The observation of the simultaneous development of a long- and a short-wave instability mode on a vortex pair

1994 ◽  
Vol 265 ◽  
pp. 289-302 ◽  
Author(s):  
P. J. Thomas ◽  
D. Auerbach
Keyword(s):  
Vortex Pair ◽  
Short Wave ◽  
Wave Mode ◽  
Core Diameter ◽  
Vortex Pairs ◽  
Long Wave ◽  
Instability Modes ◽  
Bending Mode ◽  
Theoretical Predictions ◽  
The Stability

Experiments on the stability of vortex pairs are described. The vortices (ratio of length to core diameter L/c of up to 300) were generated at the edge of a flat plate rotating about a horizontal axis in water. The vortex pairs were found to be unstable, displaying two distinct modes of instability. For the first time, as far as it is known to the authors, a long-wave as well as a short-wave mode of instability were observed to develop simultaneously on such a vortex pair. Experiments involving single vortices show that these do not develop any instability whatsoever. The wavelengths of the developing instability modes on the investigated vortex pairs are compared to theoretical predictions. Observed long wavelengths are in good agreement with the classic symmetric long-wave bending mode identified by Crow (1970). The developing short waves, on the other hand, appear to be less accurately described by the theoretical results predicted, for example, by Windnall, Bliss & Tsai (1974).

Three-dimensional instabilities of quasi-solitary waves in a falling liquid film

2014 ◽  
Vol 757 ◽  
pp. 854-887 ◽  
Author(s):  
N. Kofman ◽  
S. Mergui ◽  
C. Ruyer-Quil
Keyword(s):  
Travelling Waves ◽  
Three Dimensional ◽  
Water Film ◽  
Short Wave ◽  
Wave Mode ◽  
Long Wave ◽  
Wave Patterns ◽  
Inclination Angles ◽  
Rayleigh Taylor Instability ◽  
The Stability

AbstractThe stability of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\gamma _2$ travelling waves at the surface of a film flow down an inclined plane is considered experimentally and numerically. These waves are fast, one-humped and quasi-solitary. They undergo a three-dimensional secondary instability if the flow rate (or Reynolds number) is sufficiently high. Rugged or scallop wave patterns are generated by the interplay between a short-wave and a long-wave instability mode. The short-wave mode arises in the capillary region of the wave, with a mechanism of capillary origin which is similar to the Rayleigh–Plateau instability, whereas the long-wave mode deforms the entire wave and is triggered by a Rayleigh–Taylor instability. Rugged waves are observed at relatively small inclination angles. At larger angles, the long-wave mode predominates and scallop waves are observed. For a water film the transition between rugged and scallop waves occurs for an inclination angle around 12°.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

The instability of a pinched fluid with a longitudinal magnetic field

1958 ◽  
Vol 245 (1241) ◽  
pp. 222-237 ◽  
Keyword(s):  
Magnetic Field ◽  
Longitudinal Magnetic Field ◽  
Short Wave ◽  
Longitudinal Field ◽  
Plasma Equilibrium ◽  
Azimuthal Magnetic Field ◽  
Long Wave ◽  
Pinched Plasma ◽  
Pinch Current ◽  
The Stability

The stability of a pinched plasma equilibrium with a longitudinal magnetic field superimposed on the characteristic azimuthal magnetic field of the pinch current is studied theoretically. The linearized solutions are developed as helical perturbations of the plasma surface, and the behaviour of these is given for the different cases of uniform longitudinal, longitudinal field zero inside the plasma, and for helices of the same and opposite sense to the helix which describes the total magnetic field. Approximately, the conclusions are: that the longitudinal field has the effect of stabilizing short-wave perturbations, but that some long-wave perturbations remain unstable no matter how large the externally imposed longitudinal magnetic field.

scholarly journals On the porosity influence on stability of flow over porous medium

2020 ◽  
Vol 30 (1) ◽  
pp. 134-144
Author(s):  
K.B. Tsiberkin
Keyword(s):  
Porous Medium ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Short Wave ◽  
Wave Instability ◽  
Plane Parallel ◽  
Long Wave ◽  
The Stability ◽  
Critical Wavenumber

