Cauchy-Poisson Problem Research Articles

Jet from a very large, axisymmetric, surface-gravity wave

We demonstrate that gravity acting alone at large length scales (compared to the capillary length) can produce a jet from a sufficiently steep, axisymmetric surface deformation imposed on a quiescent, deep pool of liquid. Mechanistically, the jet owes it origin to the focusing of a concentric, surface wave towards the axis of symmetry, quite analogous to such focusing of capillary waves and resultant jet formation observed during bubble collapse at small scales. A weakly nonlinear theory based on the method of multiple scales in the potential flow limit is presented for a modal (single-mode) initial condition representing the solution to the primary Cauchy–Poisson problem. A pair of novel, coupled, amplitude equations are derived governing the modulation of the primary mode. For moderate values of the perturbation parameter $\epsilon$ (a measure of the initial perturbation steepness), our second-order theory captures the overshoot (incipient jet) at the axis of symmetry quite well, demonstrating good agreement with numerical simulation of the incompressible, Euler equation with gravity (Popinet 2014, Basilisk. http://basilisk.fr) and no surface tension. We demonstrate that the underlying wave focusing mechanism may be understood in terms of radially inward motion of nodal points of a linearised, axisymmetric, standing wave. This explanation rationalises the ubiquitous observation of such jets accompanying cavity collapse phenomena, spanning length scales from microns to several metres. Expectedly, our theory becomes inaccurate as $\epsilon$ approaches unity. In this strongly nonlinear regime, slender jets form with surface accelerations exceeding gravity by more than an order of magnitude. In this inertial regime, we compare the jets in our simulations with the inertial, self-similar, analytical solution by Longuet-Higgins (J. Fluid Mech., 1983, vol. 127, pp. 103–121) and find qualitative agreement with the same. This analysis demonstrates, from first principles, an example of a jet created purely under gravity from a smooth initial perturbation and provides support to the analytical model of Longuet-Higgins (J. Fluid Mech., 1983, vol. 127, pp. 103–121).

Uniqueness' failure for the finite element Cauchy-Poisson's problem

We focus on the ill posed data completion problem and its finite element approximation, when recast via the variational duplication Kohn–Vogelius artifice and the condensation Steklov–Poincaré operators. We try to understand the useful hidden features of both exact and discrete problems. When discretized with finite elements of degree one, the discrete and exact problems behave in diametrically opposite ways. Indeed, existence of the discrete solution is always guaranteed while its uniqueness may be lost. In contrast, the solution of the exact problem may not exist, but it is unique. We show how existence of the so called “weak spurious modes”, of the exact variational formulation, is source of instability and the reason why existence may fail. For the discrete problem, we find that the cause of non uniqueness is actually the occurrence of “spurious modes”. We track their fading effect asymptotically when the mesh size tends to zero. In order to restore uniqueness, we recall the discrete version of the Holmgren principle, introduced in Azaïez et al. (2011) [4], and we discuss the effect on uniqueness of the finite element mesh, using some graph theory basic material.

Cauchy Poisson problem for ocean with an elastic bottom

Cauchy Poisson problem for ocean with an elastic bottom

The Cauchy–Poisson Problem for Fluid with Shear Current and Non-Uniformly Compressed Ice Cover

The Cauchy–Poisson Problem for Fluid with Shear Current and Non-Uniformly Compressed Ice Cover

The Cauchy–Poisson Problem for Fluid with an Ice Cover in the Presence of Shear Current (Two-Dimensional Case)

The Cauchy–Poisson Problem for Fluid with an Ice Cover in the Presence of Shear Current (Two-Dimensional Case)

A nonlinear impulsive Cauchy–Poisson problem. Part 1. Eulerian description

Abstract

Initial Stage of the Finite-Amplitude Cauchy–Poisson Problem

The nonlinear Cauchy–Poisson problem for an incompressible inviscid fluid to start flowing under gravity is investigated analytically. The general nonlinear initial/boundary-value problem is formulated, including both an initial surface deflection and an initial velocity generated by a pressure impulse on the surface. Two subproblems are: (1) a finite-amplitude surface deflection released from rest; (2) the fluid is forced into motion by a pressure impulse on the initially horizontal surface. Solutions for these two subproblems are given to the leading order. One exact solution is given for the fully nonlinear initial-value problem, where a surface pressure impulse is applied on a surface with finite initial deflection. The concept of the highest non-breaking wave is illustrated by dipole acceleration fields at a state of gravitational release from rest. This is done for two phenomena: run-up of a non-breaking solitary wave on a sloping beach, and free nonlinear sloshing in an open container.

