Snakes and corkscrews in core–annular down-flow of two fluids

1997 ◽  
Vol 340 ◽  
pp. 297-317 ◽  
Author(s):  
YURIKO Y. RENARDY
Keyword(s):  
Small Amplitude ◽  
Annular Flow ◽  
Travelling Waves ◽  
Standing Waves ◽  
Axial Direction ◽  
Nonlinear Interactions ◽  
Azimuthal Direction ◽  
Weakly Nonlinear ◽  
Two Fluids ◽  
Axisymmetric Instability

Core–annular flow of two fluids is examined at the onset of a non-axisymmetric instability. This is a pattern selection problem: the bifurcating solutions are travelling waves and standing waves. The former travel in the azimuthal direction as well as the axial direction and would be observed as corkscrew waves. The standing waves travel in the axial direction but not in the azimuthal direction and appear as snakes. Weakly nonlinear interactions are studied to see whether one of these waves will be stable to small-amplitude perturbations. Sample situations for down-flow are discussed. The corkscrews tend to be preferred when the annulus is narrow, while snakes are more likely when the annulus is wide.

Nonlinear compressible magnetoconvection. Part 3. Travelling waves in a horizontal field

1995 ◽  
Vol 300 ◽  
pp. 287-309 ◽  
Author(s):  
D. P. Brownjohn ◽  
N. E. Hurlburt ◽  
M. R. E. Proctor ◽  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Numerical Experiments ◽  
Strong Field ◽  
Travelling Waves ◽  
Standing Waves ◽  
Weak Field ◽  
Horizontal Magnetic Field ◽  
Weakly Nonlinear ◽  
Physical Mechanisms ◽  
The Magnetic Field

We present results of numerical experiments on two-dimensional compressible convection in a polytropic layer with an imposed horizontal magnetic field. Our aim is to determine how far this geometry favours the occurrence of travelling waves. We therefore delineate the region of parameter space where travelling waves are stable, explore the ways in which they lose stability and investigate the physical mechanisms that are involved. In the magnetically dominated regime (with the plasma beta, $\hat{\beta}$ = 8), convection sets in at an oscillatory bifurcation and travelling waves are preferred to standing waves. Standing waves are stable in the strong-field regime ($\hat{\beta}$ = 32) but travelling waves are again preferred in the intermediate region ($\hat{\beta}$ = 128), as suggested by weakly nonlinear Boussinesq results. In the weak-field regime ($\hat{\beta}$ ≥ 512) the steady nonlinear solution undergoes symmetry-breaking bifurcations that lead to travelling waves and to pulsating waves as the Rayleigh number, $\circ{R}$, is increased. The numerical experiments are interpreted by reference to the bifurcation structure in the ($\hat{\beta}$, $\circ{R}$)-plane, which is dominated by the presence of two multiple (Takens-Bogdanov) bifurcations. Physically, the travelling waves correspond to slow magnetoacoustic modes, which travel along the magnetic field and are convectively excited. We conclude that they are indeed more prevalent when the field is horizontal than when it is vertical.

Direct simulation of unsteady axisymmetric core–annular flow with high viscosity ratio

1999 ◽  
Vol 391 ◽  
pp. 123-149 ◽  
Author(s):  
JIE LI ◽  
YURIKO RENARDY
Keyword(s):  
Annular Flow ◽  
Travelling Waves ◽  
High Viscosity ◽  
Time Dependent ◽  
Nonlinear Saturation ◽  
Weakly Nonlinear ◽  
Pressure Fields ◽  
Linearized Stability ◽  
Spatially Periodic ◽  
Extremal Values

Axisymmetric pipeline transportation of oil and water is simulated numerically as an initial value problem. The simulations succeed in predicting the spatially periodic Stokes-like waves called bamboo waves, which have been documented in experiments of Bai, Chen & Joseph (1992) for up-flow. The numerical scheme is validated against linearized stability theory for perfect core–annular flow, and weakly nonlinear saturation to travelling waves. Far from onset conditions, the fully nonlinear saturation to steady bamboo waves is achieved. As the speed is increased, the bamboo waves shorten, and peaks become more pointed. A new time-dependent bamboo wave is discovered, in which the interfacial waveform is steady, but the accompanying velocity and pressure fields are time-dependent. The appearance of vortices and the locations of the extremal values of pressure are investigated for both up- and down-flows.

