On the numerical solution of the turbulence energy equations for wave and tidal flows

This paper deals with the numerical solution, using finite difference methods, of the hydrodynamic and turbulence energy equations which describe wind wave and tidally induced flow. Calculations are performed using staggered and non-staggered finite difference grids in the vertical, with various time discretizations of the production and dissipation terms in the turbulence energy equations. It is shown that the time discretization of these terms can significantly influence the stability of the solution. The effect of time filtering on the numerical stability of the solution is also considered. The form of the mixing length is shown to significantly influence the bed stress in wind wave problems. A no-slip condition is applied at the sea bed, and the associated high-shear bottom boundary layer is resolved by transforming the equations onto a logarithmic or log-linear co-ordinate system before applying the finite difference scheme. A computationally economic method is developed which remains stable even when a very fine vertical grid (over 200 points) is used with a time step of up to 30 min.     

Modelling the non-linear interaction of wind and tide: Its influence on current profiles

A single-point model in the vertical is used to examine the coupling between tidal currents and wind-driven flows in shallow near-coastal regions. Calculations using both a linear slip and a no-slip condition at the sea bed clearly show that coupling between tidal and wind-driven currents cannot occur in a linear model with a time-independent eddy viscosity. However with a physically more realistic time-varying viscosity related to the flow field, coupling does occur, the magnitude of this non-linear interaction depending upon the change in eddy viscosity over a tidal cycle and the intensity of shear in the vertical. A point model in the vertical with flow induced by an oscillatory pressure gradient and an additional constant wind stress is used to examine the influence of viscosity parametrization and water depth upon this coupling. The solution in the vertical is accomplished using both a functional approach and a finite difference method. Some conclusions as to the relative merits of these approaches, particularly the use of a transformed grid in the case of high-shear surface and bed boundary layers, are made in the paper.     

An implicit three‐dimensional fully non‐hydrostatic model for free‐surface flows

An implicit method is developed for solving the complete three‐dimensional (3D) Navier–Stokes equations. The algorithm is based upon a staggered finite difference Crank‐Nicholson scheme on a Cartesian grid. A new top‐layer pressure treatment and a partial cell bottom treatment are introduced so that the 3D model is fully non‐hydrostatic and is free of any hydrostatic assumption. A domain decomposition method is used to segregate the resulting 3D matrix system into a series of two‐dimensional vertical plane problems, for each of which a block tri‐diagonal system can be directly solved for the unknown horizontal velocity. Numerical tests including linear standing waves, nonlinear sloshing motions, and progressive wave interactions with uneven bottoms are performed. It is found that the model is capable to simulate accurately a range of free‐surface flow problems using a very small number of vertical layers (e.g. two–four layers). The developed model is second‐order accuracy in time and space and is unconditionally stable; and it can be effectively used to model 3D surface wave motions. Copyright © 2004 John Wiley & Sons, Ltd.     

A non-hydrostatic numerical scheme for dispersive waves generated by bottom motion

A numerical scheme based on the staggered finite volume method is presented at the aim of studying surface waves generated by a bottom motion. We address the 2D Euler equations in which the vertical domain is resolved only by one layer. The resulting non-hydrostatic scheme is used to simulate surface waves generated by bottom motion in a water tank. Here we mimic Hammack experiments numerically, in which a bed section is moved upwards or downwards, resulting in transient dispersive waves. For an impulsive downward bottom thrust, free surface responds in terms of a negative leading wave, followed with dispersive train of waves. For an upward bottom thrust, amplitude of the leading wave decays as the wave propagates, and no wave of permanent form evolves— instead, there appears a train of solitons. In this article, we show that our numerical scheme can produce the correct wave profiles, comparable with the analytical and experimental results of Hammack. Simulations using intermediate and slow bottom motions are also presented. In addition, we perform a simulation of a wave generated by submerged landslide, that compares well against previous numerical simulations. Via this simulation, we demonstrate that our scheme can incorporate a moving wet–dry boundary algorithm in the run-up simulation.     

Non‐hydrostatic 3D free surface layer‐structured finite volume model for short wave propagation

In this paper a layer‐structured finite volume model for non‐hydrostatic 3D environmental free surface flow is presented and applied to several test cases, which involve the computation of gravity waves. The 3D unsteady momentum and mass conservation equations are solved in a collocated grid made of polyhedrons, which are built from a 2D horizontal unstructured mesh, by just adding several horizontal layers. The mesh built in such a way is unstructured in the horizontal plane, but structured in the vertical direction. This procedure simplifies the mesh generation and at the same time it produces a well‐oriented mesh for stratified flows, which are common in environmental problems. The model reduces to a 2D depth‐averaged shallow water model when one single layer is defined in the mesh. Pressure–velocity coupling is achieved by the Semi‐Implicit Method for Pressure‐Linked Equations algorithm, using Rhie–Chow interpolation to stabilize the pressure field. An attractive property of the model proposed is the ability to compute the propagation of short waves with a rather coarse vertical discretization. Several test cases are solved in order to show the capabilities and numerical stability of the model, including a rectangular free oscillating basin, a radially symmetric wave, short wave propagation over a 1D bar, solitary wave runup on a vertical wall, and short wave refraction over a 2D shoal. In all the cases the numerical results are compared either with analytical or with experimental data. Copyright © 2008 John Wiley & Sons, Ltd.     

