Simulation of the internal tide in the Strait of Bab el Mandeb (the Red Sea)

The baroclinic tide in the Strait of Bab el Mandeb is simulated by solving a three-dimensional boundary-value problem in boundary-fitted coordinates for the equations of dynamics, density constituents, and turbulence characteristics. Results related to simulations of the barotropic tide in the Red Sea and to the internal tide in the Strait of Bab el Mandeb are presented.     

Calculation of energy of the barotropic-baroclinic interaction in the strait of Bab el Mandeb

Equations for the kinetic, potential, and total energy in the curvilinear boundary-fitted coordinates are considered in the strait area and adjacent regions of the Red Sea and Gulf of Aden. The budgets of different forms of energy are presented, making it possible to assess the correctness of the formulation of the three-dimensional boundary-value problem, the accuracy of the calculation of the barotropic-baroclinic interaction in an energy norm, and the relationship between the components of energy balance. The energy fluxes in the boundary sections of the strait are estimated for the interaction of thermohaline currents with main tidal harmonics and with the total tide, as well as for the amplitude of fluctuations in boundary energy fluxes in a synodic month. The results illustrating the behavior of some dynamical characteristics and components of energy balance at vertical and horizontal sections of the strait are given.     

流速逆变张量隐式求解方法及其在航道港池流场计算中的应用

运用高分辨率的边界适应网格进行流体动力学数值计算时,如何提高计算稳定性和减少计算量成为数值求解的关键性问题.在非正交的边界适应坐标系中,每个动量方程中同时出现了两个交叉方向的水位偏导数项,给隐式求解带来困难,而显式格式下的时间步长由于受与空间步长有关的Courant-Friedrichs-Lewy条件限制,计算量成倍增加.本文从广义曲线坐标系下浅海动力学方程组出发,导出了流速的逆变张量所满足的动量方程组,使方程中的水位偏导数项变成了沿某一协变基向量方向占优的形式,方便地采用了交替方向隐式差分格式,从而提高了计算稳定性并减小了计算量.本文通过对澳门海域航道和港池中流场的计算,证实了该模式是一种进行高分辩率数值计算的有效方法.     

Numerical Tidal Current Modeling Using Fine Boundary Fitted Grids for Tidal Flats

Fine grids with small spacing in boundary-fitted coordinates are normally used to treat thecomputation of fluid dynamics for estuarine areas and tidal flats.However,the adoption of Cartesian com-ponents of velocity vectors in this kind of non-orthogonal coordinates will definitely result in a diffculty insolving implicitly the transformed momentum equations,and also complicate the wet-dry point judgementused for flood areas.To solve this problem,equations in terms of generalized contravariant velocityvectors in curvilinear coordinates are derived in the present study,by virtue of which,anAlternative-Direction-Implicit numerical scheme in non-orthogonal grids would then be easily obtained,and wet-dry point judgement would as well be largely simplified.A comparison is made between the explic-it scheme and implicit scheme,showing that the present model is accurate and numerically stable for com-putations of fluid dynamics for estuarine areas and tidal flats.     

A BFG model for calculation of tidal current and diffusion of pollutants in nearshore areas

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An implicit method using contravariant velocity components and its application to calculations in a harbour-channel area

lNTRODUCTIONAsoneofthenumericalcalculationmethodsinvolvingfluiddynamicsinnearshoreareas,theboundaryfittedgridmethodhasmanyadvantagessuchasfittingboundaries,beingsuitableforengineeringconstructionswithsmallscalesandimprovingtheaccuracybydensifyinggridpointsintheinterestedareas'Incomparisonwiththefiniteelementmethodwithnon-uniformgrids,theboundary-fittedgridmodelismorewidelyusedbecauseofitssuperioritiesinusingthematurefi-nitedifferenceschemeandinoccupyingsma1lcomputermemory(Sheng,l986;Haase.…     

Nonhydrostatic tidal dynamics in the area of a seamount

The nonhydrostatic boundary problem for an arbitrary three-dimensional domain with a seamount is considered. The problem is integrated into curvilinear boundary-fitted coordinates on a nonuniform grid. In order to identify nonhydrostatic effects the grid is condensed on the slopes of the seamount preserving a coarse resolution in the rest of the domain, where the problem is solved in the hydrostatic approximation. Calculation results for the nonhydrostatic tidal dynamics and hydrology of the Strait of Messina in the area of a seamount are presented.     

On the problem of calculating transport and admixture diffusion in an arbitrary area

A boundary-value problem is considered for the equation of transport and admixture diffusion in curvilinear coordinates conformed to the area's outline. The problem is solved numerically in a canonical domain and the initial area is reflected onto it. The method is applied to the computation of the passive admixture concentration field in the Neva inlet.Translated by Mikhail M. Trufanov.     

Numerical solution of the equations for spatial nonlinear steady-state traveling waves on the free surfaces of homogeneous and two-layer bodies of water

This study is devoted to the construction of some numerical solutions to the previously proposed model equations for three-dimensional disturbances of a small but finite amplitude in liquids of a constant depth. The forms of progressive steady-state waves (both periodic in two horizontal coordinates and solitary) are demonstrated. In a homogeneous liquid, these forms depend on the magnitudes and directions of the wave vectors, whereas, in bodies of water with a small density jump, they also depend on the ratio of layer depths.     

