Nonlinear wave packets in the Kelvin-Helmholtz instability

1979 ◽  
Vol 290 (1377) ◽  
pp. 639-681 ◽  
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Density Ratio ◽  
Nonlinear Effects ◽  
Amplitude Equation ◽  
Two Dimensions ◽  
Spatial Modulation ◽  
Linear Solution ◽  
Nonlinear Stabilization ◽  
Nonlinear Solutions

The instability of two layers of immiscible inviscid and incompressible fluids in relative motion is studied with allowance for small, but finite, disturbances and for spatial as well as temporal development. By using the method of multiple scaling, a generalized formulation of the amplitude equation is obtained, applicable to both stable and marginally unstable regions of parameter space. Of principal concern is the neighbourhood of the critical point for instability, where weakly nonlinear solutions can be found for arbitrary initial conditions. Among the analytical results, it is shown that (1) the nonlinear effects can be stabilizing or destabilizing depending on the density ratio, (2) the existence of purely spatial instability depends upon the frame of reference, the density ratio, and whether the nonlinear effects are stabilizing, (3) exact nonlinear solutions of the amplitude equation exist representing modulations of permanent form travelling faster than the signal velocity of the linear equation (in particular, a solution is found that represents a solitary wave packet), and (4) the linear solution to the impulsive initial value problem has 'fronts’ which travel with the two (multiple) values of the group velocity (the packet as a whole moves with the mean of the two values). Numerical solutions of the amplitude equation (a nonlinear, unstable Klein-Gordon equation) are also presented for the case of nonlinear stabilization. These show that the development of a localized disturbance, in one or two dimensions, is highly dependent on the precise form of the initial conditions, even when the initial amplitude is very small. The exact solutions mentioned above play an important role in this development. The numerical experiments also show that the familiar uniform solution, an oscillatory function of time only, is unstable to spatial modulation if the amplitude of oscillation is large enough.

Nonlinear temporal-spatial modulation of near-planar Rayleigh waves in shear flows: formation of streamwise vortices

1993 ◽  
Vol 256 ◽  
pp. 685-719 ◽  
Author(s):  
Xuesong Wu
Keyword(s):  
Shear Flows ◽  
Numerical Solutions ◽  
Broad Class ◽  
Nonlinear Effects ◽  
Amplitude Equation ◽  
Spatial Modulation ◽  
Linear Growth Rate ◽  
Instability Wave ◽  
Streamwise Vortices ◽  
Mean Flow

The nonlinear temporal-spatial modulation of a near-planar Rayleigh instability wave is studied. The amplitude of the wave is allowed to be a slowly varying function of spanwise position as well as of time (or streamwise variable in the spatial evolution case). It is shown that the development of the disturbance is controlled by critical-layer nonlinear effects when the linear growth rate decreases to O(ε⅖), where ε is the magnitude of the disturbance. Nonlinear interactions influence the evolution by producing spanwise dependent mean-flow distortions. The evolution is governed by an integro-partial-differential equation containing history-dependent nonlinear terms of Hickernell (1984) type. A notable feature of the amplitude equation is that the highest derivative with respect to spanwise position appears in the nonlinear terms. These terms are associated with three-dimensionality. The possible properties of the amplitude equation are discussed. Numerical solutions show that a disturbance initially centred at a spanwise position can propagate laterally to form concentrated, quasi-periodic streamwise vortices. This qualitatively captures the phenomena observed in experiments. The focusing of vorticity may be associated with a localized singularity which can occur at a finite distance downstream or within a finite time. It is noted that the amplitude equation is rather generic and applies to a broad class of shear flows which is inviscidly unstable.

scholarly journals Nonlinear higher-order spectral solution for a two-dimensional moving load on ice

2009 ◽  
Vol 621 ◽  
pp. 215-242 ◽  
Author(s):  
FÉLICIEN BONNEFOY ◽  
MICHAEL H. MEYLAN ◽  
PIERRE FERRANT
Keyword(s):  
Critical Speed ◽  
Moving Boundary ◽  
Moving Load ◽  
Phase Speed ◽  
Nonlinear Effects ◽  
Higher Order ◽  
Two Dimensions ◽  
Linear Solution ◽  
Nonlinear Solution ◽  
Bernoulli’S Equation

