The paper is devoted to an investigation of convective turbulence. A simplified approach is used for this purpose. It considers an isolated turbulent pulsation as the eigensolution to the corresponding equations of thermohydrodynamics. Turbulence is generated by nonlinear interaction of pulsations: not all interactions, but only the most probable of them are investigated. It is assumed that during convection these are interactions of cells located along the gravity vector, i.e. lying in a vertical line, and lateral interaction of the cells is ignored. This assumption allows one to consider the process of the evolution and interaction of cells as axially symmetric. It is also assumed that the vertical scales of convective cells are larger than their horizontal scales. Therefore, the Boussinesq equations simplified in accordance with the theory of vertical boundary layers can be used. The fact that buoyancy forces, in addition to diffusion, influence the increase of the vertical scales, serves as a basis for this assumption. These assumptions make it possible to obtain the analytical and numerical–analytical solutions, which qualitatively describe the evolution and interaction of convective cells of two essentially different scales: (i) centimetre-scale convective pulsations and (ii) thermals and convective clouds, and to reduce the problem to the solution of nonlinear equations (equations of the Burgers type). Two opposite tendencies are revealed, manifested in the interaction of convective cells. First, there is coagulation of cells and fine nonlinear effects associated with it, which are known from observations and supported by the theory. Secondly, there is destruction of a strong rising cell through its collision with a weak descending ‘cold’ cell. It is assumed that the destruction of cells corresponds to the absence of solutions, when some parameters reach their critical values. A numerical solution to a more accurate problem without simplifications of the vertical boundary layer serves as a basis for this hypothesis. It shows that at critical values of the parameters the process of ‘wave turnover’ begins. It is accompanied by entrainment of the motions of the cold surrounding air into a system of convection and fast dissipation of a cell. In the simplified model, this dissipation is considered to be instantaneous and is called destruction. When the cells are sufficiently strong vertically, weak random fluctuations in the fields of meteorological elements cause their destruction. These results make it possible to propose a hypothesis which relates the degree of instability of cells with the probability of their existence, and to construct functions of cell distributions.

The transition to turbulence of the flow around a circular cylinder is studied by a three-dimensional numerical simulation of the Navier–Stokes equations system in the Reynolds number range 100–300. The numerical method is second-order accurate in space and time and Neumann boundary conditions are used at the two boundaries in the spanwise direction; non-reflecting boundary conditions are specified for the outlet downstream boundary. This study predicts the frequency modulation and the formation of a discontinuity region delimited by two frequency steps within the present Reynolds number range. These features are related to the birth of streamwise vorticity and to the kinetic energy distribution in the near wake. The development of the mean dynamic quantities, the Reynolds stress correlations and the variation of their maximum values are provided in this region, where the similarity laws do not hold. The spatial evolution of the von Kármán mode and of its spectral amplitude are quantified and the variation laws of the maximum spectral amplitude and of its location as a function of Reynolds number are established. The critical Reynolds number for the appearance of the first discontinuity in the present flow system is evaluated by the fully nonlinear approach.

The damping rates, natural frequencies and amplitudes of parametrically excited, standing, water waves in a partially filled, right circular cylinder are measured and compared to existing theoretical models that assume wave slopes are small. The water surfaces were covered by insoluble monomolecular (surfactant) films of oleyl alcohol, lecithin, diolein, cholesterol, and arachidyl alcohol whose concentrations were varied from zero (clean) to saturation; wave slopes were varied from about 0.1 to 1.2. Measured damping rates increased with increasing film concentration as predicted using films of oleyl alcohol, lecithin, and diolein, even when wave slopes were about one. Measured damping rates increased with increasing film concentration as predicted, using films of cholesterol and arachidyl alcohol when wave slopes were small, but not when wave slopes were large. In fact, the measured damping rates for large-slope waves on these films were equivalent to those of waves on a clean surface. Measured natural frequencies varied as predicted for all films, but were about 5% larger. Contact-line effects are incorporated, using an empirical value for contact-line speed, to account for discrepancies between measurements and predictions of damping rates and natural frequencies. Measured steady-state amplitudes agreed well with predictions that used measured damping rates and natural frequencies in the calculations for all films except lecithin and arachidyl alcohol for which there was significant disagreement.

When droplets are expelled at a high velocity by a spray, a strong vertical air jet is induced throughout which the smallest droplets are dispersed (their Reynolds numbers associated with their relative motion being small). In our analysis we focus on the interaction between an external cross-flow and this spray jet. This interaction and the distances by which the spray jet and, over a longer distance, the large droplets are deflected are found to depend largely on the ratio of the cross-wind speed to the induced air speed U0/Uj. Using a multi-zone analysis we show that with a weak cross-flow (U0/Uj[les ]0.1), in the region immediately below the nozzle the spray entrains the external cross-flow and acts like a line sink; the streamlines close to the spray curve inwards to the centre, while further away the sink flow is weak and the streamlines follow the cross-wind. The external flow stagnates at a certain distance from the spray centreline which depends on U0/Uj. When U0/Uj[ges ]0.1 the cross-section of the spray jet and its velocity distribution change in the same way as a fluid jet in a cross-flow, whose inertia causes the deflection of the external flow around it and whose surface vorticity causes a pair of axial vortices on the downwind side of the spray. These vortices have a significant effect on the spray because they induce a back flow which reduces the tendency of the small droplets to leave the spray. When the cross-wind is strong (U0/Uj>0.3; U0[ges ]10 m s−1) the flow is too strong to be entrained; in this limit the main effect of the larger spray droplets is simply to resist the cross-flow which causes the cross-flow to slow down as it passes through the spray and to divert some of the cross-flow around the spray jet. Since the cross-flow now passes through the spray it carries the smallest droplets downwind.

