共查询到20条相似文献,搜索用时 15 毫秒
1.
Delia Ionescu-Kruse 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(9):3779-3793
We investigate the particle trajectories in a constant vorticity shallow water flow over a flat bed as periodic waves propagate on the water’s free surface. Within the framework of small amplitude waves, we find the solutions of the nonlinear differential equations system which describes the particle motion in the considered case, and we describe the possible particle trajectories. Depending on the relation between the initial data and the constant vorticity, some particle trajectories are undulating curves to the right, or to the left, others are loops with forward drift, or with backward drift, others can follow some peculiar shapes. 相似文献
2.
We study steady linear gravity waves of small amplitude travelling on a current of constant vorticity. For positive vorticity the situation resembles that of Stokes waves, but if the vorticity is large enough the particle trajectories are affected. For negative vorticity we show that there may appear internal waves and vortices, wherein the particle trajectories are not ellipses. 相似文献
3.
Dongho Chae 《Journal of Differential Equations》2006,227(2):640-651
We obtain new continuation principle of the local classical solutions of the 3D Euler equations, where the regularity condition of the direction field of the vorticiy and the integrability condition of the magnitude of the vorticity are incorporated simultaneously. The regularity of the vorticity direction field is most appropriately measured by the Triebel-Lizorkin type of norm. Similar result is also obtained for the inviscid 2D quasi-geostrophic equation. 相似文献
4.
Fattehallah Ghadi Vitoriano Ruas Mohamed Wakrim 《Journal of Computational and Applied Mathematics》2008
In this paper we present a new method to solve the 2D generalized Stokes problem in terms of the stream function and the vorticity. Such problem results, for instance, from the discretization of the evolutionary Stokes system. The difficulty arising from the lack of the boundary conditions for the vorticity is overcome by means of a suitable technique for uncoupling both variables. In order to apply the above technique to the Navier–Stokes equations we linearize the advective term in the vorticity transport equation as described in the development of the paper. We illustrate the good performance of our approach by means of numerical results, obtained for benchmark driven cavity problem solved with classical piecewise linear finite element. 相似文献
5.
We use bifurcation theory to construct small periodic gravity stratified water waves with density which depends linearly upon the pseudostream function. As a special feature the density may also decrease with depth and the waves we obtain may posses two different critical layers with cat?s eye vortices. Within the vortex, the density of the fluid has an extremum at the stagnation point. 相似文献
6.
Eugen Varvaruca 《Journal of Differential Equations》2009,246(10):4043-2949
This is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a corner of 120° or a horizontal tangent at any stagnation point about which it is supposed symmetric. Moreover, the profile necessarily has a corner of 120° if the vorticity is nonnegative near the free surface. 相似文献
7.
The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds on the free-surface profiles and on the total head are obtained under minimal assumptions about properties of solutions to the problem and the vorticity distribution. 相似文献
8.
Modulational, Benjamin-Feir, instability is studied for the down-stream evolution of surface gravity waves. An explicit solution, the soliton on finite background, of the NLS equation in physical space is used to study various phenomena in detail. It is shown that for sufficiently long modulation lengths, at a unique position where the largest waves appear, phase singularities are present in the time signal. These singularities are related to wave dislocations and lead to a discrimination between successive ‘extreme’ waves and much smaller intermittent waves. Energy flow in opposite directions through successive dislocations at which waves merge and split, causes the large amplitude difference. The envelope of the time signal at that point is shown to have a simple phase plane representation, and will be described by a symmetry breaking unfolding of the steady state solutions of NLS. The results are used together with the maximal temporal amplitude MTA, to design a strategy for the generation of extreme (freak, rogue) waves in hydrodynamic laboratories. 相似文献
9.
