The linear stability of the Stokes layers generated between a pair of synchronously oscillating parallel plates is investigated. The disturbance equations were studied using Floquet theory and pseudospectral numerical methods used to solve the resulting system. Neutral curves for an extensive range of plate separations were obtained and when the plate separation is large compared to the Stokes layer thickness the linear stability properties of the Stokes layer in a semi-infinite fluid were recovered. A detailed analysis of the damping rates of disturbances to the basic flow provides a plausible explanation of why several previous studies of the problem have failed to detect any linear instability of the flow.

To compare more faithfully with experimental work the techniques used for the channel problem were modified to allow the determination of neutral curves for axisymmetric disturbances to purely oscillatory flow in a circular pipe. Critical Reynolds numbers for the pipe flow tended to be smaller than their counterparts for the channel case but the smallest critical value was still almost twice the experimentally reported result.

The wake of a streamwise oscillating circular cylinder has been experimentally investigated over a range of oscillation amplitude and frequency ratios using laser-induced-fluorescence flow visualization, particle image velocimetry and hot-wire techniques. Five typical flow structures, referred to as S-I, S-II, A-I, A-III and A-IV, are identified. Special attention is given to the S-II mode because this flow structure is observed experimentally for the first time. It consists of two rows of binary vortices symmetrically arranged about the centreline of the wake. Each binary vortex contains two counter-rotating vortices shed from the same side of the cylinder. This flow structure corresponds to zero mean and fluctuating lift on the cylinder, which could be of engineering significance. A theoretical analysis for this flow has been conducted based on the governing equations. The solution to the two-dimensional vorticity equation suggests that the flow may be considered to be the superposition of two components, i.e. that due to a stationary cylinder in a steady uniform cross-flow and to a cylinder oscillating in fluid at rest, which are characterized by alternate and symmetric vortex shedding, respectively. The solution provides insight into the formation of the various modes of the flow structure. A semi-empirical prediction of the S-II mode structure is developed, which is in excellent agreement with experimental data as well as with previous numerical results.

In a recent paper, Segur et al. (J. Fluid Mech. vol. 539, p. 229, 2005, hereafter referred to as ${\cal S}$ showed, based on a damped version of the nonlinear Schrodinger equation (NLS), that any amount of dissipation (of a certain type) stabilizes the Benjamin–Feir instability of a modulated Stokes wave train. Their theoretical predictions are confirmed by laboratory experiments for waves of small or moderate amplitude, but not for waves of large amplitude or with relatively large perturbations. ${\cal S}$ left open questions regarding the validity of their theoretical results for these large-amplitude waves, and possibly the validity of the NLS assumptions of weak nonlinearity and narrow-bandedness. We investigate these issues using direct simulations of the primitive equations, incorporating constant and wavenumber-dependent dissipation models. For small or moderate amplitudes, our full simulations agree with the theory and experiments of ${\cal S}$. For large amplitudes, we find that it is primarily the form of the dissipation model, rather than the assumptions of NLS, that is responsible for the failure of ${\cal S}$'s theoretical predictions. Indeed, with an appropriate wavenumber-dependent dissipation model, both the full simulations and NLS obtain the correct evolution behaviour for large-amplitude waves. Finally, using direct and NLS simulations, we confirm the general conclusion of ${\cal S}$ for stabilization of the Benjamin–Feir instability over long-time wave train evolution.

