A deterministic approach to transition to turbulance in plane shear flows

Abstract

In this work, a parametrical study of the transition to turbulence in two-dimensional shear flows has been conducted. For this purpose, the solutions of the full two-dimensional Navier-Stokes equations have been investigated numerically using spectral methods. In parallel, a new spectral integration algorithm, called the Nonlinear Galerkin Method, stemming from dynamical systems theory and developed for the integration of dissipative evolution equations such as Navier-Stokes equations, has been tested and applied for the study cases. Different nonlinear Galerkin methods have been compared for this purpose with respect to each other in terms of convergence and efficiency and the improvements on the classical Galerkin spectral method have been shown numerically. Transition to turbulence has been analyzed by the parametrical investigation of qualitatively different solutions in the phase space of two-dimensional Navier-Stokes equations for bounded and unbounded shear flows with one nonhomogeneous direction. The applications were plane channel (Poiseuille) flow and oscillatory plane Poiseuille flow for the bounded flow case, and temporally growing mixing layer and plane jet flows for the unbounded flow case. With this work, we aim to contribute to the enlightening of the structure of the phase space of two-dimensional Navier-Stokes equations as well as to the testing of a new integration algorithm which seems to be promising in the direct numerical simulation of Navier-Stokes equations.