The work that follows considers the velocity profiles within the boundary layer at the wall of an arbitrarily converging funnel. The occurrence of super-velocities, i.e. components of velocity within the boundary layer exceeding their corresponding free stream component, is investigated and the relevance of such a phenomenon to the efficiency of discharge discussed.

Euromech 98 was held in Eindhoven inNovember 1977 with the authors as chairmen. The Colloquium was attended by 48 participants from a number of European countries. Bubbles play an important role in many areas of technology: propeller-induced cavitation in ship building, cavitation in fluid machinery, nucleate boiling in reactorsand similar devices, and many processes (centrifuges, mixers) in the chemical process industry. A wide variety of research, both of fundamental and of more applied nature, is going on in universities and in industrial laboratories. Representatives from all theabove areas were included among the participants and consequently the topics raised in the presentations and discussions covered a very broad field.

The detailed structure of the relaxation region in shock waves in oxygen was investigated using Blackman's experimental results. Oxygen was found to display a behaviour similar in many ways to that found previously for carbon dioxide with the relaxation frequency, as defined by the simple relaxation equation, depending on the departure from equilibrium as well as on temperature. The previous results for carbon dioxide were further analysed by means of a separate relaxation equation for each mode.

Self-similar turbulent vortex rings are investigated theoretically in the framework of the semi-empirical turbulence theory for the modified Helmholtz equation. The velocity and vorticity fields are established, as well as the transport of passive admixture by turbulent vortex rings. Turbulent vortex rings of propellant gases originating from the muzzle of a gun after a gunshot are an important phenomenon to consider in crime scene reconstruction. In this work, it is shown that this has a significant repercussion on the outcome of backward blood spatter resulting from a gunshot. Turbulent vortex rings of propellant gases skew the distribution of bloodstains on the ground and can either propel blood droplets further from the target, or even turn them backwards towards the target. This is revealed through the final bloodstain locations and the respective distributions of the number of stains and their area as a function of distance from the target for two different shooter-to-target distances. An image of the propagating muzzle gases after bullet ejection is overlaid with the predicted flow field, which reveals satisfactory agreement. Gunshot residue is an important factor in determining the events of a violent crime due to a gunshot and are considered to be entrained and transported by the propellant gases. The self-similar solutions for the flow, vorticity and concentration of gunpowder particles are predicted and the results are shown to be within the measured range of a limited set of experimental data.

The steady plane boundary-layer flows of velocity field {u(x, y), v(x, y)} induced by continuous moving surfaces are revisited in this paper. It is shown that the governing balance equations, as well as the asymptotic condition u(x, ∞) = 0 at the outer edge of the boundary layer are invariant under arbitrary translations y → y + y0(x) of the transverse coordinate y. The wall conditions, i.e. the prescribed stretching velocity u(x, 0) ≡ Uw(x) and the transpiration velocity v(x, 0) ≡ Vw(x) distributions, however, undergo in general substantial changes. The consequences of this basic symmetry property on the structure of the solution space are investigated. It is found that starting with a primary solution which describes the boundary-layer flow induced by an impermeable surface, infinitely many translated solutions can be generated which form a continuous group, the translation group of the given primary solution. The elements of this group describe boundary-layer flows induced by permeable surfaces stretching under transformed wall conditions, Uw(x) → Ũw(x) = u[x, y0(x)] and Vw(x) → Ṽw(x) = v[x, y0(x)] − y′0(x)u[x, y0(x)], respectively. In this way, starting with a known solution {u(x, y), v(x, y)} so that the inverse y0(x) = u−1(x, Ũw) of u[x, y0(x)] exists, a new solution {ũ(x, y), ṽ(x, y)} corresponding to any desired stretching velocity distribution Ũw(x) can be prepared. It also turns out that several exact solutions discovered during the latter decades are not basically new solutions, but translated counterparts of some formerly reported primary solutions. A few specific examples are discussed in detail.