The stability of incompressible fluid plane-parallel flow over a layer of a saturated porous medium is studied. The results of a linear stability analysis are described at different porosity values. The considered system is bounded by solid wall from the porous layer bottom. Top fluid surface is free and rigid. A linear stability analysis of plane-parallel stationary flow is presented. It is realized for parameter area where the neutral stability curves are bimodal. The porosity variation effect on flow stability is considered. It is shown that there is a transition between two main instability modes: long-wave and short-wave. The long-wave instability mechanism is determined by inflection points within the velocity profile. The short-wave instability is due to the large transverse gradient of flow velocity near the interface between liquid and porous medium. Porosity decrease stabilizes the long wave perturbations without significant shift of the critical wavenumber. Simultaneously, the short-wave perturbations destabilize, and their critical wavenumber changes in wide range. When the porosity is less than 0.7, the inertial terms in filtration equation and magnitude of the viscous stress near the interface increase to such an extent that the Kelvin-Helmholtz analogue of instability becomes the dominant mechanism for instability development. The stability band realizes in narrow porosity area. It separates the two branches of the neutral curve.

Solitary internal waves with oscillatory tails

1992 ◽  
Vol 242 ◽  
pp. 279-298 ◽  
Author(s):  
T. R. Akylas ◽  
R. H. J. Grimshaw
Keyword(s):  
Internal Waves ◽  
Solitary Waves ◽  
Wave Theory ◽  
Wave Mode ◽  
Lower Mode ◽  
Weakly Nonlinear ◽  
Long Wave ◽  
Special Cases ◽  
Theoretical Predictions ◽  
Solitary Internal Waves

Solitary internal waves in a density-stratified fluid of shallow depth are considered. According to the classical weakly nonlinear long-wave theory, the propagation of each long-wave mode is governed by the Korteweg–de Vries equation to leading order, and locally confined solitary waves with a ‘sech’ profile are possible. Using a singular-perturbation procedure, it is shown that, in general, solitary waves of mode n > 1 actually develop oscillatory tails of infinite extent, consisting of lower-mode short waves. The amplitude of these tails is exponentially small with respect to an amplitude parameter, and lies beyond all orders of the usual long-wave expansion. To illustrate the theory, two special cases of stratification are discussed in detail, and the amplitude of the oscillations at the solitary-wave tails is determined explicitly. The theoretical predictions are supported by experimental observations.

The instability of short waves on a vortex ring

1974 ◽  
Vol 66 (1) ◽  
pp. 35-47 ◽  
Author(s):  
Sheila E. Widnall ◽  
Donald B. Bliss ◽  
Chon-Yin Tsai
Keyword(s):  
Vortex Ring ◽  
Vortex Pair ◽  
Short Wave ◽  
The Self ◽  
Core Size ◽  
The Core ◽  
Bending Waves ◽  
Bending Mode ◽  
Radial Structure ◽  
Bending Modes

A simple model for the experimentally observed instability of the vortex ring to azimuthal bending waves of wavelength comparable with the core size is presented. Short-wave instabilities are discussed for both the vortex ring and the vortex pair. Instability for both the ring and the pair is predicted to occur whenever the self-induced rotation of waves on the filament passes through zero. Although this does not occur for the first radial bending mode of a vortex filament, it is shown to be possible for bending modes with a more complex radial structure with at least one node at some radius within the core. The previous work of Widnall & Sullivan (1973) is discussed and their experimental results are compared with the predictions of the analysis presented here.

On the stability of crystal lattices VII. Long-wave and short-wave stability for the face-centred cubic lattice

1942 ◽  
Vol 38 (1) ◽  
pp. 61-66 ◽  
Author(s):  
S. C. Power
Keyword(s):  
Short Wave ◽  
Long Waves ◽  
Three Dimensions ◽  
Linear Lattice ◽  
Crystal Lattices ◽  
Cubic Lattice ◽  
Wave Stability ◽  
Long Wave ◽  
The Face ◽  
The Stability

It is shown that the theorem stated in Born's paper, and proved for the case of a linear lattice of N equal particles under certain restrictions concerning the forces between the particles, that macroscopic stability (stability for long waves) implies microscopic stability, may be extended to three dimensions for the particular case of a face-centred cubic lattice, where the effects of all neighbours, other than the first twelve neighbours, are neglected.I take this opportunity of expressing my sincere thanks to Prof. Born for much valuable advice.

