A direct numerical simulation of a supersonic turbulent boundary layer has been performed. We take advantage of a technique developed by Spalart for incompressible flow. In this technique, it is assumed that the boundary layer grows so slowly in the streamwise direction that the turbulence can be treated as approximately homogeneous in this direction. The slow growth is accounted for by a coordinate transformation and a multiple-scale analysis. The result is a modified system of equations, in which the flow is homogeneous in both the streamwise and spanwise directions, and which represents the state of the boundary layer at a given streamwise location. The equations are solved using a mixed Fourier and B-spline Galerkin method.

Results are presented for a case having an adiabatic wall, a Mach number of M = 2.5, and a Reynolds number, based on momentum integral thickness and wall viscosity, of Reθ′ = 849. The Reynolds number based on momentum integral thickness and free-stream viscosity is Reθ = 1577. The results indicate that the Van Driest transformed velocity satisfies the incompressible scalings and a small logarithmic region is obtained. Both turbulence intensities and the Reynolds shear stress compare well with the incompressible simulations of Spalart when scaled by mean density. Pressure fluctuations are higher than in incompressible flow. Morkovin's prediction that streamwise velocity and temperature fluctuations should be anti-correlated, which happens to be supported by compressible experiments, does not hold in the simulation. Instead, a relationship is found between the rates of turbulent heat and momentum transfer. The turbulent kinetic energy budget is computed and compared with the budgets from Spalart's incompressible simulations.

A new shear-layer growth-rate compressibility-scaling parameter is proposed as an alternative to the total convective Mach number, Mc. This parameter derives from considerations of compressibility as a means of kinetic-to-thermal-energy conversion and can be significantly different from Mc for flows with far-from-unity free-stream-density and speed-of-sound ratios. Experimentally observed growth rates are well-represented by the new scaling.

A spherically-symmetric problem is considered in which a small gas bubble at the centre of a spherical flask filled with a compressible liquid is excited by small radial displacements of the flask wall. The bubble may be compressed, expanded and made to undergo periodic radial oscillations. Two asymptotic solutions have been found for the low-Mach-number stage. The first one is an asymptotic solution for the field far from the bubble, and it corresponds to the linear wave equation. The second one is an asymptotic solution for the field near the bubble, which corresponds to the Rayleigh–Plesset equation for an incompressible fluid. For the analytical solution of the low-Mach-number regime, matching of these asymptotic solutions is done, yielding a generalization of the Rayleigh–Plesset equation. This generalization takes into account liquid compressibility and includes ordinary differential equations (one of which is similar to the well-known Herring equation) and a difference equation with both lagging and leading time. These asymptotic solutions are used as boundary conditions for bubble implosion using numerical codes which are based on partial differential conservation equations. Both inverse and direct problems are considered in this study. The inverse problem is when the bubble radial motion is given and the evolution of the flask wall pressure and velocity is to be calculated. The inverse solution is important if one is to achieve superhigh gas temperatures using non-periodic forcing (Nigmatulin et al. 1996). In contrast, the direct problem is when the evolution of the flask wall pressure or velocity is given, and one wants to calculate the evolution of the bubble radius. Linear and nonlinear periodic bubble oscillations are analysed analytically. Nonlinear resonant and near-resonant periodic solutions for the bubble non-harmonic oscillations, which are excited by harmonic pressure oscillations on the flask wall, are obtained. The applicability of this approach bubble oscillations in experiments on single-bubble sonoluminescence is discussed.

First- and higher-harmonic wave loads on a vertical circular cylinder are investigated experimentally in a wave tank of small scale. The incoming waves are (periodic) Stokes waves with wave slope up to 0.24. A large set of waves which are long compared to the cylinder radius is calibrated. The first seven harmonic components of the measured horizontal force on the cylinder are analysed. The higher-harmonic forces are significantly smaller than the first-harmonic force for all wave parameters. The measurements are continued until the wave amplitude is comparable to the cylinder radius, where the second-, third- and fourth-harmonic forces become of comparable size. Comparison with existing perturbation and fully nonlinear models shows, with a few exceptions, an overall good agreement for small and moderate wave amplitude. A fully nonlinear model agrees with the experiments even up to the seventh-harmonic force for part of the amplitude range. For the large amplitudes the models mostly give conservative predictions. It is important that the distance from the wave maker to the cylinder is large in order to avoid parasitic effects in the incoming wave field. The limited width of the wave tank is not important to the results except when close to resonance.

