An example is presented of a separated flow in an unbounded domain in which, as the Reynolds number becomes large, the separated region remains of size 0(1) and tends to a non-trivial Prandtl-Batchelor flow. The multigrid method is used to obtain rapid convergence to the solution of the discretized Navier-Stokes equations at Reynolds numbers of up to 5000. Extremely fine grids and tests of an integral property of the flow ensure accuracy. The flow exhibits the separation of a boundary layer with ensuing formation of a downstream eddy and reattachment of a free shear layer. The asymptotic (’triple deck’) theory of laminar separation from a leading edge, due to Sychev (1979), is clarified and compared to the numerical solutions. Much better qualitative agreement is obtained than has been reported previously. Together with a plausible choice of two free parameters, the data can be extrapolated to infinite Reynolds number, giving quantitative agreement with triple-deck theory with errors of 20% or less. The development of a region of constant vorticity is observed in the downstream eddy, and the global infinite-Reynolds-number limit is a Prandtl-Batchelor flow; however, when the plate is stationary, the occurrence of secondary separation suggests that the limiting flow contains an infinite sequence of eddies behind the separation point. Secondary separation can be averted by driving the plate, and in this case the limit is a single-vortex Prandtl-Batchelor flow of the type found by Moore, Saffman & Tanveer (1988); detailed, encouraging comparisons are made to the vortex-sheet strength and position. Altering the boundary condition on the plate gives viscous eddies that approximate different members of the family of inviscid solutions.

The nonlinear evolution of marginally unstable wave packets in rotating pipe flow is studied. These flows depend on two control parameters, which may be taken to be the axial Reynolds number R and a Rossby number, q. Marginal stability is realized on a curve in the (R, q)-plane, and we explore the entire marginal stability boundary. As the flow passes through any point on the marginal stability curve, it undergoes a supercritical Hopf bifurcation and the steady base flow is replaced by a travelling wave. The envelope of the wave system is governed by a complex Ginzburg–Landau equation. The Ginzburg–Landau equation admits Stokes waves, which correspond to standing modulations of the linear travelling wavetrain, as well as travelling wave modulations of the linear wavetrain. Bands of wavenumbers are identified in which the nonlinear modulated waves are subject to a sideband instability.

The properties of edge waves confined by the interaction of buoyancy and Coriolis forces to the vicinity of a rigid plane boundary in a rotating, stratified, electrically conducting fluid pervaded by a magnetic field are established in some simple cases. The background shear is taken to be zero, the basic Alfvén velocity V and Brunt–Väisälä frequency N are assumed uniform, and all dissipative effects are taken to be vanishingly small. It is shown that waves trapped against the bounding wall can occur only if V is parallel to the wall. When the basic rotation vector Ω is also parallel to the wall, the hydromagnetic edge waves have a higher frequency and smaller spatial extent perpendicular to the wall than their non-hydromagnetic counterparts, but more complex behaviour is found when Ω possesses a component normal to the wall. There are conditions under which edge waves may exist even when the basic density stratification is top-heavy (i.e. when N2 < 0).

The hydrodynamic instability of a viscous incompressible flow with a free surface is studied both numerically and experimentally. While the free-surface flow is basically two-dimensional at low Reynolds numbers, a three-dimensional secondary flow pattern similar to the Taylor vorticies between two concentric cylinders appears at higher rotational speeds. The secondary flow has periodic velocity components in the axial direction and is characterized by a distinct spatially periodic variation in surface height similar to a standing wave. A numerical method, using boundary-fitted coordinates and multigrid methods to solve the Navier–Stokes equations in primitive variables, is developed to treat two-dimensional free-surface flows. A similar numerical technique is applied to the linearized three-dimensional perturbation equations to treat the onset of secondary flows. Experimental measurements have been obtained using light sheet techniques to visualize the secondary flow near the free surface. Photographs of streak lines were taken and compared to the numerical calculations. It has been shown that the solution of the linearized equations contains most of the important features of the nonlinear secondary flows at Reynolds number higher than the critical value. The experimental results also show that the numerical method predicts well the onset of instability in terms of the critical wavenumber and Reynolds number.

