Visual manifestations of intermittency in computations of three-dimensional Navier-Stokes fluid turbulence appear as the low-dimensional or “thin” filamentary sets on which vorticity and strain accumulate as energy cascades down to small scales. In order to study this phenomenon, the first task of this paper is to investigate how weak solutions of the Navier-Stokes equations can be associated with a cascade and, as a consequence, with an infinite sequence of inverse length scales. It turns out that this sequence converges to a finite limit. The second task is to show how these results scale with integer dimension and, in the light of the occurrence of thin sets, to discuss the mechanism of how the fluid might find the smoothest, most dissipative class of solutions rather than the most singular.

三维Navier-Stokes流体湍流计算中间歇性的可视化表现为低维或“细”丝状集合，在其上能量和级联减小到小规模时，涡旋和应变就累积在其中。为了研究这种现象，本文的首要任务是研究Navier-Stokes方程的弱解如何与级联相关联，从而与无限长的逆长度尺度序列相关联。事实证明，该序列收敛到有限的极限。第二项任务是显示这些结果如何按整数维缩放，并根据薄集的出现，讨论流体如何找到最平滑，耗散最大的类而不是最奇异的类的机理。