R. Dutta, J. Veeramony
In this report, generalized wave breaking equations are developed using three dimensional fully nonlinear extended Boussinesq equations to encompass rotational dynamics in wave breaking zone. The derivation for vorticity distributions are developed from Reynold based stress equations.
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We report and discuss case study simulations of the Rayleigh-Taylor instability in the Boussinesq, incompressible regime developed to turbulence. Our main focus is on a statistical analysis of density and velocity fluctuations inside of the already developed and growing in size mixing zone. Novel observations reported in the manuscript concern self-similarity of the velocity and density fluctuations spectra inside of the mixing zone snapshot, independence of the spectra of the horizontal slice level, and universality showing itself in a virtual independence of the internal structure of the mix…
In this report, Mathematical model for generalized nonlinear three dimensional wave breaking equations was de- veloped analytically using fully nonlinear extended Boussinesq equations to encompass rotational dynamics in wave breaking zone. The three dimensional equations for vorticity distributions are developed from Reynold based stress equations. Vorticity transport equations are also developed for wave breaking zone. This equations are basic model tools for numerical simulation of surf zone to explain wave breaking phenomena. The model reproduces most of the dynamics in the surf zone. Non l…
Here we suggest an alternative understanding of the surface gravity wave propagation mechanism based on the baroclinic torque, which operates to translate the interfacial vorticity anomalies at the air-water interface. We demonstrate how the non-Boussinesq term of the baroclinic torque acts against the Boussinesq one to hinder wave propagation. By standard vorticity inversion and mirror imaging, we then show how the existence of the bottom boundary affects the two types of torque. Since the opposing non-Boussinesq torque results solely from the mirror image, it vanishes in the deep water limit…
This article is devoted to exact solutions of the Boussinesq equation that models nonlinear shallow water waves. For this we use the Hirota bilinear method and differential constrains. Out solutions describe in particular the motion of the wave packets, waves on soliton and "dancing" waves. We present a simple method for multiplication of solutions of the Hirota equation which allow us to generate more complex structures from the waves.
We study the dynamics associated with the extension of turbulent convective motions from a convection zone (CZ) into a stable region (RZ) that lies below the latter. For that purpose, we have run a series of three-dimensional direct numerical simulations solving the Navier-Stokes equations under the Boussinesq approximation in a spherical shell geometry. We observe that the overshooting of the turbulent motions into the stably stratified region depends on three different parameters: the relative stability of the RZ, the transition width between the two, and the intensity of the turbulence. In …
Consideration is given to the influence of an underwater landslide on waves at the surface of a shallow body of fluid. The equations of motion which govern the evolution of the barycenter of the landslide mass include various dissipative effects due to bottom friction, internal energy dissipation, and viscous drag. The surface waves are studied in the Boussinesq scaling, with time-dependent bathymetry. A numerical model for the Boussinesq equations is introduced which is able to handle time-dependent bottom topography, and the equations of motion for the landslide and surface waves are solved …
The paper's main goal is to compare the motion of solitary surface waves resulting from two similar but slightly different approaches. In the first approach, the numerical evolution of soliton surface waves moving over the uneven bottom is obtained using single wave equations. In the second approach, the numerical evolution of the same initial conditions is obtained by the solution of a coupled set of the Boussinesq equations for the same Euler equations system. We discuss four physically relevant cases of relationships between small parameters $α,β,δ$. For the flat bottom, these cases imply t…
