Convection in a low Prandtl number fluid layer rotating about a vertical axis

Two- and three-dimensional convection flows in a horizontal layer of a low Prandtl number fluid heated from below and rotating about a vertical axis are studied numerically with a Galerkin method. Solutions for subcritical steady finite amplitude convection and convection in the form of standing oscillations are obtained. Parameter regimes that appear to be attainable in laboratory experiments have been emphasized. The stability of subcritical two-dimensional steady convection has been investigated and three-dimensional chaotic states of convection have been found.     

Non-linear convection due to compositional and thermal buoyancy in Earth's outer core

The linear stability of convection due to compositional and thermal buoyancy in Earth's outer core has been investigated. We have obtained the values of Takens-Bogdanov bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters. We have derived a non-linear two-dimensional Landau-Ginzburg equation with real coefficients near the onset of stationary convection at a supercritical pitchfork bifurcation and two non-linear one-dimensional coupled Landau-Ginzburg type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation. We have studied Nusselt number contribution from a Landau-Ginzburg equation at the onset of stationary convection. We have discussed the stability regions of standing and travelling waves. We have also discussed the occurrence of secondary instabilities such as Eckhaus, zigzag and Benjamin-Feir instabilities. We have also derived the non-linear amplitude equation near the Takens-Bogdanov bifurcation point.     

Non-Darcy natural convection from a vertical wavy surface in a porous medium

We examine the combined effect of spatially stationary surface waves and the presence of fluid inertia on the free convection induced by a vertical heated surface embedded in a fluid-saturated porous medium. We consider the boundary-layer regime where the Darcy-Rayleigh number, Ra, is very large, and assume that the surface waves have O(1) amplitude and wavelength. The resulting boundary-layer equations are found to be nonsimilar only when the surface is nonuniform and inertia effects are present; self-similarity results when either or both effects are absent. Detailed results for the local and global rates of heat transfer are presented for a range of values of the inertia parameter and the surface wave amplitude.     

Quasiperiodic and exponential transient phase waves and their bifurcations in a ring of unidirectionally coupled parametric oscillators

Phase waves rotating in a ring of unidirectionally coupled parametric oscillators are studied. The system has a pair of spatially uniform stable periodic solutions with a phase difference and an unstable quasiperiodic traveling phase wave solution. They are generated from the origin through a period doubling bifurcation and the Neimark?CSacker bifurcation, respectively. In transient states, phase waves rotating in a ring are generated, the duration of which increases exponentially with the number of oscillators (exponential transients). A power law distribution of the duration of randomly generated phase waves and the noise-sustained propagation of phase waves are also shown. These properties of transient phase waves are well described with a kinematical equation for the propagation of wave fronts. Further, the traveling phase wave is stabilized through a pitchfork bifurcation and changes into a standing wave through pinning. These bifurcations and exponential transient rotating waves are also shown in an autonomous system with averaging and a coupled map model, and they agree with each other.     

Convectively coupled wave–environment interactions

In the tropical atmosphere, waves can couple with water vapor and convection to form large-scale coherent structures called convectively coupled waves (CCWs). The effects of water vapor and convection lead to CCW–mean flow interactions that are different from traditional wave–mean flow interactions in many ways. CCW–mean flow interactions are studied here in two types of models: a multiscale model that represents CCW structures in two spatial dimensions directly above the Earth’s equator, and an amplitude model in the form of ordinary differential equations for the CCW and mean flow amplitudes. The amplitude equations are shown to capture the qualitative behavior of the spatially resolved model, including nonlinear oscillations and a Hopf bifurcation as the climatological background wind is varied. Furthermore, an even simpler set of amplitude equations can also capture some of the essential oscillatory behavior, and it is shown to be equivalent to the Duffing oscillator. The basic interaction mechanisms are that the mean flow’s vertical shear determines the preferred propagation direction of the CCW, and the CCWs can drive changes in the mean shear through convective momentum transport, with energy transfer that is sometimes upscale and sometimes downscale. In addition to CCW–mean flow interactions, also discussed are CCW–water vapor interactions, which form the basis of the Madden–Julian Oscillation (MJO) skeleton model of the first two authors. The key parameter of the MJO skeleton model is estimated theoretically and is in agreement with previously conjectured values.     

Non-linear dynamics and pattern formation in a vertical fluid layer heated from the side

We study both experimentally and numerically the convective flow in a tall vertical slot with differently heated walls. The flow is investigated for the fluid with the Prandtl number Pr=26, which is large enough to ensure the traveling waves as primary instability and small enough to prevent boundary layer convection. The flow evolution is determined on the base of the visual observations, power spectra and amplitude analysis. In the numerical simulations of two- and three-dimensional flows, we accept an assumption of an infinite fluid layer. The satisfactory agreement with experiment is observed, and the sequence of convection states is discovered. It starts with a plane-parallel flow as primary solution, which becomes unstable to two counter-propagating waves. It is followed by a tertiary three-dimensional flow in the form of wavy traveling waves. As the Grashof number is increased even further, a chaotically oscillating cellular pattern consisting of the pieces of broken waves arises. The formation of a structure in the form of the vertical rolls chaotically modulated along axes concludes this complicated picture.     