A general asymptotic formulation is presented for diffusion flames of large-activation-energy chemical reactions. In this limit chemical reaction is confined to a thin zone which, when viewed from the much larger diffusion scale, is a moving two-dimensional sheet. The formulation is not restricted to any particular configuration, and applies to conditions extending from complete combustion to extinction. The detailed structure of the reaction zone yields jump conditions that permit full determination of the combustion field on both sides of the reaction zone, as well as the instantaneous shape of the reaction sheet itself. The simplified system is subsequently used to study the intrinsic stability properties of diffusion flames and, in particular, the onset of cellular flames. We show that cellular diffusion flames form under near-extinction conditions when the reactant in the feed stream is the more completely consumed reactant, and the corresponding reactant Lewis number is below some critical value. Cell sizes at the onset of instability are on the order of the diffusion length. Predicted cell sizes and conditions for instability are therefore both comparable with experimental observations. Finally, we provide stability curves in the fuel and oxidant Lewis number parameter plane, showing where instability is expected for different values of both the initial mixture strength and the Damköhler number.

In this paper the coherent structure in the similarity region of the turbulent planar jet is examined experimentally by application of the proper orthogonal decomposition (POD). In particular, twin cross-stream rakes of X-wire probes are used to take cross-spectral measurements with different spanwise separations between the rakes and at several locations throughout the similarity region. The resulting POD spatial eigenfunctions for each of the three velocity components depend on cross-stream spatial coordinate, Strouhal number, and spanwise wavenumber. Corresponding eigenvalue distributions are obtained in Strouhal number–spanwise wavenumber space. Eigenvalue convergence is demonstrated to be rapid. When properly scaled the eigenfunctions and eigenvalues are shown to exhibit self-similarity though the streamwise location at which this commences depends on the particular velocity component. The results suggest that the flow supports a planar structure aligned in the spanwise direction as well as an essentially three-dimensional structure with asymmetrical shape in the cross-stream direction and pseudo-periodically distributed in the spanwise direction. Comparison of the single- and dual-rake implementations of the POD presented in this paper demonstrate that measurements confined to a single plane are incapable of properly extracting the planar modes. Rather, the single-rake implementation results in modes that appear to be a weighted sum of modes corresponding to different spanwise wavenumbers.

A unified view is given of the instabilities that may develop in two-layer Couette flows, as a ‘phase diagram’ in the parameter space. This view is obtained from a preliminary study of the single-fluid Couette flow over a wavy bottom, which reveals three flow regimes for the disturbances created at the bottom, each regime being characterized by a typical penetration depth of the vorticity disturbances and an effective Reynolds number. It appears that the two-layer flow exhibits the same flow regimes for the disturbances induced by the perturbed interface, and that each type of instability can be associated with a flow regime. Typical curves giving the growth rate versus wavenumber are deduced from this analysis, and favourably compared with the existing literature. In the second part of this study, we propose a mechanism for the long wavelength instability, and provide simple estimates of the wave velocity and growth rate, for channel flows and for semi-bounded flows. In particular, an explanation is given for the ‘thin-layer effect’, which is typical of multi-layer flows such as pressure driven flows or gravity driven flows, and according to which the flow is stable if the thinner layer is the less viscous, and unstable otherwise.

We propose a two-dimensional model of three-dimensional Rayleigh–Bénard convection in the limit of very high Prandtl number and Rayleigh number, as in the Earth's mantle. The model equation describes the evolution of the first moment of the temperature anomaly in the thermal boundary layer, which is assumed thin with respect to the scale of motion. This two-dimensional field is transported by the velocity that it induces and is amplified by surface divergence. This model explains the emergence of thermal plumes, which arise as finite-time singularities. We determine critical exponents for these singularities. Using a smoothing method we go beyond the singularity and reach a stage of developed convection. We describe a process of plume merging, leaving room for the birth of new instabilities. The heat flow at the surface predicted by our two-dimensional model is found to be in good agreement with available data.

Vortex sound is revisited for two-dimensional compact jets. The interactions between the flow and the rigid boundaries of the nozzle are included. Jets with steady upstream conditions and with imposed forcing are considered. A multipole expansion for the acoustic radiation is derived by the method of matched asymptotic expansion. The processes are carried out to higher order to obtain the quadrupole impingement sound in addition to the leading monopole. The emitted sound is found to be related to the vortical flow motion and the scattering of the incoming forcing from the nozzle interior. The flow motion is simulated incompressibly using a point vortex model. The discretization of the continuous vortex shedding is carefully treated to fulfil the Kutta condition and to eliminate spurious sound radiation. Numerical calculations are given for both free and forced jets. For small forcing, the leading monopole is weak and, under some circumstances, the quadrupole sound can be dominant.

