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Numerical study on the fluid dynamical equations on the basis of the Eulerian-Lagrangian formalism

基于欧拉-拉格朗日形式的流体动力学方程数值研究

基本信息

批准号：
14540203
负责人：
OHKITANI Koji
金额：
$ 1.34万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

项目摘要

We have studied the Navier-Stokes equations numerically on the basis of the Eulerian-Lagrangian formalism. In the first year we performed calculations of vortex reconnection by using two orthogonally off-set vortex tubes. The grid points used are 256^3 and 512^3. We first showed that the Jacobian determinant of the diffusive labels becomes zero because of viscous effects. To assure invertibility of labels, we reset labels as A = x when the determinant becomes close to O. It was found that the time interval over which frequent resetting takes, place corresponds to vortex reconnection. It was also found that resetting intervals are comparable to time scale of small scale turbulent motion. Thus, this method gives an objective criterion for monitoring vortex reconnection. Also, in physical space characteristics structure of iso-surfaces of vorticity |ω| and pseudo-vorticity |ζ| are compared.In the second year, Navier-Stokes turbulence was studied by applying the above methodology to numerical experiments on homogeneous isotropic turbulence. It was found that resetting is also required in the case of turbulence and that this method is effective for monitoring small-scale vortex reconnection taking place in turbulence. We next turned our attention to the relationship between the Navier-Stokes equations and the Euler equations, their inviscid counterpart. The singular perturbation nature of the relationship was characterized by using connection tensor C, a second spatial derivative of labels. It was found that the behavior of C is anomalous in the inviscid limit. More precisely, some numerical evidence was obtained which support the existence of constants A_p such that<lim>___<v→0>∫^<t_<j+1>_<t_j>||C||^2_pdt>A_p>0,||C||_p≡(1/<(2π)^3>∫|C|^pdx)^<1/p>holds for the intervals of consecutive resetting times [t_j, t_<j+1>].

我们在欧拉-拉格朗日形式主义的基础上对Navier-Stokes方程进行了数值研究。在第一年，我们用两个正交的偏置涡管进行了涡重连的计算。使用的网格点是256^3和512^3。我们首先证明了扩散标签的雅可比行列式由于粘滞效应而变为零。为了保证标签的可逆性，当行列式接近于0时，我们将标签重置为A = x。结果表明，频繁重置的时间间隔对应于旋涡重连。还发现，重置间隔与小尺度湍流运动的时间尺度相当。从而为旋涡重联的监测提供了一个客观的判据。并比较了涡度|ω|和拟涡度|ζ|等涡面在物理空间上的结构特征。第二年，将上述方法应用于均匀各向同性湍流的数值实验中，研究了Navier-Stokes湍流。研究发现，在湍流情况下也需要重置，该方法对于监测湍流中发生的小尺度涡重连是有效的。接下来，我们把注意力转向了纳维-斯托克斯方程和欧拉方程之间的关系，它们的非粘性对应物。利用连接张量C（标签的二阶空间导数）表征了该关系的奇异摄动性质。在无粘极限下，C的行为是反常的。更精确地说，得到了一些数值证据，证明了常数A_p的存在性，使得<lim>___<v→0>∫^<t_<j+1>_<t_j>||C||^2_pdt>A_p>0,||C||_p≡(1/<(2π)^3>∫|C|^pdx)^<1/p>在连续复位时间间隔内成立[t_j， t_<j+1>]。