Localized vortex/Tollmien–Schlichting wave interaction states in plane Poiseuille flow

Abstract

Strongly nonlinear three-dimensional interactions between a roll-streak structure and a Tollmien-Schlichting wave in plane Poiseuille flow are considered in this study. Equations governing the interaction at high Reynolds number originally derived by Bennett, Hall & Smith (J. Fluid Mech, vol. 223, 1991, pp. 475-495) are solved numerically. Travelling wave states bifurcating from the lower branch linear neutral point are tracked to finite amplitudes, where they are observed to localize in the spanwise direction. The nature of the localization is analysed in detail near the relevant spanwise locations, revealing the presence of a singularity which slowly develops in the governing interaction equations as the amplitude of the motion is increased. Comparisons with the full Navier-Stokes equations demonstrate that the finite Reynolds number solutions gradually approach the numerical asymptotic solutions with increasing Reynolds number.

A self-sustaining process model of inertial layer dynamics in high Reynolds number turbulent wall flows

“…As in related studies [7,8,33], this scalar can be determined by employing an iterative algorithm. Following Dempsey et al [34], we instead specify A and treat the streamwise wavenumber α as a scalar unknown that must be determined via the solution of a nonlinear eigenvalue problem. This eigenvalue problem is formulated from the steady version of equations (3.14) governing the fluctuation dynamics within the CL.…”

Exact coherent structures at extreme Reynolds number

“…In VWI and SSP theory, these 3-D motions are not turbulent fluctuations per se but wavy -that is, coherent -instability modes. For example, in the analysis by Dempsey et al (2016) the wave-induced RSD in the domain interior drives a counter-rotating cellular mean flow when the size of the waves δ = . An immediate consequence is that the waves satisfy quasi-linear equations about the O(1) streaks both in the core and in the near-wall layer.…”

“…Employing the VWI equations, Dempsey et al (2016) compute the asymptotic form of the vortex/TS wave ECS as a function of the spanwise wavenumber. The authors then compare their asymptotic solutions with ECS computed from the full NS equations at finite but large Re, demonstrating impressive agreement.…”

“…The theoretical investigation by Dempsey et al (2016) complements these strictly computational studies by providing a semi-analytical description of strongly nonlinear ECS that bifurcate from small-amplitude near-wall Tollmien-Schlichting (TS) waves in plane Poiseuille flow (PPF). Their elegant approach leverages a large-Re mathematical formalism for nonlinearly interacting 3-D wavy instabilities and 2-D streamwise vortices aptly termed vortex-wave interaction (VWI) theory.…”

“…More significantly, the asymptotic approach lays bare the essential physics of the interactions that sustain the ECS. Here, the paper by Dempsey et al (2016) is a tour-de-force, combining detailed mathematical analysis, careful numerics and deep physical insight to show in the context of PPF that ECS in the domain interior can be sustained by a near-wall viscous instability giving rise to TS waves.…”