The linear wave and geostrophic (vortex) solutions are shown to be a complete basis for physical variables $(u,v,w,ρ)$ in a rotating non-hydrostatic Boussinesq model with arbitrary stratification. As a consequence, the fluid can be unambiguously separated into linear wave and geostrophic components at each instant in time, without the need for temporal filtering. The fluid can then be diagnosed for temporal changes in wave and geostrophic coefficients at each unique wavenumber and mode, including those that inevitably occur due to nonlinear interactions. We demonstrate that this methodology ca…
The aim of this article is to derive surface wave models in the presence of surface tension and viscosity. Using the Navier-Stokes equations with a free surface, flat bottom and surface tension, we derive the viscous 2D Boussinesq system with a weak transverse variation. The assumed transverse variation is on a larger scale than along the main propagation direction. This Boussinesq system is proved to be consistent with the Navier-Stokes equations. This system is only an intermediate result that enables us to derive the Kadomtsev-Petviashvili (KP) equation which is a 2D generalization of the K…
Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of tsunami waves
本文推导并数值求解了变海底地形上的简化Boussinesq系统，通过Galerkin有限元方法模拟海啸波的产生与传播，并与线性Euler方程解析解及浅水波方程数值解进行了对比。
In this paper, we first revisit the celebrated Boussinesq approximation in stratified flows. Using scaling arguments we show that when the background shear is weak, the Boussinesq approximation yields either (i) $A_t\ll \mathcal{O}(1)$ or (ii) $Fr_c^2 \ll \mathcal{O}(1)$, where $A_t$ is the ratio of density variation to the mean density and $Fr_c$ is the ratio of the phase speed to the long wave speed. The second clause implies, contrary to the commonly accepted notion, that a flow with large density variations can also be Boussinesq. Indeed, we show that deepwater surface gravity waves are Bo…
The propagation of focused wave groups in intermediate water depth and the shoaling zone is experimentally and numerically considered in this paper. The experiments are carried out in a two-dimensional wave flume and wave trains derived from Pierson-Moskowitz and JONSWAP spectrum are generated. The peak frequency does not change during the wave train propagation for Pierson-Moskowitz waves; however, a downshift of this peak is observed for JONSWAP waves. An energy partitioning is performed in order to track the spatial evolution of energy. Four energy regions are defined for each spectrum type…
To investigate the finite time singularity in three-dimensional (3D) Euler flows, the simplified model of 3D axisymmetric incompressible fluids (i.e., two-dimensional Boussinesq approximation equations) is studied numerically. The system describes a cap-like hot zone of fluid rising from the bottom, while the edges of the cap lag behind, forming eye-like vortices. The hot liquid is driven by the buoyancy and meanwhile attracted by the vortices, which leads to the singularity-forming mechanism in our simulation. In the previous 2D Boussinesq simulations, the symmetricial initial data is used. H…
Vertical profiles for mean wind, standard deviations of velocity fluctuations, and wave-induced part of the momentum flux over a wavy fluid surface are calculated in the Cartesian coordinates on the basis of recent numerical results by Chalikov and Rainchik. Besides, calculations of spectra for surface elevation and wave-induced velocity components at this surface are carried out. Unlike the profiles typical to the near-wall turbulence, the calculated wind-velocity profiles deviate significantly from the logarithmic law throughout the entire trough-crest zone of wind waves, and the wave-induce…
On the multi-symplectic structure of Boussinesq-type systems. II: Geometric discretization
本文研究了Boussinesq型系统在数值近似中的几何离散化方法，基于多辛和哈密顿结构探讨了不同空间和时间离散策略的几何性质。
We derive and analyze in the framework of the mild-slope approximation a new double-layer Boussinesq-type model which is linearly and nonlinearly accurate up to deep water. Assuming the flow to be irrotational, we formulate the problem in terms of the velocity potential thereby lowering the number of unknowns. The model derivation combines two approaches, namely the method proposed by Agnon et al. (Agnon et al. 1999, J. Fluid Mech., 399 pp. 319-333) and enhanced by Madsen et al. (Madsen et al. 2003, Proc. R. Soc. Lond. A, 459 pp. 1075-1104) which consists in constructing infinite-series Taylor…
