Numerical Modeling of the Hyperbolic Mild-Slope Equation in Curvilinear Coordinates

The mild-slope equation is familiar to coastal engineers as it can effectively describe wave propagation in nearshore regions. However, its computational method in Cartesian coordinates often renders the model inaccurate in areas with irregular shorelines, such as estuaries and harbors. Based on the hyperbolic mild-slope equation in Cartesian coordinates, the numerical model in orthogonal curvilinear coordinates is developed. The transformed model is discretized by the finite difference method and solved by the ADI method with space-staggered grids. The numerical predictions in curvilinear co- ordinates show good agreemenl with the data obtained in three typical physical expedments, which demonstrates that the present model can be used to simulate wave propagation, for normal incidence and oblique incidence, in domains with complicated topography and boundary conditions.

Numerical Simulation of Wave-Induced Currents Combined with Parabolic Mild-Slope Equation in Curvilinear Coordinates

Researches on breaking-induced currents by waves are summarized firstly in this paper. Then, a combined numerical model in orthogonal curvilinear coordinates is presented to simulate wave-induced current in areas with curved boundary or irregular coastline. The proposed wave-induced current model includes a nearshore current module established through orthogonal curvilinear transformation form of shallow water equations and a wave module based on the curvilinear parabolic approximation wave equation. The wave module actually serves as the driving force to provide the current module with required radiation stresses. The Crank-Nicolson finite difference scheme and the alternating directions implicit method are used to solve the wave and current module, respectively. The established surf zone currents model is validated by two numerical experiments about longshore currents and rip currents in basins with rip channel and breakwater. The numerical results are compared with the measured data and published numerical results.

3D Numerical Simulation of Overbank Flow in Non-Orthogonal Curvilinear Coordinates

The velocity field in meandering compound channels with overhank flow is highly three dimensional. To date, its features have been investigated experimentally and little research has been undertaken to investigate the feasibility of reproducing these velocity fields with computer models. If computer modeling were to prove successful in this context, it could become a useful prediction technique and research tool to enhance our understanding of natural river dynamics. A 3-D k-E turbulence hydrodynamic model in curvilinear coordinates is established to simulate the overhank flow. The bodyfitted coordinate is adopted in the horizontal plane, the part grid is adopted in the vertical direction, and the wall-function method is employed to simulate the bed resistance. The model is applied to the simulation of the meandering channel with straight flood plain banks, and the main velocities and secondary velocities for both the longitudinal and cross sections are presented. Comparison and analysis show that the results of simulation are fit to reflect the results of experiment. These results show the application value of the model to 3D overhank flow.

A Numerical Model of Nonlinear Wave Propagation in Curvilinear Coordinates

For the simulation of the nonlinear wave propagation in coastal areas with complex boundaries, a numerical model is developed in curvilinear coordinates. In the model, the Boussinesq-type equations including the dissipation terms are em- ployed as the governing equations. In the present model, the dependent variables of the transformed equations are the free surface elevation and the utility velocity variables, instead of the usual primitive velocity variables. The introduction of utility velocity variables which are the products of the contravariant components of the velocity vector and the Jacobi ma- trix can make the transformed equations relatively concise, the treatment of lateral boundary conditions easier and the de- velopment of the program simpler. The predictor-corrector method and five-point finite-difference scheme are employed to discretize the time derivatives and the spatial ones, respectively. The numerical model is tested for three cases. It is found that the numerical results are in good agreement with the analytical results and experimental data.

SPACE-TIME CONSERVATION SCHEME IN NONORTHOGONAL CURVILINEAR COORDINATES

A scheme of space time conservation (STC) based on the method of space time conservation element and solution element (CE/SE) is represented in a nonorthogonal curvilinear coordinate system. The corresponding initial and boundary conditions are discussed. It is seen that in the nonorthogonal coordinates the scheme maintains the advantages of the STC method, and is noted for its simple structure, clear physical meaning, rapid calculation and high accuracy. It is easy to extend to the multidimensional flow. The numerical results for a 2D Euler equation show good agreement with those from other computational methods and the experiment.

Numerical model of flow and pollutant transportation in non-orthogonal curvilinear coordinates

The planar 2D k-ε double equations' turbulence model was adopted and transformed into non-orthogonal curvilinear coordinates. The concentration convection-diffusion was introduced to planar 2D SIMPLEC algorithm of flow in non-orthogonal curvilinear coordinates. The numerical model of pollutant transportation in non-orthogonal curvilinear coordinates was constructed. The model was applied to simulate the flow and pollutant concentration fields. In the testing concentration field, two optimal operations of contamination discharging both along bank and in the centerline at the first bend of the meandering channel were adopted. Comparison with available data showed the model developed was successful, was valuable to engineering application.

