Quantitative Comparison of Results from DNS and Nonlinear Parabolized Stability Equations for the Subharmonic Transition Process

Abstract

The laminar-turbulent transition process in a flat plate boundary layer can be viewed as a sequence of instabilities, beginning with the spatial growth of two-dimensional (2D) Tollmien-Schlichting (TS) waves. As these waves amplify to certain finite amplitudes, they evolve into a three-dimensional (3D) vortical structure, forming distinct Λ-shaped vortices. These three-dimensional disturbances rapidly grow to large amplitudes, finally causing the breakdown to turbulence. Among the possible transition mechanisms, the experiments conducted by Kachanov and Levchenko [1] provided a detailed understanding of the subharmonic resonance process, also known as the H-type transition. In this scenario, disturbances arise with twice the period and twice the wavelength of the fundamental 2D TS waves, leading to a staggered pattern of Λ-vortices (see Figure 1). The H-type transition has already been extensively studied numerically, with both direct numerical simulations (DNS) (e.g., [2]) and the more computationally efficient nonlinear parabolized stability equations (NPSE) (e.g., [3]) yielding closely aligned results in the weakly nonlinear regime. However, a detailed quantitative comparison between these two methods in the more strongly nonlinear stage of the transition process has not yet been thoroughly presented. This contribution aims to address this gap, with a particular focus on identifying the limitations of NPSE-specifically, investigating the location prior to transition where NPSE ceases to deliver accurate results.