The behaviour of the interface between two immiscible fluids of different density and viscosity in a centrifuge is investigated both analytically and experimentally. The study includes the interfacial shape and behaviour during both steady and transient operation of the centrifuge. The interface shape is investigated for rigid-body rotation and during slow steady acceleration or deceleration of the centrifuge. The dependence of the interface shape on time is investigated during a rapid spin-up to a slightly different value of the angular velocity, which then remains constant. Some aspects of interface stability are also reported.

The interaction of a pair of spherical particles of different densities and/or sizes with microscopic surface roughness sedimenting due to gravity in a viscous fluid is analysed by theory and experiment. The surface topography is modelled as a combination of small uniformly distributed bumps of uniform height and larger bumps that are more sparsely distributed. The existence of these surface asperities allows the spheres to physically contact each other, so that both hydrodynamic and solid-contact forces are important. When the angle between the line of centres and vertical is small, the spheres may rotate as a rigid body because they are not able to roll up and over a large bump. As this angle increases, however, the heavy sphere rolls and slips past the lighter sphere, and the separation between the nominal surfaces of the spheres varies between the heights of the small and large asperities. When considering many encounters, there is a distribution of nominal separations at each angle due to the distribution of initial conditions and surface topography. The average nominal separation generally increases with increasing angle between the line of centres and vertical because the normal component of gravity, which drives the spheres close together after an encounter with a large bump lifts them apart, decreases as this angle increases.

Time-dependent hydromagnetic phenomena in a rotating spherical cavity are investigated in the framework of an interior boundary-layer expansion. The interior problem is shown to contain waves whose frequencies are of order ω, Aω and A2ω, where ω is the rotation rate of the cavity and A2 = B2/4πρωA2 [Lt ] 1 is the Alfvén number. B is an imposed magnetic field, ρ the fluid density and R the radius of the cavity. The first type of wave is a modification of the hydro-dynamic inertial wave, the second is a pseudo-geostrophic wave and is involved in spin-up, and the third is related to the MAC waves of Braginskiy (1967).

It is shown that the MAC waves must satisfy more than the usual normal boundary conditions and that reference must be made to the boundary-layer solution to resolve the ambiguity regarding which conditions are to be taken. For normal liquid metals of small magnetic Prandtl number the MAC waves must satisfy full magnetic boundary conditions; only the no-slip conditions may be deferred to the boundary layers.

The boundary-layer structure is investigated in detail to display the interactions between applied field, viscosity, electrical conductivity, frequency and latitude. The decay of the pseudo-geostrophic modes, essentially the spin-up problem, is discussed for a non-axisymmetric constraining field and non-zero container conductivity. Three regimes exist, depending on container conductivity.

It is usually believed that wall slip contributes small effects to macroscopic flow characteristics. Here we demonstrate that this is not the case for the thermocapillary migration of a long bubble in a slippery tube. We show that a fraction of the wall slip, with the slip length $\lambda $ much smaller than the tube radius $R$, can make the bubble migrate much faster than without wall slip. This speedup effect occurs in the strong-slip regime where the film thickness $b$ is smaller than $\lambda $ when the Marangoni number $S= \tau _{T} R/\sigma _{0}~ (\ll 1)$ is below the critical value $S^* \sim (\lambda /R)^{1/2}$, where $\tau _{T}$ is the driving thermal stress and $\sigma _{0}$ is the surface tension. The resulting bubble migration speed is found to be $U_{b} \sim (\sigma _{0}/\mu )S^{3}(\lambda /R)$, which can be more than a hundred times faster than the no-slip result $U_{b} \sim (\sigma _{0}/\mu )S^{5}$ (Wilson, J. Eng. Math., vol. 29, 1995, pp. 205–217; Mazouchi & Homsy, Phys. Fluids, vol. 12, 2000, pp. 542–549), with $\mu $ being the fluid viscosity. The change from the fifth power law to the cubic one also indicates a transition from the no-slip state to the strong-slip state, albeit the film thickness always scales as $b\sim RS^{2}$. The formal lubrication analysis and numerical results confirm the above findings. Our results in different slip regimes are shown to be equivalent to those for the Bretherton problem (Liao, Li & Wei, Phys. Rev. Lett., vol. 111, 2013, 136001). Extension to polygonal tubes and connection to experiments are also made. It is found that the slight discrepancy between experiment (Lajeunesse & Homsy, Phys. Fluids, vol. 15, 2003, pp. 308–314) and theory (Mazouchi & Homsy, Phys. Fluids, vol. 13, 2001, pp. 1594–1600) can be interpreted by including wall slip effects.