We calculate the nonlinear response of an infinite ice sheet to a moving load in the time domain in two dimensions, using a higher-order spectral method. The nonlinearity is due to the moving boundary, as well as the nonlinear term in Bernoulli's equation and the elastic plate equation. We compare the nonlinear solution with the linear solution and with the nonlinear solution found by Parau & Dias (J. Fluid Mech., vol. 460, 2002, pp. 281–305). We find good agreement with both solutions (with the correction of an error in the Parau & Dias 2002 results) in the appropriate regimes. We also derive a solitary wavelike expression for the linear solution – close to but below the critical speed at which the phase speed has a minimum. Our model is carefully validated and used to investigate nonlinear effects. We focus in detail on the solution at a critical speed at which the linear response is infinite, and we show that the nonlinear solution remains bounded. We also establish that the inclusion of nonlinearities leads to significant new behaviour, which is not observed in the linear solution.

On the formation of Moore curvature singularities in vortex sheets

1999 ◽  
Vol 378 ◽  
pp. 233-267 ◽  
Author(s):  
STEPHEN J. COWLEY ◽  
GREG R. BAKER ◽  
SALEH TANVEER
Keyword(s):  
Complex Plane ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Effects ◽  
Vortex Sheet ◽  
Finite Amplitude ◽  
Analytical Continuation ◽  
Curvature Singularity ◽  
Singularity Formation ◽  
Vortex Sheets

Moore (1979) demonstrated that the cumulative influence of small nonlinear effects on the evolution of a slightly perturbed vortex sheet is such that a curvature singularity can develop at a large, but finite, time. By means of an analytical continuation of the problem into the complex spatial plane, we find a consistent asymptotic solution to the problem posed by Moore. Our solution includes the shape of the vortex sheet as the curvature singularity forms. Analytic results are confirmed by comparison with numerical solutions. Further, for a wide class of initial conditions (including perturbations of finite amplitude), we demonstrate that 3/2-power singularities can spontaneously form at t=0+ in the complex plane. We show that these singularities propagate around the complex plane. If two singularities collide on the real axis, then a point of infinite curvature develops on the vortex sheet. For such an occurrence we give an asymptotic description of the vortex-sheet shape at times close to singularity formation.

Nonlinear drift phase-space structures

1992 ◽  
Vol 47 (3) ◽  
pp. 401-429
Author(s):  
Sharadini Rath ◽  
P. K. Kaw
Keyword(s):  
Phase Space ◽  
Numerical Solutions ◽  
Physical Space ◽  
Nonlinear Effects ◽  
Two Dimensions ◽  
Space Structures ◽  
Stationary States ◽  
Relative Importance ◽  
Poisson System ◽  
The Magnetic Field

The collisionless Vlasov–Poisson system in the drift approximation is examined for the existence of maximum-entropy nonlinear coherent solutions in the steady state. Two major nonlinear effects are taken into account. The first is the velocity-space trapping of particles, leading to closed trajectories in phase space. The second is the physical-space trapping of particles, leading to closed trajectories in the plane perpendicular to the magnetic field. The regions of validity of these nonlinearities are discussed and their relative importance demonstrated. Numerical solutions of the equations describing the nonlinear stationary states in one and two dimensions are presented.

scholarly journals Dependence of Enstrophy Transport and Mixed Mass on Dimensionality and Initial Conditions in the Richtmyer–Meshkov Instability Induced Flows1

2020 ◽  
Vol 142 (12) ◽  
Author(s):  
Ye Zhou ◽  
Michael Groom ◽  
Ben Thornber
Keyword(s):  
Inertial Confinement Fusion ◽  
Initial Conditions ◽  
Broad Band ◽  
Initial Perturbation ◽  
Density Ratio ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Late Time ◽  
Production Term

