Nonlinear Travelling Wave Solutions Research Articles

Exact coherent states with hairpin-like vortex structure in channel flow

Hairpin vortices are widely studied as an important structural aspect of wall turbulence. The present work describes, for the first time, nonlinear travelling wave solutions to the Navier–Stokes equations in the channel flow geometry – exact coherent states (ECS) – that display hairpin-like vortex structure. This solution family comes into existence at a saddle-node bifurcation at Reynolds number $Re=666$. At the bifurcation, the solution has a highly symmetric quasi-streamwise vortex structure similar to that reported for previously studied ECS. With increasing distance from the bifurcation, however, both the upper and lower branch solutions develop a vortical structure characteristic of hairpins: a spanwise-oriented ‘head’ near the channel centreplane where the mean shear vanishes connected to counter-rotating quasi-streamwise ‘legs’ that extend toward the channel wall. At $Re=1800$, the upper branch solution has mean and Reynolds shear-stress profiles that closely resemble those of turbulent mean profiles in the same domain.

Dynamics of thin liquid films flowing down the uniformly heated/cooled cylinder with wall slippage

Coating liquid films on cylinders occurs widely in nature, material sciences, and industrial processes. A typical example is the fabrication of thin solid films, which could be achieved by drawing the heated cylinder from the liquid pool and followed by a drying process. However, in some cases, the surface of the coated cylinder is actually not smooth and exhibits hydrophobicity where the slip effect may come into play. Therefore, in this paper, we study the dynamics of thin liquid films flowing down the uniformly heated or cooled cylinder in the presence of wall slippage. The thermocapillary and slip effects on the thin film flows are examined by a thin film model. Linear stability analysis (LSA) indicates that when the cylinder is heated, thermocapillary effect enhances the Rayleigh-Plateau instability, while when the cylinder is cooled, thermocapillary effect suppresses this instability. However, we find that wall slippage always promotes the capillary instability, and when M1+2β1+3β<-13, where M and β are the reduced Marangoni number and dimensionless slip length, the system are always linearly stable. The linear wave speed given by cL=1+2β, is promoted by the wall slippage but independent of the thermocapillary effect. In addition, nonlinear traveling-wave study shows that the size and moving speed of the sliding droplets are promoted for a heated cylinder and reduced for a cooled cylinder, respectively. The size and moving speed of the sliding droplets are enhanced by wall slippage. Finally, direct numerical simulations (DNS) of the full thin film model clearly demonstrate the role of thermocapillary and slip effects, agreeing well with the linear stability analysis and the nonlinear traveling-wave solutions. Our results offer insight into the influence of wall slippage on the dynamics of thin film flows coating on cylinders in non-isothermal environments.

Finite-amplitude gravity waves in the atmosphere: travelling wave solutions

Wentzel–Kramers–Brillouin theory was employed by Grimshaw (Geophys. Fluid Dyn., vol. 6, 1974, pp. 131–148) and Achatz et al. (J. Fluid Mech., vol. 210, 2010, pp. 120–147) to derive modulation equations for non-hydrostatic internal gravity wave packets in the atmosphere. This theory allows for wave packet envelopes with vertical extent comparable to the pressure scale height and for large wave amplitudes with wave-induced mean-flow speeds comparable to the local fluctuation velocities. Two classes of exact travelling wave solutions to these nonlinear modulation equations are derived here. The first class involves horizontally propagating wave packets superimposed over rather general background states. In a co-moving frame of reference, examples from this class have a structure akin to stationary mountain lee waves. Numerical simulations corroborate the existence of nearby travelling wave solutions under the pseudo-incompressible model and reveal better than expected convergence with respect to the asymptotic expansion parameter. Travelling wave solutions of the second class also feature a vertical component of their group velocity but exist under isothermal background stratification only. These waves include an interesting nonlinear wave–mean-flow interaction process: a horizontally periodic wave packet propagates vertically while draining energy from the mean wind aloft. In the process it decelerates the lower-level wind. It is shown that the modulation equations apply equally to hydrostatic waves in the limit of large horizontal wavelengths. Aside from these results of direct physical interest, the new nonlinear travelling wave solutions provide a firm basis for subsequent studies of nonlinear internal wave instability and for the design of subtle test cases for numerical flow solvers.