钢轨轨底垂直振动模式导波检测技术的实验研究

现有铁路钢轨超声探伤车技术无法检测线路钢轨轨底缺陷,给铁路运输安全带来很大隐患.本文开展了钢轨轨底超声导波传播特性和垂直振动模式导波检测技术研究,采用半解析有限元方法分析了我国60型钢轨轨底的各振动模态导波频散曲线和波结构.应用模态锤技术对自由状态钢轨轨底垂直振动模态导波传播特性进行了实验测量,结果表明,在0～100kHz频率范围内,钢轨轨底垂直振动模态优势模式与厚度为14 mm板中的A0模式兰姆波具有等效性.进一步研究了激励频率、激励脉冲周数、传播距离对轨底垂直振动模态导波传播的影响,设计了导波斜探头,选择合适的参数在钢轨轨底激励出垂直振动模态导波并检测出了轨底的人工缺陷.本文的研究结果为线路钢轨轨底的导波检测技术奠定了一定的基础.     

A coupled boundary element–finite difference model of surface wave motion over a wall turbulent flow

An effective numerical technique is presented to model turbulent motion of a standing surface wave in a tank. The equations of motion for turbulent boundary layers at the solid surfaces are coupled with the potential flow in the bulk of the fluid, and a mixed BEM–finite difference technique is used to model the wave motion and the corresponding boundary layer flow. A mixing‐length theory is used for turbulence modelling. The model results are in good agreement with previous physical and numerical experiments. Although the technique is presented for a standing surface wave, it can be easily applied to other free surface problems. Copyright © 2005 John Wiley & Sons, Ltd.     

SINGLE-LAYER FINITE ELEMENT APPROXIMATION OF 3D NAVIER–STOKES EQUATIONS WITH FREE BOUNDARIES FOR SHALLOW WATER FLOWS

In order to simulate flows in the shallow water limit, the full incompressible Navier–Stokes equations with free boundaries are solved using a single layer of finite elements. This implies a polynomial approximation of the velocity profile in the vertical direction, which in turn distorts the wave speed. This fact is verified by numerical results: the wave speed depends on the vertical discretization. When at least two layers of finite elements are used, the boundary layer at the bottom can be simulated and the correct solution for the shallow water limit is recovered. Then this algorithm is applied to the prediction of Tsunami event.     

An ocean modelling system with turbulent boundary layers and topography: numerical description

The Sandia ocean modelling system (SOMS) is a system of three-dimensional, fully conservative, partially implicit numerical models based on primitive equations and a staggered Arakawa ‘c’ grid. A thin-shell bottom boundary layer submodel coupled to a free-stream submodel resolves boundary layers together with realistic topography. Both submodels use stretched vertical co-ordinates and an optional Mellor-Yamada level-2·5 turbulence closure. Rigid top pressures are determined by vertical integration of the conservation equations using a hydrostatic approximation. SOMS reproduces previously published results, but with notable advantages in speed and economy.     

Depth‐integrated,non‐hydrostatic model with grid nesting for tsunami generation,propagation, and run‐up

Tsunamis generated by earthquakes involve physical processes of different temporal and spatial scales that extend across the ocean to the shore. This paper presents a shock‐capturing dispersive wave model in the spherical coordinate system for basin‐wide evolution and coastal run‐up of tsunamis and discusses the implementation of a two‐way grid‐nesting scheme to describe the wave dynamics at resolution compatible to the physical processes. The depth‐integrated model describes dispersive waves through the non‐hydrostatic pressure and vertical velocity, which also account for tsunami generation from dynamic seafloor deformation. The semi‐implicit, finite difference model captures flow discontinuities associated with bores or hydraulic jumps through the momentum‐conserved advection scheme with an upwind flux approximation. The two‐way grid‐nesting scheme utilizes the Dirichlet condition of the non‐hydrostatic pressure and both the horizontal velocity and surface elevation at the inter‐grid boundary to ensure propagation of dispersive waves and discontinuities across computational grids of different resolution. The inter‐grid boundary can adapt to bathymetric features to model nearshore wave transformation processes at optimal resolution and computational efficiency. A coordinate transformation enables application of the model to small geographic regions or laboratory experiments with a Cartesian grid. A depth‐dependent Gaussian function smoothes localized bottom features in relation to the water depth while retaining the bathymetry important for modeling of tsunami transformation and run‐up. Numerical experiments of solitary wave propagation and N‐wave run‐up verify the implementation of the grid‐nesting scheme. The 2009 Samoa Tsunami provides a case study to confirm the validity and effectiveness of the modeling approach for tsunami research and impact assessment. Copyright © 2010 John Wiley & Sons, Ltd.     