Generation of Surface Waves Due to Initial Axisymmetric Surface Disturbance in Water with a Porous Bottom

A two-dimensional Cauchy Poisson problem for water with a porous bottom generated by an axisymmetric initial surface disturbance is investigated here. The problem is formulated as an initial value problem for the velocity potential describing the motion in the fluid. The Laplace and Hankel transform techniques have been used in the mathematical analysis to obtain the form of the free surface in terms of a multiple infinite integral. This integral is then evaluated asymptotically by the method of stationary phase. The asymptotic form of the free surface is depicted graphically in a number of figures for different values of the porosity parameter and for different types of initial disturbances.

Gravity–capillary jet-like surface waves generated by an underwater bubble

Jet-like surface waves generated by an electric-spark-generated underwater bubble are experimentally studied. Three different motions of jet-like surface waves are observed depending on the inception position of the bubble ($d$: 0.28–7 mm) below the free surface and the maximum radius of the bubble ($R_{m}$: 1.5–3.6 mm). When $d/R_{m}>1.3$, the surface wave shows a simple smooth hump (case 1). When $0.82<d/R_{m}<1.3$, a single droplet or multiple droplets are pinched off sequentially or simultaneously at the tip or from some points of the jet-like surface wave (case 2). Finally, when $d/R_{m}<0.82$, a series of squirting and jetting phenomena are observed at the top of the jet-like surface wave (case 3). For case 1, a proportional relationship is found between $\unicode[STIX]{x1D70C}gh/\unicode[STIX]{x0394}p$ and $(d/R_{m})^{-4.4}$, where $\unicode[STIX]{x1D70C}$ is the density of the fluid, $g$ is the gravitational acceleration and $\unicode[STIX]{x0394}p$ is the difference between the reference atmospheric pressure and the vapour pressure inside a bubble. This proportional relationship is explained semi-analytically using a scaling argument and conservation of momentum and energy, with the help of the Kelvin impulse theory. In addition, we solve the relevant axisymmetric Cauchy–Poisson problem where the initial condition is a jet-like surface wave near its maximum height. By comparing the analytical wave solution with the observed surface wave pattern, it is found that the resultant surface waves are indeed gravity–capillary waves where both the gravity and the surface tension are equally important.

Weakly nonlinear transient waves on a shear current: ring waves and skewed Langmuir rolls

We investigate the weakly nonlinear dynamics of transient gravity waves at infinite depth under the influence of a shear current varying linearly with depth. The shear field makes this problem three-dimensional and rotational in nature, but an analytical solution is permitted via integration of the Euler equations. Although similar problems were investigated in the 1960s and 70s for special cases of resonance, this is to our knowledge the first general wave interaction (mode coupling) solution derived to second order with a shear current present. Wave interactions are integrable in a spectral convolution to yield the second-order dynamics of initial value problems. To second order, irrotational wave dynamics interacts with the background vorticity field in a way that creates new vortex structures. A notable example is the large parallel vortices which drive Langmuir circulation as oblique plane waves interact with an ocean current. We also investigate the effect on wave pairs which are misaligned with the shear current to find that similar, but skewed, vortex structures are generated in every case except when the mean wave direction is precisely perpendicular to the direction of the current. This is in contrast to a conjecture by Leibovich (Annu. Rev. Fluid Mech., vol. 15, 1983, pp. 391–427). Similar nonlinear wave–shear interactions are found to also generate near-field vortex structures in the Cauchy–Poisson problem with an initial surface elevation. These interactions create further groups of dispersive ring waves in addition to those present in linear theory. The second-order solution is derived in a general manner which accommodates any initial condition through mode coupling over a continuous wave spectrum. It is therefore applicable to a range of problems including special cases of resonance. As a by-product of the general theory, a simple expression for the Stokes drift due to a monochromatic wave propagating at oblique angle with a current of uniform vorticity is derived, for the first time to our knowledge.