Baroclinic seiches in rotating basins of variable depth in the case of two-layer density stratification

We solve a plane problem of linear baroclinic seiches in closed rotating basins of variable depth with two-layer density stratification. In the long-wave approximation, we get a boundary-value problem for a system of ordinary differential equations and propose a numerical procedure for finding internal seiches. The analytic solutions of the problem are obtained for a basin of constant depth. The numerical analysis of seiches is performed for the distributions of depth corresponding to the zonal and meridional sections of the Black Sea and model basins including the cases of a shelf zone and an underwater ridge. It is shown that the baroclinic seiches become more intense in shallow-water regions and that the intense longshore currents caused by Earth’s rotation are formed in the shelf zones and over the underwater ridges.     

Simulation of unsteady oceanic cable deployment by direct integration with suppression

An improved algorithm is developed for predicting the transient response of a system of serially connected cables and bodies during unsteady deployment from a surface vessel. The governing equations of a cable-body system are derived with dependent variables of cable velocities, direction cosines and tension magnitude to form a nonlinear combined initial-value and boundary-value problem. The problem is then solved by introducing a stable Newmark-like implicit integration scheme in time and by a direct integration method with suppression of extraneous erroneous solutions. Special boundary conditions simulating actively controlled payout and slack-cable/ocean-bottom contact boundary conditions are included in the present model.     

Feasibility of Coupled Pattern Reconstruction Technique in the Detection of Internal Waves

A key problem of contemporary static and kinematic positioning is the problem of transformation of conformai coordinates of universal Mercator projection (UMP) type from a local datum (regional, national) to a global datum, for instance, the World Geodetic System 1984 (WGS 84) with reference to Boyle (1987). Such a problem is met if we use WGS 84 GPS‐derived ellipsoidal coordinates of a point for localization in a local chart of UMP type. In this article we derive and test the equations of a curvilinear datum transformation of ellipsoidal GPS coordinates in a global datum to conformai coordinates of UMP type in a local datum. The curvilinear datum transformation includes three parameters for translation, three parameters for rotation, one scale parameter, and two form parameters which account for a change in the semimajor axis and in the relative eccentricity of the reference ellipsoid.     

Interactions of cnoidal waves with cylinder arrays

The three-dimensional scattering of cnoidal waves by cylinder arrays are studied numerically by using the generalized Boussinesq equations. The boundary-fitted coordinate transformation and a dual-grid technique are used to simplify the finite-difference computation. Also, a set of open boundary conditions and an incident cnoidal wave are incorporated for time-domain simulation. The free-surface elevation and hydrodynamic forces on each cylinder are calculated to illustrate the evolution of nonlinear waves and their interactions with large cylinder arrays. Comparisons are made between the present nonlinear wave loads and those obtained from linear diffraction theory. The sheltering role played by the neighboring cylinders and the feature of wave interference are discussed.     

On the motions of the underwater remotely operated vehicle with the umbilical cable effect

In this paper, a hydrodynamic model is developed to simulate the six degrees of freedom motions of the underwater remotely operated vehicle (ROV) including the umbilical cable effect. The corresponding hydrodynamic forces on the underwater vehicle are obtained by the planar motion mechanism test technique. With the relevant hydrodynamic coefficients, the 4th-order Runge–Kutta numerical method is then adopted to solve the equations of motions of the ROV and the configuration of the umbilical cable. The multi-step shooting method is also suggested to solve the two-end boundary-value problem on the umbilical cable with respect to a set of first-order ordinary differential equation system. All operation simulations for the ROV including forward moving, ascending, descending, sideward moving and turning motions can be analyzed, either with or without umbilical cable effect. The current effect is also taken into consideration. The present results reveal that the umbilical cable indeed significantly affects the motion of the ROV and should not be neglected in the simulation.     

A Three-dimensional Tidal Model in Boundary-fitted Curvilinear Grids

In hydrodynamic models of marine and estuarine currents the use of boundary-fitted curvilinear grids not only makes the model grids fit to the coastline and bathymetry well, but also makes the kinetic boundary conditions simple and more accurate. Because of these advantages, a three-dimensional tidal model with boundary-fitted curvilinear grids has been developed to simulate both tide and current in estuarine and shelf water. The basic idea is to use a set of coupled σ-stretched and elliptic transformations to map the physical space into a corresponding transformed space such that all boundaries are coincident with co-ordinate lines and the transformed grids are rectangular. The hydrodynamic equations in the transformed space are solved in a rectangular mesh of the transformed grid system. The application of the model in the simulation of the M₂,S₂ , K₁and O₁tidal waves in the Bohai Sea, China, shows that the numerical results are in good agreement with the observations. The method of boundary-fitted curvilinear grids is effective for improving simulation accuracy of current in the estuarine and shallow seas, especially at the coastal regions where the current was usually impractical by a uniform Cartesian grid system with the shoreline and bathymetry represented by numerous stair-steps.     

