隐式完全非静压模型求解三维自由面的Navier-Stokes方程

Implicit Fully Non-hydrostatic Model for 3D Free-surface Navier-Stokes Equations

为了适应复杂环境水流模拟的需求,详细而准确地描述自由面流动问题,本文采用基于Navier-Stokes控制方程的有效的三维完全非静压模型。此模型基于变量可定义于不同位置的四边形交错网格下,利用时间空间均为二阶精度的隐格式对三维Navier-Stokes方程进行数值离散。同时,与顶层压强的动压处理方式相结合形成了完全的非静压数值离散模型。最后,通过对三维线性驻波和连接丹麦与瑞典之间海域的潮流场的模拟,结果表明它能够准确地预测水流的三维流动结构,而且计算简单高效,具有良好的数值稳定性。验证了当遇到短波高频问题时,完全非静压模型所得的结果与解析解或测量值吻合良好,与静压模型、表层静压假设的非静压模型相比具有很高的准确度,体现了非静压模型的精确性和实用性。

In order to simulate complex environment water flows and to describe free-surface flow problems accurately, based on the Navier-Stokes governing equation, an effective three-dimensional fully non-hydrostatic model is proposed. An implicit method is developed for solving the complete three-dimensional Navier-Stokes equations. With discretization, the algorithm is based upon a staggered finite difference Crank-Nicholson scheme on a Cartesian grid. A new top-layer pressure treatment is introduced so that the three-dimensional model is fully non-hydrostatic and is free of any hydrostatic assumption. The developed model is of second-order accuracy in time and space. Finally, numerical tests are made, including linear three-dimensional standing waves in a closed basin and the simulation of the flow field in the cross-sea project connecting Denmark and Sweden, which demonstrate the efficiency and accuracy of the model on simulating surface waves interacting with structures. Results show that the model can accurately simulate the structure of a three-dimensional flow, and enjoys good numerical stability. For short-wave high frequency problems, the results of fully non-hydrostatic model are in good agreement with analytic solutions or measured values. Future work on fully non-hydrostatic model computation, the computation grid as well as the computing efficiency is suggested.