Linear water waves with vorticity: Rotational features and particle paths

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.     

Existence of capillary-gravity water waves with piecewise constant vorticity

In this paper we construct small-amplitude periodic capillary-gravity water waves with a piecewise constant vorticity distribution. They describe water waves traveling on superposed linearly sheared currents that have different vorticities. This is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump. The solutions of the diffraction problem, found by using local bifurcation theory, are the desired solutions of the hydrodynamical problem.     

On frequency–amplitude dependences for surface and internal standing waves

The analytic approach proposed by Sekerzh-Zenkovich [On the theory of standing waves of finite amplitude, Dokl. Akad. Nauk USSR 58 (1947) 551–554] is developed in the present study of standing waves. Generalizing the solution method, a set of standing wave problems are solved, namely, the infinite- and finite-depth surface standing waves and the infinite- and finite-depth internal standing waves. Two-dimensional wave motion of an irrotational incompressible fluid in a rectangular domain is considered to study weakly nonlinear surface and internal standing waves. The Lagrangian formulation of the problems is used and the fifth-order perturbation solutions are determined. Since most of the approximate analytic solutions to these problems were obtained using the Eulerian formulation, the comparison of the results, as an example the analytic frequency–amplitude dependences, obtained in Lagrangian variables with the corresponding ones known in Eulerian variables has been carried out in the paper. The analytic frequency–amplitude dependences are in complete agreement with previous results known in the literature. Computer algebra procedures were written for the construction of asymptotic solutions. The application of the model constructed in Lagrangian formulation to a set of different problems shows the ability to correctly reproduce and predict a wide range of situations with different characteristics and some advantages of Lagrangian particle models (for example, the bigger radius of convergence of an expansion parameter than in Eulerian variables, simplification of the boundary conditions, parametrization of a free boundary).     

Nonlinear Wave–Current Interactions in Shallow Water

We study here the propagation of long waves in the presence of vorticity. In the irrotational framework, the Green–Naghdi equations (also called Serre or fully nonlinear Boussinesq equations) are the standard model for the propagation of such waves. These equations couple the surface elevation to the vertically averaged horizontal velocity and are therefore independent of the vertical variable. In the presence of vorticity, the dependence on the vertical variable cannot be removed from the vorticity equation but it was however shown in 1 that the motion of the waves could be described using an extended Green–Naghdi system. In this paper, we propose an analysis of these equations, and show that they can be used to get some new insight into wave–current interactions. We show in particular that solitary waves may have a drastically different behavior in the presence of vorticity and show the existence of solitary waves of maximal amplitude with a peak at their crest, whose angle depends on the vorticity. We also show some simple numerical validations. Finally, we give some examples of wave–current interactions with a nontrivial vorticity field and topography effects.     

Global existence of shock front solution to axially symmetric piston problem in compressible flow

In this paper we study the global existence of BV solution to two-dimensional piston problem in fluid dynamics. Different from previous results on related problems we remove the restriction on the strength of the leading shock and require the velocity of the piston is rather fast or the density is quite small instead. The main tool in our proof is Glimm Scheme with some improvement. To define the Glimm functional we derive more precise estimates for the interaction of elementary waves, particularly in the region near the leading shock. The paper is partially supported by National Natural Science Foundation of China 10531020, the National Basic Research Program of China 2006CB805902 and the Doctorial Foundation of National Educational Ministry 20050246001.     

Capillary waves with variable surface tension

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.     

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization.     

Shock Formation and Vorticity Creation for 3d Euler

We analyze the shock formation process for the 3D nonisentropic Euler equations with the ideal gas law, in which sound waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3, 4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the nonisentropic Euler equations which form a generic stable shock with explicitly computable blowup time, location, and direction. This is achieved by establishing the asymptotic stability of a generic shock profile in modulated self-similar variables, controlling the interaction of wave families via: (i) pointwise bounds along Lagrangian trajectories, (ii) geometric vorticity structure, and (iii) high-order energy estimates in Sobolev spaces. © 2022 Wiley Periodicals LLC.     

Computing statistics for Hamiltonian systems: A case study

We present the results of a set of numerical experiments designed to investigate the appropriateness of various integration schemes for molecular dynamics simulations. In particular, we wish to identify which numerical methods, when applied to an ergodic Hamiltonian system, sample the state-space in an unbiased manner. We do this by describing two Hamiltonian system for which we can analytically compute some of the important statistical features of its trajectories, and then applying various numerical integration schemes to them. We can then compare the results from the numerical simulation against the exact results for the system and see how closely they agree. The statistic we study is the empirical distribution of particle velocity over long trajectories of the systems. We apply four methods: one symplectic method (Störmer–Verlet) and three energy-conserving step-and-project methods. The symplectic method performs better on both test problems, accurately computing empirical distributions for all step-lengths consistent with stability. Depending on the test system and the method, the step-and-project methods are either no longer ergodic for any step length (thus giving the wrong empirical distribution) or give the correct distribution only in the limit of step-size going to zero.     

