J. J. Healey

Enhancing the absolute instability of a boundary layer by adding a far-away plate

When a solid plate, with a boundary condition of no normal flow through it, is introduced parallel to a shear layer it is normally expected to exert a stabilizing influence on any inviscid linearly unstable waves. In this paper we present an example of an absolutely unstable boundary-layer flow that can be made more absolutely unstable by the addition of a plate parallel to the original flow and far from the boundary layer itself. In particular, the addition of the plate is found to increase the growth rate of the absolute instability of the original boundary-layer flow by an order of magnitude for long waves. This phenomenon is illustrated using piecewise-linear inviscid basic-flow profiles, for which analytical dispersion relations have been derived. Long-wave stability theories have been developed in several limits clarifying the mechanisms underlying the behaviour and establishing its generic nature. The class of flows expected to exhibit this phenomenon includes a class found recently to have an exponential growth of disturbances in the wall-normal direction, owing to the approach of certain saddle-points to certain branch-cuts in the complex-wavenumber plane. The theory also suggests that a convectively unstable flow in an infinite domain can be converted, in some circumstances, into an absolutely unstable flow when the domain is made finite by the addition of a plate, however far away the plate is.

Model for unstable global modes in the rotating-disk boundary layer

Recent simulations and experiments appear to show that the rotating-disk boundary layer is linearly globally stable despite the existence of local absolute instability. However, we argue that linear global instability can be created by local absolute instability at the edge of the disk. This argument is based on investigations of the linearized complex Ginzburg-Landau equation with weakly spatially varying coefficients to model the propagation of a wavepacket through a weakly inhomogeneous unstable medium. Therefore, this mechanism could operate in a variety of locally absolutely unstable flows. The corresponding nonlinear global mode has a front at the radius of onset of absolute instability when the disk edge is far from the front. This front moves radially outwards when the Reynolds number at the disk edge is reduced. It is shown that the laminar-turbulent transition front also behaves in this manner, possibly explaining the scatter in observed transitional Reynolds numbers.

Destabilizing effects of confinement on homogeneous mixing layers

The absolute and convective instability properties of plane mixing layers are investigated for linearized inviscid disturbances. It is shown that confinement by plates parallel to the flow can enhance the absolute instability so much that even a co-flow plane mixing layer becomes absolutely unstable when the ratio of distances of the plates from the mixing layer lies in a certain range. Even when the plates are placed equidistantly from the mixing layer, a co-flow mixing layer can become absolutely unstable if the velocity profile has an asymmetry about its mid-plane. ‘Semiconfinement’, where a plate is only added to one side of the mixing layer, is also investigated. A substantial destabilization is possible when the plate is added on the side of the faster stream. Previous investigations seem only to have found absolute instability when the streams flow in opposite directions.

On the relation between the viscous and inviscid absolute instabilities of the rotating-disk boundary layer

We consider the stability of the flow produced by an infinite rotating disk. A large-Reynolds-number asymptotic theory is developed to obtain the non-parallel correction to the local absolute instability (AI) found for this flow by Lingwood, who used the parallel-flow approximation. Our asymptotic theory is based on the inviscid AI underlying the viscous AI and so is expected to give the non-parallel correction to the upper branch of Lingwoods neutral curve for the AI

Inviscid long-wave theory for the absolute instability of the rotating-disc boundary layer

This paper is concerned with the absolute instability of the boundary-layer flow produced when an infinite disc rotates in otherwise still fluid. A greater understanding of the mechanisms and properties of the absolute instability is sought through the development of an analytic theory in the inviscid long-wave limit. It is shown that the fundamental basic flow characteristic of the absolute instability is a wall-jet in the radial direction superposed with an asymptotically small cross-flow, which generates a small reverse flow outside the boundary layer in the appropriately resolved basic velocity profile. The absolute instability is produced by a modal coalescence involving the interaction of eight saddle-points of the dispersion relation. An explicit expression for the growth rate has been obtained in terms of basic flow parameters. Most curiously, the pinch-point for absolute instability is shown to become asymptotically close to a branch-cut on the imaginary axis of the complex wavenumber plane, and unstable spatial branches emanating from the pinch-point cross this imaginary axis onto a Riemann sheet of the dispersion relation composed of solutions growing exponentially with distance from the disc. The existence of such modes contradicts the expectation of monotonic exponential decay of disturbances outside a boundary layer.

