Low-dimensional models for turbulent plane Couette flow in a minimal flow unit
Low-dimensional models for turbulent plane Couette flow in a minimal flow unit
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最小流量单元中湍流平面库埃特流的低维模型
DOI：
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发表时间：
2005
影响因子：
3.7
通讯作者：
P. Holmes
中科院分区：
文献类型：
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作者：
T. R. Smith;J. Moehlis;P. Holmes
我们在最小流量单元（MFU）（其展向和流向范围足以维持湍流的域）中对湍流平面库埃特流进行建模，方法是将速度场扩展为通过数值数据的适当正交分解计算出的最佳模式的总和。常微分方程是通过将纳维-斯托克斯方程伽辽金投影到这些模式上获得的。我们首先考虑 6 模（11 维）模型，并研究包含被忽略模的损耗的影响。忽略这些，模型可以可接受地再现湍流统计数据，但无法再现动态；包括它们，我们找到了一个稳定的周期轨道，它捕获了再生循环动力学，并且与直接数值模拟非常吻合。然而，限制为少至六种模态会人为地限制流向涡流和条纹的相对大小，因此无法再现层流状态的稳定性或随着雷诺数的增加而正确解释湍流的分叉。为了解决这个问题，我们开发了基于“非耦合”特征函数的第二类模型，该模型允许流向和跨流速度分量之间的独立性。 9 模（31 维）模型生成稳态和周期状态的分岔图，与数值纳维-斯托克斯解定性一致，同时保留再生循环动态。总之，这些模型提供了经验证据，证明 MFU 湍流的“支柱”是周期性轨道，并支持剪切驱动湍流的滚动-条纹-击穿-滚动重组图。
We model turbulent plane Couette flow in the minimal flow unit (MFU) – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence – by expanding the velocity field as a sum of optimal modes calculated via proper orthogonal decomposition from numerical data. Ordinary differential equations are obtained by Galerkin projection of the Navier–Stokes equations onto these modes. We first consider a 6-mode (11-dimensional) model and study the effects of including losses to neglected modes. Ignoring these, the model reproduces turbulent statistics acceptably, but fails to reproduce dynamics; including them, we find a stable periodic orbit that captures the regeneration cycle dynamics and agrees well with direct numerical simulations. However, restriction to as few as six modes artificially constrains the relative magnitudes of streamwise vortices and streaks and so cannot reproduce stability of the laminar state or properly account for bifurcations to turbulence as Reynolds number increases. To address this issue, we develop a second class of models based on ‘uncoupled’ eigenfunctions that allow independence among streamwise and cross-stream velocity components. A 9-mode (31-dimensional) model produces bifurcation diagrams for steady and periodic states in qualitative agreement with numerical Navier–Stokes solutions, while preserving the regeneration cycle dynamics. Together, the models provide empirical evidence that the ‘backbone’ for MFU turbulence is a periodic orbit, and support the roll–streak–breakdown–roll reformation picture of shear-driven turbulence.