The majority of coastal flows are characterized by turbulence, rendering the application of shallow water equations an inadequate approach for their accurate description. This paper presents a theory for characterizing accelerated coastal flows, offering an enhanced representation of turbulent flows in the long-wave approximation. The fundamental premise of the theory is as follows: Fluctuating velocities present in the water flow are found to correlate and generate additional motion (or fluctuating motion), which is associated with a velocity of the same order and regularity as the mean flow velocity. The resulting high-order long-wave model is both stable and efficient, comprising classical shallow water equations and additional equations that describe the transport of kinetic energy due to turbulence. The model's detailed eigenstructure and Rankine-Hugoniot relations, steady state solutions are presented in this paper. The proposed theory shows promise for addressing a variety of engineering problems, including roll waves, hydraulic jumps, dam breaks, and flooding. The derived model can accurately address the problem of the mixing layer interaction with a free surface and its transition into a turbulent surface jet. Recent application to sediment transport gives interesting results and good numerical convergences.
Comment on the paper "The third-order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom" by M. Fokou, T.C. Kofané, A. Mohamadou and E. Yomba, Eur. Phys. J. Plus, 132, 410 (2017)
“Third-order摄动Korteweg-de Vries方程for shallow water waves with a non-flat bottom”一文[1]的作者声称，他们已经导出了不平坦海底情况下的完整的三阶摄动KdV方程。我们表明，作者的推导是不一致的，因为事实上，他们只考虑了一些三阶校正，而不是所有的。此外，我们表明，一个一致的三阶扰动Korteweg-de Vries方程的浅水波与非平底不能推导出一个一般形式的底函数。
In this paper, we consider a family of piecewise constant solutions of the quasi-geostrophic shallow-water (QGSW) equation. We derive the contour dynamics equation of the QGSW front, which is a nonlinear, nonlocal dispersive equation, and prove the global existence of the solutions when the initial data is sufficiently close to a flat front.
The fundamental dispersion relation of transverse electro-magnetic waves in a cold collisionless plasma is formally equivalent to the two dimensional dispersion relation of inertio-gravity waves in a rotating shallow water system, where the Coriolis frequency can be identified with the plasma frequency, and the shallow water gravity wave phase speed plays the role of the speed of light. Here we examine this formal equivalence in the governing linearised equations, and compare between the propagation wave mechanisms in these seemingly unrelated physical systems.
In this paper, we shall implement KAM theory in order to construct a large class of time quasi-periodic solutions for an active scalar model arising in fluid dynamics. More precisely, the construction of invariant tori is performed for quasi-geostrophic shallow-water equations when the {\it Rossby deformation length} belongs to a massive Cantor set. As a consequence, we construct pulsating vortex patches whose boundary is localized in a thin annulus for any time.
In this article, we present a predictive model for the amplitude of impulse waves generated by the collapse of a granular column into a water layer. The model, which combines the spreading dynamics of the grains and the wave hydrodynamics in shallow water, is successfully compared to a large dataset of laboratory experiments, and captures the influence of the initial parameters while giving an accurate prediction. Furthermore, the role played on the wave generation by two key dimensionless numbers, i.e., the global Froude number and the relative volume of the immersed deposit, is rationalized.…
The modulation equations for Stokes waves in shallow water coupled to wave generated mean flow, derived in Whitham (1967), based on an averaged Lagrangian are revisited. Firstly, it is shown that they can be recast into two coupled classical shallow water equations, with modified gravity having the sign of the Whitham index: sign($ω_0'' ω_2$). Secondly, it is shown that the amplitude of the mean flow and amplitude of the wave are, in general, independent. Thirdly, the implications of the coalescing characteristics, whose unfolding is associated with the Benjamin-Feir instability, are studied.
