The dimension of attractors underlying periodic turbulent Poiseuille flow
The dimension of attractors underlying periodic turbulent Poiseuille flow
复制标题
周期性湍流泊肃叶流的吸引子维数
DOI：
--
复制
发表时间：
1992
影响因子：
3.7
通讯作者：
John Kim
中科院分区：
文献类型：
--
作者：
L. Keefe;P. Moin;John Kim
采用粗粒度（16 × 33 × 8）数值模拟，计算出压力梯度雷诺数为3200时，湍流周期Poiffille流的吸引子的李雅普诺夫维数Dλ的下限约为352。这些结果是在流向和展向周期为1.6π的空间域上获得的，对应于壁单元雷诺数为80。比较李雅普诺夫指数谱从这个和一个更高的分辨率（16 × 33 × 16）在同一域上的模拟表明，这些频谱具有一个通用的形状时，适当的缩放。利用这些标度性质，以及来自更高分辨率（32 × 33 × 32）模拟的部分指数谱，我们认为在给定的计算域上，吸引子运动的实际维数约为780。中等分辨率的计算建立了这个尺寸作为一个强大的下限在这个计算域，而在最高分辨率计算的部分指数谱提供了一些证据，在完全解决湍流的吸引子尺寸是不可能大得多。这些计算表明，这种周期性的湍流剪切流是确定性的混沌，一个奇怪的吸引子不根本解决方案的Navier-Stokes方程在这样的流动。然而，测量的维度的大小使任何认为这种湍流的全球动力学可以归因于几个自由度的相互作用的概念无效。动力系统理论首次对充分发展的湍流的复杂性进行了测量;人们发现答案高得令人生畏。
Using a coarse grained (16 × 33 × 8) numerical simulation, a lower bound on the Lyapunov dimension, Dλ, of the attractor underlying turbulent, periodic Poiseuille flow at a pressure-gradient Reynolds number of 3200 has been calculated to be approximately 352. These results were obtained on a spatial domain with streamwise and spanwise periods of 1.6π, and correspond to a wall-unit Reynolds number of 80. Comparison of Lyapunov exponent spectra from this and a higher-resolution (16 × 33 × 16) simulation on the same domain shows these spectra to have a universal shape when properly scaled. Using these scaling properties, and a partial exponent spectrum from a still higher-resolution (32 × 33 × 32) simulation, we argue that the actual dimension of the attractor underlying motion on the given computational domain is approximately 780. The medium resolution calculation establishes this dimension as a strong lower bound on this computational domain, while the partial exponent spectrum calculated at highest resolution provides some evidence that the attractor dimension in fully resolved turbulence is unlikely to be substantially larger. These calculations suggest that this periodic turbulent shear flow is deterministic chaos, and that a strange attractor does underly solutions to the Navier–Stokes equations in such flows. However, the magnitude of the dimension measured invalidates any notion that the global dynamics of such turbulence can be attributed to the interaction of a few degrees of freedom. Dynamical systems theory has provided the first measurement of the complexity of fully developed turbulence; the answer has been found to be dauntingly high.