Asymptotic Suction Boundary Layer: Alternative Linear and Weakly Non-Modal Stability Modes - a New Route to Large-Scale Turbulent Structures

渐进吸力边界层：替代线性和弱非模态稳定模式 - 大规模湍流结构的新途径

• 批准号: 316376675
• 负责人: Professor Dr.-Ing. Martin Oberlack
• 金额: --
• 依托单位: Fachgebiet Strömungsdynamik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/316376675?language=en
• 关键词: Asymptotic; Suction; Boundary; Layer; Alternative

Turbulent simulations of asymptotic suction boundary layer (ASBL) have shown that very large-scale motion are observed being rather different e.g. from roles in turbulent Couette flow. The structures of ASBL have a very strong influence even on the mean velocity and seem to be responsible for effects such as significant change of the von Karman constant or the wake region.Presently a combined theoretical and numerical approach is proposed. In part A, new symmetry based non-modal (NM) linear and weakly non-linear stability modes will be computed analytically and, thereafter, in part B validated numerically. Further, the purpose of the numerical simulation is to track the stability modes beyond their theoretical basis up into a fully non-linear regime, where in particular two key questions are to be answered (i) if modes may persist even in a fully non-linear regime, and (ii) if they correspond to the large-scales expected from previous investigations.For the theoretical part A, stability theory, it is to note, that the modal Ansatz of stability theory rests on three symmetries, i.e. translation in space and time and scaling of the dependent variable. In a series or publications, the applicant has shown that for a broad variety of canonical shear flows such as Couette, Poiseuille, pipe or Taylor-Couette flow the linearized Navier-Stokes equations admit at least one additional symmetry, which, in turn, results in very different NM type of eigenfunctions. Most of the new NM eigenfunctions exhibit algebraic behavior in time, though not limited to the initial state as in transient growth theory.Specifically for the ASBL a new symmetry has been derived, which results in new NM type of eigenfunctions with a stability/instability behavior which is double exponential in time. In particular, the interplay of NM eigenfunctions will be investigated employing Fokas method. In recent years this method has experienced an impressive growth as it comprehensively extends classical methods to solve linear partial differential equations.Further, a weakly non-linear stability analysis based on approximate groups is intended, which rests on the idea of merging symmetry analysis and perturbation theory. Compared to the classical approaches, the major advantage of using approximate groups is, that the employed perturbative series is not assumed a priori. However, it is an outcome of the analysis, and results in a tailor-made series for the problem under investigation. The objective is to understand the non-linear structures, which are expected to be responsible for some of the results observed in simulations.The objective of the final step of the stability part of the proposal will be to numerically track the theoretical findings beyond its theoretical limits. This, however, is not only to push theoretical results beyond its limits but also to numerically follow the computed modes and resulting linear/non-linear structures deep into a fully non-linear regime.

渐近吸力边界层（ASBL）的湍流模拟表明，观察到的大尺度运动与湍流库埃特流（Couette flow）的作用相当不同。ASBL的结构甚至对平均速度也有非常强的影响，似乎是造成冯卡门常数或尾迹区显著变化的原因。目前提出了一种理论与数值相结合的方法。在A部分，新的基于对称的非模态（NM）线性和弱非线性稳定模式将进行解析计算，然后在B部分进行数值验证。此外，数值模拟的目的是跟踪超出其理论基础的稳定模式，直到完全非线性状态，其中特别需要回答两个关键问题：(i)模式是否可能在完全非线性状态下持续存在，以及（ii）它们是否符合先前调查所期望的大尺度。对于理论部分A，稳定性理论，需要注意的是，稳定性理论的模态Ansatz依赖于三个对称性，即时空平移和因变量的标度。在一系列或出版物中，申请人已经表明，对于各种典型剪切流，如Couette， Poiseuille，管道或Taylor-Couette流，线性化的Navier-Stokes方程至少承认一个额外的对称性，这反过来又导致非常不同的NM型本征函数。大多数新的NM特征函数在时间上表现出代数行为，尽管不限于瞬态增长理论中的初始状态。特别是对于ASBL，导出了一种新的对称性，从而得到了具有稳定/不稳定双指数特性的新型NM型特征函数。特别地，NM特征函数的相互作用将采用Fokas方法进行研究。近年来，这种方法得到了令人印象深刻的发展，因为它全面扩展了求解线性偏微分方程的经典方法。在此基础上，提出了一种基于近似群的弱非线性稳定性分析方法。与经典方法相比，使用近似群的主要优点是所采用的微扰级数不是先验假设的。然而，它是分析的结果，并为正在调查的问题提供量身定制的系列。我们的目标是理解非线性结构，这是在模拟中观察到的一些结果的原因。提案稳定性部分最后一步的目标将是对超出理论极限的理论发现进行数值跟踪。然而，这不仅是推动理论结果超越其极限，而且在数值上遵循计算模式和由此产生的线性/非线性结构深入到完全非线性状态。