On the approximation of horizontal gradients in sigma co-ordinates for bathymetry with steep bottom slopes

Nowadays the simulation of free surface flow and transport in rivers, estuaries and seas is often based upon three-dimensional modelling systems. Most of these three-dimensional modelling systems use sigma co-ordinates in the vertical. By the use of the sigma transformation the water column can be divided into the same number of layers independently of the water depth. Especially for steep bottom slopes combined with vertical stratification of the density, sigma-transformed grids impose numerical problems for the accurate approximation of horizontal gradients. This paper deals with algorithms for the approximation in sigma co-ordinates of the horizontal diffusive fluxes of temperature and salinity and for the approximation of the horizontal pressure gradients. The approximation of the horizontal diffusive fluxes is based upon a finite volume method. The approximation of the pressure gradients is directly related to the approximation of the diffusive fluxes. Artificial vertical diffusion and artificial flow due to truncation errors are minimized. The method described in this paper is not hampered by the so-called ‘hydrostatic consistency condition’. This will be illustrated by numerical experiments.     

BOUNDARY CONDITIONS IN SHALLOW WATER MODELS—AN ALTERNATIVE IMPLEMENTATION FOR FINITE ELEMENT CODES

Finite element solution of the shallow water wave equations has found increasing use by researchers and practitioners in the modelling of oceans and coastal areas. Wave equation models, most of which use equal-orderC⁰ interpolants for both the velocity and the surface elevation, do not introduce spurious oscillation modes, hence avoiding the need for artificial or numerical damping. An important question for both primitive equation and wave equation models is the interpretation of boundary conditions. Analysis of the characteristics of the governing equations shows that for most geophysical flows a single condition at each boundary is sufficient, yet there is not a consensus in the literature as to what that boundary condition must be or how it should be implemented in a finite element code. Traditionally (partly because of limited data), surface elevation is specified at open ocean boundaries while the normal flux is specified as zero at land boundaries. In most finite element wave equation models both of these boundary conditions are implemented as essential conditions. Our recent work focuses on alternative ways to numerically implement normal flow boundary conditions with an eye towards improving the mass-conserving properties of wave equation models. A unique finite element formulation using generalized functions demonstrates that boundary conditions should be implemented by treating normal fluxes as natural conditions with the flux interpreted as external to the computational domain. Results from extensive numerical experiments show that the scheme does conserve mass for all parameter values. Furthermore, convergence studies demonstrate that the algorithm is consistent, as residual errors at the boundary diminish as the grid is refined.     

三维非均匀介质中弹性波传播的数值模拟

提出了一种三维非均匀介质中弹性波传播数值模拟的方法,文中称为三维格子法。该算法是二维格子法（一种二维非均匀介质中P－SV波传播的数值模拟算法）向三维非均匀介质情况的推广。在空间离散上该文方法与有限元方法类似,容许根据连续体的形状和介质分界面任意剖面网格,且自然满足自由表面边界条件。不同于常规有限差分法在各个节点上满足动力学微分方程,该算法通过满足各节点周围格子的整体平衡（积分平衡方程）来对问题进行求解,三维格子法所需的计算机内存及计算耗时与同阶精度的规则网格有限差分法相当。算例表明,该文提出的三维格子法具有较高的精度且可很好地模拟三维复杂形状地表对弹性波的反射和绕射。     

Nanofluid bioconvection: interaction of microorganisms oxytactic upswimming, nanoparticle distribution, and heating/cooling from below

The aim of this paper is to develop a theory describing the onset of convection instability (called here nanofluid bioconvecion) that is induced by simultaneous effects produced by oxytactic microorganisms, nanoparticles, and vertical temperature variation. The theory is developed for the situation when the nanofluid occupies a shallow horizontal layer of finite depth. The layer is defined as shallow as long as oxygen concentration at the bottom of the layer is above the minimum concentration required for the bacteria to be active (to actively swim up the oxygen gradient). The lower boundary of the layer is assumed rigid, while at the upper boundary both situations when the boundary is rigid or stress free are considered. Physical mechanisms responsible for the slip velocity between the nanoparticles and the base fluid, such as Brownian motion and thermophoresis, are accounted for in the model. A linear instability analysis is performed, and the resulting eigenvalue problem is solved analytically using the Galerkin method.     