In this paper we analyze the stability of equilibrium manifolds of hyperbolic shallow water moment equations. Shallow water moment equations describe shallow flows for complex velocity profiles which vary in vertical direction and the models can be seen as extensions of the standard shallow water equations. Equilibrium stability is an important property of balance laws that determines the linear stability of solutions in the vicinity of equilibrium manifolds and it is seen as a necessary condition for stable numerical solutions. After an analysis of the hyperbolic structure of the models, we i…
A basic shallow water system with variable topography is analyzed from the point of view of a Lagrangian derivation of momentum, energy, and pseudomomentum balances. A two-dimensional action and associated momentum equation are derived. The latter is further manipulated to derive additional equations for energy and pseudomomentum. This revealed structure emphasizes broken symmetries in space and a reference configuration, and preserved symmetry in time.
We present a geometrical demonstration for persistence properties for a bi-Hamiltonian system modelling waves in a shallow water regime. Both periodic and non-periodic cases are considered and a key ingredient in our approach is one of the Hamiltonians of the system. As a consequence of our developments we improve recent works dealing with unique continuation properties for shallow water equations, as well as we provide a novel way to prove that the unique compactly supported solution of the system is necessarily the zero function.
Riemann problems at geometric discontinuities are a classic and fascinating issue of hydraulics. In the present paper, the complete solution to the Riemann problem of the one-dimensional Shallow water Equations at monotonic width discontinuities is presented. This solution is based on the assumption that the relationship between the states immediately to the left and to the right of the discontinuity is a stationary weak solution of the one-dimensional variable-width Shallow water Equations. It is demonstrated that the solution to the Riemann problem always exists and it is unique, but there a…
The group properties of the shallow-water equations with the complete Coriolis force is the subject of this study. In particular we apply the Lie theory to classify the system of three nonlinear partial differential equations according to the admitted Lie point symmetries. For each case of the classification problem the one-dimensional optimal system is determined. The results are applied for the derivation of new similarity solutions.
A simple numerical scheme for the 2D shallow-water system
本文提出了一种求解二维浅水方程的简单数值格式。受一维SWES数值逼近研究的启发，Audusse et al.（2015）将问题从一维扩展到二维，并保持了应用的简单性。新方案被实施到代码TELEMAC-2D [tel 2d，2014]中，并进行了几次测试以验证在静止平衡态和不同类型的流态（即，河流状态、从河流到急流状态的跨临界流、具有水跃的跨临界流）。通过灵敏度分析检验了格式的收敛性。
Despite the sphericity of the collapsing stellar core, the birth conditions of neutron stars can be highly non spherical due to a hydrodynamical instability of the shocked accretion flow. Here we report the first laboratory experiment of a shallow water analogue, based on the physics of hydraulic jumps. Both the experiment and its shallow water modeling demonstrate a robust linear instability and nonlinear properties of symmetry breaking, in a system which is one million times smaller and about hundred times slower than its astrophysical analogue.
Using Levi-Civita's theory of ideal fluids, we derive the complex Korteweg-de Vries (KdV) equation, describing the complex velocity of a shallow fluid up to first order. We use perturbation theory, and the long wave, slowly varying velocity approximations for shallow water. The complex KdV equation describes the nontrivial dynamics of all water particles from the surface to the bottom of the water layer. A crucial new step made in our work is the proof that a natural consequence of the complex KdV theory is that the wave elevation is described by the real KdV equation. The complex KdV approach…
Extended shallow water wave equations are derived, using the method of asymptotic expansions, from the Euler (or water wave) equations. These extended models are valid one order beyond the usual weakly nonlinear, long wave approximation, incorporating all appropriate dispersive and nonlinear terms. Specifically, first we derive the extended Korteweg-de Vries (KdV) equation, and then proceed with the extended Benjamin-Bona-Mahony and the extended Camassa-Holm equations in (1+1)-dimensions, the extended cylindrical KdV equation in the quasi-one dimensional setting, as well as the extended Kadomt…
We investigate the numerical artifact known as a carbuncle, in the solution of the shallow water equations. We propose a new Riemann solver that is based on a local measure of the entropy residual and aims to avoid carbuncles while maintaining high accuracy. We propose a new challenging test problem for shallow water codes, consisting of a steady circular hydraulic jump that can be physically unstable. We show that numerical methods are prone to either suppress the instability completely or form carbuncles. We test existing cures for the carbuncle. In our experiments, only the proposed method …
