Spatial Stability of Incompressible Attachment-Line Flow

Linear stability analysis of incompressible attachment-line flow is presented within the spatial framework. The system of perturbation equations is solved using spectral collocation. This system has been solved in the past using the temporal approach and the current results are shown to be in excellent agreement with neutral temporal calculations. Results amenable to direct comparison with experiments are then presented for the case of zero suction. The global solution method utilized for solving the eigenproblem yields, aside from the well-understood primary mode, the full spectrum of least-damped waves. Of those, a new mode, well separated from the continuous spectrum is singled out and discussed. Further, relaxation of the condition of decaying perturbations in the far-field results in the appearance of sinusoidal modes akin to those found in the classical Orr-Sommerfeld problem. Finally, the continuous spectrum is demonstrated to be amenable to asymptotic analysis. Expressions are derived for the location, in parameter space, of the continuous spectrum, as well as for the limiting cases of practical interest. In the large Reynolds number limit the continuous spectrum is demonstrated to be identical to that of the Orr-Sommerfeld equation

A resolution of Stewartson's quarter-infinite plate problem

We revisit a problem originally considered by Stewartson in 1961: the incompressible, high-Reynolds-number flow past a quarter-infinite plate, with a leading edge that is perpendicular to, and a side edge that is parallel to, an undisturbed oncoming freestream. Particular emphasis is placed on the key region close to the side edge, where the flow is (superficially) three-dimensional, although the use of similarity variables reduces the dimensionality of the problem down to two. As noted by Stewartson, this problem has several intriguing features; it includes singularities and is also of a mixed parabolic type, with edge conditions influencing the solution in both directions across the flow domain. These features serve to greatly complicate the (numerical) solution process (the problem is of course also highly non-linear), and computation was clearly infeasible in 1961. In the present paper, a detailed computational study is presented, answering many of the questions that arose from the 1961 study. We present detailed numerical results together with asymptotic analyses of the key locations in the flow

Pulsatile Jets

We consider the evolution of high-Reynolds-number, planar, pulsatile jets in an incompressible viscous fluid. The source of the jet flow comprises a mean-flow component with a superposed temporally periodic pulsation, and we address the spatiotemporal evolution of the resulting system. The analysis is presented for both a free symmetric jet and a wall jet. In both cases, pulsation of the source flow leads to a downstream short-wave linear instability, which triggers a breakdown of the boundary-layer structure in the nonlinear regime. We extend the work of Riley,Sanchez-Sans & Watson (J. Fluid Mech., vol. 638, 2009, p. 161) to show that the linear instability takes the form of a wave that propagates with the underlying jet flow, and may be viewed as a (spatially growing) weakly non-parallel analogue of the (temporally growing) short-wave modes identified by Cowley, Hocking & Tutty (Phys. Fluids, vol. 28, 1985, p. 441). The nonlinear evolution of the instability leads to wave steepening, and ultimately a singular breakdown of the jet is obtained at a critical downstream position. We speculate that the form of the breakdown is associated with the formation of a �pseudo-shock� in the jet, indicating a failure of the (long-length scale) boundary-layer scaling. The numerical results that we present disagree with the recent results of Riley et al. (2009) in the case of a free jet, together with other previously published works in this area

Long's vortex revisited

We reconsider exact solutions to the Navier--Stokes equations that describe a vortex in a viscous, incompressible fluid. This type of solution was first introduced by Long (1958) and is par ameterised by an inverse Reynolds number $\epsilon$. Long's attention (and that of many subsequent investigators) was centred upon the `quasi-cylindrical' (QC) case corresponding to $\epsilon = 0$. We show that the limit $\epsilon \to 0$ is not straightforward, and that it reveals other solutions to this fundamental exact reduction of the Navier--Stokes system (which are not of QC form). Through careful numerical investigation, supported by asymptotic descriptions, we identify new solutions and describe the full parameter space that is spanned by $\epsilon$ and the pressure at the vortex core. Some erroneous results that exist in the literature are corrected

Boundary layers in a dilute particle suspension

The general problem of a boundary-layer flow carrying a dilute, mono-disperse suspension of small particles (together with gravitational effects) is considered. The problem is modelled using the �dusty-gas� equations, which are a coupled equation set linking the fluid motion to that of the particle motion (both of which are modelled as continua). A number of qualitatively distinct potential scenarios are predicted. These include a variety of boundary-layer breakdowns, and the formation of shock transitions in the distribution of the particulate phase (together with the development of particle-free zones). Numerical results predicting these differing behaviours are confirmed through local asymptotic analyses of the governing equations. Although we consider a general class of boundary layer, our results are compared and contrasted with previous studies of specific cases, most notably the constant freestream fluid velocity case (akin to the �clean� Blasius boundary layer). In the case of a boundary-layer flow driven by a linearly retarding free stream (the analogue of the classical �Howarth� boundary-layer problem), the effects of the particle phase are surprisingly seen to (slightly) delay the separation of the boundary layer

On the spatial development of a dusty wall jet

We consider the flow of an incompressible particle-laden fluid through the application of the so-called �dusty-gas� equations, which treat the fluid/particle suspension as two continua. The two phases are described by their individual field equations and interact through a Stokes-drag mechanism. The particular flow we consider is of boundary-layer type, corresponding to the downstream development of a Glauert-type jet adjacent to a horizontal boundary (the inclusion of the particulate phase requires the flow to be non-self-similar). We solve the governing boundary-layer equations through a numerical spatial marching technique in the three distinct cases of (i) weak gravitational influence, (ii) a jet �above� a wall under the action of gravity and (iii) a jet �below� a wall under the action of gravity. The qualitative and quantitative features of the three cases are quite different and are presented in detail. Of particular interest is the development of a stagnation point in the particle velocity field at a critical downstream location in case (i), the development of fluid/particle flow reversal in case (ii) and the development of �shock� solutions and particle-free regions in case (iii). Asymptotic descriptions are given of the critical phenomena, which support the numerical results. It is found that inclusion of a Saffman force has no substantial effect on either the location or structure of the stagnation-point region

On the boundary layer arising in the spin-up of a stratified fluid in a container with sloping walls

In this paper we consider the boundary layer that forms on the sloping walls of a rotating container (notably a conical container), filled with a stratified fluid, when flow conditions are changed abruptly from some initial (uniform) state. The structure of the solution valid away from the cone apex is derived, and it is shown that a similarity-type solution is appropriate. This system, which is inherently nonlinear in nature, is solved numerically for several flow regimes, and the results reveal a number of interesting and diverse features. In one case, a steady state is attained at large times inside the boundary layer. In a second case, a finite-time singularity occurs, which is fully analysed. A third scenario involves a double boundary-layer structure developing at large times, most significantly including an outer region that grows in thickness as the square-root of time. We also consider directly the nonlinear fully steady solutions to the problem, and map out in parameter space the likely ultimate flow behaviour. Intriguingly, we find cases where, when the rotation rate of the container is equal to that of the main body of the fluid, an alternative nonlinear state is preferred, rather than the trivial (uniform) solution. Finally, utilizing Laplace transforms, we re-investigate the linear initial-value prob- lem for small differential spin-up studied by MacCready & Rhines (1991), recovering the growing-layer solution they found. However, in contrast to earlier work, we find a critical value of the buoyancy parameter beyond which the solution grows exponentially in time, consistent with our nonlinear results