Details are given of an implicit six-point finite-difference scheme for solving two-temperature, laminar, boundary-layer flows not in chemical equilibrium in ionizing argon. The analysis extends previous work by considering the radiation-energy loss and the chemical reactions due to atom-atom and electron-atom collisions in the ionizing boundary-layer and free-stream flow. Also included are variations in transport properties based on known elastic-scattering cross-sections, effects of chemical reactions, radiation-energy loss and the electric-sheath wall boundary conditions. The results are compared with dual-wavelength interferometric boundary-layer data obtained by using a Mach-Zehnder interferometer 23 cm in diameter with the UTIAS 10 × 18 cm Hypervelocity Shock Tube for shocks of initial Mach numbers Ms ∼ 13 and 16 moving into argon at a pressure p0 ∼ 5 torr and temperature T0 ∼ 297 °K. Considering the difficulties involved in solving such complex plasma flows, satisfactory agreement was obtained between analytic and experimental total-density profiles and electron-number-density profiles for the case Ms ∼ 16 and good agreement for Ms ∼ 13.

Laser-Doppler velocity measurements in water are reported for the flow in the plane of symmetry of a turbulent spot. The unsteady mean flow, defined as an ensemble average, is fitted to a conical growth law by using data at three streamwise stations to determine the virtual origin in x and t. The two-dimensional unsteady stream function is expressed as $\psi = U^2_{\infty}tg(\xi, \eta) $ in conical similarity co-ordinates ζ = x/U∞t and η = y/U∞t. In these co-ordinates, the equations for the unsteady particle displacements reduce to an autonomous system. This system is integrated graphically to obtain particle trajectories in invariant form. Strong entrainment is found to occur along the outer part of the rear interface and also in front of the spot near the wall. The outer part of the forward interface is passive. In terms of particle trajectories in conical co-ordinates, the main vortex in the spot appears as a stable focus with celerity 0·77U∞. A second stable focus with celerity 0·64U∞ also appears near the wall at the rear of the spot.

Some results obtained by flow visualization with a dense, nearly opaque suspension of aluminium flakes are also reported. Photographs of the sublayer flow viewed through a glass wall show the expected longitudinal streaks. These are tentatively interpreted as longitudinal vortices caused by an instability of Taylor-Görtler type in the sublayer.

A theoretical study is made of the motion of a small sphere in an incompressible viscous fluid bounded by two plane walls. The analysis is based on the Stokes equations of motion and the method of reflexions is employed to obtain the lowest-order wall effect on the sphere, which is allowed either to rotate or not as it translates in an arbitrary direction. The effect of two plane walls on the force and torque (or angular velocity) experienced by the sphere is not accurately estimated by superposing the contributions from the two individual walls.

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.

Unsteady laminar compressible boundary-layer flow with variable properties at a three-dimensional stagnation point for both cold and hot walls has been studied for the case when the velocity of the incident stream varies arbitrarily with time. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. Computations have been carried out for two particular unsteady free-stream velocity distributions: (i) an accelerating stream and (ii) a fluctuating stream. The results indicate that the variation of the density-viscosity product across the boundary layer, the wall temperature and the nature of stagnation point significantly affect the skin friction and heat transfer.

We present a phenomenological model of intermittency called the β-model and related to the Novikov-Stewart (1964) model. The key assumption is that in scales ∼ l02−n only a fraction βn of the total space has an appreciable excitation. The model, the idea of which owes much to Kraichnan (1972, 1974), is dynamical in the sense that we work entirely with inertial-range quantities such as velocity amplitudes, eddy turn-over times and energy transfer. This gives more physical insight than the traditional approach based on probabilistic models of the dissipation.

The β-model leads in an elementary way to the concept of the self-similarity dimension D, a special case of Mandelbrot's (1974, 1976) ‘fractal dimension’. For three-dimensional turbulence, the correction B to the $\frac{5}{3}$ exponent of the energy spectrum is equal to $\frac{1}{3}(3 - D)$ and is related to the exponent μ of the dissipation correlation function by $B = \frac{1}{3}\mu $ (0.17 for the currently accepted value). This is a borderline case of the Mandelbrot inequality $B \leqslant \frac{1}{3}\mu $. It is shown in the appendix that this inequality may be derived from the Navier-Stokes equation under the strong, but plausible, assumption that the inertial-range scaling laws for second- and fourth-order moments have the same viscous cut-off.

The predictions of the β-model for the spectrum and for higher-order statistics are in agreement with recent conjectures based on analogies with critical phenomena (Nelkin 1975) but generally diasgree with the 1962 Kolmogorov lognormal model. However, the sixth-order structure function 〈δv6(l)〉 and the dissipation correlation function 〈ε(r)ε(r + 1)〉 are related by \[ \langle \delta v^6(l)\rangle /l^2\sim\langle \epsilon({\bf r})\epsilon({\bf r}+{\bf l})\rangle \] in both models. We conjecture that this relation is model independent.

