Andrew G. Walton

Nonlinear interaction of near-planar TS waves and longitudinal vortices in boundary-layer transition

The nonlinear interactions that evolve between a planar or nearly planar Tollmien-Schlichting (TS) wave and the associated longitudinal vortices are considered theoretically for a boundary layer at high Reynolds number. The vortex flow is either induced by the TS nonlinear forcing or is input upstream, and similarly for the nonlinear wave development. Three major kinds of nonlinear spatial evolution, Types 1-3, are found. Each can start from secondary instability and then become nonlinear, Type 1 proving to be relatively benign but able to act as a pre-cursor to the Types 2, 3 which turn out to be very powerful nonlinear interactions. Type 2 involves faster stream-wise dependence and leads to a finite-distance blow-up in the amplitudes, which then triggers the full nonlinear 3-D triple-deck response, thus entirely altering the mean-flow profile locally. In contrast, Type 3 involves slower streamwise dependence but a faster spanwise response, with a small TS amplitude thereby causing an enhanced vortex effect which, again, is substantial enough to entirely alter the meanflow profile, on a more global scale. Streak-like formations in which there is localized concentration of streamwise vorticity and/or wave amplitude can appear, and certain of the nonlinear features also suggest by-pass processes for transition and significant changes in the flow structure downstream. The powerful nonlinear 3-D interactions 2, 3 are potentially very relevant to experimental findings in transition.

The nonlinear instability of thread--annular flow at high Reynolds number

The surgical technique of thread injection of medical implants is modelled by the axial pressure-gradient-driven flow between concentric cylinders with a moving core. The nonlinear stability of the basic flow is analysed theoretically at asymptotically large Reynolds number and it is found that non-axisymmetric nite-amplitude neutral modes can be supported over a wide range of thread radii and injection velocities. The axial force on the thread is calculated and it is found to be signicantly less than that predicted by undisturbed-flow theory, in agreement with thread{annular experiments. Thread injection is a newly devised surgical technique which enables porous medical implants to be placed inside the body in a minimally invasive way, thus reducing surgical trauma. The thread is stored on a spool and injected within a fluid by applying an axial pressure gradient to the cylindrical container holding the liquid and the thread (gure 1). The thread velocity V is controlled by a motor. It is clearly desirable for the thread to be injected smoothly and to not suer lateral deviation: it is therefore important that the flow is kept laminar. For this reason the transition to turbulence of the basic thread{annular flow and its dependence upon Reynolds number, thread radius and injection velocity are of great practical interest. In a recent experimental paper Frei, L¨ & Wintermantel (2000, hereinafter referred to as FLW) modelled the thread injection process by using a cylindrical rubber lament to represent the thread. The lament was allowed to move concentrically through a steel cylindrical pipe (representing the injection vessel) lled with water. They measured various quantities including the axial force on the thread due to pressure gradient and viscous eects. On comparing the results with the theoretical predictions that arise from the exact Navier{Stokes solution for axial flow between concentric cylinders they discovered that the force measured in the experiments was always signicantly less than that predicted by their theory. This observation forms the motivation for the current study in which we investigate whether this discrepancy could be caused by a nonlinear instability of the basic thread{annular flow. There have been many theoretical studies of the stability of so-called core{annular flows in which two fluids with dierent properties occupy a single pipe (e.g. the temporal studies of Preziosi, Chen & Joseph 1989 and Huang & Joseph 1995 and the spatio-temporal approach of Shen & Li 1996 and Chen & Lin 2002). In recent years there has been less attention focused on the case where the inner cylinder is a solid body rather than a fluid. The exact solution of the Navier{Stokes equations

FLOW PAST A TWO- OR THREE-DIMENSIONAL STEEP-EDGED ROUGHNESS

Flow past a single small planar or three–dimensional roughness mounted on a smooth surface is investigated theoretically for various edge steepnesses, the oncoming planar motion being within a boundary layer or other near–wall shear. Nonlinear edge properties at large Reynolds numbers largely control the flow responses at the three–dimensional roughness wing–tips and the impacts of separation(s), among other features. From analysis and computation, criteria are found for the generation of nonlinear upstream influence, downstream influence and separations, for two– and three–dimensional roughnesses, as well as wing–tip separations. In particular, it is predicted that with a severe edge (e.g. a 90° forward–facing step) the ratio of the upstream separation distance over the roughness edge height is a constant times ReW1/4 in two dimensions, the constant being approximately 0.142 and the Reynolds number ReW being based on the roughness edge height and the incident velocity slope at the surface. In three dimensions ReW is multiplied by sin ψ as expected physically, where ψ is the tangent angle of the roughness planform. The ratio prediction above is very general, applying not only for any incident shear flow, but also for any front–edge geometry. Other separation and reattachment properties, extensions and a comparison with an experiment, are also discussed.

