A detailed numerical study has been conducted on the effect of unsteadiness on the dynamics of counterflowing strained diffusion methane/oxygen/nitrogen flames. The modelling included the solution of the unsteady conservation equations of mass, momentum, energy, and species along the stagnation streamline in an opposed jet using detailed descriptions of chemistry and transport. The unsteadiness was introduced by independently imposing sinusoidal variations of the reactant velocity, concentration and temperature at the exits of the nozzles. The results demonstrate that the flame's response is quasi-steady at low frequencies, while at higher frequencies the amplitudes of the induced oscillations are reduced and phase shifted with respect to the imposed signal. At still higher frequencies, the flame no longer responds to the oscillations in the external field. A rigorous physical explanation of the frequency response was provided from first principles by identifying that oscillations imposed at the nozzle exits result in reactant concentration and temperature oscillations at the outer edge of the preflame diffusive zones. The diffusion attenuates the oscillations in this zone in a manner analogous to the velocity attenuation in Stokes’ second problem. The validity of the analogy was confirmed by examining flames over a wide range of frequencies and initial conditions. The current analysis also provided a criterion for the cutoff frequency separating the quasi-steady and transient regimes – information which can be useful in the establishment of laminar flamelet libraries.

Heat and mass transport in the bulk of a suspension or fixed bed in the limit of high Péclet number is controlled by hydrodynamic diffusion. This hydrodynamic diffusion is caused by the stochastic velocity field induced by the randomly distributed particles. However, the no-slip boundary conditions require that the hydrodynamic diffusivity vanish at boundaries of the medium. Thus, molecular diffusion must aid the transport in a thin boundary layer near the solid boundary. Quantitative results are derived for the hydrodynamic diffusivity, boundary layer thickness, and net resistance to transport across a dilute fixed fibrous bed in the direction transverse to the mean flow. A scaling analysis is presented for transport in the direction of the gradient of the mean velocity in a sheared suspension of neutrally buoyant spheres. This scaling analysis is able to explain the qualitative variations of the transport resistance with the Péclet number and the ratio of the suspension thickness to the particle radius.

A new exact solution of the nonlinear shallow-water equations is presented. The solution corresponds to divergent and non-divergent free oscillations in an infinite straight channel of parabolic cross-section on the rotating Earth. It provides a description of the one-dimensional subclass of shallow-water flows in paraboloidal basins considered by Ball (1964), Thacker (1981), Cushman-Roisin (1987) and others in which the velocity field varies linearly and the free-surface displacement varies quadratically with the spatial coordinates. In contrast to the previous exact solutions describing divergent oscillations in circular and elliptic paraboloidal basins, the oscillation frequency of the divergent oscillation in the parabolic channel is found to depend, in part, on the amplitudes of the relative vorticity and free-surface curvature. This result is consistent with Thacker's (1981) numerical finding that when the free surface in parabolic channel flow is curved, the oscillation frequency depends on the amplitude of the motion. Solutions for parcel trajectories are also presented. The exact solution provides a rare description of a class of nonlinear flows and is potentially valuable as a validation test for numerical shallow-water models in Eulerian and Lagrangian frameworks.

A high-frequency part of the acoustic noise spectrum excited by laser-driven cavitation in liquid is investigated theoretically. It is assumed that the liquid is inviscid and compressible and the surface tension may be neglected. The specific heat ratio is taken to be 5/3. It is shown that, in the first approximation, the spectral density of the acoustic energy emitted by a cavity explosion varies as the -4/7 power of the frequency and asymptotically decreases by ∼ 3.4 dB/octave.

Hamiltonian approximation methods yields approximate dynamical equations that apply to nearly geostrophic flow at scales larger than the internal Rossby deformation radius. These equations incorporate fluid inertia with the same order of accuracy as the semi-geostrophic equations, but are nearly as simple (in appropriate coordinates) as the equations obtained by completely omitting the inertia.

Thermal convection with a continuous finite bandwidth of modes in a porous layer with horizontal walls at different mean temperatures is considered when a spatially non-uniform temperature is prescribed at the lower wall. The nonlinear problem of three-dimensional convection for values of the Rayleigh number close to the classical critical value is solved by using multiple scales and perturbation techniques. The preferred flow solutions are determined by a stability analysis. It is found that for the case of near-resonant wavelength excitation regular or non-regular solutions in the form of superposition of small-scale multi-modal solutions with large-scale multimodal (or non-modal) amplitude can become preferred, provided the wave vectors of the solutions are contained in the set of wave vectors due to the modal form of the boundary imperfections and the form of the large-scale part is the same as that due to the boundary imperfections. For the case of non-resonant wavelength excitation some three-dimensional solutions in the form of superposition of small-scale multi-modal solutions with large-scale multi-modal (or non-modal) amplitudes can be preferred, provided that the wavelength of the small-scale modulation is not too small. Large-scale flow structures are quite different from the small-scale flow structures in a number of cases and, in particular, they can exhibit kinks and can be non-modal in nature. The resulting flow patterns are affected accordingly, and they can provide quite unusual and non-regular three-dimensional preferred patterns. In particular, they are multiples of irregular rectangular patterns, and they can be non-periodic.