In this paper analytical models have been developed for all the practical ranges of the ratio of the jet speed to the cross-wind speed. This enables spray drift to be calculated. These models require very little computer time and can be run interactively. Spray droplet trajectories can be plotted straightforwardly for both axisymmetric and flat-fan sprays.

The linear stability of incompressible boundary-layer flow of dusty gas on a semi-infinite flat plate is considered. The particles are assumed to be under the action of the Stokes drag only. The problem is reduced to the solution of the modified Orr–Sommerfeld equation (Saffman 1962). This is solved numerically using two approaches: directly by orthonormalization method, and by perturbation method at small particle mass content. The stability characteristics are calculated for both mono- and polydisperse particles.

The dust suppresses the instability waves for a wide range of the particle size. The most efficient suppression takes place when the relaxation length of the particle velocity is close to the wavelength of the Tollmien–Schlichting (TS) wave. The reduction in growth rate per unit dust content is approximately ten times greater than the characteristic value of the growth rate for a clean gas.

For monosized dust the complex frequency of the TS wave changes in a discontinuous way. As a result a domain in the space of independent parameters arises where two discrete TS modes exist and a domain where no TS mode may exist. For polydisperse dust with a discrete distribution in particle size the number of breaks in the dependence equals the number of particle sizes. For the continuous distribution in particle size the dependence of the complex-frequency on Reynolds number and wavenumber is continuous. The eigenfunction becomes a non-smooth function of the normal coordinate in this case.

Some comments are made about the role of the lift force acting on the particles for the problem in question.

Recent work on single-bubble sonoluminescence (SBSL) has shown that many features of this phenomenon, especially the dependence of SBSL intensity and stability on experimental parameters, can be explained within a hydrodynamic approach. More specifically, many important properties can be derived from an analysis of bubble wall dynamics. This dynamics is conveniently described by the Rayleigh–Plesset (RP) equation. Here we derive analytical approximations for RP dynamics and subsequent analytical laws for parameter dependences. These results include (i) an expression for the onset threshold of SL, (ii) an analytical explanation of the transition from diffusively unstable to stable equilibria for the bubble ambient radius (unstable and stable sonoluminescence), and (iii) a detailed understanding of the resonance structure of the RP equation. It is found that the threshold for SL emission is shifted to larger bubble radii and larger driving pressures if surface tension is increased, whereas even a considerable change in liquid viscosity leaves this threshold virtually unaltered. As an enhanced viscosity stabilizes the bubbles to surface oscillations, we conclude that the ideal liquid for violently collapsing, surface-stable SL bubbles should have small surface tension and large viscosity, although too large viscosity (ηl[ges ]40ηwater) will again preclude collapses.

The significance of drop-fluid viscosity on the effective rheological properties and on the dynamics of the microstructure of mono-disperse suspensions of two-dimensional liquid drops with constant interfacial tension is investigated by means of numerical simulations at vanishing Reynolds number, using the boundary integral method for Stokes flow. Three important features of the numerical method are the computation of the doubly-periodic Green's function and associated stress tensor by tabulation and interpolation, the iterative solution of a deflated integral equation for the interfacial velocity, and the repositioning of the drop interfaces at close proximity to avoid artificial coalescence. In the first part of the simulations, the interaction of two intercepting drops in simple shear flow is studied in an extended range of conditions, and the results are used to quantify the behaviour and develop insights into the physics of dilute systems. In the second part of the simulations, the motion of a random suspension of 25 drops repeated periodically in the two spatial directions is studied at the areal fraction ϕ=0.30, drop fluid to ambient fluid viscosity ratio λ=1 or 10, and drop capillary number Ca=0.10 or 0.30, a total of four combinations. It is found that the rheological properties of the suspension and the average drop deformation and orientation depend on the values of λ and Ca in a subtle fashion. As the viscosity of the drops is raised, the drop-centre pair distribution function undergoes a transition from a liquid-like to a rigid-particle-like behaviour, and particle aggregation and cluster formation become more important. For λ=10, the results are in excellent qualitative, and in some cases quantitative, agreement with those presented in previous studies for mono-layered suspensions of rigid spheres. The drop self-diffusivity is computed and its dependence on λ and Ca is discussed, although the results carry some uncertainty owing to the moderate number of drops within each periodic cell.

