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

在欧拉-拉格朗日形式的基础上，我们数值研究了N-S方程。在第一年，我们使用两个相互垂直的偏置涡管进行了涡流重联的计算。所使用的网格点分别为256^3和512^3。我们首先证明了由于粘性效应，扩散标记的雅可比行列式变为零。为了保证标号的可逆性，当行列式接近0时，我们将标号重置为A=x。研究发现，频繁重置发生的时间间隔对应于涡旋重联。研究还发现，重置时间间隔与小尺度湍流运动的时间尺度相当。因此，该方法为涡旋重联的监测提供了一个客观判据。在物理空间特征上，比较了涡度|ω|和伪涡度|ζ|等值面的结构。第二年，将上述方法应用于均匀各向同性湍流的数值实验，研究了N-S湍流。结果表明，在湍流情况下也需要重置，这种方法对于监测湍流中发生的小尺度涡旋重联是有效的。接下来，我们将注意力转向纳维斯托克斯方程和欧拉方程之间的关系，欧拉方程是它们的无粘性对应方程。利用标号的二阶空间导数--连接张量C刻画了关系的奇异摄动性质。研究发现，在无粘极限下，C的行为是反常的。更确切地说，得到了一些数值证据，这些证据支持连续重置时间区间[t_j，t_&lt；j+1&gt；]存在that&lt；lim&gt；___&lt；v→0&gt；∫^&lt；t_&lt；j+1&gt；_&lt；t_j&gt；||C||^2_pdt&gt；A_p&gt；0，||C||_p≡(1/&lt；(2π)^3&gt；∫|C|^pdx)^&lt；1/p&gt；holds这样的常数A_p。

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>].