Asymptotic stability of a polar vortex perturbed by harmonic waves describing atmospheric gravity waves circulating in an equatorial plane of a spherical planet

We examine stability of the vortex that represents one particular class of exact solution of a a nonlinear shallow water model describing atmospheric gravity waves circulating in an equatorial plane of a spherical planet. The mathematical model is represented by a two-dimensional free boundary Cauchy–Poisson problem on the nonstationary motion of a perfect uid around a solid circle with a sufficiently large radius so that the gravity is directed to the center of the circle. It is shown that the model admits two functionally independent nonlinear systems of shallow water equations. Two essential parameters that control stability of the vortex for both systems are identified. The order of their importance is analyzed and it is shown that one of the systems is more resistant to small perturbations and remains stable for larger range of these two parameters.

Exact and asymptotic solutions of the Cauchy–Poisson problem with localized initial conditions and a constant function of the bottom

In the paper, the asymptotic solutions for a problem of Cauchy–Poisson type with localized initial conditions are constructed. The bottom of the basin under consideration which was constant before the perturbation, is instantly perturbed at the initial time moment by a spatially localized function. Simplifications of the corresponding formulas are presented inside and outside the vicinity of the leading front, as well as in the case of a special choice of the initial condition. It is shown that, in the vicinity of the leading front, the asymptotic solution coincides with the asymptotic solution of the linear Boussinesq equation.

Initial surface disturbance on a shear current: The Cauchy-Poisson problem with a twist

We solve for the first time the classical linear Cauchy-Poisson problem—the time evolution of an initial surface disturbance—when a shear current of uniform vorticity is present beneath the surface. The solution is general, including the effects of gravity, surface tension, and constant finite depth. The particular case of an initially Gaussian disturbance of width b is studied for different values of three system parameters: a “shear Froude number” \documentclass[12pt]{minimal}\begin{document}$S\sqrt{b/g}$\end{document}Sb/g (S is the vorticity), the Bond number and the depth relative to the initial perturbation width. Different phase and group velocity in different directions yield very different wave patterns in different parameter regimes when the shear is strong, and the well-known pattern of diverging ring waves in the absence of shear can take on very different qualitative behaviours. For a given shear Froude number, both finite depth and nonzero capillary effects are found to weaken the influence of the shear on the resulting wave pattern. The various patterns are analysed and explained in light of the shear-modified dispersion relation.

Cauchy-poisson problem for a two-layer fluid with an inertial surface

This paper is concerned with the generation of waves due to initial disturbances at the upper surface of a two-layer fluid, as the upper layer is covered by an inertial surface and the lower layer extends infinitely downwards. The inertial surface is composed of thin but uniform distribution of non-interacting material. In the mathematical analysis, the Fourier and Laplace transform techniques have been utilized to obtain the depressions of the inertial surface and the interface in the form of infinite integrals. For initial disturbances concentrated at a point, the inertial surface depression and the interface depression are evaluated asymptotically for large time and distance by using the method of stationary phase. They are also depicted graphically for two types of initial disturbances and appropriate conclusions are made.

On the use of the finite fault solution for tsunami generation problems

The present study is devoted to the problem of tsunami wave generation. The main goal of this work is two-fold. First of all, we propose a simple and computationally inexpensive model for the description of the sea bed displacement during an underwater earthquake, based on the finite fault solution for the slip distribution under some assumptions on the dynamics of the rupturing process. Once the bottom motion is reconstructed, we study waves induced on the free surface of the ocean. For this purpose we consider three different models approximating the Euler equations of the water wave theory. Namely, we use the linearized Euler equations (we are in fact solving the Cauchy-Poisson problem), a Boussinesq system and a novel weakly nonlinear model. An intercomparison of these approaches is performed. The developments of the present study are illustrated on the 17 July 2006 Java event, where an underwater earthquake of magnitude 7.7 generated a tsunami that inundated the southern coast of Java.

