Fig. 1 (Color online) Sketch of the study area

To determine the real-time navigable flow conditions, a navigable flow condition simulation system was developed. The functional modules of the system include data monitoring, flow calculation, database,and visualization modules. The structure of the system is illustrated in Fig. 2. The data monitoring module is used to obtain the upstream inflow and water level data, and provides the initial calculation parameters for the flow calculation module. These data are obtained from monitoring terminals installed in hydrological stations (see Fig. 1). In this study reach,the hydrological data is refreshed once per hour. The flow calculation module reads the terrain data from the database module and the boundary data acquired by the monitoring module, and outputs navigable flow condition information from the parallel simulation model. When the simulation has been finished, the results are stored in the database module. This module also stores the terrain data of the waterway, the boundary data from the data monitoring module, and the simulation results from the flow calculation module. The visualization module takes the flow simulation results from the database module, and feeds data back to the user in a vivid way, providing a reference for the navigation decision. The main focus of this paper is to study the implementation of the flow calculation module.

2. Method

2.1 2-D hydrodynamic model

2.1.1 Control equation

In modeling the waterway flow, the horizontal scale is larger than the vertical depth scale. Therefore,ignoring the impact of the Coriolis force and wind, the 2-D shallow water equation is used as the control equation. The conservation of the 2-D shallow water equation is as follows:the coordinate system used in the equation is the plane Cartesian coordinate system, where u and v represent the velocity components in the x and y directions, respectively, g represents gravitational acceleration, Sox=-?Zb/?x , Soy=-?Zb/?y represent the bottom slope term of the x and y directions,respectively, Zbrepresents the bottom elevation,represent the friction slope of the x and y directions,respectively, and n is the Manning coefficient.

To adapt to the irregular terrain boundary, the study area is divided into an unstructured quadrilateral mesh, and the finite volume method is used for the numerical discretization of this mesh (Fig. 3).

The integral of the mesh elements is calculated as

Fig. 2 Schematic overview of the navigable flow condition simulation system

Fig. 3 Schematic diagram of the finite volume discretization

Using Gauss? formula to convert the volume integral to a line integral along the boundary of the mesh cell, we obtain

where i is the edge of each mesh element, Δliis the edge length of the mesh element,iF is the numerical flux of edge i, niis the external normal unit vector of edge i, and ΔAkis the area of the mesh.

Roe?s scheme for the Riemann approximate solution is used to solve the interfacial numerical flux

where URand ULare the right and left conservative variables of the interface, respectively, and J is the Jacobian matrix of the Roe scheme.

To allow the model to adapt to the complicated terrain, the bottom slope source term is processed by the bottom slope characteristic decomposition method

whereis the average velocity of the Roe scheme,=0.5[(Zb)L+(Zb)R], with (Zb)L, (Zb)Rdenoting the bottom elevations of the left and right sides of the interface, respectively, and(j=1,2,3) is the eigenvector of. The symbolic function sign() is expressed as:

In this paper, an explicit scheme is adopted for the solution of these equations. Thus, the final discrete equation on the mesh is as follows

2.1.2 Calculation process

Once the calculation has started, the model reads the mesh data and the flow boundary data, initializing the calculation parameters and providing initial values for solving the flux and source terms. The mesh equations are then solved with the explicit finite volume method, which enables the state of the current moment of each mesh to be associated with that of the previous moment of the adjacent meshes. Thus, the solution of the flux and source terms for each mesh is independent, and we can obtain the state information(i.e., the water depth and velocity components in the x and y directions) of each mesh at time T+Δt as long as the interface flux, bottom slope source term,and friction source term are solved according to the state information of each mesh at time T. When the calculation is finished, state information is obtained for each mesh at time Tmax. Finally, state information for each mesh at Tmaxis output and stored in the database. The calculation process of the model is illustrated in Fig. 4.

2.2 Parallel model implementation based on openmp

2.2.1 Basic concepts of openmp

文章来源：《水动力学研究与进展》 网址: http://www.sdlxyjyjzzz.cn/qikandaodu/2021/0114/463.html

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