Nonlinear Propagation of Positron-Acoustic Periodic Travelling Waves in a Magnetoplasma with Superthermal Electrons and Positrons

The nonlinear propagation of positron acoustic periodic (PAP) travelling waves in a magnetoplasma composed of dynamic cold positrons, superthermal kappa distributed hot positrons and electrons, and stationary positive ions is examined. The reductive perturbation technique is employed to derive a nonlinear Zakharov–Kuznetsov equation that governs the essential features of nonlinear PAP travelling waves. Moreover, the bifurcation theory is used to investigate the propagation of nonlinear PAP periodic travelling wave solutions. It is found that kappa distributed hot positrons and electrons provide only the possibility of existence of nonlinear compressive PAP travelling waves. It is observed that the superthermality of hot positrons, the concentrations of superthermal electrons and positrons, the positron cyclotron frequency, the direction cosines of wave vector k along the z-axis, and the concentration of ions play pivotal roles in the nonlinear propagation of PAP travelling waves. The present investigation may be used to understand the formation of PAP structures in the space and laboratory plasmas with superthermal hot positrons and electrons.

Travelling wave states in pipe flow

In this paper, we have found two new nonlinear travelling wave solutions in pipe flows. We investigate possible asymptotic structures at large Reynolds number $R$ when wavenumber is independent of $R$ and identify numerically calculated solutions as finite $R$ realizations of a nonlinear viscous core (NVC) state that collapses towards the pipe centre with increasing $R$ at a rate $R^{-1/4}$. We also identify previous numerically calculated states as finite $R$ realizations of a vortex wave interacting (VWI) state with an asymptotic structure similar to the ones in channel flows studied earlier by Hall &amp; Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In addition, asymptotics suggests the possibility of a VWI state that collapses towards the pipe centre like $R^{-1/6}$, though this remains to be confirmed numerically.

Nonlinear Dynamics and Exact Traveling Wave Solutions of the Higher-Order Nonlinear Schrödinger Equation with Derivative Non-Kerr Nonlinear Terms

By using the method of dynamical system, the exact travelling wave solutions of the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms are studied. Based on this method, all phase portraits of the system in the parametric space are given with the aid of the Maple software. All possible bounded travelling wave solutions, such as solitary wave solutions, kink and anti-kink wave solutions, and periodic travelling wave solutions, are obtained, respectively. The results presented in this paper improve the related previous conclusions.

Exact coherent states and connections to turbulent dynamics in minimal channel flow

Several new families of nonlinear three-dimensional travelling wave solutions to the Navier–Stokes equation, also known as exact coherent states, are computed for Newtonian plane Poiseuille flow. The symmetries and streak/vortex structures are reported and their possible connections to critical layer dynamics are examined. While some of the solutions clearly display fluctuations that are localized around the critical layer (the surface on which the streamwise velocity matches the wave speed of the solution), for others this connection is not as clear. Dynamical trajectories along unstable directions of the solutions are computed. Over certain ranges of Reynolds number, two solution families are shown to lie on the basin boundary between laminar and turbulent flow. Direct comparison of nonlinear travelling wave solutions to turbulent flow in the same channel is presented. The state-space dynamics of the turbulent flow is organized around one of the newly identified travelling wave families, and in particular the lower-branch solutions of this family are closely approached during transient excursions away from the dominant behaviour. These observations provide a firm dynamical-systems foundation for prior observations that minimal channel turbulence displays time intervals of ‘active’ turbulence punctuated by brief periods of ‘hibernation’ (see, e.g., Xi &amp; Graham, Phys. Rev. Lett., vol. 104, 2010, 218301). The hibernating intervals are approaches to lower-branch nonlinear travelling waves. Representing these solutions on a Prandtl–von Kármán plot illustrates how their bulk flow properties are related to those of Newtonian turbulence as well as the universal asymptotic state called maximum drag reduction (MDR) found in viscoelastic turbulent flow. In particular, the lower- and upper-branch solutions of the family around which the minimal channel dynamics is organized appear to approach the MDR asymptote and the classical Newtonian result respectively, in terms of both bulk velocity and mean velocity profile.

Exact Travelling Wave Solutions for Stochastic Fractional Hirota-Satsuma Coupled KdV Equations

A modied fractional sub-equation method is used for constructing exact travelling wave solutions of nonlinear stochastic fractional partial dierential equations. The main idea of this method is to take full advantage of the fractional Riccati equation, which has many exact solutions. Moreover, white noise functional solutions are obtained for the Wick-type stochastic fractional Hirota-Satsuma coupled KdV equations via Hermite transform and white noise analysis. These solutions include stochastic exponential decay, soliton, and periodic wave solutions.

