圆锥型水轮旋转现象的数学模型及其仿真

Mathematical model and simulation on rotating phenomenon of conical water wheel

推导圆锥型水轮旋转现象的数学模型并对其进行仿真研究。根据圆锥型水轮旋转现象构造其物理模型,通过动量矩定理、质量守恒方程等物理学知识,利用微元法、微积分以及傅里叶展开等数学方法,建立力矩平衡方程、质量守恒方程,推导出圆锥型水轮旋转现象的数学模型为三维非线性微分方程组。运用MATLAB软件绘制模型的分叉图和Lyapunov指数进行数值仿真和分析,展示模型的混沌行为。根据数值仿真结果,分析解释圆锥型水轮混沌旋转现象:当注水率较小时,模型系统是稳定的,此时对应水轮静止或匀速旋转;当注水率增大时,模型系统发生分叉,表明水轮发生不稳定旋转;当注水率进一步增大时,模型系统发生混沌行为,表明水轮出现混沌旋转。

The mathematical model of rotating phenomenon of conical water wheel is deduced and simulated.According to the rotating phenomenon of conical water wheel,the physical model is constructed.Based on the physical knowledge such as moment of momentum theorem and mass conservation equation,the mathematical methods such as differential element method,calculus and Fourier expansion are used to establish the moment balance equation and mass conservation equation.The mathematical model of rotating phenomenon of conical water turbine is deduced as three-dimensional nonlinear differential equations.The bifurcation diagram and Lyapunov exponent of the model are drawn by MATLAB software for numerical simulation and analysis to show the chaotic behavior of the model.Based on the results of numerical simulation,the chaotic rotation of conical turbine is analyzed and explained.When the water injection rate is small,the model system is stable,and the corresponding water wheel is stationary or rotating at constant speed.When the water injection rate increases,the model system bifurcates,which indicates that the water wheel rotates unsteadily.When the water injection rate further increases,the chaotic behavior of the model system occurs,which indicates that the water turbine appears chaotic rotation.