The turbulent separated-flow region occurring at a compression corner under certain circumstances at supersonic speed has been examined in the light of recent improvements to base pressure theory (McDonald 1964). This base pressure theory is further extended from what could be termed a single-layer model of the re-attaching boundary layer to a two-layer model, thus enabling the inviscid shock configuration which occurs at the corner to be determined. Application of this analysis to some experimental results indicates a substantial measure of agreement.

While this analysis has been framed for estimating the scale of the corner interaction, the extension can of course be applied to increase the range of initial boundary-layer thicknesses to which McDonald's analysis is applicable. An example of such an application is shown to be in good agreement with experiment.

The behaviour of turbulent flows within the single-layer quasi-geostrophic (Charney–Hasegawa–Mima) model is shown to be strongly dependent on the Rossby deformation wavenumber $\lambda$ (or free-surface elasticity). Herein, we derive a bound on the inverse energy transfer, specifically on the growth rate ${\rm d}\ell/{\rm d} t$ of the characteristic length scale $\ell$ representing the energy centroid. It is found that ${\rm d}\ell/{\rm d} t\le2\|q\|_\infty/(\ell_s\lambda^2)$, where $\|q\|_\infty$ is the supremum of the potential vorticity and $\ell_s$ represents the potential enstrophy centroid of the reservoir, both invariant. This result implies that in the potential-energy-dominated regime ($\ell\ge\ell_s\,{\gg}\,\lambda^{-1}$), the inverse energy transfer is strongly impeded, in the sense that under the usual time scale no significant transfer of energy to larger scales occurs. The physical implication is that the elasticity of the free surface impedes turbulent energy transfer in wavenumber space, effectively rendering large-scale vortices long-lived and inactive. Results from numerical simulations of forced-dissipative turbulence confirm this prediction.

We investigate the instability and nonlinear saturation of temperature-stratified Taylor–Couette flows in a finite height cylindrical gap and calculate angular momentum transport in the nonlinear regime. The model is based on an incompressible fluid in Boussinesq approximation with a positive axial temperature gradient applied. While both ingredients, the differential rotation as well as the stratification due to the temperature gradient, are stable themselves, together the system becomes subject of the stratorotational instability and a non-axisymmetric flow pattern evolves. This flow configuration transports angular momentum outwards and will therefore be relevant for astrophysical applications. The belonging coefficient of β viscosity is of the order of unity if the results are adapted to the size of an accretion disk. The strength of the stratification, the fluid's Prandtl number and the boundary conditions applied in the simulations are well suited too for a laboratory experiment using water and a small temperature gradient around 5 K. With such a set-up the stratorotational instability and its angular momentum transport could be measured in an experiment.

The turbulent separated flow through an asymmetric diffuser with and without manipulation of incoming turbulence-driven mean secondary vortices (MSVs) from a rectangular duct is investigated by large-eddy simulations. The simulations carried out for two diffuser geometries reveal that by introducing a small amount of mean-flow kinetic energy via the MSVs into the flow, the complex three-dimensional separation behaviour and pressure recovery can be effectively controlled. Manipulated MSVs were found to enhance cross-sectional transport of high-momentum fluid, which determined the location, shape, and size of the separation bubble. The integral effect was a delay or expedition in the onset of separation. This change strongly affected the conversion of mean-flow kinetic energy to pressure, in particular for the front part of the diffuser. In addition, a substantial reduction in total pressure loss could be achieved. The manipulation of the MSVs is an efficient mechanism for performance enhancement in the cases investigated. The results have important implications for both control and statistical modelling of turbulent separated flow in rectangular diffusers.

