A new approach to the well-known spin-up from rest problem is proposed based on a search for self-similarity. The Wedemeyer (1963) model is first tested for spin-down to rest and then used for spin-up. The general-form solution is found and is shown to tend to a self-similar limiting stage. Experimental results in a cylinder of small height-to-diameter ratio are analysed to demonstrate this self-similarity in a certain range of external parameters.

This paper presents a study of the convection in acetone due jointly to the thermocapillary (Marangoni) and thermogravitational effects. The liquid (acetone) is submitted to a horizontal temperature difference. Experiments and numerical simulations both show the existence of three different states: monocellular steady states, multicellular steady states and spatio-temporal structures. The results are discussed and compared with the linear stability analysis of Smith & Davis (1983).

Rayleigh-Bénard convection in the presence of a plane Couette flow is investigated by numerical computations. From earlier work it is well known that longitudinal rolls are preferred at the onset of convection and that at Prandtl numbers of the order unity or less these rolls become unstable with respect to the wavy instability which introduces wavy distortions perpendicular to the axis of the rolls. In the present analysis the three-dimensional flows arising from these distortions are studied and their stability is considered. A main result is the subcritical existence of three-dimensional flows at Rayleigh numbers far below the critical value for onset of convection.

The nature of the flow during impulsive spin-down to rest in a cylindrical cavity is studied experimentally. Flow visualization using reflective flakes and laser-Doppler velocimetry are the tools of this investigation. The velocimeter is configured to measure simultaneously the azimuthal velocity component at two arbitrarily separated locations within the cylinder. The Ekman number is about 10−5 and the flow is unstable. The mean angular velocity decreases non-uniformly and monotonically. The velocity fluctuation amplitudes and frequencies decrease steadily. A novel data analysis is used to study the velocity fluctuations, which are neither stationary nor uniform. The assumptions of this analysis are the validity of Taylor's hypothesis of frozen-eddy transport and the ergodicity of the process following that rescaling. The fluctuations are equally dominant during all phases of the spin-down process when scaled with the current mean velocity. The root-mean-squared intensity measurements in the core (r/R < 0.4) suggest an r−1 dependence while a uniform value is observed in the buffer region (0.4 < r/R < 0.8). Flow visualizations and spatial velocity correlations indicate that the flow in the core consists of vortices having axes parallel to the rotation axis and extending throughout the height of the cylinder. The power spectra of the velocity fluctuations, after amplitude scaling with the current mean velocity and Taylor's scaling in time, suggest a –2.6 power dependence on the wavenumber k. The flow in the latter phases tends to a single vortex.

Spatially growing mixing layers are simulated numerically using a two-dimensional vortex method. Special attention is paid to the effect of double-frequency forcing on the development of a mixing layer. Two different types of forcing are considered: superposition of a fundamental frequency (F) on one of its subharmonics (Case I), and superposition of two frequencies of a resonance type, F ± ΔF (Case II). The effects of forcing amplitude and relative phase shift between the two forcing frequencies are also examined. Instantaneous plots of discrete vortices and various statistics up to the second-order moment are obtained to see the variation of coherent structures. Results show that the number of merging vortices and thus the growth of a mixing layer can be effectively controlled by double-frequency forcing if forcing frequencies, phase shifts and forcing amplitudes are suitably selected.

Long nonlinear waves in the two-layer flow of an inviscid, incompressible fluid are considered. Both the free surface and the interface of the two fluids are unknown free boundaries. The flow is forced by an obstruction on the bottom and/or an external pressure (usually called the wind stress) on the free surface. Away from the critical depth ratio between the surface layer and the internal layer of the fluids, the weak nonlinearity is of second order. Then there exists a balance between the dispersion and the second-order nonlinearity. The first-order asymptotic approximations of both free-surface elevation and interface elevation satisfy forced Korteweg-de Vries equations (fKdV). Because of the existence of two modes, the total number of types of solutions is double that for the single-layer flow. There are two hydraulic falls and four solitary waves for positive forcing. The first hydraulic fall and the first two solitary waves correspond to the fast mode. The remaining ones correspond to the slow mode. The first hydraulic fall was numerically found by Forbes (1989) by directly integrating the Laplace equation with nonlinear boundary conditions. There are two types of forcing according to the length of the base of the obstruction. One type is called local and the length of its base has the same scale as the height of the forcing. The expression of the forcing in the fKdV is given by the Dirac delta function. Based upon our long-wave scale the horizontal laboratory lengthscale is shrunk by ε½, while the vertical laboratory lengthscale is unchanged. Hence the semicircular bumps in the papers by Vanden-Broeck (1987), and Forbes (1988, 1989) are all considered to be local in the present paper. For the locally forced cases, stationary problems have been solved analytically. We have derived analytical expressions for the upstream speeds UL and Uc, at which the hydraulic falls can occur and solitary waves respectively cease to exist. The formulae are reduced to those due to Miles (1986) in the case of a single-layer flow and very small forcing. A full comparison among the asymptotic, computational and experimental results is provided. The comparison shows that the difference is less than 10% for UL and Uc in most of the parameter range where the asymptotic method is valid, the computational scheme converges and experiments were conducted (see figure 10). The second type of forcing is called non-local and the length of the base of the obstruction has the same scale as the wavelength. The existence and behaviour of the stationary solutions of the non-locally forced fKdV are described. Surprisingly, there can be more than four solitary waves sustained on the site of some negative forcing.