In this paper, we first establish a strong convergence criterion of approximate solutions for the 3D steady incompressible Euler equations. For axisymmetric flows, under the assumption that the vorticity is of one sign and uniformly bounded in L1 space, we obtain a sufficient and necessary condition for the strong convergence in of approximate solutions. Furthermore, for one-sign and L1-bounded vorticity, it is shown that if a sequence of approximate solutions concentrates at an isolated point in (r,z)-plane, then the concentration point can appear neither in the region near the axis (including the symmetry axis itself) nor in the region far away from the axis. Finally, we present an example of approximates solutions which converge strongly in by using Hill's spherical vortex. 相似文献
10.
Mats Ehrnström 《Journal of Mathematical Analysis and Applications》2008,339(1):636-643
We consider steady symmetric gravity water waves on finite depth with constant vorticity and a monotone surface profile between crests and troughs. The problem is transformed into one concerning the vertical velocity. A representation formula for the stream function in terms of the surface and the vorticity is presented, and we show that the surface can be determined from the vertical velocity. 相似文献
11.
The classical problem of capillary waves propagating at a constant velocity at the surface of a fluid of infinite depth is reexamined. The surface tension is assumed to vary along the free surface. The problem is solved numerically by series truncation. It is shown that the properties of the waves are qualitatively similar to those of waves with constant surface tension and that there are nonsymmetric waves with variable surface tension. 相似文献
12.
Joseph Majdalani Sjoerd W. Rienstra 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(2):289-308
The bidirectional vortex refers to the bipolar, coaxial swirling motion that can be triggered, for example, in cyclone separators
and some liquid rocket engines with tangential aft-end injectors. In this study, we present an exact solution to describe
the corresponding bulk motion in spherical coordinates. To do so, we examine both linear and nonlinear solutions of the momentum
and vorticity transport equations in spherical coordinates. The assumption will be that of steady, incompressible, inviscid,
rotational, and axisymmetric flow. We further relate the vorticity to some power of the stream function. At the outset, three
possible types of similarity solutions are shown to fulfill the momentum equation. While the first type is incapable of satisfying
the conditions for the bidirectional vortex, it can be used to accommodate other physical settings such as Hill’s vortex.
This case is illustrated in the context of inviscid flow over a sphere. The second leads to a closed-form analytical expression
that satisfies the boundary conditions for the bidirectional vortex in a straight cylinder. The third type is more general
and provides multiple solutions. The spherical bidirectional vortex is derived using separation of variables and the method
of variation of parameters. The three-pronged analysis presented here increases our repertoire of general mean flow solutions
that rarely appear in spherical geometry. It is hoped that these special forms will permit extending the current approach
to other complex fluid motions that are easier to capture using spherical coordinates. 相似文献
13.
Jun Cheng Jie Cang Shi-Jun Liao 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,60(3):450-478
A train of periodic deep-water waves propagating on a steady shear current with a vertical distribution of vorticity is investigated
by an analytic method, namely the homotopy analysis method (HAM). The magnitude of the vorticity varies exponentially with
the magnitude of the stream function, while remaining constant on a particular streamline. The so-called Dubreil–Jacotin transformation
is used to transfer the original exponentially nonlinear boundary-value problem in an unknown domain into an algebraically
nonlinear boundary-value problem in a known domain. Convergent series solutions are obtained not only for small amplitude
water waves on a weak current but also for large amplitude waves on a strong current. The nonlinear wave-current interaction
is studied in detail. It is found that an aiding shear current tends to enlarge the wave phase speed, sharpen the wave crest,
but shorten the maximum wave height, while an opposing shear current has the opposite effect. Besides, the amplitude of waves
and fluid velocity decay over the depth more quickly on an aiding shear current but more slowly on an opposing shear current
than that of waves on still water. Furthermore, it is found that Stokes criteria of wave breaking is still valid for waves
on a shear current: a train of propagating waves on a shear current breaks as the fiuid velocity at crest equals the wave
phase speed. Especially, it is found that the highest waves on an opposing shear current are even higher and steeper than
that of waves on still water. Mathematically, this analytic method is rather general in principle and can be employed to solve
many types of nonlinear partial differential equations with variable coefficients in science, finance and engineering. 相似文献
14.