In this study we investigated the capabilities of the mesh-free, Lagrangian particle method (Smoothed Particle Hydrodynamics, SPH) to simulate the detailed hydrodynamic processes generated by both spilling and plunging breaking waves within the surf zone. The weakly-compressible SPH code DualSPHysics was applied to simulate wave breaking over two distinct bathymetric profiles (a plane beach and fringing reef) and compared to experimental flume measurements of waves, flows, and mean water levels. Despite the simulations spanning very different wave breaking conditions (including an extreme case…
Shear flows and their suppression at large aspect ratio. Two-dimensional simulations of a growing convection zone
本研究通过二维模拟探究生长对流区的纬向流演化，发现对流层纵横比小于2时出现纬向流且演化受Pr数调控，大纵横比域可抑制纬向流，为相关对流过程的二维研究提供了可能。
Phase-resolved Modeling Of Surf Zone Wave Transformation Over Idealized Rough Bottoms
本研究针对粗糙岩石海床冲浪带波浪变换建模的局限性，将体冠层阻力（BCD）参数化方法应用于三维非静压相位分辨SYMPHONIE模型，通过实验验证了其对波浪耗散过程的准确捕捉能力，为海岸过程研究提供了更可靠的建模框架。
Visco-potential free-surface flows and long wave modelling
本文提出粘势自由表面流公式，包含局部和非局部耗散项，推导出非局部长波方程，并分析色散关系及孤立波衰减。
Peregrine's system revisited
本文重新审视传统称为Peregrine系统的长波模型，提出了一种改进的控制方程，使其在弱非线性波条件下与初始模型渐近相似，同时保留了完整水波问题的额外对称性，从而在滑动区具有良好的散射项，允许高效稳定的上溯计算。
Motivated by the dynamics in the deep interiors of many stars, we study the interaction between overshooting convection and the large-scale poloidal fields residing in radiative zones. We have run a suite of 3D Boussinesq numerical calculations in a spherical shell that consists of a convection zone with an underlying stable region that initially compactly contains a dipole field. By varying the strength of the convective driving, we find that, in the less turbulent regime, convection acts as turbulent diffusion that removes the field faster than solely molecular diffusion would do. However, i…
A novel deep learning technique called Physics Informed Neural Networks (PINNs) is adapted to study steady groundwater flow in unconfined aquifers. This technique utilizes information from underlying physics represented in the form of partial differential equations (PDEs) alongside data obtained from physical observations. In this work, we consider the Dupuit-Boussinesq equation, which is based on the Dupuit-Forchheimer approximation, as well as a recent more complete model derived by Di Nucci (2018) as underlying models. We then train PINNs on data obtained from steady-state analytical soluti…
This paper presents numerical simulations of a bichromatic wave group propagating and breaking over a fixed breaker bar. The simulations are performed using a newly stabilized Reynolds-averaged Navier Stokes (RANS) two-equation turbulence closure, which solves the longstanding problem of over-production of turbulence beneath surface waves in the nearly potential flow region prior to breaking. This model has previously been tested on small-scale spilling breaking regular waves, whereas in this work focus is on full (rather than model) scale application, wave groups (rather than regular waves) a…
Evolution of perturbed long nonlinear plane, ring and hybrid surface waves
研究通过cKdV型和KPII方程建模扰动长非线性表面波的演化，揭示了大局部化波在内向传播波中的瞬态行为，对 rogue wave 生成机制有贡献。
Phase-resolved modelling of wave transformation in the surf zone over idealised rough bottoms
为理解岩石环境海岸洪水过程，本研究基于Symphonie代码在3D非静压相位分辨波浪模型中实现粗糙度诱导耗散参数化，对比实验室实验评估海底应力法和冠层拖曳法性能并提出建议。
The dynamical response of edge waves under the influence of self-gravity is examined in an idealized two-dimensional model of a proto-stellar disc, characterized in steady state as a rotating vertically infinite cylinder of fluid with constant density except for a single density interface at some radius r0. The fluid in basic state is prescribed to rotate with a Keplerian profile $Ω_k(r)\sim r^{-3/2}$ modified by some additional azimuthal sheared flow. A linear analysis shows that there are two azimuthally propagating edge waves, kin to the familiar Rossby waves and surface gravity waves in te…
A single incompressible, inviscid, irrotational fluid medium bounded by a free surface and varying bottom is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types taking into account the effect of the slowly varying bottom. The arising KdV equation with variable coefficients is studied numerically when the initial condition is in the form of the one soliton solution …
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