The dynamics of thermocapillary flows in differentially heated cylindrical liquid bridges is investigated numerically using a mixed finite volume/pseudo-spectral method to solve the Navier–Stokes equations in the Boussinesq approximation. For large Prandtl numbers (Pr = 4 and 7) and sufficiently high Reynolds numbers, the axisymmetric basic flow is unstable to three-dimensional hydrothermal waves. Finite-amplitude azimuthally standing waves are found to decay to travelling waves. Close to the critical Reynolds number, the former may persist for long times. Representative results are explained by computing the coefficients in the Ginzburg–Landau equations for the nonlinear evolution of these waves for a specific set of parameters. We investigate the nonlinear phenomena characteristic of standing and pure travelling waves, including azimuthal mean flow and time-dependent convective heat transport. For Pr [Lt ] 1 the first transition from the two-dimensional basic flow to the three-dimensional stationary flow is inertial in nature. Particular attention is paid to the secondary transition leading to oscillatory three-dimensional flow, and this mechanism is likewise independent of Pr. The spatial and temporal structure of the perturbation flow is analysed in detail and an instability mechanism is proposed based on energy balance calculations and the vorticity distribution.

The linear stability of a circular vortex interacting with two plane acoustic waves propagating in opposite directions is investigated. When the wavelength is large compared to the size of the vortex, the core is subjected to time-periodic compressions and strains. A stability analysis is performed with the geometrical optics approximation, which considers short-wavelength perturbations evolving along the trajectories of the basic flow. On the vortex core, the problem is reduced to a single Hill–Schrödinger equation with periodic or almost-periodic potential, the solution to which grows exponentially when parametric resonances occur. On interacting with the acoustic waves, the circular vortex is thus unstable to three-dimensional perturbations.

In this paper we perform high-resolution one-dimensional time-dependent numerical simulations of detonations for which the underlying steady planar waves are of the pathological type. Pathological detonations are possible when there are endothermic or dissipative effects in the system. We consider a system with two consecutive irreversible reactions A→B→C, with an Arrhenius form of the reaction rates and the second reaction endothermic. The self-sustaining steady planar detonation then travels at the minimum speed, which is faster than the Chapman–Jouguet speed, and has an internal frozen sonic point at which the thermicity vanishes. The flow downstream of this sonic point is supersonic if the detonation is unsupported or subsonic if the detonation is supported, the two cases having very different detonation wave structures. We compare and contrast the long-time nonlinear behaviour of the unsupported and supported pathological detonations. We show that the stability of the supported and unsupported steady waves can be quite different, even near the stability boundary. Indeed, the unsupported detonation can easily fail while the supported wave propagates as a pulsating detonation. We also consider overdriven detonations for the system. We show that, in agreement with a linear stability analysis, the stability of the steady wave is very sensitive to the degree of overdrive near the pathological detonation speed, and that increasing the overdrive can destabilize the wave, in contrast to systems where the self-sustaining wave is the Chapman–Jouguet detonation.

Experiments were carried out to investigate the drop formation behaviour of a hydrophobic hydrate-forming liquid, HCFC-141b (CH3CCl2F), at a single nozzle in a water stream under hydrate-formable thermodynamic conditions. Attention was focused on the relation between the clathrate-hydrate formation and the drop formation. It was observed that two discrete hydrate crusts grow along the liquid–liquid interface; one forms a frontal cap and the other forms a cylindrical root on each growing drop before its detachment from the nozzle. Most of the latter crust remains at the tip of the nozzle after the detachment of the drop so that it grows into a bell-shaped or nearly cylindrical funnel composed of hydrate deposits in the course of successive growth/detachment of drops. The size of these drops is dependent on the instantaneous diameter of the hydrate-funnel tip rather than the diameter of the nozzle itself. Thus, the size of the drops successively released into the water stream generally varies synchronously with quasi-periodical alternation of growth and breaking of the hydrate funnels. The growth and breaking of the hydrate funnels and the resultant drop-size variation are significantly dependent on the system temperature (or the system subcooling from the liquid/liquid/hydrate equilibrium temperature), the nozzle diameter, and the velocity of the drop-forming liquid through the nozzle.