Unbounded incompressible fluid in solid-body rotation is subjected to spatially uniform strain rates that are sinusoidal in time and of arbitrarily large amplitude. The exact governing equations for the evolution of plane-wave disturbances to this time-periodic flow are linear, as for related steady flows. Attention focuses mainly on the in viscid problem, since incorporation of viscosity is straightforward.

Plane-wave disturbances to axisymmetric flows are governed by a Hill's equation, or equivalently, a pair of first-order equations, to which Floquet theory is applied. Analytical and computational results show several instability bands, the first few of which can exhibit large growth rates. The exact governing equations for plane-wave disturbances to non-axisymmetric flows are similarly derived; but, as these are not singly periodic, results are given only for small-amplitude periodic forcing. As the non-axisymmetric strain produces a periodic elliptical distortion of the flow, a modified elliptical-instability mechanism joins that present in axisymmetric cases.

Despite necessary idealizations, the analysis and results shed light on the stability of periodically strained vortices in a turbulent environment and in geophysical contexts.

Experiments have been carried out to quantify the mixing induced by the interaction of a weak shock wave with a cylindrical volume of a gas (helium) that is lighter than its surroundings (air). In these experiments a round laminar jet was used to produce the light-gas cylinder, and planar laser-induced fluorescence (PLIF), utilizing a fluorescent tracer (biacetyl) mixed with the helium, was used to visualize the flow. These techniques provide a higher quality of flow visualization than that obtained in previous investigations. In addition, the PLIF technique could be used for the measurement of species concentration. The distortion of the helium cylinder produced by the passing shock wave was found to be similar to that displayed by images from previous experimental and computational investigations. The downstream displacement of several points on the boundary of the light-gas cylinder are measured and agree reasonably well with the results of earlier experimental and theoretical studies as well. Because the mixing process causes the helium originally contained within the cylinder to be dispersed into the surrounding air, the PLIF image area inside the contour at one half the maximum concentration of the fluorescent tracer decreases as the two gases mixed. The change in this area is used as a measure of the mixing rate, and it is found that the time rate of change of this area divided by the area of the initial jet is approximately −0.7 × 103 s−1.

A higher harmonic resonance with wavenumber ratio of 1:3 is found to take place in Rayleigh-Bénard convection under rigid-rigid boundary conditions. Bifurcation diagrams for two-dimensional motion are obtained for various values of the Prandtl number P. It is found that a pure mode and mixed mode solutions exist as nonlinear equilibrium states of primary roll solutions for relatively high-Prandtl-number fluids (P ≥ 0.13) while the pure mode, mixed modes, travelling wave and modulated wave solutions exist for relatively low-Prandtl-number fluids (P ≤ 0.12).

The spatial distribution of the mean velocity in a two-dimensional turbulent wall jet was measured for a variety of nozzle Reynolds numbers. It was determined that the bulk of the flow is self-similar and it depends on the momentum flux at the nozzle and on the viscosity and density of the fluid. The width of the nozzle which was commonly used to reduce these data has no part in the similarity considerations as has already been suggested by Narasimha et al. (1973). This type of self-similarity can be easily applied to determine the skin friction, which can otherwise only be determined with considerable difficulty. It was also shown that the ‘law of the wall’ applies only to the viscous sublayer. The Reynolds stress in the inviscid, inner portion of the flow is not constant thus the assumption of a ‘constant stress layer’ is not applicable. The applicability and universality of the ‘outer scaling law’ (i.e. Coles’ law of the wake) has been verified throughout the inviscid inner portion of the wall jet. The logarithmic velocity distribution cannot be derived by making the usual assumptions based on the constancy of the Reynolds stresses or on the thinness of the logarithmic region relative to the thickness of the inner layer.