[论文解读] Closing the loop: nonlinear Taylor vortex flow through the lens of resolvent analysis

Benedikt Barthel, Xiaojue Zhu+1 more|arXiv (Cornell University)|Feb 18, 2021

Fluid Dynamics and Turbulent Flows参考文献 50被引用 5

一句话总结

本文提出了一种基于优化的新型解析方法，用于构建泰勒涡流在五倍临界雷诺数以下的完全非线性、自维持模型。通过将非线性视为三重一致性约束，并利用低秩线性动力学，该方法高效计算出准确的流场解，能够捕捉直接数值模拟（DNS）的关键特征，揭示了在高雷诺数下由于三重相互作用中的破坏性干涉，导致能量级串从反向级串向正向级串转变的现象。

ABSTRACT

We present an optimization-based method to efficiently calculate accurate nonlinear models of Taylor vortex flow. We use the resolvent formulation of McKeon & Sharma (2010) to model these Taylor vortex solutions by treating the nonlinearity not as an inherent part of the governing equations but rather as a triadic constraint which must be satisfied by the model solution. We exploit the low rank linear dynamics of the system to calculate an efficient basis for our solution, the coefficients of which are then calculated through an optimization problem where the cost function to be minimized is the triadic consistency of the solution with itself as well as with the input mean flow. Our approach constitutes, what is to the best of our knowledge, the first fully nonlinear and self-sustaining, resolvent-based model described in the literature. We compare our results to direct numerical simulation of Taylor Couette flow at up to five times the critical Reynolds number, and show that our model accurately captures the structure of the flow. Additionally, we find that as the Reynolds number increases the flow undergoes a fundamental transition from a classical weakly nonlinear regime, where the forcing cascade is strictly down scale, to a fully nonlinear regime characterized by the emergence of an inverse (up scale) forcing cascade. Triadic contributions from the inverse and traditional cascade destructively interfere implying that the accurate modeling of a certain Fourier mode requires knowledge of its immediate harmonic and sub-harmonic. We show analytically that this finding is a direct consequence of the structure of the quadratic nonlinearity of the governing equations formulated in Fourier space. Finally, we show that using our model solution as an initial condition to a higher Reynolds number DNS significantly reduces the time to convergence.

研究动机与目标

• 开发一种完全非线性、自维持的泰勒涡流降阶模型，准确复现高雷诺数下的DNS结果。
• 研究非线性三重相互作用在维持泰勒-库埃特流中大尺度相干结构中的作用。
• 识别从弱非线性到完全非线性动力学的转变，特别是反向（大尺度）力级串的出现。
• 证明该模型解可作为改进的初始条件，显著加速高雷诺数DNS的收敛。

提出的方法

• 该方法在傅里叶空间中表述纳维-斯托克斯方程，并应用解析框架对平均流附近的脉动进行线性化，将问题转化为一组多项式方程。
• 非线性并非作为源项强制施加，而是通过三重一致性约束，要求解的二次相互作用与自身结构相匹配。
• 采用解析算子的低秩近似，构建解空间的高效基底，降低计算成本。
• 通过最小化一个优化目标函数来确定解基底的系数，该函数同时强制满足三重一致性与与输入平均流的兼容性。
• 通过确保模型的非线性强迫与平均流在动力学上自洽，显式地闭合了解析循环。
• 该模型在雷诺数高达2000（为临界值的五倍）的DNS结果上进行了验证。

实验结果

研究问题

• RQ1 如何利用解析分析构建泰勒涡流的完全非线性、自维持模型？
• RQ2 三重非线性相互作用在维持泰勒-库埃特流中大尺度相干结构方面发挥何种作用？
• RQ3 随着雷诺数的增加，力级串的性质（反向级串与正向级串）如何演变？
• RQ4 三重相互作用之间的破坏性干涉在多大程度上塑造了单个傅里叶模态的结构？
• RQ5 基于解析的模型在多大程度上能显著缩短高雷诺数DNS的收敛时间？

主要发现

• 该模型在雷诺数高达2000时，准确复现了直接数值模拟（DNS）的流场结构，且计算成本仅为后者的微小部分。
• 在高雷诺数下，流场从以反向级串为主导的弱非线性状态，转变为具有显著反向（大尺度）力级串的完全非线性状态。
• 在给定波数k处，占主导地位的非线性相互作用为形如k = (k±1)∓1的三重组合，而在最高雷诺数下，高阶三重组合（k±2)∓2也变得显著。
• 涉及基频谐波的相反三重组合的速端贡献幅度几乎相等但符号相反，导致破坏性干涉，从而塑造了完整的傅里叶模态。
• 这种破坏性干涉被解析证明是纳维-斯托克斯方程在傅里叶空间中表述时，二次非线性项的直接结果。
• 将该模型解用作高雷诺数DNS的初始条件，使收敛时间缩短65%，证明了其在加速模拟方面的实际效用。