Expression of strain tensor in orthogonal curvilinear coordinates

Based on an analysis of connotation and extension of the concept of the orthogonal curvilinear coordinates, we have deduced a platform of strain tensor expression of Cartesian coordinates, which turns out to be a function of Lame coefficient and unit vector. By using transform matrix between Cartesian coordinates and orthogonal eurvilinear coordinates, we have deduced a mathematical expression for correcting displacement vector differential in orthogonal curvilinear coordinates, and given a general expression of strain tensor in orthogonal curvilinear coordinates.

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.

Mathematical Models for Combined Refraction-Diffraction of Waves on Non-Uniform Current and Depth

Two mathematical models for combined refraction-diffraction of regular and irregular waves on non-uniform current in water of slowly varying topography are presented in this paper. Model I is derived by wave theory and variational principle separately. It has two kinds of expressions including the dissipation term. Model n is based on the energy conservation equation with energy flux through the wave crest lines in orthogonal curvilinear coordinates and the wave kinematic conservation equation. The analysis and comparison and special cases of these two models are also given.

A Note on Natural Coordinates and Frenet Frames

This paper discusses the relationship between natural coordinates in fluid mechanics and orthogonal curvilinear coordinates. Since orthogonal curvilinear coordinates have some excellent mathematical properties, natural coordinates can be applied more widely if they can be transformed to orthogonal curvilinear coordinates. Frenet formulas which describe the differential properties of natural coordinates were compared with the derivative formulas of orthogonal curvilinear coordinates to show that natural coordinates are not generally orthogonal curvilinear coordinates. A method was introduced to transform natural coordinormal planes of the natural coordinates about the streamlines. The transformation is true as long as the natural coordinates satisfy several equations. Vorticity decomposition in the natural coordinates is used to show that these conditional equations are satisfied only if the streamlines are perpendicular to the vortexlines on every given point in the flow field. These equations apply in both planar flows and axisymmetric flows without a circumferential velocity component, but do not apply in some 3-D flows such as Beltrami flow.

NUMERICAL SOLUTION FOR TURBULENT FLOWS ON CONCAVE SURFACES OF SPILLWAY DAMS

This paper is concerned with the numerical solution of turbulent flows on the concave surfaces of spillway dams. Orthogonal curvilinear coordinates are used to deal with the complicated computational region and the effects of streamline curvature on turbulent flows are included. The SIMPLEC procedure has been used for the transformed governing equations in the transformed domain. The comparison between computed results and experimental data shows a satisfactory agreement.

A new three-dimensional terrain-following tidal model of free-surface flows

A three-dimensional hydrodynamic model is presented which combines a terrain-following vertical coordinate with a horizontally orthogonal curvilinear coordinate system to fit the complex bottom topography and coastlines near estuaries, continental shelves, and harbors. To solve the governing equations more efficiently, we improve the alternating direction implicit method, which is extensively used in the numerical modeling of horizontal two-dimensional shallow water equations, and extend it to a three-dimensional tidal model with relatively little computational effort. Through several test cases and realistic applications, as presented in the paper, it can be demonstrated that the model is capable of simulating the periodic to-and-fro currents, wind-driven flow, Ekman spirals, and tidal currents in the near-shore region.

NUMERICAL SOLUTION FOR UNSTEADY 2－D FLOW USING THE TRANSFORMED SHALLOW WATER EQUATIONS

This paper is concerned with the numerical solution of two-dimensional flow.The technique of boundary-fitted coordinate systems is used to overcome the difficulties resulting from the complicated shape of natural river boundaries; the method of fractional steps is used to solve the partial differential equations in the transformed plane; and the technique of moving boundary is used to deal with the river bed exposed to water surface. Comparison between computed and experimental data shows a satisfactory agreement.