which explains the type of stall inception pattern observed in a particular compressor. Measurements from a single stage low-speed compressor and from a multistage high-speed compressor are presented in support of the model.

The effect of magnetic fields on the nonlinear instability of two superposed magnetic streaming fluids, each of a finite depth

1995 ◽  
Vol 73 (3-4) ◽  
pp. 163-173 ◽  
Author(s):  
Abdel Raouf F. Elhefnawy
Keyword(s):  
Magnetic Field ◽  
Multiple Scale ◽  
Finite Depth ◽  
Nonlinear Instability ◽  
Helmholtz Instability ◽  
Perturbed System ◽  
Stability Diagrams ◽  
The Stability ◽  
Normal Magnetic Field ◽  
Horizontal Interface

The nonlinear Kelvin–Helmholtz instability of a horizontal interface separating two flowing superposed magnetic fluids of finite depths is described in the presence of a normal magnetic field. The fluids are taken to be incompressible and inviscid and the motion is assumed to be irrotational. The method of multiple-scale perturbations is used to obtain two nonlinear Schrödinger equations describing the behaviour of the perturbed system. The stability of the system is discussed both theoretically and numerically and the stability diagrams are obtained. The nonlinear cutoff magnetic field that separates the regions of instability from those of stability is also obtained.

Three-Dimensional Viscous Potential Electrohydrodynamic Kelvin–Helmholtz Instability Through Vertical Cylindrical Porous Inclusions With Permeable Boundaries

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Galal M. Moatimid ◽  
Mohamed A. Hassan
Keyword(s):  
Dispersion Relation ◽  
Three Dimensional ◽  
Wave Number ◽  
Helmholtz Instability ◽  
Large Wave ◽  
Porous Layers ◽  
Azimuthal Wave ◽  
Azimuthal Wave Number ◽  
Streaming Velocity ◽  
The Stability

In this paper, the electrohydrodynamic three-dimensional Kelvin–Helmholtz instability of a cylindrical interface with heat and mass transfer between liquid and vapor phases is studied. The liquid and the vapor are saturated, two coaxial cylindrical porous layers, and the suction/injection velocities for the fluids at the permeable boundaries are also taken into account. The dispersion relation is derived and the stability analysis is discussed for various parameters. It is found that the streaming velocity has a destabilizing effect, while the axial electric field has a stabilizing one. The suction for both the liquid and the steam has a destabilizing effect in contrast with the injection at both boundaries. The flow through porous structure is more stable than the pure flow. The case of the axisymmetric (for zero value of the azimuthal wave number m) and asymmetric (for nonzero value of the azimuthal wave number m) disturbances at large wavelength (at the wave number k→0) are always stable. Meanwhile, it is the same dispersion relation for the plane geometry at large wave number. Finally, our results are corroborated by comparing them with the previous published results.

Influence of optimally amplified streamwise streaks on the Kelvin–Helmholtz instability

2018 ◽  
Vol 838 ◽  
pp. 478-500 ◽  
Author(s):  
Mathieu Marant ◽  
Carlo Cossu
Keyword(s):  
Initial Conditions ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Basic Flow ◽  
Sensitivity Analyses ◽  
Mean Flow ◽  
Helmholtz Instability ◽  
Flow Distortion ◽  
Streamwise Streaks ◽  
The Stability

The optimal energy amplifications of streamwise-uniform and spanwise-periodic perturbations of the hyperbolic-tangent mixing layer are computed and found to be very large, with maximum amplifications increasing with the Reynolds number and with the spanwise wavelength of the perturbations. The optimal initial conditions are streamwise vortices and the most amplified structures are streamwise streaks with sinuous symmetry in the cross-stream plane. The leading suboptimal perturbations have opposite (varicose) symmetry. When forced with finite amplitudes these perturbations modify the characteristics of the Kelvin–Helmholtz instability. Maximum temporal growth rates are reduced by optimal sinuous perturbations and are slightly increased by varicose suboptimal ones. In contrast, the onset of absolute instability is delayed by varicose suboptimal perturbations and is slightly promoted by sinuous optimal ones. We show that if, instead of the computed fully nonlinear basic-flow distortions, the stability analysis is based on a shape assumption for the flow distortions, then opposite effects on the flow stability are predicted in most of the considered cases. These strong differences are attributed to the spanwise-uniform component of the nonlinear basic-flow distortion which, we conclude, should be systematically included in sensitivity analyses of the stability of two-dimensional basic flows to three-dimensional basic-flow perturbations. We finally show that the leading-order quadratic sensitivity of the eigenvalues to the amplitude of the streaks is preserved if the effects of the mean flow distortion are included in the sensitivity analysis.

