Modeling mode interactions in boundary layer flows via the Parabolized Floquet Equations

TL;DR

This paper introduces the Parabolized Floquet Equations (PFE), a new linearized modeling approach to analyze mode interactions in boundary layer flows, accurately capturing harmonic growth and transition mechanisms.

Contribution

The paper develops a novel PFE framework combining linear PSE and Floquet decomposition to study mode interactions in boundary layers, validated against DNS and experimental data.

Findings

PFE accurately predicts harmonic growth in boundary layer transitions.

The method captures mode interactions consistent with DNS and experiments.

Effective for analyzing H-type transition and streak development.

Abstract

In this paper, we develop a model based on successive linearization to study interactions between different modes in boundary layer flows. Our method consists of two steps. First, we augment the Blasius boundary layer profile with a disturbance field resulting from the linear Parabolized Stability Equations (PSE) to obtain the modified base flow; and, second, we draw on Floquet decomposition to capture the effect of mode interactions on the spatial evolution of flow fluctuations via a sequence of linear progressions. The resulting Parabolized Floquet Equations (PFE) can be conveniently advanced downstream to examine the interaction between different modes in slowly varying shear flows. We apply our framework to two canonical settings of transition in boundary layers; the H-type transition scenario that is initiated by exponential instabilities, and streamwise elongated laminar streaks that are triggered by the lift-up mechanism. We demonstrate that the PFE capture the growth of various harmonics and provide excellent agreement with the results obtained in direct numerical simulations and in experiments.