The receptivity phenomenon, which is the process of environmental disturbances initially entering the boundary layers and generating disturbance waves, is one of the important but not well understood mechanisms involving laminar–turbulent transition of hypersonic flows. This paper presents a numerical simulation study of the receptivity to weak free-stream fast acoustic waves for a Mach 7.99 axisymmetric flow over a $7^\circ$ half-angle blunt cone. In hypersonic boundary-layer flow over a blunt cone, the process of receptivity to free-stream disturbances is altered considerably by the presence of a bow shock and an entropy layer. In the present study, both steady and unsteady flow solutions are obtained by computing the full Navier–Stokes equations with a fifth-order shock-fitting finite-difference scheme, which is able to account for the effects of bow-shock/free-stream-disturbance interaction accurately. The current numerical results for the steady base flow are compared with previous experimental and numerical results. In addition, a normal-mode linear stability analysis is used to identify the main components of boundary-layer disturbances generated by forcing free-stream fast acoustic waves. It is found that neither the first mode nor the second-mode instability waves are excited by free-stream fast acoustic waves in the early region along the cone surface, although the Mack modes can be unstable there. Instead, the second mode is excited downstream of the second-mode Branch I neutral stability point. The delay of the second-mode excitation is because the hypersonic boundary-layer receptivity is governed by a two-step resonant interaction process: (i) resonant interactions between the forcing waves and a stable boundary-layer wave mode I near the leading-edge region; and (ii) resonant interactions between the induced stable mode I and the unstable second Mack mode downstream. The same receptivity mechanism also explains the results that no first Mack mode components are generated by the current receptivity process because there is no resonant interaction between fast acoustic waves and the first Mack mode.

This paper presents analytical and numerical results for separated stratified inviscid flow over and around an isolated mountain in the limit of small Froude number. The vertical density profile consists of a lower strongly stratified layer whose depth is just less than that of the mountain. It is separated from a semi-infinite upper stably stratified layer by a thin, highly stable, inversion layer. The paper aims to provide, for this particular profile, a thorough analysis of the three-dimensional separated flow over a mountain top with strong stratification. The Froude numbers $F$ and $F_I$ of the lower layer and the interface are small with $F_I\,{\ll}\, F\,{\ll}\,1$, but the upper-layer Froude number is arbitrary. The flow at each height in the lower layer is governed by the two-dimensional Euler equations and moves horizontally around the mountain. It is given by a modification of a previous model using Kirchhoff free-streamline theory for the separated flow region downstream of the mountain. The pressure variations associated with the lower-layer flow are of the same order as the dynamic head and induce significant displacements of the inversion layer. When the inversion is near the top of the mountain these deflections are of the same order as the height of the projecting part of the mountain top and combine with the flow over the mountain top to excite vertically propagating internal waves in the upper layer. The resultant pressure field, vertical stream surface displacements, and surface streamlines in the upper layer are described consistently in the hydrostatic limit. Many of the features of the upper flow, including the perturbations of the critical dividing streamlines, are similar to those in flows with uniform stable stratification at low Froude number. Comparisons are made with experiments and approximate models for these summit flows based on the assumption that the dividing streamlines have small vertical displacement.

We study the time-independent free-surface flow which forms when a fluid drains out of a container, a so-called bathtub vortex. We focus on the bathtub vortex in a rotating container and describe the free-surface shape and the complex flow structure using photographs of the free surface, flow visualizations, and velocity measurements. We find that the velocity field in the bulk of the fluid agrees with predictions from linear Ekman theory for the boundary layer at the bottom, and we discuss the limitations of linear Ekman theory for the source–sink flow in the experiment. We introduce a radial expansion approximation of the central vortex core and reduce the model to a single first-order equation. We solve the equation numerically and find that the axial velocity depends linearly on height whereas the azimuthal velocity is almost independent of height. We discuss the model of the bathtub vortex introduced by Lundgren (J. Fluid Mech. vol. 155, 1985, p. 381) and compare it with our experiment. We find that the measured velocities and surface profiles are described well by the model when Ekman upflow and surface tension effects are included.