A simple mechanism of parametric excitation of internal gravity waves in a uniformly stably stratified flow under the inviscid Boussinesq approximation is presented. It consists in an oscillating planar irrotational strain field with frequency $\omega$ disturbed by three-dimensional plane waves. When the amplitude of the strain is weak, the problem is reduced to a Mathieu equation and a condition for parametric resonance is easily deduced. For a large-amplitude strain field equations are solved numerically with Floquet theory. In both cases, it is shown that parametric instabilities are excited when stratification is large enough, that is when $N > \frac{1}{2}\omega$, where $N$ is the Brunt–Väisälä frequency of the flow. On the other hand, when $N \leq \frac{1}{2}\omega$, the flow is shown to be stable for any periodic background excitation thanks to a theorem by Joukowski. Therefore, stratification promotes instability. In the strongly stratified case $N\,{\gg}\,\omega$, resonant waves satisfy the Billant–Chomaz self-similarity law and the resulting instabilities develop inside correlated quasi-horizontal layers. After discussion of the viscous effects, the theory of the paper is applied to the stability of an elliptical vortex in a rotating stratified medium.

We study the coupled problem of a liquid bridge connected to a porous surface and an impermeable surface, where the gap between the surfaces is an externally controlled function of time. The relative motion between the surfaces influences the pressure distribution and geometry of the liquid bridge, thus affecting the shape of liquid penetration into the porous material. Utilizing the lubrication approximation and Darcy’s phenomenological law, we obtain an implicit integral relation between the relative motion between the surfaces and the shape of liquid penetration. A method to control the shape of liquid penetration is suggested and illustrated for the case of conical penetration shapes with an arbitrary cone opening angle. We obtain explicit analytic expressions for the case of constant relative speed of the surfaces as well as for the relative motion between the surfaces required to create conical penetration shapes. Our theoretical results are compared with experiments and reasonable agreement between the analytical and experimental data is observed.

Experiments are described in which the populations of metastable levels in ionizing argon are measured through spatially resolved hook interferometry. The results are compared with the present model for shock-induced ionization and a recently proposed mechanism to explain observed flow instabilities. It is found that the experimental measurements support the presently accepted model, which states that electron–atom collisions play the dominant role in the excitation process, but contradicts recent proposals which predict a rapid build-up of anomalously high metastable populations through atom–atom collisions.

The stability of an infinite layer of fluid of uniform thickness at rest with two horizontal boundaries is investigated when the difference between the temperatures at the top and bottom boundaries has a component which is a fluctuating periodic function of time in addition to a constant part. When both boundaries are free small fluctuations have a stabilizing effect on the layer, while large fluctuations tend to make it less stable, consistently with the numerical results of Yih & Li (1972); the effect is attributable entirely to the variation of the temperature gradient with time. An approximate relation between the mean Rayleigh number and the amplitude of the fluctuations is found which separates stable situations from unstable ones. This is compared with the criteria deduced by Homsy (1974) using energy arguments for disturbances of any amplitude.

An asymptotic analysis of the compressible turbulent boundary-layer equations is carried out for large Reynolds numbers and mainstream Mach numbers of O(1). A self-consistent two-layer asymptotic structure is described wherein the time-mean velocity and total enthalpy are logarithmic within the overlap zone but in terms of the Howarth–Dorodnitsyn variable; the proposed structure leads to a compressible law of the wall for high-speed turbulent flows with surface heat transfer. Simple outer-region algebraic turbulence models are formulated to reflect the effects of compressibility. To test the proposed asymptotic structure and turbulence models, a set of self-similar outer-region profiles for velocity and total enthalpy is obtained for constant-pressure flow and for constant wall temperature; these are combined with wall-layer profiles to form a set of composite profiles valid across the entire boundary layer. A direct comparison with experimental data shows good agreement over a wide range of conditions for flows with and without surface heat transfer.