Upper edge of chaos and the energetics of transition in pipe flow

Abstract

In the past two decades, our understanding of the transition to turbulence in shear flows with linearly stable laminar solutions has greatly improved. Regarding the susceptibility of the laminar flow, two concepts have been particularly useful: the edge states and the minimal seeds. In this nonlinear picture of the transition, the basin boundary of turbulence is set by the edge state's stable manifold and this manifold comes closest in energy to the laminar equilibrium at the minimal seed. We begin this paper by presenting numerical experiments in which three-dimensional perturbations are too energetic to trigger turbulence in pipe flow but they do lead to turbulence when their amplitude is reduced. We show that this seemingly counter-intuitive observation is in fact consistent with the fully nonlinear description of the transition mediated by the edge state. In order to understand the physical mechanisms behind this process, we measure the turbulent kinetic energy production and dissipation rates as a function of the radial coordinate. Our main observation is that the transition to turbulence relies on the energy amplification away from the wall, as opposed to the turbulence itself, whose energy is predominantly produced near the wall. This observation is further supported by the similar analyses on the minimal seeds and the edge states. Furthermore, we show that the time-evolution of production-over-dissipation curves provide a clear distinction between the different initial amplification stages of the transition to turbulence from the minimal seed. arXiv:1912.09270v1 [physics.flu-dyn] 19 Dec 2019 2

Designing a minimal baffle to destabilise turbulence in pipe flows

“…The peak is thus located in the lower part of the buffer layer (5 < y + < 30), where the majority of the turbulent-kinetic-energy production is found to occur in DNS of wall-bounded shear flows (Kim, Moin & Moser 1987;Pope 2000). Note also that for the same Reynolds number Re = 3000 studied here, Budanur et al (2020) found that the maximum of the turbulent-kinetic-energy production approaches r ≈ 0.9 as the flow becomes fully turbulent.…”

A probabilistic protocol for the assessment of transition and control

“…Our results for plane Couette flow in a small domain indicate that the laminarization probability decreases as the kinetic energy of the initial perturbation from the laminar flow is increased. It may increase back to 1 for sufficiently large initial energy levels as a consequence of the location of the upper edge of chaos (Budanur et al 2020). We identified that the laminarization probability decreases as the Reynolds number is increased, which reflects the contraction of the basin of attraction of the laminar flow.…”

“…The role of the non-zero waiting time t turb is to ensure that an event where the flow exceeds E turb and decays immediately after is not counted as transitional. Such events associated with high-energy perturbations may be due to the crossing of the upper edge of chaos (Budanur et al 2020). We compute P lam (E ( j) ), the laminarization probability of a random initial perturbation (RP) of energy E ( j) .…”

Stabilising pipe flow by a baffle designed using energy stability

“…Inspired by Kühnen et al (2018a), Marensi, Willis & Kerswell (2019) investigated the nonlinear non-modal instability (Kerswell 2018) of a flattened velocity profile in pipe flow, finding that a flattened velocity profile can significantly increase the minimal energy of disturbance that triggers turbulence. Moreover, flattening the mean profile significantly reduces the turbulence energy production in the bulk region, which is shown to be crucial for transition (Budanur et al 2020). Marensi et al (2019) used a baffle modelled by a linear drag force f = −A × B(z)u (Giannetti & Luchini 2007), where z is the axial coordinate in the dimensionless momentum equation…”

“…Moreover, flattening the mean profile significantly reduces the turbulence energy production in the bulk region, which is shown to be crucial for transition (Budanur et al. 2020). Marensi et al.…”