Jun Cheng Jie Cang Shi-Jun Liao 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,7(1):450-478
A train of periodic deep-water waves propagating on a steady shear current with a vertical distribution of vorticity is investigated
by an analytic method, namely the homotopy analysis method (HAM). The magnitude of the vorticity varies exponentially with
the magnitude of the stream function, while remaining constant on a particular streamline. The so-called Dubreil–Jacotin transformation
is used to transfer the original exponentially nonlinear boundary-value problem in an unknown domain into an algebraically
nonlinear boundary-value problem in a known domain. Convergent series solutions are obtained not only for small amplitude
water waves on a weak current but also for large amplitude waves on a strong current. The nonlinear wave-current interaction
is studied in detail. It is found that an aiding shear current tends to enlarge the wave phase speed, sharpen the wave crest,
but shorten the maximum wave height, while an opposing shear current has the opposite effect. Besides, the amplitude of waves
and fluid velocity decay over the depth more quickly on an aiding shear current but more slowly on an opposing shear current
than that of waves on still water. Furthermore, it is found that Stokes criteria of wave breaking is still valid for waves
on a shear current: a train of propagating waves on a shear current breaks as the fiuid velocity at crest equals the wave
phase speed. Especially, it is found that the highest waves on an opposing shear current are even higher and steeper than
that of waves on still water. Mathematically, this analytic method is rather general in principle and can be employed to solve
many types of nonlinear partial differential equations with variable coefficients in science, finance and engineering. 相似文献
15.
Bogdan-Vasile Matioc 《Monatshefte für Mathematik》2014,174(3):459-475
We study periodic capillary and capillary-gravity waves traveling over a water layer of constant vorticity and finite depth. Inverting the curvature operator, we formulate the mathematical model as an operator equation for a compact perturbation of the identity. By means of global bifurcation theory, we then construct global continua consisting of solutions of the water wave problem which may feature stagnation points. A characterization of these continua is also included. 相似文献
16.
The finite difference approximation of a nonstationary pseudo-advected
vorticity equation is proved to yield generalized solutions to the
two-dimensional stationary Euler equations with nonvanishing
vorticity. This result is obtained by the simultaneous limiting of
lattice scale and time.Received: 15 May 2002 相似文献
17.
Bin Cheng 《Journal of Differential Equations》2011,250(2):690-709
We prove global existence and asymptotic behavior of classical solutions for two dimensional inviscid rotating shallow water system with small initial data subject to the zero relative vorticity condition. One of the key steps is a reformulation of the problem into a symmetric quasilinear Klein-Gordon system with quadratic nonlinearity, for which the global existence of classical solutions is then proved with combination of the vector field approach and the normal form method. We also probe the case of general initial data and reveal a lower bound for the lifespan that is almost inversely proportional to the size of the initial relative vorticity. 相似文献
18.
Bo Tian He Li Yi-Tian Gao 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(5):783-790
The Broer–Kaup system is among the important integrable models for the shallow water waves. For a (2+1)-dimensional Broer–Kaup system and with symbolic computation, we present some similarity solutions, which are expressible in terms of the Jacobian elliptic functions and second Painlevé transcendent. Our results are in agreement with the Painlevé conjecture.Received: February 26, 2003; revised: August 11, 2003 相似文献
19.
In this paper we consider the steady water wave problem for waves that possess a merely \(L_r\) -integrable vorticity, with \(r\in (1,\infty )\) being arbitrary. We first establish the equivalence of the three formulations – the velocity formulation, the stream function formulation, and the height function formulation – in the setting of strong solutions, regardless of the value of \(r\) . Based upon this result and using a suitable notion of weak solution for the height function formulation, we then establish, by means of local bifurcation theory, the existence of small-amplitude capillary and capillary–gravity water waves with an \(L_r\) -integrable vorticity. 相似文献
20.
Walter Craig David Lannes Catherine Sulem 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2012
This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems. 相似文献