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

我们研究了Navier-Stokes方程数值的基础上的欧拉-拉格朗日形式主义。在第一年中，我们使用两个正交偏置的涡管进行了涡重联的计算。使用的网格点是256^3和512^3。我们首先表明，雅可比行列式的扩散标签成为零，因为粘性效应。为了确保标签的可逆性，当行列式接近O时，我们将标签重置为A = x。结果发现，在频繁复位的时间间隔，发生对应于涡重联。研究还发现，重置间隔与小尺度湍流运动的时间尺度相当。因此，该方法提供了一个监测涡旋重联的客观判据。在物理空间特征上，涡度等值面的结构|ω|伪涡度|ζ|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Navier-Stokes湍流。结果表明，在湍流情况下也需要重新设置，这种方法对监测湍流中发生的小尺度涡重联是有效的。接下来，我们把注意力转向了纳维尔-斯托克斯方程和欧拉方程之间的关系。奇异摄动性质的关系的特点是使用连接张量C，标签的二阶空间导数。结果表明，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，支持常数A_p的存在，<lim>使得__&lt;v→0&gt;^&lt;t_&lt;j+1&gt;_<t_j>||C|| ^2_pdt&gt;A_p&gt;0，||C||_p（1/&lt;（2π）^3&gt;| C| ^pdx）^&lt;1/p&gt;对于连续复位时间间隔[t_j，t_&lt;j+1&gt;]成立。

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>].