R. I. Bowles

The standing hydraulic jump : theory, computations and comparisons with experiments

In this theoretical and computational study of the flow of a liquid layer, under the influence of surface tension and gravity most notably, the nonlinear equations governing an interaction between viscous effects and the effects of surface tension, gravity and streamline curvature for the limit of large Reynolds numbers are derived. The aim is to make a comparison between the predictions of this theory and the experiments of Craik et al. on the axisymmetric hydraulic jump. Such a jump is commonly encountered in the everyday context of the initial filling of a kitchen sink, for example, and it is found in the present work that initially all the effects listed above can play a primary role in practice in the local jump phenomenon. As a first step here, the flow of the layer over a small obstacle is considered. It is seen that as surface tension becomes increasingly significant the upstream influence becomes more wave-like. Second, calculations and analysis of the nonlinear free interaction are presented and show wave-like behaviour upstream, followed downstream by a depth profile not unlike that in the typical hydraulic jump. The effects of gravity dominate those of surface tension downstream. Finally, comparisons are made with the experiments and show fair quantitative agreement, supporting the present proposition that these hydraulic jumps are caused by boundary-layer separation due to a viscous-in viscid interaction forced by downstream boundary conditions on, in this case, a fully developed, high-Froude-number liquid layer.

Short-scale break-up in unsteady interactive layers: local development of normal pressure gradients and vortex wind-up

Following the finite-time collapse of an unsteady interacting boundary layer (step 1), shortened length and time scales are examined here in the near-wall dynamics of transitional-turbulent boundary layers or during dynamic stall. The next two steps are described, in which (step 2) normal pressure gradients come into operation along with a continuing nonlinear critical-layer jump and then (step 3) vortex formation is induced typically. Normal pressure gradients enter in at least two ways, depending on the internal or external flow configuration. This yields for certain internal flows an extended KdV equation with an extra nonlinear integral contribution multiplied by a coefficient which is proportional to the normal rate of change of curvature of the velocity profile locally and whose sign turns out to be crucial. Positive values of the coefficient lead to a further finite-time singularity, while negative values produce a rapid secondary instability phenomenon. Zero values in contrast allow an interplay between solitary waves and wave packets to emerge at large scaled times, this interplay eventually returning the flow to its original, longer, interactive, boundary-layer scales but now coupled with multiple shorter-scale Euler regions. In external or quasi-external flows more generally an extended Benjamin–Ono equation holds instead, leading to a reversal in the roles of positive and negative values of the coefficient. The next step, 3, typically involves the strong wind-up of a local vortex, leading on to explosion or implosion of the vortex. Further discussion is also presented, including the three-dimensional setting, the computational implications, and experimental links.

On the spiking stages in deep transition and unsteady separation

Numerical simulations of a large-amplitude nonlinear two-dimensional train of Tollmien-Schlichting waves are performed first and show the development of short-scaled structures or spikes. A careful description of the spiking process and its subsequent development is given describing the generation of localized maxima in the streamwise pressure distribution and associated vortices, spikes in a perturbation velocity trace and the emergence of strong wall-normal pressure gradients. A high-Reynolds-number asymptotic theory has previously been developed by two of the authors which aims to describe this spiking process. The current work is the first to give a comparison between this theory and planar Navier-Stokes computations. We give a brief description of the theory showing how normal pressure gradients become active and their role in the generation of the streamwise pressure distribution and its subsequent effects such as vortex generation and wall-layer vorticity eruptions. The presentation is given with close qualitative reference to the simulations, so giving credence to the relevance of the theoretical account and an interpretation of that account in physical terms. The comparison, although primarily qualitative, is successful in that it is possible to identify the physical processes highlighted by the theory in the computations, so clarifying the complex fluid motions and suggesting directions for further research. The paper concludes with firstly a discussion of how the results of the simulations and the theory could be used to give an understanding of similar processes seen in unsteady planar separation and secondly their relevance to the strongly three-dimensional processes at work in deep transition experiments.