The formation of vortex rings generated by an impulsively started jet in the presence of uniform background co-flow is studied experimentally to extend previous results. A piston–cylinder mechanism is used to generate the vortex rings and the co-flow is supplied through a transparent shroud surrounding the cylinder. Digital particle image velocimetry (DPIV) is used to measure the development of the ring vorticity and its eventual pinch off from the generating jet for ratios of the co-flow to jet velocity ($R_{v})$ in the range 0 – 0.85. The formation time scale for the ring to obtain maximal circulation and pinch off from the generating jet, called the formation number ($F$), is determined as a function of $R_{v}$ using DPIV measurements of circulation and a generalized definition of dimensionless discharge time or ‘formation time’. Both simultaneous initiation and delayed initiation of co-flow are considered. In all cases, a sharp drop in $F$ (taking place over a range of 0.1 in $R_{v}$) is centred around a critical velocity ratio ($R_{crit}$). As the initiation of co-flow was delayed, the magnitude of the drop in $F$ and the value of $R_{crit}$ decreased. A kinematic model based on the relative velocities of the forming ring and jet shear layer is formulated and correctly predicts vortex ring pinch off for $R_{v} \,{>}\, R_{crit}$. The results of the model indicate the reduction in $F$ at large $R_{v}$ is directly related to the increased convective velocity provided to the ring by the co-flow.

The shape of the interface between two superimposed layers in a two-dimensional channel confined between a planar and a corrugated or indented wall is investigated in the limit of Stokes flow. A perturbation analysis for walls with small-amplitude sinusoidal corrugations reveals that an insoluble surfactant amplifies the deformation of the interface and causes a negative drift in the phase shift under most conditions. The effect is most significant at moderate capillary numbers and for corrugations whose wavelength is large compared to the thickness of the adjacent layer lining the wavy wall. The precise effect of the surfactant depends on the ratio of the fluid viscosities, proximity of the interface to the planar wall, capillary number, and wavelength of the corrugations. When the interface is near the plane wall, introducing surfactant reduces the interfacial amplitude and causes a positive phase shift with respect to the wavy wall. As the interface further approaches the plane wall, the interfacial wave tends to become in phase with the wavy wall, reflecting its unshifted topography. In the second part of this study, a boundary integral method is implemented to compute Stokes flow over a wall with an arbitrary periodic profile, and results are presented for sinusoidal walls and planar walls containing a periodic sequence of square and circular depressions or projections. The numerical results reveal that the linear perturbation theory overestimates the deformation of the interface over a wavy wall, and illustrate the nature of shear-driven film flow over a planar wall with indented topography.

Optimal perturbations in compressible, non-parallel boundary layers are considered here. The flows past a flat plate and past a sphere are analysed. The governing equations are derived from the linearized Navier-Stokes equations by employing a scaling that relies on the presence of streamwise vortices, which are well-known for being responsible for the ‘lift-up’ effect. Consequently, the energy norm of the inlet perturbation encompasses the wall-normal and spanwise velocity components only. The effect of different choices of the energy norm at the outlet is studied, testing full (all velocity components and temperature) and partial (streamwise velocity and temperature only) norms. Optimal perturbations are computed via an iterative algorithm completely derived in the discrete framework. The latter simplifies the derivation of the adjoint equations and the coupling conditions at the inlet and outlet.

Results for the flat plate show that when the Reynolds number is of the order of $10^3$, a significant difference in the energy growth is found between the cases of full and partial energy norms at the outlet. The effect of the wall temperature is in agreement with previous parallel-flow results, with cooling being a destabilizing factor for both flat plate and sphere. Flow divergence, which characterizes the boundary layer past the sphere, has significant effects on the transient growth phenomenon. In particular, an increase of the sphere radius leads to a larger transient growth, with stronger effects in the vicinity of the stagnation point. In the range of interesting values of the Reynolds number that are typical of wind tunnel tests and flight conditions for a sphere, no significant role is played by the wall-normal and streamwise velocity components at the outlet.

We study the capillary instability of a liquid film (thickness $h_0$) coating a horizontal cylindrical tube (radius $R_0$). We show experimentally that the instability only occurs if $h_0/R_0>0.3(R_0/a)^2$, where $a$ is the capillary length. If this criterion is not fulfilled, the liquid film does not destabilize into an array of drops, owing to the gravitational drainage.