The stability of a helical vortex filament

1972 ◽  
Vol 54 (4) ◽  
pp. 641-663 ◽  
Author(s):  
Sheila E. Widnall
Keyword(s):  
Short Wave ◽  
Mutual Inductance ◽  
The Self ◽  
Vortex Filament ◽  
Wave Instability ◽  
Long Wave ◽  
Amplification Rate ◽  
Helical Vortex ◽  
Induced Motion ◽  
The Stability

The stability of a helical vortex filament of finite core and infinite extent to small sinusoidal displacements of its centre-line is considered. The influence of the entire perturbed filament on the self-induced motion of each element is taken into account. The effect of the details of the vorticity distribution within the finite vortex core on the self-induced motion due to the bending of its axis is calculated using the results obtained previously by Widnall, Bliss & Zalay (1970). In this previous work, an application of the method of matched asymptotic expansions resulted in a general solution for the self-induced motion resulting from the bending of a slender vortex filament with an arbitrary distribution of vorticity and axial velocity within the core.The results of the stability calculations presented in this paper show that the helical vortex filament has three modes of instability: a very short-wave instability which probably exists on all curved filaments, a long-wave mode which is also found to be unstable by the local-induction model and a mutual-inductance mode which appears as the pitch of the helix decreases and the neighbouring turns of the filament begin to interact strongly. Increasing the vortex core size is found to reduce the amplification rate of the long-wave instability, to increase the amplification rate of the mutual-inductance instability and to decrease the wavenumber of the short-wave instability.

Instability of coupled geostrophic density fronts and its nonlinear evolution

2008 ◽  
Vol 613 ◽  
pp. 309-327 ◽  
Author(s):  
EMILIE SCHERER ◽  
VLADIMIR ZEITLIN
Keyword(s):  
Nonlinear Evolution ◽  
Wave Mode ◽  
Water Model ◽  
Shallow Water Model ◽  
Long Wave ◽  
Classical Stability ◽  
Rotating Shallow Water ◽  
Zero Potential ◽  
The Stability ◽  
Stability Results

Instability of coupled density fronts, and its fully nonlinear evolution are studied within the idealized reduced-gravity rotating shallow-water model. By using the collocation method, we benchmark the classical stability results on zero potential vorticity (PV) fronts and generalize them to non-zero PV fronts. In both cases, we find a series of instability zones intertwined with the stability regions along the along-front wavenumber axis, the most unstable modes being long wave. We then study the nonlinear evolution of the unstable modes with the help of a high-resolution well-balanced finite-volume numerical scheme by initializing it with the unstable modes found from the linear stability analysis. The most unstable long-wave mode evolves as follows: after a couple of inertial periods, the coupled fronts are pinched at some location and a series of weakly connected co-rotating elliptic anticyclonic vortices is formed, thus totally changing the character of the flow. The characteristics of these vortices are close to known rodon lens solutions. The shorter-wave unstable modes from the next instability zones are strongly concentrated in the frontal regions, have sharp gradients, and are saturated owing to dissipation without qualitatively changing the flow pattern.

The stability of short waves on a straight vortex filament in a weak externally imposed strain field

1976 ◽  
Vol 73 (4) ◽  
pp. 721-733 ◽  
Author(s):  
Chon-Yin Tsai ◽  
Sheila E. Widnall
Keyword(s):  
Strain Field ◽  
Circular Cross Section ◽  
Short Wave ◽  
Vortex Filament ◽  
Linear Stability Theory ◽  
Short Waves ◽  
Constant Vorticity ◽  
Long Wave ◽  
Intersection Points ◽  
The Stability

The stability of short-wave displacement perturbations on a vortex filament of constant vorticity in a weak externally imposed strain field is considered. The circular cross-section of the vortex filament in this straining flow field becomes elliptical. It is found that instability of short waves on this strained vortex can occur only for wavelengths and frequencies at the intersection points of the dispersion curves for an isolated vortex. Numerical results show that the vortex is stable at some of these points and unstable at others. The vortex is unstable at wavelengths for which ω = 0, thus giving some support to the instability mechanism for the vortex ring proposed recently by Widnall, Bliss & Tsai (1974). The growth rate is calculated by linear stability theory. The previous work of Crow (1970) and Moore & Saffman (1971) dealing with long-wave instabilities is discussed as is the very recent work of Moore & Saffman (1975).

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