强制动边界问题的光滑粒子流体动力学研究
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摘要
固体驱动的动边界在流体机械中比较常见,也是流体机械一个非常重要的研究领域。为了区别自由面、两相流交界面以及液体扩散边界,本文将固体边界驱动的动边界定义为强制动边界。作为一种拉格朗日形式的无网格粒子法,光滑粒子流体动力学(Smooth Particle Hydrodynamics,简称SPH)方法不受网格限制,具有自适应性、拉格朗日性质和粒子性质,非常适合处理流动问题。
本文将SPH方法应用到固体驱动的动边界问题的数值模拟中,经研究发现传统的Lennard-Jones形式的边界力模型具有很强的非线性因素,容易引起发散。因此采用一种线性的边界处理方法,研究其对SPH数值模拟稳定性的影响。
构建一种弹簧阻尼边界力模型,该模型不仅能保证边界力与距离的线性关系,还能使边界力与速度相关联,且具有较好的长程稳定性(模拟时间较长时仍具有稳定性)。将该边界力模型成功应用到固体驱动的动边界问题中,为SPH方法的边界处理提供了一种可行方法。
本文分析了如何将SPH方法应用到动边界流动问题中,并介绍了使用FORTRAN语言编写的SPH程序。分别采用Lennard-Jones形式的边界力模型和弹簧阻尼边界力模型进行边界处理,采用不变光滑长度和可变光滑长度进行光滑长度计算,对运动形式由平动到旋转的三种动边界模型,循序渐进地进行了SPH数值模拟,并验证了弹簧阻尼边界力模型的可行性、长程稳定性和正确性。
通过对典型算例的数值模拟,得出结论：弹簧阻尼边界力模型比传统的Lennard-Jones形式的边界力模型具有更好的长程稳定性,更适合处理固体驱动的动边界问题；与Fluent软件模拟结果的现象和趋势基本吻合,说明了该边界方法的正确性；另外,研究发现采用可变光滑长度能够提高SPH方法的计算精度。
The solid-driven moving boundary is common in Fluid Machinery, and also an important area of research. Different from the definition of the free surface, the interface of two-phase flow and the liquid diffusion boundary, the solid-driven moving boundary is defined as the forced moving boundary. As a Lagrangian form of meshfree particle method, Smooth Particle Hydrodynamics (SPH) method is mesh-free, adaptive, Lagrangian with particle nature, and is very suited to deal with fluid flow problems.
SPH method was applied to the numerical simulation of the solid-driven moving boundary problem, and the studies found that the traditional Lennard-Jones boundary force model is strongly non-linear and easily leads to inconvergence. Therefore, a linear boundary force model was utilized to study the impact on the stability of the SPH numerical simulation.
The spring-damping boundary force model was constructed, which has better long-term stability. It not only can guarantee the boundary force to be a linear relationship with the distance, but also associates the boundary force with the velocity. By successfully applying the model to solid-driven moving boundary flow problem, a feasible method is provided for the boundary treatment of the SPH method.
This thesis analyzed how the SPH method is applied to the moving boundary flow problem, and introduced the SPH program written in FORTRAN language. Lennard-Jones and spring-damping model were applied respectively as boundary treatment, and constant smoothing length and variable smoothing length were also applied respectively. Three moving boundary models, which are from translation to rotation, were simulated in a gradual way. At last, the feasibility and long-term stability of the spring-damping model were validated.
With the numerical simulations of the three typical examples, we concluded that: Compared with the Lennard-Jones boundary force model, the spring-damping model is long-term stable and is better to deal with the solid-driven moving boundary flow problem; The comparisons with the results of the Fluent indicated the correctness of the boundary treatment; In addition, the studies found that the variable smoothing length can improve the accuracy of the SPH method.
本文将SPH方法应用到固体驱动的动边界问题的数值模拟中,经研究发现传统的Lennard-Jones形式的边界力模型具有很强的非线性因素,容易引起发散。因此采用一种线性的边界处理方法,研究其对SPH数值模拟稳定性的影响。
构建一种弹簧阻尼边界力模型,该模型不仅能保证边界力与距离的线性关系,还能使边界力与速度相关联,且具有较好的长程稳定性(模拟时间较长时仍具有稳定性)。将该边界力模型成功应用到固体驱动的动边界问题中,为SPH方法的边界处理提供了一种可行方法。
本文分析了如何将SPH方法应用到动边界流动问题中,并介绍了使用FORTRAN语言编写的SPH程序。分别采用Lennard-Jones形式的边界力模型和弹簧阻尼边界力模型进行边界处理,采用不变光滑长度和可变光滑长度进行光滑长度计算,对运动形式由平动到旋转的三种动边界模型,循序渐进地进行了SPH数值模拟,并验证了弹簧阻尼边界力模型的可行性、长程稳定性和正确性。
通过对典型算例的数值模拟,得出结论：弹簧阻尼边界力模型比传统的Lennard-Jones形式的边界力模型具有更好的长程稳定性,更适合处理固体驱动的动边界问题；与Fluent软件模拟结果的现象和趋势基本吻合,说明了该边界方法的正确性；另外,研究发现采用可变光滑长度能够提高SPH方法的计算精度。
The solid-driven moving boundary is common in Fluid Machinery, and also an important area of research. Different from the definition of the free surface, the interface of two-phase flow and the liquid diffusion boundary, the solid-driven moving boundary is defined as the forced moving boundary. As a Lagrangian form of meshfree particle method, Smooth Particle Hydrodynamics (SPH) method is mesh-free, adaptive, Lagrangian with particle nature, and is very suited to deal with fluid flow problems.