Ice sheets are susceptible to the formation of ice streams, or narrow bands of fast-flowing ice whose high velocities are caused by rapid sliding at the contact between ice and the underlying bed. Based on recent geophysical work which has shown that the sliding motion of ice streams may be described by a Coulomb friction law, we investigate how the location of ice streams depends on the geometry of an ice sheet and on the mechanical properties of the underlying bed. More generally, this problem is relevant to the flow of thin films with Coulomb (or ‘solid’) friction laws applied at their base. By analogy with friction problems in elasticity, we construct a variational formulation for the free boundary between ice streams, where bed failure occurs, and the surrounding ice ridges, where there is little or no sliding. This variational problem takes the form of a non-coercive variational inequality, and we show that solutions exist provided a force and moment balance condition is satisfied. In that case, solutions are also unique except under certain specialized circumstances which are unlikely to arise for a real ice sheet. Further, we show how the variational formulation of the ice flow problem can be exploited to calculate numerical solutions, and to simulate the effect of changing ice geometry and bed friction on the location and velocities of streaming flow. Lastly, we study the effect of ice-shelf buttressing on the flow of ice streams whose spatial extent is determined by the yield stress distribution of the bed. In line with previous studies of ice-shelf buttressing, we find that the removal of an ice shelf can cause an ice stream feeding the ice shelf to speed up considerably, which underlines the important role ice shelves may play in controlling the dynamics of marine ice sheets.

We report on a newly discovered bifurcation that occurs in dry grains flowing down a chute with a wavy bottom. We find that the bifurcation outwardly resembles the long-known fluid analogue of inviscid channel flow over a wavy bottom reported in 1886 by Lord Kelvin; however, in detail, the two situations differ significantly. We compare three distinct states seen in the granular system: a ‘regular’ flow in phase with the bottom wave; an ‘antiregular’ flow that is out of phase; and a ‘flat’ flow in which the surface slides nearly uniformly downhill. Additionally, we discuss evidence that sustained subsurface circulation in the granular bed, accompanied by strong fluctuations in flow velocities, can appear in granular flows over wavy surfaces.

Wide-field particle image velocimetry measurements were performed in a Mach 2 turbulent boundary layer to study the characteristics of large-scale coherence at two wall-normal locations ($y/\delta\,{=}\,0.16$ and 0.45). Instantaneous velocity fields at both locations indicate the presence of elongated streamwise strips of uniform low- and high-speed fluid (length$\,{>}\,8\delta$). These long coherent structures exhibit strong similarities to those that have been found in incompressible boundary layers, which suggests an underlying similarity between the incompressible and supersonic regimes. Two-point correlations of streamwise velocity fluctuations show coherence over a longer streamwise distance at $y/\delta\,{=}\,0.45$ than at $y/\delta\,{=}\,0.16$, which indicates an increasing trend in the streamwise length scale with wall-normal location. The spanwise scale of these uniform-velocity strips increases with increasing wall-normal distance as found in subsonic boundary layers. The large-scale coherence observed is consistent with the very large-scale motion (VLSM) model proposed by Kim & Adrian (Phys. Fluids, vol. 11, 1999, p. 417) for incompressible boundary layers.

By adding minute concentrations of a high-molecular-weight polymer, liquid jets or bridges collapsing under the action of surface tension develop a characteristic shape of uniform threads connecting spherical fluid drops. In this paper, high-precision measurements of this beads-on-string structure are combined with a theoretical analysis of the limiting case of large polymer relaxation times and high polymer extensibilities, for which the evolution can be divided into two distinct regimes. For times smaller than the polymer relaxation time over which the beads-on-string structure develops, we give a simplified local description, which still retains the essential physics of the problem. At times much larger than the relaxation time, we show that the solution consists of exponentially thinning threads connecting almost spherical drops. Both experiment and theoretical analysis of a one-dimensional model equation reveal a self-similar structure of the corner where a thread is attached to the neighbouring drops.

