Domain Effects in the Finite / Infinite Time Stability Properties of a Viscous Shear Flow Discontinuity (original) (raw)

Stability of shear flows near a boundary

arXiv: Analysis of PDEs, 2020

This book is devoted to the study of the linear and nonlinear stability of shear flows and boundary layers for Navier Stokes equations for incompressible fluids with Dirichlet boundary conditions in the case of small viscosity. The aim of this book is to provide a comprehensive presentation to recent advances on boundary layers stability. It targets graduate students and researchers in mathematical fluid dynamics and only assumes that the readers have a basic knowledge on ordinary differential equations and complex analysis. No prerequisites are required in fluid mechanics, excepted a basic knowledge on Navier Stokes and Euler equations, including Leray's theorem. This book consists of three parts. Part I is devoted to the presentation of classical results and methods: Green functions techniques, resolvent techniques, analytic functions. Part II focuses on the linear analysis, first of Rayleigh equations, then of Orr Sommerfeld equations. This enables the construction of Green f...

A Necessary and Sufficient Instability Condition for Inviscid Shear Flow

Studies in Applied Mathematics, 1999

We derive a condition that is necessary and sufficient for the instability of inviscid, twodimensional, plane parallel, shear flow with equilibrium velocity profiles that are monotonic, real analytic, functions of the cross stream coordinate. The analysis, which is based upon the Nyquist method, includes a means for delineating the possible kinds of bifurcations that involve the presence of the continuous spectrum, including those that occur at nonzero wavenumber. Several examples are given.

On the stability of the simple shear flow of a Johnson–Segalman fluid

Journal of Non-Newtonian Fluid Mechanics, 1998

We solve the time-dependent simple shear flow of a Johnson-Segalman fluid with added Newtonian viscosity. We focus on the case where the steady-state shear stress/shear rate curve is not monotonic. We show that, in addition to the standard smooth linear solution for the velocity, there exists, in a certain range of the velocity of the moving plate, an uncountable infinity of steady-state solutions in which the velocity is piecewise linear, the shear stress is constant and the other stress components are characterized by jump discontinuities. The stability of the steady-state solutions is investigated numerically. In agreement with linear stability analysis, it is shown that steady-state solutions are unstable only if the slope of a linear velocity segment is in the negative-slope regime of the shear stress/shear rate curve. The time-dependent solutions are always bounded and converge to a stable steady state. The number of the discontinuity points and the final value of the shear stress depend on the initial perturbation. No regimes of self-sustained oscillations have been found.

Temporal Stability of Boundary-Free Shear Flows: The Effects of Diffusion

Theoretical and Computational Fluid Dynamics, 1999

The stability of boundary-free shear flow is studied for the case of variable viscosity due to binary diffusion across the shear layer. This leads to the main difficulty of this investigation, the direct coupling of the momentum and species equations in both the base state calculations as well as the stability analysis.

On the role of viscosity in shear instabilities

Physics of Fluids, 1998

This paper aims at investigating the viscous corrections to mode selection and associated growth rate in the inflectional instability of shear layers. While, in the inviscid limit, the most unstable mode and its growth rate are fully determined by the initial thickness of the layer 2L and the velocity jump 2u it experiences, we show here that these quantities are modified by a factor $11@(a/0.2)/Re# 2 % 21/2 in the small Reynolds number Re5uL/n limit, with a a constant depending on the detailed shape of the initial velocity profile. This result agrees well with early numerical computations of Betchov and Szewczyk @Phys. Fluids 6, 1391~1963!# and its interest is discussed in several different contexts.

Divergent versus Nondivergent Instabilities of Piecewise Uniform Shear Flows on the f Plane

Journal of Physical Oceanography, 2009

The linear instability of a piecewise uniform shear flow is classically formulated for nondivergent perturbations on a 2-D barotropic mean flow with linear shear, bounded on both sides by semi-infinite half-planes where the mean flows are uniform. The problem remains unchanged on the f-plane, since for non-divergent perturbations the instability is driven by vorticity gradient at the edges of the inner, linear shear region, whereas the vorticity itself does not affect it. The instability of the unbounded case is recovered when the outer regions of uniform velocity are bounded, provided that the widths of these regions are at least twice as wide as the inner region of non-zero shear. The numerical calculations demonstrate that this simple scenario is greatly modified when the perturbations' divergence and the variation of the mean height (that balances geostrophically the mean flow) are retained in the governing equations. Although a finite deformation radius exists on the shallow water f-plane, the mean vorticity gradient that governs the instability in the non-divergent case remains unchanged, so that it is not obvious how the instability is modified by the inclusion of divergence in the numerical solutions of the equations.

Finite-amplitude instability of parallel shear flows

Journal of Fluid Mechanics, 1967

A formal expansion method for analysis of the non-linear development of an oblique wave in a parallel flow is presented. The present approach constitutes an extension and modification of the method of Stuart and Watson. Results are obtained for plane Poiseuille flow, and for a combination of plane Poiseuille and plane Couette flow. The Poiseuille flow exhibits finite-amplitude subcritical instability, and relatively weak but finite disturbances markedly reduce the critical Reynolds number. The combined flow, which becomes stable to infinitesimal disturbances at all Reynolds numbers when the Couette component is sufficiently great, remains unstable to finite disturbances.

Stability of shear flow with density gradient and viscosity

1968

The stability of the mixing region between co-flowing streams of different velocity and density has never been adequately investigated. The reason for this is that a general similarity solution for velocity and density profiles has not been available until recently. In this work, the solution method of Iessen (1948') for the homogeneous case was extended to the heterogeneous case in an attempt to find a neutral stability curve for the more complex case. The extension was based on the recent similarity solution obtained by the authors. A branch line of the neutral stability curve was found but curves with non-zero amplification and damping factors fell on the same side of the neutral stability curves.

Global linear stability analysis of weakly non-parallel shear flows

Journal of Fluid Mechanics, 1993

The global linear stability of incompressible, two-dimensional shear flows is investigated under the assumptions that far-field pressure feedback between distant points in the flow field is negligible and that the basic flow is only weakly non-parallel, i.e. that its streamwise development is slow on the scale of a typical instability wavelength. This implies the general study of the temporal evolution of global modes, which are time-harmonic solutions of the linear disturbance equations, subject to homogeneous boundary conditions in all space directions. Flow domains of both doubly infinite and semi-infinite streamwise extent are considered and complete solutions are obtained within the framework of asymptotically matched WKBJ approximations. In both cases the global eigenfrequency is given, to leading order in the WKBJ parameter, by the absolute frequency ω0(Xt) at the dominant turning pointXtof the WKBJ approximation, while its quantization is provided by the connection of soluti...