Spatio-temporal instability in mixed convection boundary layers

In this work we analyse the stability properties of the flow over an isothermal, semi-infinite vertical plate, placed at zero incidence to an otherwise uniform stream at a different temperature. Near the leading edge the boundary layer resembles Blasius flow, but further downstream it approaches that of pure buoyancy-driven flow. A coordinate transformation that describes in a smooth way the evolution between these two limiting similarity states, where the viscous and buoyancy forces are respectively dominant, is used to calculate the basic flow. The stability of this flow has been investigated by making the parallel flow approximation, and using an accurate spectral method on the resulting stability equations. We show how the stability modes discussed by other authors can be followed continuously between the forced and free convection limits; in addition, new instability modes not previously reported in the literature have been found. A spatio-temporal stability analysis of these modes has been carried out to distinguish between absolute and convective instabilities. It seems that absolute instability can only occur when buoyancy forces are opposed to the free stream and when there is a region of reverse flow

Wave packet analysis and break{up length calculations for an accelerating planar liquid jet

This paper examines the process of transition to turbulence within an accelerating planar liquid jet. By calculating the propagation and spatial evolution of disturbance wave packets generated at a nozzle where the jet emerges, we are able to estimate break{up lengths and break{up times for dierent magnitudes of acceleration and dierent liquid to air density ratios. This study uses a basic jet velocity prole which has shear layers in both the air and the liquid either side of the uid interface. The shear layers are constructed as functions of velocity which behave in line with our CFD simulations of injecting Diesel jets. The non{dimensional velocity of the jet along the jet centre{line axis is assumed to take the form V (t) = tanh(at) where the parameter a determines the magnitude of the acceleration. We compare the fully unsteady results obtained by solving the unsteady Rayleigh equation, to those of a quasi{steady jet to determine when the unsteady eects are signicant,

Long-wave theory for a new convective instability with exponential growth normal to the wall

A linear stability theory is presented for the boundary-layer flow produced by an infinite disc rotating at constant angular velocity in otherwise undisturbed fluid. The theory is developed in the limit of long waves and when the effects of viscosity on the waves can be neglected. This is the parameter regime recently identified by the author in a numerical stability investigation where a curious new type of instability was found in which disturbances propagate and grow exponentially in the direction normal to the disc, (i.e. the growth takes place in a region of zero mean shear). The theory describes the mechanisms controlling the instability, the role and location of critical points, and presents a saddle-point analysis describing the large-time evolution of a wave packet in frames of reference moving normal to the disc. The theory also shows that the previously obtained numerical solutions for numerically large wavelengths do indeed lie in the asymptotic long-wave regime, and so the behaviour and mechanisms described here may apply to a number of cross-flow instability problems.

Wave-envelope steepening in the Blasius boundary-layer

Abstract Previous experiments into the evolution of small amplitude disturbances to the Blasius boundary layer have shown that modulated waves become nonlinear at lower amplitudes than unmodulated waves. In this paper we propose a mechanism that may account for this behaviour. It involves a wave-envelope steepening scenario analogous to water-wave overturning and shock formation. Larger amplitude parts of a modulated wave travel at a different speed to lower amplitude parts, due to the proposed nonlinear mechanism, leading to an asymmetry between the steepness of decaying and growing sections. These effects occur in the higher order Ginzburg–Landau equation, so this may be a useful model for the process. Results from a windtunnel experiment, and a direct numerical simulation, will be presented and analysed for this effect. Both show a clear progressive asymmetry developing as the amplitude, and hence nonlinearity, are increased. Comparison between the experiment and simulation highlights key differences between two- and three-dimensional nonlinear evolutions.

Characterizing boundary-layer instability at finite Reynolds numbers

Abstract When the Reynolds number is treated as an asymptotically large number in a boundary-layer stability analysis, it is possible to identify Reynolds number scalings at which different terms in the governing equations balance. These balances determine which physical mechanisms are operating under which circumstances and enable a systematic treatment of the various parameter regimes to be carried out. Linear waves can grow by viscous mechanisms if the velocity profile is noninflexional and both viscous and inviscid instabilities are present for inflexional profiles. When disturbances become nonlinear the resulting dynamics depend upon the type of instability and, in particular, on whether the critical layer lies within the viscous wall layer or is separate from it. Therefore, the classification of a wave as viscous or inviscid is important to the theory of transition. However, at finite Reynolds numbers the boundaries separating different types of instability become blurred. Moreover, certain asymptotic theories are known to give poor quantitative agreement with experiment while others remain untested by detailed experimental comparison. This paper is concerned with identifying the domains where the different asymptotic theories are most relevant so as to facilitate their comparison with experiment. Numerical solutions of the Orr-Sommerfeld equation are presented that indicate that in wind-tunnel experiments the instability driving transition is esssentially viscous even for adverse pressure gradient boundary layers, and that inviscid instability waves would be difficult to observe. For an inviscid wave near the upper branch there could be a lower frequency viscous wave at the same point in the flow with amplitude 50 times larger in a typical practical situation.