On the generation and maintenance of waves and turbulence in simulations of free-surface turbulence

One of the challenges in numerical simulation of wave–turbulence interaction is the precise setup and maintenance of wave and turbulence fields. In this paper, we investigate techniques for the generation and suppression of specific surface wave modes, the generation of turbulence in an inhomogeneous physical domain with a wavy boundary-fitted grid, and the generation and maintenance of waves and turbulence during the complex wave–turbulence interaction process. We apply surface pressure to generate and suppress waves. Based on the solution of linearized Cauchy–Poisson problem, we derive three pressure expressions, which lead to a δ -function method, a time-segment method, and a gradual method. Numerical experiments show that these methods generate waves as specified and eliminate spurious waves effectively. The nonlinear wave effect is accounted for with a time-relaxation method. For turbulence generation, we extend the linear forcing method to an inhomogeneous physical domain with a curvilinear computational grid. Effects of force distribution and computational grid distortion are examined. For wave–turbulence interaction, we develop an algorithm to instantaneously identify specific progressive and standing waves. To precisely control the wave amplitude in a complex turbulent flow field, we further develop an energy controlling method. Finally, a simulation example of wave–turbulence interaction is presented. Results show that turbulence has unique features in the presence of waves. Velocity fluctuations are found to be strongly dependent on the wave phase; variations of these fluctuations are explained by the pressure–strain correlation associated with the wave-induced strain field.

Simple asymptotic solution of the Cauchy-Poisson problem for head waves

By using a modification of the method suggested in one of the works in 1984 by V. A. Borovikov together with M. Ya. Kel’bert, a simple asymptotic solution of the Cauchy-Poisson problem for the axially symmetric initial perturbation of power-law type is derived. It is shown that the solution thus derived describes the wave profiles of the head waves with high accuracy, in the cases of both very strong and weak dispersion.

Estimate for the errors of asymptotic solutions of the Cauchy-Poisson problem

Asymptotic solutions of a Cauchy-Poisson problem, which are known in general form, are considered for Gaussian axially symmetric initial elevations of the free surface of fluid. Errors for the approximation of asymptotic solutions are presented in dependence on large dimensionless time. These errors were found in approximation of infinitely deep fluid and for the cases in which the ratio of the typical radius of the initial perturbation to the depth of the fluid belongs to the interval between a half and eight.

Energy of tsunami waves generated by bottom motion

In the vast literature on tsunami research, few articles have been devoted to energy issues. A theoretical investigation on the energy of waves generated by bottom motion is performed here. We start with the full incompressible Euler equations in the presence of a free surface and derive both dispersive and non-dispersive shallow-water equations with an energy equation. It is shown that dispersive effects appear only at higher order in the energy budget. Then we solve the Cauchy–Poisson problem of tsunami generation for the linearized water-wave equations. Exchanges between potential and kinetic energies are clearly revealed.

The initial stage of dam-break flow

The liquid flow and the free surface shape during the initial stage of dam breaking are investigated. The method of matched asymptotic expansions is used to derive the leading-order uniform solution of the classical dam-break problem. The asymptotic analysis is performed with respect to a small parameter which characterizes the short duration of the stage under consideration. The second-order outer solution is obtained in the main flow region. This solution is not valid in a small vicinity of the intersection point between the initially vertical free surface and the horizontal rigid bottom. The dimension of this vicinity is estimated with the help of a local analysis of the outer solution close to the intersection point. Stretched local coordinates are used in this vicinity to resolve the flow singularity and to derive the leading-order inner solution, which describes the formation of the jet flow along the bottom. It is shown that the inner solution is self-similar and the corresponding boundary-value problem can be reduced to the well-known Cauchy–Poisson problem for water waves generated by a given pressure distribution along the free surface. An analysis of the inner solution reveals the complex shape of the jet head, which would be difficult to simulate numerically. The asymptotic solution obtained is expected to be helpful in the analysis of developed gravity-driven flows.