A method for studying waves with spatially localized envelopes in a class of nonlinear partial differential equations

A methodology for investigating stationary and travelling waves with spatially localized envelopes is presented. The nonlinear governing partial differential equations considered possess a constant first integral of motion, and are separable in space and time when the small parameter of the problem is set to zero. To study stationary waves, a coordinate transformation on the governing nonlinear partial differential equation is imposed which eliminates the time dependence from the problem. An amplitude modulation function is then introduced to express the response of the system at an arbitrary point as a nonlinear function of a reference response. Analytic approximations to the amplitude modulation function are developed by expressing it in power series, and asymptotically solving sets of singular functional equations at the various orders of approximation. Travelling solutions may be computed from stationary ones, by imposing appropriate Lorentz transformations. As an application of the methodology, stationary and travelling breathers of a nonlinear partial differential equation are analytically computed.     

Effect of the stiffness and inertia of a rib reinforcement on localized bending waves in semi-infinite strips

This paper reports an analytical and numerical study of localized bending waves in a thin elastic isotropic semi-infinite strip with a rib reinforment. Such waves can be considered as spatially non-uniform bending perturbations localized near free edges, similar to Rayleigh waves decaying exponentially with the distance. From the analysis of localized bending waves in thin elastic structures it may be possible to infer the presence of imperfections, cracks or inclusions, in the structure. In this study, a semi-infinite strip, simply supported on two edges and reinforced on the free one, is considered. A general solution is presented and the conditions under which localized bending waves exist are derived. The mathematical model, accompanied by numerical simulations, reveals that the presence of a reinforcement rib can suppress localized bending waves. This effect is mainly due to the stiffness coming from the rib, rather than from its added inertia terms.     

The evolution and interaction of Marangoni-Bénard solitary waves

Marangoni-Bénard convection is the process by which oscillatory waves are generated on an interface due to a change in surface tension. This process, which can be mass or temperature driven is described by a perturbed Korteweg-de Vries (KdV) equation. The evolution and interaction of solitary waves generated by Marangoni-Bénard convection is examined. The solitary wave with steady-state amplitude, which occurs when the excitation and friction terms of the perturbed KdV equation are in balance is found to second-order in the perturbation parameter. This solitary wave has a fixed amplitude, which depends on the coefficients of the perturbation terms in the governing equation. The evolution of a solitary wave of arbitrary amplitude to the steady-state amplitude is also found, to first-order in the perturbation parameter. In addition, by using a perturbation method based on inverse scattering, it is shown that the interaction of two solitary waves is not elastic with the change in wave amplitude determined. Numerical solutions of the perturbed KdV equation are presented and compared to the asymptotic solutions.     

Sur la nature de la transition à l'instationnaire d'un écoulement de convection naturelle en cavité différentiellement chauffée à grands écarts de température

Natural convection of air inside a rectangular cavity, differentially heated under large temperature gradients, is considered. The low Mach approximation equations are those obtained by Paolucci allowing for filtering of sound waves. Transition to unsteadiness is studied with numerical simulation, with a finite volume code based on a fractional time step method derived from projection methods used for incompressible flows. When the fluid physical properties are prescribed constants, transition to unsteadiness follows the classical scheme of a Hopf bifurcation. The transition is quite different when viscosity obeys Sutherland's law while the Prandtl number is kept constant. There is evidence of hysteresis, therefore the transition seems to be subcritical. In the vicinity of the transition, on the large amplitude branch, an intermittent solution is observed, with periodic bursts separating quasi-steady states.     

Localization of two-dimensional non-linear strain waves in a plate

The influence of an external medium on the evolution of two-dimensional long non-linear strain waves in an elastic plate is studied. The governing non-linear equations for longitudinal and shear waves are obtained. A threshold value of the external medium parameter is found that separates the existence of either one-dimensional (or plane) localized strain wave or two-dimensional localized strain wave. A considerable increase in the amplitude of the wave is found during the formation of the two-dimensional localized strain wave from an arbitrary initial pulse.     

Numerical analysis on flow dynamics and heat transfer of falling liquid films with interfacial waves

Flow dynamics and heat transfer of falling liquid films with interfacial waves flowing on a vertical plate have been studied with originally proposed numerical simulation method. To discretize basic equations a staggered grid fixed on a physical space is employed. A small amplitude disturbance generated at inflow boundary develops to a solitary wave which consists of a large amplitude roll wave and small amplitude capillary waves. Instantaneous streamwise velocity profiles at the wave crest and trough are very different from a laminar flow. A circulation flow occurs in the roll wave and it affects temperature distributions, especially the strong effect is observed for high Prandtl number liquids. The interfacial wave enhances the heat transfer by two kinds of effects which are a film thinning effect and a convection effect. The dominating effect depends on the Prandtl number. Received on 23 December 1998     

Structuring of suspended sediments in periodic vortex flow over a vortex ripple

The formation of spatially ordered structures in a suspended sediment under the action of two-dimensional standing surface gravity waves is studied experimentally for the first time in a rectangular vessel oscillating in the vertical direction. The parameters of the structured regions in vessels with individual vortex ripples and groups of ripples are found for the first and second wave modes. Isolated structured regions of the suspended sediment appear over the bottom topography and gradually reach the free surface. The corresponding spatial horizontal scales are determined by the sand ripple dimensions, while the vertical scale of the clouds increases with time. In all experiments, the structures formed remained unchanged during the whole interval of the fluid wave motion and disappeared when the parametric excitation of the waves stopped.     