The weakly nonlinear interfacial stability of a core–annular flow in a corrugated tube

2002 ◽  
Vol 466 ◽  
pp. 149-177 ◽  
Author(s):  
HSIEN-HUNG WEI ◽  
DAVID S. RUMSCHITZKI
Keyword(s):  
Surface Tension ◽  
Annular Flow ◽  
Complex Geometry ◽  
Travelling Waves ◽  
Immiscible Fluids ◽  
Base Flow ◽  
Interfacial Stability ◽  
Chaotic Motions ◽  
Weakly Nonlinear ◽  
Corrugated Tube

A core–annular flow, the concurrent axial flow of two immiscible fluids in a circular tube or pore with one fluid in the core and the other in the wetting annular region, is frequently used to model technologically important flows, e.g. in liquid–liquid displacements in secondary oil recovery. Most of the existing literature assumes that the pores in which such flows occur are uniform circular cylinders, and examine the interfacial stability of such systems as a function of fluid and interfacial properties. Since real rock pores possess a more complex geometry, the companion paper examined the linear stability of core–annular flows in axisymmetric, corrugated pores in the limit of asymptotically weak corrugation. It found that short-wave disturbances that were stable in straight tubes could couple to the wall's periodicity to excite unstable long waves. In this paper, we follow the evolution of the axisymmetric, linearly unstable waves for fluids of equal densities in a corrugated tube into the weakly nonlinear regime. Here, we ask whether this continual generation of new disturbances by the coupling to the wall's periodicity can overcome the nonlinear saturation mechanism that relies on the nonlinear (kinematic-condition-derived) wave steepening of the Kuramoto–Sivashinsky (KS) equation. If it cannot, and the unstable waves still saturate, then do these additional excited waves make the KS solutions more likely to be chaotic, or does the dispersion introduced into the growth rate correction by capillarity serve to regularize otherwise chaotic motions?We find that in the usual strong surface tension limit, the saturation mechanism of the KS mechanism remains able to saturate all disturbances. Moreover, an additional capillary-derived nonlinear term seems to favour regular travelling waves over chaos, and corrugation adds a temporal periodicity to the waves associated with their periodical traversing of the wall's crests and troughs. For even larger surface tensions, capillarity dominates over convection and a weakly nonlinear version of Hammond's no-flow equation results; this equation, with or without corrugation, suggests further growth. Finally, for a weaker surface tension, the leading-order base flow interface follows the wall's shape. The corrugation-derived excited waves appear able to push an otherwise regular travelling wave solution to KS to become chaotic, whereas its dispersive properties in this limit seem insufficiently strong to regularize chaotic motions.

Nonlinear evolution, travelling waves, and secondary instability of sheared-film flows with insoluble surfactants

2007 ◽  
Vol 594 ◽  
pp. 125-156 ◽  
Author(s):  
DAVID HALPERN ◽  
ALEXANDER L. FRENKEL
Keyword(s):  
Small Amplitude ◽  
Marangoni Number ◽  
Landau Equation ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Nonlinear Pdes ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Long Wave ◽  
Interfacial Surfactant