Finally, some possible directions for further numerical and experimental work on intermittency are indicated.

Nonlinear thermosolutal convection is investigated using the mean-field approximation (Herring 1963; Busse 1970). The boundary-layer method for rigid boundaries is used by assuming a large Rayleigh number R for different ranges of the diffusivity ratio τ and the solute Rayleigh number RS. The heat and solute fluxes F and FS are determined for the values of the wavenumbers αn which optimize F. The αN mode (where N denotes the total number of modes) is shown to have a solute-layer thickness of order $\tau^{\frac{1}{3}}\delta _{tN}$ (δtN denoting the temperature-layer thickness in the αN mode), and it is also proved that it is only for this mode that the solute concentration affects the boundary-layer structure. Solutions are possible if K = FSRS/FR < 1. For K ≥ 1, the stabilizing effect of solute is unimportant and there can be infinitely many modes. However, as K → 1 −, N, αn, F and FS decrease rapidly and the maximizing convection is suppressed entirely by the solute concentration. A simple interpretation of the model for the diffusive system leads also to the results for the salt-finger system.

The response of a Hill's spherical vortex to an irrotational axisymmetric small perturbation is examined on the assumption that viscous diffusion of vorticity is negligible. The problem of determining the response is first reduced to a system of differential equations for the evolution of the Legendre coefficients of the disturbance stream function. This system is then solved approximately and it is shown that, if the initial disturbance is such as to make the vortex prolate spheroidal, the vortex detrains a fraction ⅗ε of its original volume, where ε is the fractional extension of the axis of symmetry in the imposed distortion. The detrained fluid forms a thin spike growing from the rear stagnation point. If the vortex is initially oblate, irrotational fluid is entrained at the rear stagnation point to the interior of the vortex.

An inconsistency in a previous stability analysis of a cylindrical vortex sheet is resolved. It is found that disturbances with azimuthal wavenumbers m = 1 and 2 are stable, whereas previously the stability of these modes was uncertain. It has been hypothesized that the multiple-vortex phenomenon is initiated by such an instability (Ward 1972). Since double vortices (m = 2) arise, we are led to consider a central downdraft surrounded by a uniform updraft; for a non-zero vertical wavenumber γ, the modes m = 1 and 2 are destabilized. Our theory is supported by the observation that double vortices tend to form as intertwining spirals (i.e. they have vertical structure).

The initial-value problem for a one-dimensional gravity wave of amplitude a and characteristic length l in water of depth d is examined for 0 < a/d [Lt ] d2/l2 [Lt ] 1. A preliminary reduction leads to a Korteweg-de Vries (KdV) equation in which the nonlinear term is O(ε) relative to the linear terms, where ε = 3al2/4d3 [Lt ] 1 is a measure of nonlinearity/dispersion. The linear approximation (ε ↓ 0) is found to be valid if and only if $\epsilon\tau^{\frac{1}{3}}\ll 1 $ where $\tau = \frac{1}{2}(d/l)^2(gd)^{\frac{1}{2}}({\rm time})/l $ is the slow time in the KdV equation. The asymptotic solution of the KdV equation is obtained with the aid of inverse-scattering theory and is found to comprise not only a decaying wave train that is qualitatively similar to that predicted by the linear approximation, but also a soliton of amplitude 3V2/4d3 = O(εa) if V > 0, where V is the cross-sectional area of the initial displacement, or of amplitude = O(ε3a) if V = 0 (there is no soliton if V < 0). This soliton is fully evolved, and dominates the solution, only for $\epsilon \tau^{\frac{1}{3}} \gg 1$ if V > 0 or $\epsilon^2\tau^{\frac{1}{3}}\gg 1 $ if V = 0, but nonlinearity already has significant effects for $\epsilon\tau^{\frac{1}{3}} = O(1)$.

Non-parallel shear flows of an inviscid, incompressible, density-stratified fluid are considered. The stability is studied in terms of the possibility of complete mixing within a horizontal layer of given thickness. It is assumed that the energy interchange between the mixed region and the external fluid can be neglected. It is also assumed that the required turbulent energy is greater than the energy needed to invert the region to be mixed. The term ‘stable’ as used here means that the kinetic energy released by making the velocity constant over the layer thickness is not sufficient to provide the required turbulent energy for that layer thickness.

If each horizontal fluid plane has a translational velocity of the same magnitude and shear is produced by rotation of the velocity vector with increasing height, then ‘stability’ increases strongly with increasing layer thickness.