SPH method was applied to the numerical simulation of the solid-driven moving boundary problem, and the studies found that the traditional Lennard-Jones boundary force model is strongly non-linear and easily leads to inconvergence. Therefore, a linear boundary force model was utilized to study the impact on the stability of the SPH numerical simulation.
The spring-damping boundary force model was constructed, which has better long-term stability. It not only can guarantee the boundary force to be a linear relationship with the distance, but also associates the boundary force with the velocity. By successfully applying the model to solid-driven moving boundary flow problem, a feasible method is provided for the boundary treatment of the SPH method.
This thesis analyzed how the SPH method is applied to the moving boundary flow problem, and introduced the SPH program written in FORTRAN language. Lennard-Jones and spring-damping model were applied respectively as boundary treatment, and constant smoothing length and variable smoothing length were also applied respectively. Three moving boundary models, which are from translation to rotation, were simulated in a gradual way. At last, the feasibility and long-term stability of the spring-damping model were validated.
With the numerical simulations of the three typical examples, we concluded that: Compared with the Lennard-Jones boundary force model, the spring-damping model is long-term stable and is better to deal with the solid-driven moving boundary flow problem; The comparisons with the results of the Fluent indicated the correctness of the boundary treatment; In addition, the studies found that the variable smoothing length can improve the accuracy of the SPH method.
引文
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[4]顾元通,丁桦.无网格法及其最新进展[J].力学进展,2005,35(3)：323-337.
[5]刘更,刘天祥,谢琴.无网格法及其应用[M].西安：西北工业大学出版社,2005.
[6]江顺亮,朱复华.“挤出过程计算机模拟”[M].机械工业出版社,2005.1.
[7]Gingold R. A., Monaghan J. J., Smoothed particle hydrodynamics:theory and application to nonspherical stars[J]. Monthly. Nat. R. Astron. Soc,1977,181:375.
[8]Liu G. R., Liu M. B..光滑粒子流体动力学—一种无网格粒子法[M].湖南：湖南大学出版社,2005.10.
[9]张健,陆利蓬,刘恩洲.SPH方法在溃坝流动模拟中的应用[J].自然科学进展,2006,16(10)：1326-1330.
[10]龚凯,刘桦,王本龙.SPH固壁边界处理方法的改进[J].力学季刊,2008,29(4)：507-514.
[11]王红,陈红全,马志华,等.动边界问题无网格算法及其动态点云[J].南京航空航天大学学报,2009,41(3)：296-301.
[12]蒲赛虎,陈红全.处理动边界问题的无网格/直角网格混合算法[J].南京航空航天大学学报,2010,42(4)：477-482.
[13]周星,许厚谦.计算含动边界非定常流动的无网格算法[J].力学与实践,2010,32(3)：16-21.
[14]周星,许厚谦.计算大位移动边界非定常流动的无网格算法.力学与实践[J],2010,32(6)：8-12.
[15]汤文辉,毛益明.SPH数值模拟中固壁边界的一种处理方法[J].国防科技大学理学院,2001,23(6)：54-57.
[16]董添文,黄兴元,江顺亮.两种边界条件设置方法在ISPH中的比较[J].塑性工程学报,2010,17(6)：136-142.
[17]杨波,欧阳洁,蒋涛,等.PTT黏弹性流体的光滑粒子动力学方法模拟[J].力学学报,2011,43(4)：667-673.
[18]孙中国,席光.搅拌器内部流动的无网格法三维数值模拟[J].工程热物理学报,2010,31(12)：2027-2030.
[19]J. J. Monaghan, J. B. Kajtar. SPH particle boundary forces for arbitrary boundaries[J]. Computer Physics Communications,2009,180:1811-1820.
[20]唐少武,冯振兴.关于无单元法的若干注记[J].力学进展,2003,33(4)：560-561.
[21]张雄,刘岩.无网格法[M].北京：清华大学出版社,2004.
[22]Fan X. J., Tanner R. I., Zheng R.. Smoothed particle hydrodynamics simulation of non-Newtonian moulding flow [J]. Non-Newtonian Fluid Mech,2010, (165):219-226.
[23]曹文炅,周照耀,何毅,等.压铸充型过程的SPH方法建模及数值模拟[J].华南理工大学学报(自然科学版),2011,39(1)：100-105.
[24]Cleary P.,Prakash M, Ha J. Novel applications of smoothed particle hydrodynamics (SPH) in metal forming[J]. Journal of Materials Processing Technology,2006,177(3):41-48.
[25]张小华,欧阳洁,张林.圆管聚合物热流中黏性耗散分析的无网格模拟[J].化工学报,2007,58(1)：15-20.
[26]Ellero M., Tanner R. I.,SPH simulations of transient viscoelastic flow at low Reynolds number [J]. J. Non-Newt. Fluid Mech.2005,(132) 61.
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