Modeling sea dynamics and turbulent zones on high spatial resolution nested grids

The problem of simulating sea dynamics in areas comprising near-shore zones and zones of high turbulence is considered. A mathematical model and the numerical algorithm of its solving are formulated. The model is based on the equations for nonhydrostatic dynamics and includes (k-ε) and (k-ω) parameterization of turbulent processes. The equations of the model are written in a σ-coordinate system. The numerical algorithm for solving the problem is based on the use of implicit schemes owing to the splitting with respect to the physical processes and space coordinates. The model calculations were performed for four nested sea basins with different spatial resolution: the Baltic Sea (3.7-km space resolution), the Gulf of Finland (1.85-km resolution), the Tallinn-Helsinki area (560-m resolution), and Tallinn Bay (93-m resolution). The results of the experiment show that the model well simulates the processes of enhanced turbulent activity in the near-shore zones that affect the local features of the sea characteristics.     

Hadley and Rossby regimes in a simple model of convection of a rotating fluid

This paper analyzes the properties of solutions to the equations describing the motion of a stratified fluid in the class of velocity and temperature fields linear in coordinates. For an ideal fluid, these equations, on the one hand, are exact for the corresponding hydrodynamic problem and, on the other hand, are identical to the equations of motion for a heavy top. In a conservative case, the equations of motion of a top share common solutions with the equations of geophysical fluid dynamics and reproduce motions similar to those existing in the theory of the large-scale atmospheric circulation. This study considers the effects of viscosity and heat conduction in the fluid, which are, in a sense, similar to the effect of friction in the case of a top. The influence of deflections of the vectors of gravity and external rotation from their standard directions for a plane-parallel atmosphere is also considered. The regimes of motions that are described by the starting equations and approximations commonly used to model the atmospheric general circulation (the quasi-geostrophic approximation) are analyzed. It is shown that these equations correctly describe the Hadley and Rossby circulation regimes and transitions between them that are observed in numerical and laboratory experiments. Particular attention is given to the consistency between different regimes of the exact equations and their quasi-geostrophic approximations, which is manifested for small Rossby numbers and is generally absent for large Rossby numbers. The asymptotic behaviors of the curves of transition between the Hadley and Rossby regimes under the conditions of breaking the external symmetry of flows are obtained. These asymptotics explain the corresponding transition boundaries for the regimes observed in the known experiments in annuluses.     

Fluid-induced inertia and damping in vibrating offshore structures

The problem addressed in this paper is that of the fluid-structure interaction in offshore structures for which effects due to fluid viscosity are negligible. A method for evaluating the dynamic response of structures is described in which the generalized coordinates for the analysis are associated with the ‘dry’ modes of the structure. All hydrodynamic actions, including those induced by motions of the structure, are retained on the right-hand side of the equations of motion. This formulation requires the determination of generalized added mass and damping matrices associated with motions of the structure in its first few ‘dry’ modes. By these means, free surface and three dimensional fluid flow effects are retained in the analysis. Examples of the resulting frequency dependent matrices, computed using the boundary integral method, are presented for some common structural forms, and the validity of this approach is demonstrated through some experiments on a deforming column structure. Finally, results for the wave-excited dynamic response of a typical offshore structure obtained using (i) conventional strip theory ‘wet’ modes, and (ii) ‘dry’ modes, are compared.     

Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure

A numerical method for non-hydrostatic, free-surface, irrotational flow governed by the nonlinear shallow water equations including the effects of vertical acceleration is presented at the aim of studying surf zone phenomena. A vertical boundary-fitted grid is used with the water depth divided into a number of layers. A compact finite difference scheme is employed for accurate computation of frequency dispersion requiring a limited vertical resolution and hence, capable of predicting the onset of wave breaking. A novel wet–dry algorithm is applied for a proper handling of moving shoreline. Mass and momentum are strictly conserved at discrete level while the method only dissipates energy in the case of wave breaking. The numerical results are verified with a number of tests and show that the proposed model using two layers without ad-hoc assumptions enables to resolve propagating nonlinear shoaling, breaking waves and wave run-up within the surf and swash zones in an efficient manner.     

Gravitational mode calculation of basins discretized by orthogonal curvilinear grids

This paper presents a simple and straightforward method for carrying out the direct numerical solution of the eigenvalue problem associated to the homogeneous linear shallow-water equations expressed using orthogonal curvilinear coordinates, when ‘adiabatic’ boundary conditions apply. These equations, together with the boundary conditions, define a self-adjoint problem in the continuum. The method presented here, which is thought for calculating the 2-D theoretical gravity modes of both natural and artificial basins, relies on a change of basis of the dependent variable vector. This preliminary transformation makes it, in fact, possible to formulate two different numerical approaches which guarantee the self-adjoint property of the discrete form of the system consisting of the governing equations and the boundary conditions. The method is tested using a square and a fully circular domain, both of which allow comparisons with well-known analytical and numerical solutions. Discretizing the physical domain of a fully circular basin by a cylindrical coordinate grid makes it possible to show the actual efficiency of the method in calculating the theoretical gravity modes of basins discretized by a boundary-following coordinate grid which allows laterally variable resolution.