自由表面流动问题数值方法的理论研究及应用
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摘要
自由表面流动问题的数值模拟一直是水动力学和计算流体力学研究领域中的一个热点和难点,研究自由表面流动问题的数值方法几乎涉及到了计算流体动力学的所有理论方法和前沿技术。在船舶水动力学、航天、水利工程、机械工程、石油化工及土木工程中,均存在大量的自由表面流动问题。因此,对自由表面流动问题数值方法的研究具有重要的理论研究意义和实际工程意义。
本文首先对当今比较流行的自由表面流动问题数值方法的理论进行全面回顾和系统研究。详细论述了基于欧拉网格的Hirt和Nichols、FLAIR、Youngs、FCT这四种用于界面重构的VOF(流体体积)方法、基于无结构网格FVM(有限体积方法)的Level Set方法,和无网格的SPH(光滑粒子流体动力学)方法的基本理论。
通过数值模拟平移流场、旋转流场及剪切流场这三个追踪自由表面的经典算例,对基于欧拉网格的VOF方法、基于无结构网格的Level Set方法和无网格的SPH方法进行了比较分析,以自由表面追踪的效果为依据定性地选择比较好的数值方法,从而将SPH方法作为进一步研究的对象。
研究了适用于广义流体动力学的SPH控制方程及其相关的数值技术。由拉格朗日形式的流体控制方程推导出SPH形式下的流体控制方程,应用人工粘性的SPH表达式来模拟求解粘性问题,应用人工压缩率并通过选择对应的状态方程将不可压缩流体模拟为拟不可压缩流体来进行求解分析。论述了固壁边界条件和自由边界条件的处理、时间积分与时间步长的确定方法,特别是自由表面粒子搜索的方法。
在广义流体动力学的SPH方法基础上,考虑Riemann(黎曼)解法,采用了一种适合强间断性表面的改进的自由表面粒子确定方法,用于确定自由表面的位置,并给出了流体表面张力的计算方法。采用FORTRAN语言编制了计算程序并应用于算例-水珠溅落液面问题,实现了水珠溅落液面问题的SPH方法数值模拟,并将计算结果与Level Set方法作对比。
应用SPH方法对不同领域的实际工程的自由表面流动问题进行数值求解,成功实现了对水利工程领域的大坝表孔泄流问题和船舶水动力学领域的液舱晃荡问题的数值模拟。
The numerical simulation of free surface flow has been a hot and difficult problem in the field of hydrodynamics and computational fluid dynamics research. The numerical simulation of the free surface flow involved almost all the theory of computational fluid dynamics methods and cutting-edge technology. the question of Free surface flow has been existed in the engineering field, such as ship hydrodynamics, aerospace, hydraulic engineering, mechanical engineering, petrochemical engineering and civil engineering, and so on., Therefore, the problem of free surface flow numerical method of research was important and significance both in theory and in practical engineering.
Firstly, the popular numerical methods of the free surface flows were reviewed and investigated systematically. The VOF (Volume of Fluid) method for the interface reconstruction based on the Euler grid, which was containd by Hirt and Nichols method, FLAIR method, Youngs method, FCT method, and Level Set Method based on unstructured grid FVM (Finite Volume Method), and SPH(Smoothed Partical Hydrodynamics) based on the meshfree method, all of which, the basic theory were elaborated in detail.
The numerical methods aboved were compared and analyzed by simulating the three classic examples, which were simple convection flow field, a rotational flow field, and a shear flow field.And selected a finer method from aboved methods by the effect of free surface tracking, and the finer mthod was SPH, which would be discussed in this paper.
SPH control equations which can be applied to generalized hydrokinetics and related numerical techniques were studied. SPH control equations were derived from Lagrangian equations. Viscosity problem was simulated and solved by applying SPH expression which contained viscosity. Incompressible fluid was converted to approximated incompressible fluid to be solved and analyzed by using artificial compressibility and selecting corresponding state equation. The solution of the solid wall boundary conditions and free boundary conditions, time integration method and time step identification, especially the method for searching free surface particles, all of which were discussed.
Based on the condition about Riemann condition and SPH method of the generalized hydrokinetics, a new method of how to search the free surface particles and how to calculate free surface tension was determined. SPH numerical simulation was enforced by a Fortran program to simulate the problem of a drop impacting free liquid surface, and its results were compared with the level set result.