Two-dimensional convection in a horizontal fluid layer with spatially periodic boundary temperatures

Two-dimensional thermal convection in a fluid layer confined between two horizontal rigid walls kept at spatially periodic temperatures is investigated by direct numerical simulations. With increasing the Rayleigh number, convection evolves from a steady state to a temporally chaotic flow. It is observed that the transition to the chaos occurs via quasi-periodic states with two or three basic frequencies or via sequences of period-doubling bifurcations, according to the boundary temperature distributions.     

Mathematical model of spiral waves propagating in the aorta

The spiral waves in the viscous incompressible fluid flow within an arterial vessel modeled by a thin elastic isotropic shell are studied. Asymptotic expansions are constructed for two types of spiral waves. The first type is spiral long wall waves generated (owing to the viscous fluid no-slip at the inner shell wall) by the longitudinal and twist harmonic waves that propagate along the wall. For these waves the amplitude distribution over the vessel cross-section has the form of a boundary layer localized near the inner shell surface. The second is short small-amplitude waves that practically fill the entire vessel cross-section. It is shown that for the short waves the transfer mechanismis the steady-state flow, the role of the longitudinal wall waves and the elastic characteristics of the shell being in this case insignificant.     

Wave–wave interactions in a periodic chain with quadratic nonlinearity

Two waves are studied using perturbation analysis for their interactions in an one-dimensional periodic structure with quadratic nonlinearity. A first-order multiple-scales analysis along with numerical simulations on the full chain are used to understand the interaction of two waves when one is the sub- or super-harmonic of the other. The strength of quadratic nonlinearity affects the rate at which the energy is exchanged between the two waves. Depending on parameters and energy states, the interactions between the waves are periodic or whirling and result in quasi-periodic combined propagating waves with either phase drifts or weakly phase-locking properties. The analysis suggests the possibility of the existence of emergent wave harmonics. Due to quadratic nonlinearity, a very small amplitude subharmonic or superharmonic wave mode can drift in its phase, and then burst out with a larger amplitude as it circumnavigates a separatrix. Depending on the parameters and wave numbers, the amplitude of this emergent wave burst can have varying significance.     

Nonlinear faraday waves in a parametrically excited circular cylindrical container

IntroductionIn 1 83 1 ,Faraday[1]reportedhisexperimentalobservationofsurfacewavesindifferentfluidscoveringahorizontalplatesubjectedtoaverticalvibration ,andheobservedthesurfacestandingwavesoffluidsliketheteethofaveryshortcoarsecomb .Heremarksthatthesesurfacewaveshaveafrequencyequaltoonehalfthatoftheexcitation .ThisisthefamousFaradayexperiment.WedesignatethosefluidsurfacewavesformedbyverticallyexcitationandhaveafrequencyequaltoonehalfthatoftheexcitationasFaradaywaves.FollowingthisproblemMatth…     

Plane Problem of Vibrations of an Elastic Floating Plate under Periodic External Loading

The Wiener–Hopf technique is used to construct an analytical solution of the problem of vibrations of a semiinfinite elastic floating plate under periodic external loading. The solution is obtained in explicit form ignoring draft. The dependences of the amplitudes of surface waves and iceplate deflection on the loading distribution and frequency, ice thickness, and liquid depth are studied numerically. It is established that for some types of acting load, no waves propagate in the plate and liquid and the plate vibrations are standing waves localized near the loading region. An example of such vibrations is given and a condition for the occurrence of localized vibrations is found.     

Effect of a Standing Acoustic Wave on the Development of Large-Scale Convection in a Horizontal Layer

The effect of a standing acoustic wave on the development of long-wave convective perturbations in a horizontal layer with thermally insulated boundaries is investigated. The main two-dimensional flow is determined. A nonlinear amplitude equation with spatially-periodic coefficients is derived for investigating the stability of the main flow and secondary convection flows in the neighborhood of the stability threshold. The intensity of the acoustic field is assumed to be low. It is shown that the acoustic action leads to destabilization of the layer. Plane and three-dimensional perturbations are critical at large and small Prandtl numbers, respectively. Nonlinear one-dimensional steady-state solutions of the amplitude equation are obtained and their stability is investigated.     

Hot-wire anemometry in acoustic waves

 Experimental techniques developed for the measurement of the acoustic velocity oscillation without a superimposed steady flow in gases using a hot-wire anemometer are reported. The techniques developed include amplitude and phase calibrations in standing waves.