Symmetry-based stability theory in fluid mechanics

The present work deals with the stability theory of fluid flows. The central subject is the question under which circumstances a flow becomes unstable. Instabilities are a frequent trigger of laminar-turbulent transitions. Stability theory helps to explain the emergence of structures, e.g. wave-like perturbation patterns. In this context, the use of Lie symmetries allows the classification of existing and the construction of new solutions within the framework of linear stability theory. In addition, a new nonlinear eigenvalue problem (NEVP) is presented, whose derivation is completely based on Lie symmetries. In classical linear stability theory, a normal ansatz is used for perturbations. Another ansatz that has been shown in early work is the Kelvin mode ansatz. In the work of Nold and Oberlack (2013) and Nold et al. (2015) it was shown that these ansätze can be traced back to the Lie symmetries of the linearized perturbation equations. Interestingly, knowledge of the symmetries also allows for the construction of new ansatz functions that go beyond the known ansätze. For a plane rotational shear flow, in addition to the normal mode ansatz, an algebraic mode ansatz with algebraic behavior in time t^s (eigenvalue s) can be constructed. The flow is stable according to Rayleigh's inflection point criterion, which is also confirmed by the algebraic mode ansatz. Furthermore, exact solutions of the eigenfunctions can be found and new stable modes can be determined by asymptotic methods. Thereby, spiral-like structures of the vorticity can be recognized, which propagate in the region with time. Another key result of this work is the formulation and solution of an NEVP based on the Lie symmetries of the Euler equation. It can is shown that an NEVP can be formulated for a class of flows with a constant velocity gradient. These include, for example, linear shear flows, strained flows, and rotating flows. The NEVP for linear shear flows shows a relation to experimental data from turbulent shear flows. It can be theoretically shown that the turbulent kinetic energy scales exponentially with the eigenvalue of the NEVP. The eigenvalue is determined numerically using a parallel spectral solver. Initially, nonlinear terms are neglected. The determined eigenvalues are in the range of known literature values for turbulent shear flows. Furthermore, the NEVPs for plane flows with pure rotation and pure strain are solved. It is shown that the flow is invariant to rotation, while oscillatory eigenfunctions are found in the case of strain. In addition, an algorithm to solve the NEVP including the nonlinear terms is presented. The results allow an exciting insight into a new stability theory and form the basis for further investigation and understanding of the full nonlinear dynamics of the fluid flows based on the NEVP