SPH method was applied to simulate free surface flow in the practical engineering field. And the two numerical simulation was finished by SPH, which are the liquid sloshing in ship hydrodynamics field and the discharge through crest orifice on dam in the hydraulic engineering field.
本文首先对当今比较流行的自由表面流动问题数值方法的理论进行全面回顾和系统研究。详细论述了基于欧拉网格的Hirt和Nichols、FLAIR、Youngs、FCT这四种用于界面重构的VOF(流体体积)方法、基于无结构网格FVM(有限体积方法)的Level Set方法,和无网格的SPH(光滑粒子流体动力学)方法的基本理论。
通过数值模拟平移流场、旋转流场及剪切流场这三个追踪自由表面的经典算例,对基于欧拉网格的VOF方法、基于无结构网格的Level Set方法和无网格的SPH方法进行了比较分析,以自由表面追踪的效果为依据定性地选择比较好的数值方法,从而将SPH方法作为进一步研究的对象。
研究了适用于广义流体动力学的SPH控制方程及其相关的数值技术。由拉格朗日形式的流体控制方程推导出SPH形式下的流体控制方程,应用人工粘性的SPH表达式来模拟求解粘性问题,应用人工压缩率并通过选择对应的状态方程将不可压缩流体模拟为拟不可压缩流体来进行求解分析。论述了固壁边界条件和自由边界条件的处理、时间积分与时间步长的确定方法,特别是自由表面粒子搜索的方法。
在广义流体动力学的SPH方法基础上,考虑Riemann(黎曼)解法,采用了一种适合强间断性表面的改进的自由表面粒子确定方法,用于确定自由表面的位置,并给出了流体表面张力的计算方法。采用FORTRAN语言编制了计算程序并应用于算例-水珠溅落液面问题,实现了水珠溅落液面问题的SPH方法数值模拟,并将计算结果与Level Set方法作对比。
应用SPH方法对不同领域的实际工程的自由表面流动问题进行数值求解,成功实现了对水利工程领域的大坝表孔泄流问题和船舶水动力学领域的液舱晃荡问题的数值模拟。
The numerical simulation of free surface flow has been a hot and difficult problem in the field of hydrodynamics and computational fluid dynamics research. The numerical simulation of the free surface flow involved almost all the theory of computational fluid dynamics methods and cutting-edge technology. the question of Free surface flow has been existed in the engineering field, such as ship hydrodynamics, aerospace, hydraulic engineering, mechanical engineering, petrochemical engineering and civil engineering, and so on., Therefore, the problem of free surface flow numerical method of research was important and significance both in theory and in practical engineering.
Firstly, the popular numerical methods of the free surface flows were reviewed and investigated systematically. The VOF (Volume of Fluid) method for the interface reconstruction based on the Euler grid, which was containd by Hirt and Nichols method, FLAIR method, Youngs method, FCT method, and Level Set Method based on unstructured grid FVM (Finite Volume Method), and SPH(Smoothed Partical Hydrodynamics) based on the meshfree method, all of which, the basic theory were elaborated in detail.
The numerical methods aboved were compared and analyzed by simulating the three classic examples, which were simple convection flow field, a rotational flow field, and a shear flow field.And selected a finer method from aboved methods by the effect of free surface tracking, and the finer mthod was SPH, which would be discussed in this paper.
SPH control equations which can be applied to generalized hydrokinetics and related numerical techniques were studied. SPH control equations were derived from Lagrangian equations. Viscosity problem was simulated and solved by applying SPH expression which contained viscosity. Incompressible fluid was converted to approximated incompressible fluid to be solved and analyzed by using artificial compressibility and selecting corresponding state equation. The solution of the solid wall boundary conditions and free boundary conditions, time integration method and time step identification, especially the method for searching free surface particles, all of which were discussed.
Based on the condition about Riemann condition and SPH method of the generalized hydrokinetics, a new method of how to search the free surface particles and how to calculate free surface tension was determined. SPH numerical simulation was enforced by a Fortran program to simulate the problem of a drop impacting free liquid surface, and its results were compared with the level set result.
SPH method was applied to simulate free surface flow in the practical engineering field. And the two numerical simulation was finished by SPH, which are the liquid sloshing in ship hydrodynamics field and the discharge through crest orifice on dam in the hydraulic engineering field.
引文
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