Numerical solution of the time-dependent incompressible Navier–Stokes equations by piecewise linear finite elements

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.     

Mathematical modelization of equatorial waves

Our purpose is to calculate waves propagating along the equator in an oceanic domain and the influence of a characteristic mean equatorial circulation on the nature of these waves. Equations satisfied by perturbations of currents and temperature are of the Navier-Stokes type and have been linearized around a stationary solution. Existence and uniqueness of the solution have been proved. Numerical experiments have been carried out and provided us with time-dependent values. The excited waves are exhibited by Fourier analysis of these time series.     

Capillary surfaces of constant mean curvature in a right solid cylinder

In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the solid cylinder, we obtain some results of symmetry by using the Alexandrov reflection method. When the mean curvature is zero, we give sufficient conditions to conclude that the surface is part of a plane or a catenoid.     

Global smooth solutions of the compressible Navier-Stokes equations with density-dependent viscosity

In this paper we study a free boundary problem for the viscous, compressible, heat conducting, one-dimensional real fluids. More precisely, the viscosity is assumed to be a power function of density, i.e., μ(ρ)=ρα, where ρ denotes the density of fluids and α is a positive constant. In addition, the equations of state include and are more general than perfect flows which only depend linearly on temperature. The global existence (uniqueness) of smooth solutions is established with for general, large initial data, which improves the previous results. Moreover, it is also shown that the solutions will not develop vacuum, mass concentration or heat concentration in a finite time provided the initial data are bounded and smooth, and do not contain vacuum.     

Regularity criteria for the 3D generalized MHD equations in terms of vorticity

We prove regularity criteria for the 3D generalized MHD equations. These criteria impose assumptions on the vorticity only. In addition, we also prove a result of global existence for smooth solution under some special conditions.     

On subsonic compressible flows in a two-dimensional duct

This paper concerns subsonic flows passing a two-dimensional duct for the steady compressible Euler system. If the Bernoulli constant is uniform in the flow field, the density at the entry and both the pressures at the entrance and the exit are given, we show that the problem is generally ill-posed; but if we give the pressure at the exit with a constant difference, then under the same other conditions as above we establish the existence of subsonic flows.     

Global existence of diffusive-dispersive traveling waves for general flux functions

We establish a global existence of traveling waves for diffusive-dispersive conservation laws for locally Lipschitz flux functions. Using Lyapunov stability techniques, we reduce the global problem of finding traveling waves to considering local behaviors of a stable trajectory of the saddle point.     

Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion

We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with anisotropic viscosity acting only in the horizontal direction, which arises in ocean dynamics models. Global well-posedness for this system was proven by Danchin and Paicu; however, an additional smoothness assumption on the initial density was needed to prove uniqueness. They stated that it is not clear whether uniqueness holds without this additional assumption. The present work resolves this question and we establish uniqueness without this additional assumption. Furthermore, the proof provided here is more elementary; we use only tools available in the standard theory of Sobolev spaces, and without resorting to para-product calculus. We use a new approach by defining an auxiliary “stream-function” associated with the density, analogous to the stream-function associated with the vorticity in 2D incompressible Euler equations, then we adapt some of the ideas of Yudovich for proving uniqueness for 2D Euler equations.     

Numerical simulation of the performance of a snow fence with airfoil snow plates by FVM

The aim of this work is to analyze the efficiency of a snow fence with airfoil snow plates to avoid the snowdrift formation, to improve visibility and to prevent blowing snow disasters on highways and railways. In order to attain this objective, it is necessary to solve particle transport equations along with the turbulent fluid flow equations since there are two phases: solid phase (snow particles) and fluid phase (air). In the first place, the turbulent flow is modelled by solving the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible viscous flows through the finite volume method (FVM) and then, once the flow velocity field has been determined, representative particles are tracked using the Lagrangian approach. Within the particle transport models, we have used a particle transport model termed as Lagrangian particle tracking model, where particulates are tracked through the flow in a Lagrangian way. The full particulate phase is modelled by just a sample of about 15,000 individual particles. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated using a simple integration method to calculate the behaviour of the particles as they traverse the flow domain. Finally, the conclusions of this work are exposed.