Distinct Organizational States of Fully Developed Turbulent Pipe Flow

Organizational states of turbulence are identified through novel analysis of large scale pipe flow experiments at a Reynolds number of 35 000. The distinct states are revealed by an azimuthal decomposition of the two-point spatial correlation of the streamwise velocity fluctuation. States with dominant azimuthal wave numbers corresponding to k_{θ}=2,3,4,5,6 are discovered and their structure revealed as a series of alternately rotating quasistreamwise vortices. Such organizational states are highly reminiscent of the nonlinear traveling wave solutions previously identified at Reynolds numbers an order of magnitude lower.

Shear-imposed falling film

AbstractThe study of a film falling down an inclined plane is revisited in the presence of imposed shear stress. Earlier studies regarding this topic (Smith, J. Fluid Mech., vol. 217, 1990, pp. 469–485; Wei, Phys. Fluids, vol. 17, 2005a, 012103), developed on the basis of a low Reynolds number, are extended up to moderate values of the Reynolds number. The mechanism of the primary instability is provided under the framework of a two-wave structure, which is normally a combination of kinematic and dynamic waves. In general, the primary instability appears when the kinematic wave speed exceeds the speed of dynamic waves. An equality criterion between their speeds yields the neutral stability condition. Similarly, it is revealed that the nonlinear travelling wave solutions also depend on the kinematic and dynamic wave speeds, and an equality criterion between the speeds leads to an analytical expression for the speed of a family of travelling waves as a function of the Froude number. This new analytical result is compared with numerical prediction, and an excellent agreement is achieved. Direct numerical simulations of the low-dimensional model have been performed in order to analyse the spatiotemporal behaviour of nonlinear waves by applying a constant shear stress in the upstream and downstream directions. It is noticed that the presence of imposed shear stress in the upstream (downstream) direction makes the evolution of spatially growing waves weaker (stronger).

Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models

Fundamental diagrams of vehicular traffic flow are generally multi-valued in the congested flow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical Payne-Whitham model or the inhomogeneous Aw-Rascle-Zhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multi-valued parts in the fundamental diagram correspond precisely to jamiton-dominated solutions. This study shows that transitions from function-valued to set-valued parts in a fundamental diagram arise naturally in well-known second order models. As a particular consequence, these models intrinsically reproduce traffic phases.

Self-sustained nonlinear waves in traffic flow

In analogy to gas-dynamical detonation waves, which consist of a shock with an attached exothermic reaction zone, we consider herein nonlinear traveling wave solutions to the hyperbolic ("inviscid") continuum traffic equations. Generic existence criteria are examined in the context of the Lax entropy conditions. Our analysis naturally precludes traveling wave solutions for which the shocks travel downstream more rapidly than individual vehicles. Consistent with recent experimental observations from a periodic roadway [Y. Sugiyama, N. J. Phys. 10, 033001 (2008)], our numerical calculations show that nonlinear traveling waves are attracting solutions, with the time evolution of the system converging toward a wave-dominated configuration. Theoretical principles are elucidated by considering examples of traffic flow on open and closed roadways.

Non-travelling wave solutions of two (2 + 1)-dimensional nonlinear mathematical physics equations

The overall objective of this paper is to find and analyze the new nonlinear travelling wave solutions of two nonlinear mathematical physics equations: the (2 + 1)-dimensional Perturbed-KdV equation and the (2 + 1)-dimensional breaking soliton equation. In this paper, based on symbolic computation and the further improved F-expansion method, many new non-travelling wave solutions of the two equations are obtained and their properties are studied by analyzing their figures.

The Pertubative and Poincare Stability of the Travelling Wave Solution of the Logistic Growth with a Non-Linear Advection Model

We have obtained new evidence for the stability of traveling wave solutions for the perturbation growth model nonlinear logistic beradveksi simple. We also show that the solution with negative velocity close to zero critical point is stable, while a solution which is close to another crucial point is only stable if the velocity coordinates of the wave is greater than the product of the limit and parameter logistic curve adveksinya references. Keywords: logistic growth beradveksi nonlinear traveling wave solutions; the stability of the perturbation; the stability of the Poincare.

