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，观察到的非常大尺度的运动与湍流Couette流中的作用相当不同。ASBL的结构有很强的影响，甚至对平均速度，似乎是负责的效果，如显着改变的冯卡门常数或尾流区。目前提出了一个结合的理论和数值方法。在A部分中，将分析计算新的基于对称性的非模态（NM）线性和弱非线性稳定性模式，然后在B部分中进行数值验证。此外，数值模拟的目的是跟踪超出其理论基础的稳定性模式进入完全非线性状态，其中特别要回答两个关键问题：（i）模式是否可以甚至在完全非线性状态下持续存在，以及（ii）它们是否对应于从先前研究中预期的大尺度。对于理论部分A，稳定性理论，应注意，稳定性理论的模态分析依赖于三种对称性，即空间和时间的平移以及因变量的标度。在一系列出版物中，申请人已经表明，对于各种各样的正则剪切流，例如Couette流、Poiffille流、管道流或Taylor-Couette流，线性化的Navier-Stokes方程允许至少一个额外的对称性，这又导致非常不同的NM型本征函数。大多数新的NM型本征函数在时间上表现出代数行为，尽管不像瞬态增长理论那样仅限于初始状态，特别是对于ASBL，导出了一种新的对称性，这导致了新的NM型本征函数在时间上具有双指数的稳定/不稳定行为。特别是，NM本征函数的相互作用将采用Fokas方法进行研究。近年来，该方法在求解线性偏微分方程中得到了广泛的应用，并将对称性分析与扰动理论相结合，提出了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，所采用的扰动级数不是先验假设的。然而，它是分析的结果，并导致为正在调查的问题量身定制的系列。目的是了解非线性结构，这是预计负责模拟中观察到的一些结果。该提案的稳定性部分的最后一步的目标将是数值跟踪超出其理论极限的理论结果。然而，这不仅是为了推动理论结果超出其极限，而且在数值上遵循计算模式和所得线性/非线性结构深入到完全非线性状态。

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.