On the theory of reflection from a wire grid parallel to an interface between homogeneous media

Summary The reflection from a wire grid parallel to a plane interface is considered. The respective media are homogeneous and either or both can be dissipative. The grid is composed of thin equi-spaced wires of finite conductivity. The plane wave solution for arbitrary incidence is then generalized for cylindricalwave excitation. The energy absorbed from a magnetic line source by a grid situated on the surface of a dissipative half-space is treated in some detail. This latter problem is a two-dimensional analogy of a vertical antenna with a radial wire ground system.     

Modeling acoustic waves in locally enhanced meshes with a staggered-grid finite difference approach

Finite difference is a well-suited technique for modeling acoustic wave propagation in heterogeneous media as well as for imaging and inversion. Typically, the method aims at solving a set of partial differential equations for the unknown pressure field by using a regularly spaced grid. Although finite differences can be fast and cheap to implement, the accuracy of the solution is always restricted by the computational resources. This is a fundamental key point to treat when dealing with large-scale problems. In this work, we present and test a method that uses a non-uniform distribution of grid points to improve on accuracy or to reduce the required computational resources. The applied grid is generated through a coordinate transformation. Differential geometry and generalized coordinates are used to handle and analyze the effect of using a non-uniform grid. Results obtained with the presented method show that the applied transformation as well as the number of points-per-wavelength influences the stability and dispersion in the solution. We exploit this observation to locally improve the accuracy of our simulations. The work presented in this paper allows us to conclude that differential geometry for finite differences can be used to reduce dispersion and hence improve the accuracy when modeling acoustic wave propagation in heterogeneous media. In addition, it can be used to avoid oversampling through the optimization of the number of grid nodes required to have an accurate solution or just honor to the boundaries.     

On uniformly accurate high‐order Boussinesq difference equations for water waves

A new accurate finite‐difference (AFD) numerical method is developed specifically for solving high‐order Boussinesq (HOB) equations. The method solves the water‐wave flow with much higher accuracy compared to the standard finite‐difference (SFD) method for the same computer resources. It is first developed for linear water waves and then for the nonlinear problem. It is presented for a horizontal bottom, but can be used for variable depth as well. The method can be developed for other equations as long as they use Padé approximation, for example extensions of the parabolic equation for acoustic wave problems. Finally, the results of the new method and the SFD method are compared with the accurate solution for nonlinear progressive waves over a horizontal bottom that is found using the stream function theory. The agreement of the AFD to the accurate solution is found to be excellent compared to the SFD solution. Copyright © 2005 John Wiley & Sons, Ltd.     

An integrated modelling system for coastal area dynamics

Two-dimensional initial-boundary value problems are considered for the shallow water equations and the equation of advection and dispersion of pollutants. The problems are solved in curvilinear boundary fitted co-ordinates. The transformed equations are integrated on a regular grid by the semi-implicit and implicit finite difference methods. Based on the numerical method, the integrated modelling system Cardinal for coastal area dynamics and pollution processes is developed for application on personal computers. Examples of computations are given.     

FLAT-PLATE BOUNDARY-LAYER FLOWS INDUCED BY DUSTY SHOCK WAVE

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A coupled-mode model for water wave scattering by vertically sheared currents in variable bathymetry regions

A coupled-mode model is developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, sheared current with linear vertical velocity profile, over general bottom topography. The wave potential is represented by a series of local vertical modes containing the propagating and evanescent modes, plus additional terms accounting for the satisfaction of the boundary conditions. Using the above representation, in conjunction with a variational principle, a coupled system of differential equations on the horizontal plane is derived, with respect to the unknown modal amplitudes. In the case of small-amplitude waves, a linearized version of the above coupled-mode system is obtained, extending previous analysis by Belibassakis et al. (2011) to the propagation of water waves over variable bathymetry regions in the presence of vertically sheared currents. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system reduces to a one-equation model, that is shown to extend known mild-slope mild vertical shear equation concerning wave–current interaction over slowly varying topography. After additional simplifications, the latter model is shown to be compatible with the extended mild-slope mild-shear equation by Touboul et al. (2016). Results are presented for various representative test cases demonstrating the usefulness of the present coupled mode system and the importance of various terms in the modal expansion, and compared against experimental data collected in wave flume validating the present method. The analytical structure of the present system facilitates extensions to model non-linear effects and applications concerning wave scattering by inhomogeneous currents in coastal regions with general 3D bottom topography.