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

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Couette流中的作用有很大的不同。ASBL的结构甚至对平均速度都有很大的影响，而且似乎对von Karman常数或尾流区的显着变化等影响也有很大影响。在A部分，将对新的基于对称性的非模线性和弱非线性稳定模式进行解析计算，然后在B部分进行数值验证。此外，数值模拟的目的是将稳定性模式从其理论基础追踪到完全非线性区域，其中特别需要回答两个关键问题：(I)即使在完全非线性区域内模式是否仍然存在，以及(Ii)它们是否与先前研究所预期的大尺度相对应。对于理论部分A，稳定性理论，值得注意的是，稳定性理论的模式分析依赖于三个对称性，即空间和时间上的平移以及因变量的尺度。在一系列或多篇文献中，申请人已经证明，对于各种各样的正则剪切流，如Couette流、Poiseuille流、PIPE流或Taylor-Couette流，线性化的Navier-Stokes方程至少允许一个额外的对称性，这反过来又导致非常不同的NM类型的特征函数。大多数新的NM特征函数在时间上表现出代数行为，但并不像瞬变增长理论中那样局限于初始状态。具体地，对于ASBL，导出了一个新的对称性，这导致了新的NM型特征函数在时间上具有双指数的稳定/不稳定行为。特别地，我们将用Fokas方法研究NM本征函数之间的相互作用。近年来，该方法得到了长足的发展，将经典方法推广到线性偏微分方程组的求解中，并基于对称性分析和摄动理论相结合的思想，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的。然而，这是分析的结果，并为正在调查的问题量身定做了一系列结果。我们的目标是了解非线性结构，预计这将是在模拟中观察到的一些结果的原因。本提案稳定性部分最后一步的目标将是从数值上跟踪超出其理论极限的理论结果。然而，这不仅是为了将理论结果推向它的极限，而且也是为了在数值上跟随计算模式以及由此产生的线性/非线性结构深入到完全非线性区域。

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.