The nonlinear development of the interfacial-surfactant instability is studied for the semi-infinite plane Couette film flow. Disturbances whose spatial period is close to the marginal wavelength of the long-wave instability are considered first. Appropriate weakly nonlinear partial differential equations (PDEs) which couple the disturbances of the film thickness and the surfactant concentration are obtained from the strongly nonlinear lubrication-approximation PDEs. In a rescaled form each of the two systems of PDEs is controlled by a single parameter C, the ‘shear-Marangoni number’. From the weakly nonlinear PDEs, a single Stuart–Landau ordinary differential equation (ODE) for an amplitude describing the unstable fundamental mode is derived. By comparing the solutions of the Stuart–Landau equation with numerical simulations of the underlying weakly and strongly nonlinear PDEs, it is verified that the Stuart–Landau equation closely approximates the small-amplitude saturation to travelling waves, and that the error of the approximation converges to zero at the marginal stability curve. In contrast to all previous stability work on flows that combine interfacial shear and surfactant, some analytical nonlinear results are obtained. The Hopf bifurcation to travelling waves is supercritical for C < Cs and subcritical for C > Cs, where Cs is approximately 0.29. This is confirmed with a numerical continuation and bifurcation technique for ODEs. For the subcritical cases, there are two values of equilibrium amplitude for a range of C near Cs, but the travelling wave with the smaller amplitude is unstable as a periodic orbit of the associated dynamical system (whose independent variable is the spatial coordinate). By using the Bloch (‘Floquet’) disturbance modes in the linearized PDEs, it transpires that all the small-amplitude travelling-wave equilibria are unstable to sufficiently long-wave disturbances. This theoretical result is confirmed by numerical simulations which invariably show the large-amplitude saturation of the disturbances. In view of this secondary instability, the existence of small-amplitude periodic solutions (on the real line) bifurcating from the uniform flow at the marginal values of the shear-Marangoni number does not contradict the earlier conclusions that the interfacial-surfactant instability has a strongly nonlinear character, in the sense that there are no small-amplitude attractors such that the entire evolution towards them is captured by weakly-nonlinear equations. This suggests that, in general, for flowing-film instabilities that have zero wavenumber at criticality, the saturated disturbance amplitudes do not always have to decrease to zero as the control parameter approaches its value at criticality.

scholarly journals Travelling and standing waves in magnetoconvection

1993 ◽  
Vol 441 (1913) ◽  
pp. 649-658 ◽  
Keyword(s):  
Numerical Solutions ◽  
Travelling Waves ◽  
Standing Waves ◽  
Periodic Boundary Conditions ◽  
Perturbation Techniques ◽  
Oscillatory Convection ◽  
Weakly Nonlinear ◽  
Modulated Waves ◽  
Steady Shearing ◽  
Rayleigh Numbers

The problem of Boussinesq magnetoconvection with periodic boundary conditions is studied using standard perturbation techniques. It is found that either travelling waves or standing waves can be stable at the onset of oscillatory convection, depending on the parameters of the problem. When travelling waves occur, a steady shearing flow is present that is quadratic in the amplitude of the convective flow. The weakly nonlinear predictions are confirmed by comparison with numerical solutions of the full partial differential equations at Rayleigh numbers 10% above critical. Modulated waves (through which stability is transferred between travelling and standing waves) are found near the boundary between the regions in parameter space where travelling waves and standing waves are preferred.

Resonant capillary–gravity interfacial waves

1994 ◽  
Vol 265 ◽  
pp. 303-343 ◽  
Author(s):  
P. Christodoulides ◽  
F. Dias
Keyword(s):  
Normal Form ◽  
Fundamental Mode ◽  
Second Harmonic ◽  
Dimensional Space ◽  
Travelling Waves ◽  
Density Ratio ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Two Fluids ◽  
Near Resonance

