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

• Applicant: Professor Dr.-Ing. Martin Oberlack
• Subject Area: Fluid Mechanics
• Term: from 2016 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 316376675

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1881: Turbulent Superstructures
• International Connection: Spain
• Cooperation Partner: Professor Dr. Sergio Hoyas, Ph.D.