Curve integral with path independent in orthogonal curvilinear coordinate system

It is explored that the line integral is a path independent in two or three arbitrary dimensional orthogonal curvilinear coordinate systems, which is based on the integral condition with the path independent in two or three dimensional rectangular coordinate systems. Firstly, according to the coordinate transformation, the condition that the line integral is the path independent in the polar coordinate system is obtained easily from the Green's theorem in two-dimensional rectangular coordinate system and the condition is extended to arbitrary two-dimension orthogonal curvilinear coordinates. Secondly, through the coordinate transformation relationship and the area projection method, the Stokes formula in three-dimensional rectangular coordinate system is promoted to the spherical coordinate system and cylindrical coordinate system, and the condition that the line integral is a path independent is obtained. Furthermore, the condition is extended to arbitrary three-dimension orthogonal curvilinear coordinates. Lastly, the conclusions are made.

Generalization of the Second Order Vector Potential Formulation for Arbitrary Non-Orthogonal Curvilinear Coordinates Systems from the Covariant Form of Maxwell's Equations

A great number of semi-analytical models, notably the representation of electromagnetic fields by integral equations are based on the second order vector potential (SOVP) formalism which introduces two scalar potentials in order to obtain analytical expressions of the electromagnetic fields from the two potentials. However, the scalar decomposition is often known for canonical coordinate systems. This paper aims in introducing a specific SOVP formulation dedicated to arbitrary non-orthogonal curvilinear coordinates systems. The electromagnetic field representation which is derived in this paper constitutes the key stone for the development of semi-analytical models for solving some eddy currents moelling problems and electromagnetic radiation problems considering at least two homogeneous media separated by a rough interface. This SOVP formulation is derived from the tensor formalism and Maxwell’s equations written in a non-orthogonal coordinates system adapted to a surface characterized by a 2D arbitrary aperiodic profile.

Electrodynamics in Curvilinear Coordinates and the Equation of a Geodesic Line

The creation of the theory of relativity, which discovered the equivalence of mass and energy, showed that the concept of a point charge, used in the formulation of Coulomb’s law, one of the basic laws of classical electrodynamics, contradicts the famous formula establishing the equivalence of mass and energy. But the discovery of quarks makes it possible to present classical electrodynamics in a form free from the indicated contradiction. In the article, having considered the electromagnetic field in a curvilinear coordinate system, a theory has been created that expands our understanding of the electromagnetic field, the nature of quarks, the nature of strong interaction, and the connection between strong interaction and electromagnetic interaction. This theory is based on the principle of equivalence of an electromagnetic field to a free material particle formulated in the article and the law of formation of elementary particles from an electromagnetic field that follows from it.

Staggered Calculation of Turbulent Flows Over a 3D Bluff Body Using Curvilinear Nonorthogonal Coordinates

A finite difference method is developed to predict turbulent flows over 3D bluffbodies. The K-ε turbulence model with Launder and Spalding's wall treatment isemployed. The solution alsorithm is based on a body fitted nonorthogonalcurvilinear eourdinate system and a stagsered grid arrangement. The covariantvelocity components are chosen as dependent variables. Convective fluxes aredescribed by the Power haw Scheme. The grids are generated with an ellipticgrid generator using control functions. Results obtained are compared withexporiment measurements and other calculations.

Expression of strain tensor in orthogonal curvilinear coordinates

基于涵义的分析和直角的曲线的坐标的概念的扩展，我们推出了笛卡儿的坐标的紧张张肌表示的一个平台，它结果是瘸的系数和单位向量的一个函数。由使用在笛卡儿的坐标和直角的曲线的坐标之间的变换矩阵，我们为改正在直角的曲线的坐标微分的排水量向量推出了一个数学表达式，并且在直角的曲线的坐标给紧张张肌的一般表情。

Algebraic-Stress Turbulent Model of Planar 2-D Pollutant Transportation in Curvilinear Coordinates

The shallow-water equations and pollutant convective-diffusive equation are transformed into curvilinear coordinate system. The anisotropic algebraic-stress turbulent model is introduced to simulate the turbulence items, and algebraic-stress turbulent model of planar 2-D pollutant convective-diffusive in curvilinear coordinates is build. The meandering channel with measured data of concentration in lab is adopted to validate the model, and the distribution figure of pollutant concentration field calculated through this model and that of the k-ε model, which show the model is superior to k-ε turbulent model in dealing with anisotropy character of flow.

仿射空间拉普拉斯算子探讨

为了进一步研究提升惯性导航系统的精度,对拉普拉斯算子进行讨论:拉普拉斯算子是个微分算子,拉普拉斯方程又名调和方程、位势方程,求解拉普拉斯方程是物理学和力学等领域经常遇到的一类重要数学问题;基于曲线坐标系和张量理论,分别给出n维仿射空间中的拉普拉斯算子表达式,并证明不同表述的等价性;最后,基于格林定理和变方方法,分别给出3维仿射空间拉普拉斯算子的表达式。

NUMERICAL SIMULATION OF 2D HORIZONTAL COOLING WATER DISCHARGE IN GENERALIZED CURVILINEAR COORDINATE

2D horizontal model is one of the major mathematical methods for the research on cooling water discharge from the power plant. In this paper, the shallow water equations are transformed under the generalized curvilinear coordinate system and the elliptic differential equations are used to generate curvilinear grids, so a model in generalized curviline ar coordinate is presented to simulate 2D horizontal cooling water, Governing equations of the model are discretized by finite volume method, and non-staggered grids and SIMPLE algorithm are introduced to simplify the program during the discretization. This model is used to simulate the movement of cooling water in a simplified meandering channel and a natural channel, calculating results indicate this model can correctly reflect the movement rules of cooling water, which verifies the model can be applied in engineering practice.