Abstract This paper presents a comparative study of the enstrophy budget and mixed mass between two- and three-dimensional flows induced by Richtmyer–Meshkov instability (RMI). Specifically, the individual contributions to the enstrophy budget due to the production from baroclinicity and from vortex stretching (which vanishes in two-dimensional (2D) flow) are delineated. This is enabled by a set of two- and three-dimensional computations at Atwood 0.5 having both narrow- and broad-band perturbations. A further three-dimensional (3D) computation is conducted at Atwood 0.9 using an identical narrowband perturbation to the Atwood 0.5 case to examine the sensitivity to density ratio. The mixed mass is also considered with the goal to obtain insight on how faithfully a simplified calculation performed in two dimensions can capture the mixed mass for an inertial confinement fusion (ICF) or other practical application. It is shown that the late time power law decay of variable density enstrophy is substantially different in two and three dimensions for the narrowband initial perturbation. The baroclinic production term is negligible in three dimensions (aside from the initial shock interaction), as vortex stretching is larger by two orders of magnitude. The lack of vortex stretching considerably reduces the decay rate in both narrowband and broadband perturbations in two dimensions. In terms of mixed mass, the lack of vortex stretching reduces the mixed mass in two dimensions compared to three in all cases. In the broadband cases, the spectral bandwidth in the 2D case is wider; hence, there is a longer time period of sustained linear growth which reduces the normalized mixed mass further.

How strain and spin may make a star bi-polar

2014 ◽  
Vol 746 ◽  
pp. 332-367 ◽  
Author(s):  
Lawrence K. Forbes
Keyword(s):  
Point Source ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Effects ◽  
Solution Technique ◽  
Ambient Fluid ◽  
Apparent Contradiction ◽  
Spectral Solution ◽  
Linearized Theory ◽  
The One

AbstractA previous study by Forbes (ANZIAM J., vol. 53, 2011, pp. 87–121) has argued that, when a light fluid is injected from a point source into a heavier ambient fluid, the interface between them is most unstable to perturbations at the lowest spherical mode. This means that, regardless of initial conditions, the outflow from a point source eventually becomes a one-sided jet. However, two-sided (bi-polar) outflows are nevertheless often observed in astrophysics, in apparent contradiction to this prediction. While there are many possible explanations for this fact, the present paper considers the effect of a straining flow in the ambient fluid. In addition, solid-body rotation in the inner fluid is also accounted for, in a Boussinesq viscous model. It is shown analytically that there are circumstances under which straining flow alone is sufficient to convert the one-sided jet into a genuine bi-polar outflow, in linearized theory. This is confirmed in a numerical solution of a viscous model of the flow, based on a spectral solution technique that accounts for nonlinear effects. Rotation can also generate flows that are two-sided, and this is likewise revealed through an asymptotic analysis and numerical solutions of the nonlinear equations.

Nonlinear Sloshing in Fixed and Vertically Excited Containers

2002 ◽  
Author(s):  
Jannette B. Frandsen ◽  
Alistair G. L. Borthwick
Keyword(s):  
Perturbation Theory ◽  
Free Surface ◽  
Small Perturbation ◽  
Numerical Solutions ◽  
Vertical Direction ◽  
Nonlinear Effects ◽  
Breaking Waves ◽  
Surface Flow ◽  
Nonlinear Behaviour ◽  
Computational Domain

Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model analyses two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free-surface and bed boundary is applied to map the moving free surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. The small-amplitude free-surface predictions in the fixed and vertically excited tanks compare well with 2nd order small perturbation theory. For stable steep waves in the vertically excited tank, the free-surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow exponentially, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep non-breaking waves. It is computationally quick and accurate, and there is no need for free surface smoothing because of the σ-transformation.