Transition theory and experimental comparisons on (I) amplification into streets and (II) a strongly nonlinear break-up criterion

The two stages I, II are studied by using recent nonlinear theory and then compared with the experiments of Nishioka et al. (1979) on the transition of plane Poiseuille flow. The first stage I starts at low amplitude from warped input, which is deformed through weakly nonlinear interaction into a blow-up in amplitude and phase accompanied by spanwise focusing into streets. This leads into the strongly nonlinear stage II. It holds for a broad range of interactive boundary layers and related flows, to all of which the nonlinear break-up criterion applies. The experimental comparisons on I, II for channel flow overall show encouraging quantitative agreement, supporting recent comparisons (in the boundary-layer setting) of the description of stage I in Stewart & Smith (1992) with the experiments of Klebanoff & Tidstrom (1959) and of the break-up criterion of Smith (1988 a) with the computations of Peridier et al (1991a, b).

On boundary-layer transition in transonic flow

Boundary-layer transition in transonic external flow is addressed theoretically. The transonic area is rich in different flow structures, and transition paths, and the work has wide potential application in transonic aerodynamics, including special reference to the example of flow transition over an engine nacelle. The investigation is intended partly to aid, compare with, and detect any limitations of, a quasi-parallel empirical methodology for design use in the area, especially with respect to the transonic range, and partly to develop an understanding and possible control of the nonlinear natural or by-pass properties of the compressible transition present. The mechanisms behind three major factors, (a) substantial external-flow deceleration, (b) rapid boundary-layer thickening, (c) three-dimensional nonlinear interactions, are identified; these three are involved in the specific application above and in more general configurations, depending on the disturbance background present. It is found also that some similarities exist with the phenomenon of buffeting on transonic airfoils, and the relevant physics and governing equations throughout are identified. Sensitive nonlinear effects are important in all the factors (a)-(c), especially a resonance linkage between shock buffeting and boundary-layer thickening, and nonlinearly enhanced three-dimensional growth triggered by slight three-dimensional warping for instance, peculiar to the transonic range. The latter enhanced growth is perhaps the most significant finding. The implications, in the general setting as well as for the nacelle-flow context in particular, are also presented.

Upstream influence and the form of standing hydraulic jumps in liquid-layer flows on favourable slopes

Steady planar flow of a liquid layer over an obstacle is studied for favourable slopes. First, half-plane Poiseuille flow is found to be a non-unique solution on a uniformly sloping surface since eigensolutions exist which are initially exponentially small far upstream. These have their origin in a viscous–inviscid interaction between the retarding action of viscosity and the hydrostatic pressure from the free surface. The cross-stream pressure gradient caused by the curvature of the streamlines also comes into play as the slope increases. As the interaction becomes nonlinear, separation of the liquid layer can occur, of a breakaway type if the slope is sufficiently large. The breakaway represents a hydraulic jump in the sense of a localized relatively short-scaled increase in layer thickness, e.g. far upstream of a large obstacle. The solution properties give predictions for the shape and structure of hydraulic jumps on various slopes. Secondly, the possibility of standing waves downstream of the jump is addressed for various slope magnitudes. A limiting case of small gradient, governed by lubrication theory, allows the downstream boundary condition to be included explicitly. Numerical solutions showing the free-surface flow over an obstacle confirm the analytical conclusions. In addition the predictions are compared with the experimental and computational results of Pritchard et al. (1992), yielding good qualitative and quantitative agreement. The effects of surface tension on the jump are also discussed and in particular the free interaction on small slopes is examined for large Bond numbers.

Stall, Transition and Turbulence: a Tribute to JDAW (invited)

The article is in salute to the late Dave Walker (JDAW) especially concerning dynamic stall, boundary layer transition and turbulent structures /modelling for near-wall flows. Much new work is described as well as some of the current impact of JDAWs research work in terms of theoretical, analytical and computational understanding and prediction. The article considers in turn the nonlinear interacting layer with eruption and spiking, normal pressure gradients and vortex wind-up; comparisons with direct simulations; alternative critical layer dynamics during spiking and deep transition or stall; and distinct three-dimensional nonlinear properties.

Vortex-wave interaction in a strong adverse pressure gradient

The wide-vortex / Tollmein-Schlichting-wave three-dimensional interaction equations are considered in the limit of a strong adverse pressure gradient driving the boundary-layer flow. The asymptotic structure that emerges enables simplification of the equations and results in a partial differential equation governing directly the three-dimensional skin-friction field coupled with the effects of the wave forcing. A numerical scheme is developed to solve this system and the results are compared with an analytic solution valid for short distances after the onset of the interaction, among other things.