Drag-reduction experiments with combined injection of high-molecular-weight long-chained polymers and microbubbles were conducted on a 3.1 m long flat plate model in the 1.22 m diameter water tunnel at the Applied Research Laboratory of the Pennsylvania State University. Combined gas injection upstream of polymer injection produced, over a wide range of test conditions, higher levels of drag reduction than those obtained from the independent injection of polymer or microbubbles alone. These increased levels of drag reduction with combined injection were often greater than the product of the drag reductions obtained by the independent constituents, defined as synergy. We speculate that the synergy is a result of the gas-layer-induced extension of the polymer-alone initial diffusion zone in combination with the increased drag reduction by microbubbles. This increased length of the initial zone layer, consistent with high drag reduction, can significantly increase the persistence of the drag reduction and may improve the outlook for practical application.

Prandtl's secondary mean motions of the second kind are driven by the variation of Reynolds stresses near resistive boundaries. In the flows considered here the turbulence is generated away from the boundary in the absence of a mean flow and then impacts onto a rigid surface placed into the flow at $t\,{=}\,0$. The initial development of the distorted flow is obtained using the linear approximation and the statistical analysis of rapid distortion theory, following Hunt & Graham (1978) assuming homogeneous stationary high-Reynolds-number turbulence with an integral length scale $L_\infty$ and r.m.s. velocity $v_\infty\prime$. First, the effects of axisymmetric anisotropy and of different forms of the spectra are analysed for turbulence impinging onto a plane surface lying at an angle $\alpha$ to the unit vector $\bf e$ of the axis of symmetry of the energy spectrum tensor $\Phi_{ij}(\mbox{{\boldmath k}})$. $R$ is defined as the ratio of the largest to smallest variances of the velocity components. The surface blocking leads to gradients of Reynolds shear stresses normal to the surface in the source layer $B^{(s)}$ with thickness of order $L_\infty$ and thence to a mean velocity $U(t) \,{\sim}\,{-}tv_\infty\prime^2 \sin 2\alpha (1\,{-}\,1/R) /L_\infty$ along the slope in the opposite direction of the projection of ${\boldmath e}$ onto the plane (i.e. in the direction ($\mbox{{\boldmath e}}\wedge\mbox{{\boldmath n}}$)$\wedge\mbox{{\boldmath n}}$ where ${\boldmath n}$ is the normal into the flow). $U$ is greatest near the surface where $y \,{\ll}\, L_\infty$. As a result of shear stresses being induced by the mean velocity gradient, a steady flow results over a time scale $T_L\,{=}\,L_\infty/v_\infty\prime$ – an order of magnitude estimate for the steady-state mean velocity is thence $U(t/T_L \,{\to}\, \infty) \,{\sim}\, v_\infty\prime(\sin 2\alpha (1\,{-}\,1/R))^{1/2}$. Secondly, the effect of a curved surface is studied by analysing isotropic turbulence near an undulating surface of wavelength $\Lambda$ and amplitude $H$, with a low slope so that $H \,{\ll}\, \Lambda$. The boundary condition of zero normal velocity at the curved surface generates larger irrotational fluctuations in the troughs, smaller fluctuations over the crest, and shear stresses over the slopes. The curl of the gradients of Reynolds normal and shear stresses within $B^{(s)}$ cause the growth of a mean vorticity which induces a mean velocity of order $-tv_\infty\prime^2/L_\infty$ within $B^{(s)}$ and a weaker recirculating velocity of order $-tv_\infty\prime^2/\Lambda$ in a deeper wave layer, $B^{(w)}$, with thickness of order $\Lambda$ outside $B^{(s)}$. The wavelength of the mean motion is $\Lambda$, with downward motions over the troughs and upward motion over the crest. As in the first case, a steady flow is predicted when $t/T_L \,{\gg}\, 1$. Anisotropic free-stream turbulence also induces mean motions on undulating surfaces with the same wavelength $\Lambda$ as that of the undulating surface, but the directions of these mean motions can be towards or away from the troughs/crests depending on the orientation of the anisotropy of the free stream. Flow visualization experiments conducted in a mixing box with oscillating anisotropic and isotropic grids demonstrated the existence of these mean flows and that they reach a steady state with an intensity and length scale comparable to those predicted. These results are also consistent with numerical simulation of Krettenauer & Schumann (1992) of convective turbulence over an undulating surface.