On three-dimensional packets of surface waves

1974 ◽  
Vol 338 (1613) ◽  
pp. 101-110 ◽  
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Partial Differential Equations ◽  
Plane Waves ◽  
Three Dimensional ◽  
Phase Variation ◽  
Finite Depth ◽  
Amplitude Variation ◽  
The Stability ◽  
Small Disturbances

In this note we use the method of multiple scales to derive the two coupled nonlinear partial differential equations which describe the evolution of a three-dimensional wave-packet of wavenumber k on water of finite depth. The equations are used to study the stability of the uniform Stokes wavetrain to small disturbances whose length scale is large compared with 2π/ k . The stability criterion obtained is identical with that derived by Hayes under the more restrictive requirement that the disturbances are oblique plane waves in which the amplitude variation is much smaller than the phase variation.

The stability of axisymmetric menisci

1973 ◽  
Vol 275 (1253) ◽  
pp. 489-528 ◽  
Keyword(s):  
Three Dimensional ◽  
Axisymmetric Deformation ◽  
Critical Conditions ◽  
Stability Criteria ◽  
Stable States ◽  
Energy Profiles ◽  
Second Derivatives ◽  
Numerical Interpolation ◽  
The Stability ◽  
Stability Type

The conditions that govern the equilibrium and stability of a meniscus have been obtained from the first and second derivatives of the energy of the meniscus when it undergoes axisymmetric deformation. The energy of forming a meniscus is defined in thermodynamic terms and methods are given for calculating the free energy of a mensicus in the perturbed and unperturbed state. The stable, critically stable and unstable equilibrium states of a meniscus are all defined in terms of an energy profile, that is, the variation of energy with degree of perturbation. The variational problem of defining parameters for a critically stable meniscus is solved graphically by using a three-dimensional cluster of energy profiles, and it is shown that certain properties of the meniscus, notably volume or pressure, reach limiting values at critical conditions. Four types of stability are considered for each of three forms of axisymmetric menisci. The stability types are those limited by volume or pressure, in conjunction with limitation by the size of the supporting solid surface or the angle of contact. The three forms of menisci are pendant drops, sessile drops and rod-in-free-surface menisci. Detailed stability criteria are given for each of the twelve different combinations of stability type and meniscus form. The stability criteria of this study are all derived by numerical interpolation methods applied to the tables of equilibrium meniscus shapes - they are thus theoretical. Where possible they have been compared with experiment and with other studies, and are found to predict critically stable states with an accuracy greater than that likely to be found in the normal course of experiments.

Kelvin--Helmholtz instability of a horizontal interface between a finite subsonic gas and a finite magnetic liquid

1998 ◽  
Vol 76 (5) ◽  
pp. 361-374 ◽  
Author(s):  
K Zakaria
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Solid Surfaces ◽  
Wave Number ◽  
Magnetic Liquid ◽  
Helmholtz Instability ◽  
Perturbed System ◽  
Progressive Waves ◽  
The Stability ◽  
Horizontal Interface

The nonlinear Kelvin-Helmholtz instability of a horizontal interface between a magnetic inviscid incompressible liquid and an inviscid laminar subsonic gas is investigated. The gas and the liquid are assumed to have finite thicknesses. The applied magnetic field is parallel to the solid surfaces of the considered system. The method of multiple scales is used to obtain two nonlinear Schrodinger equations describing the behaviour of the perturbed system. The stability of the progressive waves is discussed theoretically. The nonlinear cutoff wave number is obtained, where the stability conditions of the standing waves are obtained. A numerical example is applied to discuss the stability diagrams.PACS Nos.: 51.60 and 47.20

A Liquid Metal Flow Between Two Coaxial Cylinders System

2019 ◽  
Vol 11 (1) ◽  
pp. 79-86
Author(s):  
Abdelkrim Merah ◽  
Ridha Kelaiaia ◽  
Faiza Mokhtari
Keyword(s):  
Couette Flow ◽  
Metal Flow ◽  
Three Dimensional ◽  
Liquid Sodium ◽  
Taylor Vortex ◽  
Turbulent Regime ◽  
Coaxial Cylinders ◽  
Concentric Cylinders ◽  
Ideal Tool ◽  
The Stability

Abstract The Taylor-Couette flow between two rotating coaxial cylinders remains an ideal tool for understanding the mechanism of the transition from laminar to turbulent regime in rotating flow for the scientific community. We present for different Taylor numbers a set of three-dimensional numerical investigations of the stability and transition from Couette flow to Taylor vortex regime of a viscous incompressible fluid (liquid sodium) between two concentric cylinders with the inner one rotating and the outer one at rest. We seek the onset of the first instability and we compare the obtained results for different velocity rates. We calculate the corresponding Taylor number in order to show its effect on flow patterns and pressure field.

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