Session GD: Shear Layer Instabilities

Revisiting the stability analysis of miscible shear flow in the inertialess regime

Inertia is responsible for the shear instability between two immiscible fluids of different viscosities (growth rate proportional to Re). For the two miscible fluids case, molecular diffusion leads to a transition layer of intermediate viscosity between the two fluids. For moderate Peclet number (small mixing layer) and finite Reynolds number, it has been shown, (P. Ern et al JFM 496, 2003) that mixing stabilizes the inertial instability. Here, we investigate numerically the stability of miscible viscous flow in the inertialess regime (Re=0). Surprisingly, we do observe that this system can also be unstable at intermediate wavenumber depending on the position and the thickness of the pseudo-interface. By solving the LSA for a modeled flow consisting of three layers of immiscible fluids (without surface tension) with different viscosities at Re=0. This simplified model showes that the instability observed in the miscible case is due to the viscosity stratification of the base state.   

Entrainment mechanisms in a two-fluid channel flow

The temporal evolution of an initially laminar two-fluid channel flow is investigated using linear stability analysis and direct numerical simulation. The two fluids are miscible with dissimilar densities and viscosities. The thickness of one of the fluid layers is much smaller than that of the other, with the denser and more viscous fluid comprising the thin layer. Two distinct entrainment mechanisms are observed in the DNS calculations, one of which results in significantly more entrainment and mixing between the two fluids as well as a greater degree of vorticity generation in the flow. These two mechanisms are a result of different initial conditions. The initial conditions are supplied by an Orr-Sommerfeld-type linear stability analysis. The observed entrainment mechanisms correspond to the two least stable modes present in the two-fluid channel flow, with the perturbations applied at a finite amplitude. In the range of parameters investigated, changes in viscosity and density ratios are found to affect the flow only in a quantitative sense without altering the basic entrainment mechanisms.   

Projection of spatial shear layers in a symmetry-reduced space

Symmetry reduction technique has been successfully combined with traditional Proper Orthogonal Decomposition/Galerkin projection method to obtain models with very low dimension for temporal shear layers (Wei and Rowley, 2008). This study extends the approach to spacial shear layers. A scaling variable is introduced to factor out the downstream viscous growth, and therefore results in a projection model with fewer basis functions. However, Navier-Stokes equations have different characteristics along time and space. Firstly, we need to remove the stream-wise ellipticity to strictly develop a dynamic system evolving downstream. Secondly, the spatial developing term is from nonlinear convection instead of linear time-advancement for temporal case. These two problems are addressed in this study, and 2- and 4-mode models are developed respectively to capture single-frequency behavior (e.g. vortex roll-up) and double-frequency behavior (e.g. vortex pairing).   

On unstable modes in plane Couette flow

In this talk we report the finding of spectrally unstable linear modes in plane Couette flow, which are the solutions of the corresponding Orr-Sommerfeld equation on semi-infinite and finite two-dimensional channels, as motivated by standard experimental setups. These modes represent an absolute instability, which takes place for any non-zero Reynolds number. However, a finite non-zero critical Reynolds number does exist when considering a subset of these unstable modes, which suggests that probably not all these modes exist in real experiments as well as their subset (and thus critical Reynolds number) varies from experiment to experiment.   

Laminar-turbulent boundary in plane Couette flow

We apply the iterated edge state tracking algorithm to study the boundary between laminar and turbulent dynamics in plane Couette flow at Re=400. Perturbations that are not strong enough to become fully turbulent nor weak enough to relaminarize tend towards a hyperbolic coherent structure in state space, termed the edge state, which seems to be unique up to obvious continuous shift symmetries. The results reported here show that in cases where a fixed point has only one unstable direction, as for the lower branch solution in plane Couette flow, the iterated edge tracking algorithm converges to this state. They also show that choice of initial state is not critical, and that essentially arbitrary initial conditions can be used to find the edge state.   

Localized critical perturbations for turbulence transition in plane Couette flow

The transition to turbulence in linearly stable shear flows requires perturbations of finite amplitude. The shape and size of critical perturbations can be determined using the edge tracking algorithm (PRL {\bf 96}, 174101 (2006); PRL {\bf 99}, 034502 (2007)). In small, periodically continued domains critical perturbations that extend throughout the domain have been identified. Here we determine for wide and long domains localized critical perturbations (edge states) that correspond to minimal disturbances required to trigger turbulence. The edge states are localized in spanwise direction, in downstream direction or in both directions. The smallest structures are dominated by a pair of downstream vortices.   

The Edge of Chaos for Plane Couette Flow in Long Channels

For plane Couette flow, classical linear stability theory predicts that the laminar state is asymptotically stable for all Reynolds numbers, yet turbulence may be achieved both experimentally and numerically via finite amplitude perturbations. We study the boundary which separates laminar and turbulent dynamics in phase space, called the \textit{edge of chaos}. We implement an iterative edge tracking algorithm to find solutions near the boundary from arbitrarily chosen initial conditions. We study the \textit{edge of chaos} for fixed wall-normal and spanwise lengths but variable streamwise lengths, and determine how the edge evolves as a function of channel length.   

Equilibria, traveling waves, and periodic orbits of plane Couette flow

Equilibrium, traveling wave, and periodic orbit solutions of pipe and plane Couette flow can now be computed precisely at Reynolds numbers above the onset of turbulence. These invariant solutions capture the complex dynamics of rolls and streaks and provide a framework for understanding turbulent wall-bounded shear flows as dynamical systems. We present a number of newly computed equilibria, traveling waves, and periodic orbits of plane Couette flow, classify their symmetry groups, and observe how frequently they are visited by turbulent dynamics. What emerges is a picture of low-Reynolds turbulence as a walk among a set of weakly unstable invariant solutions.   

Turbulent stripe in direct numerical simulation of transitional channel flow

We present a series of ``large-scale'' DNS conducted in two types of plane channel flows---Poiseuille (PF) and Couette flows (CF), considering the subcritical-transition regime. Thorough the simulations in a periodic domain with streamwise and spanwise lengths of $327\delta \times 128\delta$ ($\delta$: the channel half width) or larger, we show that stripe patterns of oblique bands, alternating between turbulence and laminar flow, are the intrinsic regime in the reverse transition (from uniform turbulence to laminar) for both PF and CF. The pattern is oriented obliquely to the streamwise direction at an angle of 20--$25^{\circ}$ and similar to the one that takes place in a Taylor-Couette flow between counter-rotating cylinders, where spiral turbulence is observed by Coles (JFM 21, 1965). The emergence of turbulent stripes in CF has been found experimentally by Prigent et al. (Phys. Rev. Lett. 89, 2002) and analyzed numerically by Barkley \& Tuckerman (JFM 576, 2007). However, the mechanism responsible for the stabilization of the stipe pattern is still an open question. In this work, we will focus on similarity and difference between the turbulent stipes in PF and CF, with emphasis on the wavelength and the angle of its modulation.