A semi-analytical method has been developed to solve for the inviscid incompressible flow induced by a heavily loaded actuator disk with non-uniform loading. The solution takes the contraction of the slipstream fully into account. The method is an extension of the analytical theory of Conway (1995) for the linearized actuator disk and is exact for an incompressible perfect fluid. The solutions for the velocities and stream function are given as one-dimensional integrals of expressions containing complete elliptic integrals. Any load distribution with bounded radial gradient can be treated. Results are presented here for both contra-rotating and normal propellers. For the special case of a contra-rotating propeller with a parabolic velocity profile in the ultimate wake, the vorticity in the slipstream is shown to be the same as in the analytically tractable spherical vortex of Hill (1894) and the related family of steady vortices explored by Fraenkel (1970, 1972) and Norbury (1973).

Streak breakdown caused by a spanwise inflectional instability is one phase of the following transition scenarios, which occur in plane Poiseuille and Couette flow. The streamwise vortex scenario is described by

formula here

The oblique wave scenario is described by

formula here

The purpose of this paper is to investigate the streak breakdown phase of the above scenarios by a linear stability analysis and compare threshold energies for transition for the above scenarios with those for transition initiated by Tollmien-Schlichting waves (TS), two-dimensional optimals (2DOPT), and random noise (N) at subcritical Reynolds numbers.

We find that if instability occurs, it is confined to disturbances with streamwise wavenumbers α0 satisfying 0<αmin< [mid ]α0[mid ]<αmax. In these unstable cases, the least stable mode is located near the centre of the channel with a phase velocity approximately equal to the centreline velocity of the mean flow. Growth rates for instability increase with streak amplitude. For Couette flow streak breakdown is inhibited by mean shear. Using the linear stability analysis, we determine lower bounds on threshold amplitude for transition for scenario (SV) that are consistent with thresholds determined by direct numerical simulations.

In the final part of the paper we show that the threshold energies for transition in Poiseuille flow at subcritical Reynolds numbers for scenarios (SV) and (OW) are two orders of magnitude lower than the threshold for transition initiated by Tollmien–Schlichting waves (TS) and an order of magnitude lower than that for (2DOPT). Scenarios (SV) and (OW) occur on a viscous time scale. However, even when transition times are taken into account, the threshold energy required for transition at a given time for (SV) and (OW) is lower than that for the (TS) and (2DOPT) scenarios at Reynolds number 1500.

Optimal and robust control theories are used to determine effective, estimator-based feedback control rules for laminar plane channel flows that effectively stabilize linearly unstable flow perturbations at Re=10 000 and linearly stable flow perturbations, characterized by mechanisms for very large disturbance amplification, at Re=5000. Wall transpiration (unsteady blowing/suction) with zero net mass flux is used as the control, and the flow measurement is derived from the wall skin friction. The control objective, beyond simply stabilizing any unstable eigenvalues (which is relatively easy to accomplish), is to minimize the energy of the flow perturbations created by external disturbance forcing. This is important because, when mechanisms for large disturbance amplification are present, small-amplitude external disturbance forcing may excite flow perturbations with sufficiently large amplitude to induce nonlinear flow instability.

The control algorithms used in the present work account for system disturbances and measurement noise in a rigorous fashion by application of modern linear control techniques to the discretized linear stability problem. The disturbances are accounted for both as uncorrelated white Gaussian processes ([Hscr ]2 or ‘optimal’ control) and as finite ‘worst case’ inputs which are maximally detrimental to the control objective ([Hscr ]∞ or ‘robust’ control). Root loci and transient energy growth analyses are shown to be inadequate measures to characterize overall system performance. Instead, appropriately defined transfer function norms are used to characterize all systems considered in a consistent and relevant manner. In order to make a parametric study tractable in this high-dimensional system, a convenient new scaling to the estimation problem is introduced such that three scalar parameters {γ, α, [lscr ]} may be individually adjusted to achieve desired closed-loop characteristics of the resulting systems. These scalar parameters may be intuitively explained, and are defined such that the resulting control equations retain the natural dual structure between the control parameter, [lscr ], and the estimation parameter, α. The performance of the present systems with respect to these parameters is thoroughly investigated, and comparisons are made to simple proportional schemes where appropriate.

Aspects of the solution to the linearized water-wave problem involving a pair of surface-piercing cylinders in oblique waves and infinite water depth are examined. In particular, the solution is proved to be unique for certain geometrical arrangements and wave parameters depending on whether the wave frequency is above or below the cut-off frequency. Outside these regions of uniqueness are constructed examples of non-uniqueness using ideas developed in McIver (1996) in the normal incidence case. Although non-uniqueness examples are obtained numerically, we are able to prove the existence of non-uniqueness under the assumption that the wave obliqueness is small.

The stability of uniform straining flow in a semi-infinite body of viscous fluid subjected to surface cooling is examined. The viscosity of the fluid is assumed to be a prescribed function of temperature. If the viscosity variations caused by the cooling are sufficiently large the straining flow is linearly unstable to a mode in which the rate of extension of the viscous thermal boundary layer becomes localized. The parameters of the problem are the viscosity contrast in the fluid and a dimensionless measure of the rate of strain relative to the rate of cooling. The conditions under which instability occurs are determined and the physical mechanisms responsible are examined. The results are applied to discuss the formation of some surface features in lava flows.