Properties of strongly nonlinear vortex/Tollmien-Schlichting-wave interactions

An analytical and computational study is presented on solution properties of strongly nonlinear vortex/wave interactions involving Tollmien/Schlichting waves, in boundary-layer transition. The longitudinal vortex part, i.e. the total mean flow, is governed by a three-dimensional vortex system but coupled, through an effective spanwise slip condition at the surface, with the accompanying wave part, so that both the vortex and the wave parts are unknowns. Terminal forms of the space-marching or time-marching problem are proposed first, yielding either a lift-off separation singularity or a strong-attachment singularity. Second, a similarity version of the complete system is addressed numerically and analytically. This leads to a number of interesting solution features as the typical wave pressure is increased into the strongly nonlinear regime. In particular, lift-off separation and attachment forms seem to emerge which are analogous with those proposed above. The flow developments beyond the terminal forms are discussed, together with the links of the work with recent computational results and, tentatively, with experimental observations including the creation of lambda vortices (as a form of lift-off separation).

Stability of circular Poiseuille-Couette flow to axisymmetric disturbances

The stability of circular Poiseuille–Couette flow to axisymmetric disturbances is investigated theoretically. First, the governing circular Orr–Sommerfeld equation for linear perturbations is formulated and analysed asymptotically at large values of the Reynolds number. The existence of multiple regions of instability is predicted and their dependence upon radius ratio and inner cylinder velocity is determined explicitly. These findings are confirmed when the linear problem is solved numerically at finite Reynolds number and multiple neutral curves are found. The relevance of these results to the thread injection of medical implants is discussed, and it is shown how the linear modes are connected to nonlinear amplitude-dependent modes at high Reynolds number that exist for

The linear and nonlinear stability of thread-annular flow

O(1)

The stability of nonlinear neutral modes in Hagen–Poiseuille flow

values of the inner cylinder velocity.

The temporal evolution of neutral modes in the impulsively started flow through a circular pipe and their connection to the nonlinear stability of Hagen{Poiseuille flow

The surgical technique of thread injection of medical implants is modelled by the axial pressure-gradient-driven flow between concentric cylinders with a moving core. The linear stability of the flow to both axisymmetric and asymmetric perturbations is analysed asymptotically at large Reynolds number, and computationally at finite Reynolds number. The existence of multiple regions of instability is predicted and their dependence upon radius ratio and thread velocity is determined. A discrepancy in critical Reynolds numbers and cut-off velocity is found to exist between experimental results and the predictions of the linear theory. In order to account for this discrepancy, the high Reynolds number, nonlinear stability properties of the flow are analysed and a nonlinear, equilibrium critical layer structure is found, which leads to an enhanced correction to the basic flow. The predictions of the nonlinear theory are found to be in good agreement with the experimental data.

Singularity formation in the strongly nonlinear wide-vortex/Tollmien{Schlichting-wave interaction equations

The nonlinear stability of Hagen–Poiseuille flow through a pipe of circular crosssection subjected to non–symmetric disturbances is studied asymptotically at large Reynolds number. By introducing unsteady effects into the nonlinear critical layer, an evolution equation for the disturbance amplitude is derived and is found to possess an equilibrium solution first identified by Smith and Bodonyi. This solution is shown to provide a threshold amplitude, above which, on this scaling, perturbations experience finite–time blow–up, while, below the threshold, disturbances of all wavenumbers decay to zero.

Strongly nonlinear vortex-Tollmien-Schlichting- wave interactions in the developing flow through a circular pipe

In the middle of the nineteenth century an exact solution to the Navier{Stokes equations was found for the steady flow of a fluid through a straight pipe of circular cross-section. The solution has become known as Hagen{Poiseuille flow (HPF) in honour of the two researchers who independently discovered the experimental law relating the axial pressure gradient along the pipe to the mean velocity of the flow. If a carefully controlled experiment is performed it is indeed possible to realize HPF: however in many practical situations the disturbance environment is such that this solution only exists over a nite range of Reynolds number or in some situations not at all. Attention therefore focused on the study of the stability of this flow, with the rst approach incorporating the eects of viscosity due to Sexl (1927), who found the flow to be linearly stable to axisymmetric disturbances at high Reynolds number. Since this early work there have been many theoretical investigations (e.g. Gill 1965, 1973; Davey & Drazin 1969), and numerical studies (Salwen & Grosch 1972; Garg & Rouleau 1972). For a more complete list the reader is referred to the paper by Draad, Kuiken & Nieuwstadt (1998). The general conclusion of these studies is that HPF is stable to all linear disturbances and that the least-damped modes at high Reynolds number have azimuthal wavenumbers N = 0 (axisymmetric) and N = 1. This latter mode is also found to give the largest amplication through transient growth (Schmid & Henningson 1994).