Evolution of clusters of sedimenting low-Reynolds-number particles with Oseen interactions

2008 ◽  
Vol 603 ◽  
pp. 63-100 ◽  
Author(s):  
G. SUBRAMANIAN ◽  
DONALD L. KOCH
Keyword(s):  
Reynolds Number ◽  
Velocity Field ◽  
Aspect Ratio ◽  
Evolution Equation ◽  
Similarity Solution ◽  
Initial Conditions ◽  
Radial Distance ◽  
Two Dimensions ◽  
Number Density ◽  
Cluster Radius

A theoretical framework is developed to describe, in the limit of small but finite Re, the evolution of dilute clusters of sedimenting particles. Here, Re =aU/ν is the particle Reynolds number, where a is the radius of the spherical particle, U its settling velocity, and ν the kinematic viscosity of the suspending fluid. The theory assumes the disturbance velocity field at sufficiently large distances from a sedimenting particle, even at small Re, to possess the familiar source--sink character; that is, the momentum defect brought in via a narrow wake behind the particle is convected radially outwards in the remaining directions. It is then argued that for spherical clusters with sufficiently many particles, specifically with N much greater than O(R0U/ν), the initial evolution is strongly influenced by wake-mediated interactions; here, N is the total number of particles, and R0 is the initial cluster radius. As a result, the cluster first evolves into a nearly planar configuration with an asymptotically small aspect ratio of O(R0U/N ν), the plane of the cluster being perpendicular to the direction of gravity; subsequent expansion occurs with an unchanged aspect ratio. For relatively sparse clusters with N smaller than O(R0U/ν), the probability of wake interactions remains negligible, and the cluster expands while retaining its spherical shape. The long-time expansion in the former case, and that for all times in the latter case, is driven by disturbance velocity fields produced by the particles outside their wakes. The resulting interactions between particles are therefore mutually repulsive with forces that obey an inverse-square law. The analysis presented describes cluster evolution in this regime. A continuum representation is adopted with the clusters being characterized by a number density field (n(r, t)), and a corresponding induced velocity field (u (r, t)) arising on account of interactions. For both planar axisymmetric clusters and spherical clusters with radial symmetry, the evolution equation admits a similarity solution; either cluster expands self-similarly for long times. The number density profiles at different times are functions of a similarity variable η = (r/t1/3), r being the radial distance away from the cluster centre, and t the time. The radius of the expanding cluster is found to be of the form Rcl (t) = A (ν a)1/3N1/3t1/3, where the constant of proportionality, A, is determined from an analytical solution of the evolution equation; one finds A = 1.743 and 1.651 for planar and spherical clusters, respectively. The number density profile in a planar axisymmetric cluster is also obtained numerically as a solution of the initial value problem for a canonical (Gaussian) initial condition. The numerical results compare well with theoretical predictions, and demonstrate the asymptotic stability of the similarity solution in two dimensions for long times, at least for axisymmetric initial conditions.

Oscillatory flow past a circular cylinder in a rotating frame

1997 ◽  
Vol 345 ◽  
pp. 101-131
Author(s):  
M. D. KUNKA ◽  
M. R. FOSTER
Keyword(s):  
Circular Cylinder ◽  
Steady Flow ◽  
Stagnation Point ◽  
Oscillatory Flow ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Temporal Integration ◽  
Oncoming Flow ◽  
Flow Past ◽  
Rear Stagnation Point

Because of the importance of oscillatory components in the oncoming flow at certain oceanic topographic features, we investigate the oscillatory flow past a circular cylinder in an homogeneous rotating fluid. When the oncoming flow is non-reversing, and for relatively low-frequency oscillations, the modifications to the equivalent steady flow arise principally in the ‘quarter layer’ on the surface of the cylinder. An incipient-separation criterion is found as a limitation on the magnitude of the Rossby number, as in the steady-flow case. We present exact solutions for a number of asymptotic cases, at both large frequency and small nonlinearity. We also report numerical solutions of the nonlinear quarter-layer equation for a range of parameters, obtained by a temporal integration. Near the rear stagnation point of the cylinder, we find a generalized velocity ‘plateau’ similar to that of the steady-flow problem, in which all harmonics of the free-stream oscillation may be present. Further, we determine that, for certain initial conditions, the boundary-layer flow develops a finite-time singularity in the neighbourhood of the rear stagnation point.

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

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