微分求积模拟二维流体中流函数约束的施加方法研究

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摘要采用微分求积法数值求解流函数-涡度方程来模拟二维流体时会遇到流函数的超约束问题，即虽然流函数方程为二阶偏微分方程，但在每个固体边界上都存在两个约束条件：一个Dirichlet条件和一个Neumann条件。以二维驱动方腔流动为例，对该问题进行深入分析，进而提出一种新的超约束处理方法，即在边界涡度的计算中考虑Neumann条件，而仅将Dirichlet条件施加于流函数方程。数值结果显示该方法可行，且计算效率较高。同时给出前人提出的单层法和双层法进行比较。试算表明单层法对于网格数的奇偶性很敏感，不适于处理本文问题。对比本文方法与双层法发现：前者计算精度较高，且由于回避了超约束问题而更加方便于使用。

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	王通1
	何涛1
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	曹曙阳2

关键词 ：微分求积；流函数-涡度方程；方腔流；边界条件；超约束

Abstract：The 2D lid-driven cavity flow was simulated by applying the differential quadrature method to solve the stream function-vorticity equations.There were two boundary conditions,one Dirichlet and one Neumann,for the stream function equation at each solid boundary though the stream function equation was just second order.Analysis on this over- specified problem was carried out,based on which a new applying method was proposed: the Neumann condition was considered in calculating the vorticity at the boundary while only the Dirichlet condition was applied in the stream function equation.Validity of this method was verified by comparing its numerical results with benchmark data.Two other existing methods,the one-layer approach and the two-layer approach were shown as contrasts.Trial calculations indicate that the one-layer approach is sensitive to the parity of grid numbers and is not suitable for the present problem.Comparisons between the new method and the two-layer approach show that the former is not only more accurate but also more convenient to be used in practice for avoiding the over-specified problem.

Key words： differential quadrature method stream function-vorticity equations cavity flow boundary condition over-specified

收稿日期: 2016-04-12 出版日期: 2017-04-15

引用本文:

王通1，何涛1,3，曹曙阳2. 微分求积模拟二维流体中流函数约束的施加方法研究[J]. 振动与冲击, 2017, 36(8): 173-178.
WANG Tong1,HE Tao1,3,CAO Shuyang2. Methods on applying stream-function restraints in differential quadrature modelling of two-dimensional flow. JOURNAL OF VIBRATION AND SHOCK, 2017, 36(8): 173-178.

链接本文:

http://jvs.sjtu.edu.cn/CN/ 或 http://jvs.sjtu.edu.cn/CN/Y2017/V36/I8/173

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