Two-dimensional space-periodic cabillary–gravity waves at the interface between two fluids of different densities are considered when the second harmonic and the fundamental mode are near resonance. A weakly nonlinear analysis provides the equations (normal form), correct to third order, that relate the wave frequency with the amplitudes of the fundamental mode and of the second harmonic for all waves with small energy. A study of the normal form for waves which are also periodic in time reveals three possible types of space- and time-periodic waves: the well-known travelling and standing waves as well as an unusual class of three-mode mixed waves. Mixed waves are found to provide a connection between standing and travelling waves. The branching behaviour of all types of waves is shown to depend strongly on the density ratio. For travelling waves the weakly nonlinear results are confirmed numerically and extended to finite-amplitude waves. When slow modulations in time of the amplitudes are considered, a powerful geometrical method is used to study the resulting normal form. Finally a discussion on modulational stability suggests that increasing the density ratio has a stabilizing effect.

scholarly journals Experimental travelling waves identification in mechanical structures

2017 ◽  
Vol 24 (1) ◽  
pp. 152-167
Author(s):  
Izhak Bucher ◽  
Ran Gabai ◽  
Harel Plat ◽  
Amit Dolev ◽  
Eyal Setter
Keyword(s):  
Energy Flux ◽  
Power Flow ◽  
Reactive Power ◽  
Mechanical Energy ◽  
Travelling Waves ◽  
Electrical Power ◽  
Normal Modes ◽  
Standing Waves ◽  
Amplitude Curve ◽  
Physical Point

Vibrations are often represented as a sum of standing waves in space, i.e. normal modes of vibration. While this can be mathematically accurate, the representation as travelling waves can be compact and more appropriate from a physical point of view, in particular when the energy flux along the structure is meaningful. The quantification of travelling waves assists in computing the energy being transferred and propagated along a structure. It can provide local as well as global information about the structure through which the mechanical energy flows. Presented in this paper is a new method to quantify the fraction of mechanical power being transmitted in a vibration cycle at a specific direction in space using measured data. It is shown that the method can detect local defects causing slight non-uniformity of the energy flux. Equivalence is being made with the electrical power factor and electromagnetic standing waves ratio, commonly employed in such cases. Other methods to perform experiment based wave identification in one-dimension are compared with the power flow based identification. Including a signal processing approach that fits an ellipse to the complex amplitude curve and Hilbert transform for obtaining the local phase and amplitude. A new representation of the active and reactive power flow is developed and its relationship to standing waves ratio is demonstrated analytically and experimentally.

scholarly journals Existence and linearized stability of solitary waves for a quasilinear Benney system

2016 ◽  
Vol 146 (3) ◽  
pp. 547-564 ◽  
Author(s):  
João-Paulo Dias ◽  
Mário Figueira ◽  
Filipe Oliveira
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Compact Support ◽  
Travelling Waves ◽  
Standing Waves ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Linearized Stability ◽  
Stability Of Solitary Waves ◽  
Image Position

We prove the existence of solitary wave solutions to the quasilinear Benney systemwhere , –1 < p < +∞ and a, γ > 0. We establish, in particular, the existence of travelling waves with speed arbitrarily large if p < 0 and arbitrarily close to 0 if . We also show the existence of standing waves in the case , with compact support if – 1 < p < 0. Finally, we obtain, under certain conditions, the linearized stability of such solutions.

Three-dimensional tidal sand waves

2009 ◽  
Vol 618 ◽  
pp. 1-11 ◽  
Author(s):  
PAOLO BLONDEAUX ◽  
GIOVANNA VITTORI
Keyword(s):  
Stability Analysis ◽  
Nonlinear Interaction ◽  
Small Amplitude ◽  
Three Dimensional ◽  
Time Development ◽  
Resonance Mechanism ◽  
Weakly Nonlinear ◽  
Sand Waves ◽  
Tidal Sand ◽  
Harmonic Components

The process which leads to the formation of three-dimensional sand waves is investigated by means of a stability analysis which considers the time development of a small-amplitude bottom perturbation of a shallow tidal sea. The weakly nonlinear interaction of a triad of resonant harmonic components of the bottom perturbation is considered. The results show that the investigated resonance mechanism can trigger the formation of a three-dimensional bottom pattern similar to that observed in the field.

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