MATHEMATICAL MODEL FOR 2-D TIDAL FLOW AND WATER QUALITY WITH ORTHOGONAL CURVILINEAR COORDINATES

This paper presents a numerical method for simulating the 2-D tidal flow andwater quality with the orthogonal curvilinear coordinates. In order to overcome the computationaldifficulties in natural rivers, such as the complicated boundary figures, the great disparitybetween length and width of computational domain, etc. , orthogonal boundary-filled grid was used.The irregular domain in physical plane was transformed into a rectangular domain in a transformedplane, and the depth-averaged momentum equations and mass equation were given and discretized basedon the alternating direction implicit finite difference scheme in curvilinear coordinates. Theapplication of the presented method was illustrated by an example of analyzing the Yangtze River inthe vicinity of Nanjing city. A fair agreement between the measured data and computed resultsdemonstrates the validity of the developed method.

NUMERICAL SIMULATION OF 2-D TIDAL FLOW AND WATER QUALITY UNDER THE CURVILINEAR COORDINATES

This paper presents a numerical method to simulate the 2-D tidal flow and water quality under the curvilinear coordinates. In order to overcome the computational difficulties in natural rivers, such as the complicated boundary figures, the great disparity between length and width of computational domain, etc. , boundary-fitted grid is used, the irregular domain in physical plane is transformed into a rectangular domain in transformed plane, and the depth-averaged momentum equations and mass equation are rewritten and discretized based on the finite volume techniques in curvilinear coordinates. Practical application of the method is illustrated by an example for the Dachangzhen Section of the Yangtze River. A fair agreement between the values measured and computed demonstrates the validity of the method developed.

Vorticity vector-potential method based on time-dependent curvilinear coordinates for two-dimensional rotating flows in closed configurations

In this study, a vorticity vector-potential method for two-dimensional viscous incompressible rotating driven flows is developed in the time-dependent curvilinear coordinates. The method is applicable in both inertial and non-inertial frames of reference with the advantage of a fixed and regular calculation domain. The numerical method is applied to triangle and curved triangle configurations in constant and varying rotational angular velocity cases respectively. The evolutions of flow field are studied. The geostrophic effect, unsteady effect and curvature effect on the evolutions are discussed.

WATER WAVE SIMULATION IN CURVILINEAR COORDINATES USING A TIME-DEPENDENT MILD SLOPE EQUATION

The purpose of this article is to model the detailed progress of wave propagation in curvilinear coordinates with an effective time-dependent mild slope equation. This was achieved in the following approach, firstly deriving the numerical model of the equation, i.e., Copeland＇s hyperbolic mild-slope equation, in orthogonal curvilinear coordinates based on principal of coordinate transformation, and then finding the numerical solution of the transformed model by use of the Alternative Directions Implicit （ADI） method with a space-staggered grid. To test the curvilinear model, two cases of a channel with varying cross section and a semi-circular channel were studied with corresponding analytical solutions. The model was further investigated through a numerical simulation in Ponce de Leon Inlet, USA. Good agreement is reached and therefore, the use of the present model is valid to calculate the progress of wave propagation in areas with curved shorelines, nearshore breakwaters and other complicated geometries.

A three-dimensional k-ε-k_p model in curvilinear coordinates for sediment movement and bed evolution

To aim at the substitution of the magnitude and direction of water flow movement near bed for those of bed load transport in solid-liquid two-phase one-fluid model, and to simulate the effect of secondary flow on transverse bed load transport in channel bends and the effect of bed slope on bed load trans- port in a better way, a three-dimensional k-ε-kp solid-liquid two-phase two-fluid model in curvilinear coordinates is solved numerically with a finite-volume method on an adaptive grid for studying wa- ter-sediment movements and bed evolution in a 120° channel bend. Numerical results show that the trajectories of solid-phase deviate from those of liquid-phase in the channel bend, and the deviation increases with the increase of the particle diameters. The calculated bed deformation by the k-ε-kp model is in better agreement with measured bed deformation than those by one-fluid model. It is proved that the k-ε-kp model can simulate the effect of secondary flow on lateral bed load transport with the higher accuracy than the one-fluid model.

The Patch Test Condition in Curvilinear Coordinate System——Formulation and Application

The constant stress patch test condition has widely been studied in terms of Cartesian coordinates. However, such kind of conditions cannot be used for curvilinear coordinates. In this paper, certain forms of the patch test are established in terms of the general and the orthogonal curvilinear systems respectively. For the axisymmetric problem, the related patch test condition is formulated and used to construct an axisymmetric incompatible element. In addition, some new approaches to avoid spurious shear stress are presented.