This paper investigates the Bénard problem in a binary mixture of dilute gases in which an imposed vertical temperature gradient induces a concentration gradient owing to the thermal diffusion effect. The transfer equations are derived by first-order perturbation theory which leads to instability criteria. Numerical results indicate that instability will set in only as stationary convection. This is distinctly different from the cases of liquids and concentrated gases, in which the thermal diffusion (or Soret) effect gives rise to oscillatory instability. It is disclosed in the study that the destabilization of the dilute gas-mixture layer is enhanced by an increase in the thermal diffusion ratio and/or the molecular weight ratio of the species.

The paper is concerned with the interaction between the boundary layer on a smooth body surface and the outer inviscid compressible flow in the vicinity of a sonic point. First, a family of local self-similar solutions of the Kármán–Guderley equation describing the inviscid flow behaviour immediately outside the interaction region is analysed; one of them was found to be suitable for describing the boundary-layer separation. In this solution the pressure has a singularity at the sonic point with the pressure gradient on the body surface being inversely proportional to the cubic root dpw/dx ∼ (−x)−1/3 of the distance (−x) from the sonic point. This pressure gradient causes the boundary layer to interact with the inviscid part of the flow. It is interesting that the skin friction in the boundary layer upstream of the interaction region shows a characteristic logarithmic decay which determines an unusual behaviour of the flow inside the interaction region. This region has a conventional triple-deck structure. To study the interactive flow one has to solve simultaneously the Prandtl boundary-layer equations in the lower deck which occupies a thin viscous sublayer near the body surface and the Kármán–Guderley equations for the upper deck situated in the inviscid flow outside the boundary layer. In this paper a numerical solution of the interaction problem is constructed for the case when the separation region is entirely contained within the viscous sublayer and the inviscid part of the flow remains marginally supersonic. The solution proves to be non-unique, revealing a hysteresis character of the flow in the interaction region.

Small perturbations of a parallel shear flow U(z) in an inviscid, incompressible, stably stratified fluid of density ρ(z) are considered, for which the principal measure of stability is the Richardson number, R. For an arbitrary velocity and density profile we discuss the problem of determining whether a curve of neutral stability has adjacent unstable regions in an (α, R) plane, where α is the wave-number of the disturbance. Neutral curves bounding unstable regions are then obtained for a triangular jet flow in conjunction with various density distributions. A comparison is also made between the stability characteristics of jet and shear flows with corresponding density structures.

Journal of Fluid Mechanics, vol. 380 (1999), pp. 1–38

I am indebted to Professor Prosperetti and to Dr Ren for finding a significant error in my detailed analysis of the temperature and pressure fluctuations in a pulsating particle. As they point out, the error invalidates the results of that analysis for the temperature fluctuations in gas bubbles. Those results are withdrawn. The error also affects, although in a less significant manner, the results for the temperature fluctuations in pulsating liquid drops with specific heat ratio significantly different from 1. This includes the droplets chosen to illustrate the results, namely toluene droplets. However, correct results for both bubbles and droplets may be easily obtained and this is done here.

The error found by Prosperetti and Ren (see Prosperetti & Ren 2001) exists in both the internal and external solutions, but because the external fluid is in both cases a liquid having a very small value of (γf − 1), the error has no effect there. Thus, fortuitously, the external solution is valid, provided the correct values of the interfacial temperature and pressure fluctuations are used. In addition, the simple solution presented in Appendix B, as well as the general expressions for derived quantities, such as those for the energy dissipation rates and the attenuation coefficients, are not affected by the error, provided, again, that the values of the temperature and pressure ratios appearing in them are properly evaluated. This is done below for both drops and bubbles by complementing the analysis of Appendix B with the polytropic approximation. The required steps are straightforward.

A new theory for the emergence of dispersion in shallow water hydrodynamics in two horizontal space dimensions is presented. Starting with the key properties of uniform flow in open-channel hydraulics, it is shown that criticality is the key mechanism for generating dispersion. Modulation of the uniform flow then leads to model equations. The coefficients in the model equations are related precisely to the derivatives of the mass flux, momentum flux and mass density. The theory gives a new perspective – from the viewpoint of hydraulics – on how and why key shallow water models like the Korteweg–de Vries equation and Kadomtsev–Petviashvili equations arise in the theory of water waves.