The nonlinear stability of falling film flow down an inclined flat plane is investigated when an electric field acts normal to the plane. A systematic asymptotic expansion is used to derive a fully nonlinear long-wave model equation for the scaled interface, where higher-order terms must be retained to make the long-wave approximation valid for long times. The effect of the electric field is to introduce a non-local term which comes from the potential region above the liquid film. This term is always linearly destabilizing and produces growth rates proportional to the cubic power of the wavenumber – surface tension is included and provides a short wavelength cutoff. Even in the absence of an electric field, the fully nonlinear equation can produce singular solutions after a finite time. This difficulty is avoided at smaller amplitudes where the weakly nonlinear evolution is governed by an extension of the Kuramoto–Sivashinsky equation. This equation has solutions which exist for all time and allows for a complete study of the nonlinear behaviour of competing physical mechanisms: long-wave instability above a critical Reynolds number, short-wave damping due to surface tension and intermediate growth due to the electric field. Through a combination of analysis and extensive numerical experiments, we find parameter ranges that support non-uniform travelling waves, time-periodic travelling waves and complex nonlinear dynamics including chaotic interfacial oscillations. It is established that a sufficiently high electric field will drive the system to chaotic oscillations, even when the Reynolds number is smaller than the critical value below which the non-electrified problem is linearly stable. A particular case of this is Stokes flow.

This paper addresses how the turbulent flow field in submerged wall jets responds to an abrupt change from smooth to rough beds. Experiments were conducted for submerged wall jets having different submergence factors and jet Froude numbers. The bed configurations investigated consisted of different combinations of the lengths of smooth beds and the roughness of rough beds. The vertical profiles of time-averaged velocity components, turbulence intensity components and Reynolds stress were detected by an acoustic Doppler velocimeter at different streamwise distances; and the horizontal distributions of bed shear stress were estimated from the Reynolds stress profiles. The flow field displays the decay of jet velocity due to abrupt changes from smooth to rough beds. The boundary layer grows more quickly with increase in roughness of rough beds. The change in bed roughness induces an increased depression of the free surface over the smooth bed. The Reynolds and bed shear stresses are also computed by solving the Navier–Stokes equations. The response of the turbulent flow characteristics of submerged wall jets to abrupt changes from smooth to rough beds is analysed from the point of view of similarity, growth of the length scale, and decay of the velocity and turbulence characteristics scales. The significant observation is that the flow in the fully developed zone is plausibly self-preserving on both smooth and rough beds. Also, the use of a common length scale makes it possible to collapse all the flow data onto a single band; and there is a gradual variation of flow at the junction of the smooth and rough beds.

The equilibrium scour profiles downstream of a smooth apron due to submerged wall jets are computed from the threshold condition of the sediment particles on the scoured bed. Use of the modified bed shear stress for the downstream variation of scoured bed permits the computation of the equilibrium scour profiles. The time-variation of maximum scour depth is computed from the bed shear stress with a modification for the time dependence. The agreement between the results obtained from the model and the experimental data is satisfactory.

Approximations to the scattering of linear surface gravity waves on water of varying quiescent depth are investigated by means of a variational approach. Previous authors have used wave modes associated with the constant depth case to approximate the velocity potential, leading to a system of coupled differential equations. Here it is shown that a transformation of the dependent variables results in a much simplified differential equation system which in turn leads to a new multi-mode ‘mild-slope’ approximation. Further, the effect of adding a bed mode is examined and clarified. A systematic analytic method is presented for evaluating inner products that arise and numerical experiments for two-dimensional scattering are used to examine the performance of the new approximations.