A mathematical example displaying features of turbulence
A mathematical example displaying features of turbulence
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DOI：
10.1002/cpa.3160010401
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发表时间：
1948-12
影响因子：
3
通讯作者：
E. Hopf
中科院分区：
文献类型：
--
作者：
E. Hopf
在开始研究所讨论的例子之前，我们想对实际的流体力学问题做一些介绍性的评论，特别是关于解的未来行为的已知和猜测。考虑在给定外力作用下，给定材料边界内的不可压缩均质粘性流体。边界条件和外力假定为静止的，即与时间无关。为此，这些墙本身并不一定是静止的。在几何边界作为一个整体保持静止的情况下，部分材料墙可以在静止的运动中移动。实例是以规定的恒定速度旋转的两个同心圆柱体之间的流体，或者是两个平行平面之间的流体，这两个平面在自身内部以给定的恒定速度平移。关于外力的平稳性，我们可以引用流过有压降(视为外力)的无限长管道的情况。在这种情况下，压降需要是一个与时间无关的给定常数。在这些条件下理论上可能的流体的每一次运动都满足纳维-斯托克斯方程(p=1)。
Before entering upon the study of the example in question we want to make some introductory remarks about the actual hydrodynamic problems, in particular, about what is known and what is conjectured concerning the future behavior of the solutions. Consider an incompressible and homogeneous viscous fluid within given material boundaries under given exterior forces. The boundary conditions and the outside forces are assumed to be stationary, i.e. independent of time. For that, it is not necessary that the walls be at rest themselves. Parts of the material walls may move in a stationary movement provided that the geometrical boundary as a whole stays a t rest. An instance is a fluid between two concentric cylinders rotating with prescribed constant velocities or a fluid between two parallel planes which are translated within themselves with given constant velocities. As to the stationarity of the exterior forces we may cite the case of a flow through an infinitely long pipe with a pressure drop (regarded as an outside force). In this case the pressure drop is required to be a given constant independent of time. Each motion of the fluid that is theoretically possible under these conditions satisfies the Navier-Stokes equations ( p = 1)