Dynamic interaction between a boundary layer of air and a liquid film is investigated in this paper. The low air-to-film-viscosity ratio is considered in which case the boundary layer is quasi-steady on the time scale within which interfacial waves develop. The base flow consists of a boundary layer that drags a film of constant shear. Linear analysis, in the context of triple-deck theory, predicts the formation of a wavepacket of capillary waves that advances and spreads with time. The Froude number of de-/anti-icing fluids or water interacting with air falls well within the supercritical regime, i.e. Fr > FrCr. Numerical simulations of such flow systems were performed in the context of triple-deck theory, and they do not exhibit wave saturation or formation of uniform wavetrains. The long-term interaction is mainly dependent on film inertia as this is characterized by parameter = (μ/μf)2(ρf/ρ), which involves film and air viscosity and density ratios, and the dimensionless film thickness, H0, and shear, λ, provided by the base flow. Weakly nonlinear analysis taking into consideration mean drift, i.e. generation of long waves, due to self-interaction of the linear wave to O(ϵ2) in amplitude of the initial disturbance, reveals resonance between the wavepacket predicted by linear theory and long waves when the group velocity of the former happens to coincide with the phase velocity, H0λ, of long interfacial waves. Numerical simulations with anti-icing fluids and water verify this pattern. In both cases, long waves eventually dominate the dynamics and, as they are modulated with time, they lead to soliton-type structures. Anti-icing fluids eventually exhibit oscillatory spikes whose mean value never exceeds 2H0, roughly. Water films exhibit a single spike that keeps growing, thus generating a large separation bubble.

We introduce a finite element model for neutrally buoyant particle suspensions of cylinders at zero Reynolds number and infinite Péclet number in the purely hydrodynamic limit, which allows us to access a high-accuracy fluid velocity field at any time during the simulation. We use the diffusive strip method to characterize the development of the concentration field in the fluid phase of sheared suspensions from initial thin filaments, and characterize the structures that form with their fractal dimension. We find that the growth of the fractal dimension of the filaments scales with the increase of mean square displacement in the fluid phase. Further, we measure the concentration distribution of tracers in the fluid phase, as well as the shear-induced self-diffusion coefficient in both the solid phase and the fluid phase. We demonstrate that the shear-induced self-diffusion coefficient is slightly larger in the fluid phase at infinite Péclet number. Finally, we investigate enhanced mass diffusivity in the fluid phase by systematic measurements of the shear-induced self-diffusion coefficient in the fluid phase for a wide range of fluid tracer Péclet numbers. We find that the functional dependence $D_{s}/D=1+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6FC}}Pe^{\unicode[STIX]{x1D701}}$ (where $D_{s}$ is the shear-induced self-diffusion coefficient, $D$ is the molecular diffusivity and $\unicode[STIX]{x1D719}$ is the particle volume fraction) fits the observations fairly well. We measure the constants $\unicode[STIX]{x1D6FD}=2.98\pm 0.39$, $\unicode[STIX]{x1D6FC}=1.61\pm 0.26$ and $\unicode[STIX]{x1D701}=0.900\pm 0.031$.

The growth rates of gravity waves are computed from linear, inviscid stability theory for wind velocity profiles that are representative of the mean flow in a turbulent boundary layer. The energy transfer to the waves is largely concentrated in an angle (to the wind) interval that broadens with increasing wind speed and narrows with increasing wavelength. At sufficiently high wind speeds and sufficiently short wavelengths, the waves of maximum growth rate propagate at an oblique angle to the wind. The connection with bimodal directional distributions of observed spectra is discussed.

The drag force acting on a circular cylinder fluctuating erratically in a viscous fluid is measured with a laser–cantilever force transducer. The experimental results compare very well with the theory.

Two-dimensional free-surface flow over a spatially periodic channel bed topography is examined using a steady periodically forced Korteweg–de Vries equation. The existence of new forced solitary-type waves with periodic tails is demonstrated using recently developed non-autonomous dynamical-systems theory. Bound states with two or more co-existing solitary waves are also identified. The solution space for varying amplitude of forcing is explored using a numerical method. A rich bifurcation structure is uncovered and shown to be consistent with an asymptotic theory based on small forcing amplitude.

Axisymmetric steady motion of an inhomogeneous rotating fluid is considered. A system of equations, with appropriate boundary conditions, controlling the smooth interior fields is derived under the assumption of small dissipation and small side-boundary conductance. It is argued that this system, being derived without linearization of the equations, might form the basis of valuable numerical analysis. Assuming sufficiently weak forcing, i.e. high insulation of the non-horizontal boundaries, a linear system is derived. An explicit solution is presented and discussed for a particularly simple and important case.