Exact nonlinear travelling waves in rotating systems

Linear and nonlinear exact travelling wave solutions for an exponentially stratified, incompressible, rotating fluid is studied using both numerical and analytical techniques. We observe that these waves are not periodic in general. However, waves propagating strictly either in the horizontal direction or in the vertical direction are periodic.

Three-dimensional traveling-wave solutions in plane Couette flow

Nonlinear three-dimensional time-dependent solution branches are obtained in plane Couette flow modified by plane Poiseuille flow component. It is found that as the Poiseuille component is added a branch of time-dependent solutions is produced from the time-independent solution branch in plane Couette flow, and that there exists a second branch of time-dependent solutions in the form of a closed loop inside the primary time-dependent solution branch. The second branch intersects the line of zero plane Poiseuille flow component at two points with nonvanishing phase velocity for higher Reynolds numbers, creating shape preserving nonlinear traveling-wave solutions in plane Couette flow.

An invariant manifold approach for studying waves in a one-dimensional array of non-linear oscillators

We consider a one-dimensional linear spring-mass array coupled to a one-dimensional array of uncoupled pendula. The principal aim of this study is to investigate the non-linear dynamics of this large-scale system in the limit of weak non-linearities, i.e. when the (fast) non-linear pendulum effects are small compared to the underlying (slow) linear dynamics of the linear spring-mass chain. We approach the dynamics in the context of invariant manifolds of motion. In particular, we prove the existence of an invariant manifold containing the (predominantly) slow dynamics of the system, with the fast pendulum dynamics providing small perturbations to the motions on the invariant manifold. By restricting the motion on the slow invariant manifold and performing asymptotic analysis we prove that the non-linear large-scale system possesses propagation and attenuation zones (PZs and AZs) in the frequency domain, similarly to the corresponding zones of the linearized system. Inside PZs non-linear travelling wave solutions exist, whereas in AZs only attenuating waves are permissible.

Nonlinear coupling of a superluminal electromagnetic wave to a relativistic electron beam

The nonlinear propagation of a superluminal, linearly polarized electromagnetic wave in the presence of a relativistic cold electron beam is investigated. At large amplitudes the wave couples to the electron-beam plasma mode owing to two important nonlinear effects, namely the relativistic variation of the electron mass and the excitation of longitudinal space charge fields by strong v × B forces. The nonlinear propagation equations for the coupled electromagnetic and longitudinal waves are derived within the context of a relativistic cold-fluid model. Nonlinear travelling-wave solutions are sought to describe the saturated state of the coupled system. Using Hamiltonian techniques, a wide variety of solutions are obtained and their characteristics discussed.

Arbitrary amplitude circularly polarized electromagnetic waves in electron positron magnetoplasmas

A class of exact nonlinear traveling wave solutions of the relativistic cold two-fluid model describing a one dimensional magnetized electron-positron plasma is presented. The solutions, which are circularly polarized transverse electromagnetic waves of arbitrary amplitude, propagate along the magnetic field and can be classified into nonlinear fast and slow waves.

Nonlinear effects involving whistler wave propagation in the magnetosphere

We consider nonlinear effects associated with the propagation of a quasi‐monochromatic whistler wave packet along the inhomogeneous geomagnetic field. We show that under the specific conditions of the Siple experiment the whistler packet is subject to the modulational instability, resulting in longitudinal and transverse modulation of the envelope amplitude and the excitation of quasi‐neutral density perturbations akin to ion acoustic waves. In particular, we examine the behavior of the wave packet in the vicinity of the region where the group velocity of the packet is a maximum. In general, the wave packet is subject to the competing effects of dispersive broadening and the nonlinear tendency to steepen. In the vicinity of the region of interest the dispersive broadening is small, and the tendency for steepening prevails. This results in the creation of anomolously narrow solitary waves. We analyze the situation analytically using a modification of the nonlinear Schrodinger equation. We evaluate the conditions required for the onset of the modulational instability and find its typical growth rate. We then construct nonlinear traveling wave solutions. These take the form of solitons, which move toward the rear edge of the wave packet. Because the dispersive effects are small, these solitons are narrow structures with steep boundaries. We then analyze the possibility that these solitons act as drivers for triggered emission. The interaction of resonant particles with the wave at the soliton leads to particle bunching in velocity. The resulting current serves as a source term for monochromatic radiation, the frequency of which changes with time as the bunches move in the inhomogeneous geomagnetic field.