We present a detailed numerical study of the interaction of a weak shock wave with an isolated cylindrical gas inhomogeneity. Such interactions have been studied experimentally in an attempt to elucidate the mechanisms whereby shock waves propagating through random media enhance mixing. Our study concentrates on the early phases of the interaction process which are dominated by repeated refractions and reflections of acoustic fronts at the bubble interface. Specifically, we have reproduced two of the experiments performed by Haas & Sturtevant: a Mach 1.22 planar shock wave, moving through air, impinges on a cylindrical bubble which contains either helium or Refrigerant 22. These flows are modelled using the two-dimensional compressible Euler equations for a two-component fluid (air-helium or air–Refrigerant 22). Utilizing a novel shock-capturing scheme in conjunction with a sophisticated mesh refinement algorithm, we have been able to reproduce numerically the intricate mechanisms that were observed experimentally, e.g. transition from regular to irregular refraction, cusp formation and shock wave focusing, multi-shock and Mach shock structures, and jet formation. The level of agreement lends credibility to a number of observations that can be made using information from the simulations for which there is no experimental counterpart. Thus we can now present an updated description for the dynamics of a shock-bubble interaction which goes beyond that provided by the original experiments.

The plane unsteady problem of the deflection of a solid, slightly curved plate in collision with an ideal weakly compressible liquid is considered. In order to describe the impact process, the acoustic approximation and the method of normal modes are used. The analysis is focused on the supersonic stage of the impact when the liquid surface remains undisturbed outside the contact spot between the solid plate and the liquid. However, the positions of the contact points are unknown in advance, in contrast to the case of undeformable body impact, and have to be found together with the liquid flow, the pressure distribution, and the bottom deformations. It was shown that the duration of the supersonic stage depends on the entering body elasticity. A spray jet is formed earlier and the stage at which the liquid compressibility is a governing factor is shorter than under rigid-body impact. It is revealed that the elastic plate deflection is quite small and can be satisfactorily approximated by a few modes. On the other hand, the calculation of the bending stress distribution needs a much greater number of normal modes. The pressure distribution over the contact region is quite difficult to find by the mode method; an alternative approach is suggested.

A study is made of the formation of a nonlinear critical layer (first found by Benney & Bergeron 1969 and Davis 1969) in homogeneous and weakly stratified incompressible shear flows as initially small unstable disturbances develop, whose growth rate is so small that all the evolution proceeds in the ‘quasi-steady’ regime. It is shown that such an evolution can be described from start to finish analytically using a pair of evolution equations for the wave amplitude and phase which involve universal functions of the familiar Haberman (1972) parameter, λ, that characterizes the relative importance of the dissipation and nonlinearity.

In addition to the function of a ‘logarithmic phase jump’ that was introduced and investigated by Haberman (1972), the evolution equations generally also contain other functions of λ. In this paper we introduce and study (numerically and analytically) another three such functions.

Many studies of fragmentation of liquid drops at supersonic Mach numbers report the appearance of large amounts of mist. Photographs from other studies, which do not mention mist at all, strongly suggest that copious amounts of mist are formed at the earliest stages of fragmentation. In this paper, we present arguments and calculations which indicate that this mist is formed from condensed vapour arising from the flash vaporization of the hot and low-pressure liquid on the leeside of the drop. Low leeside pressures are produced by the rarefaction of the gas, the acceleration of the drop, and the high tensions generated by rapid stretching of the stripped liquid. The droplet temperature may rise because of heat transfer from the hot gas to thin drop filaments, and by viscous heating due to rapid deformation.

We develop herein the similarity form of a non-Boussinesq gaseous plume. This is done by obtaining the equation for the conservation of enthalpy flux, and using it and the continuity equation to demonstrate that the flux of density deficiency is conserved, and not the flux of buoyancy as is the case for Boussinesq plumes. We then use this conservation relation to describe the form of the similarity solution in the non-Boussinesq case. The similarity solution is then used to derive the theoretical form of the entrainment velocity across the plume edge, which is seen to be in agreement with the ‘modified entrainment assumption’ suggested empirically from experiments by Ricou & Spalding (1961).

A closed tube containing a BZT-fluid is driven by an applied velocity near and at resonant frequencies. A small-rate theory is shown to predict the existence of stable periodic expansion and compression shocks in a resonant frequency band. The significant effects of wave interaction and thermoviscous damping are demonstrated.

Detailed observations of the air flow velocity, pressure and Reynolds stresses above water waves in a wave flume are presented. The static pressure fluctuations induced by the waves are observed following a new procedure that eliminates acoustical contamination by the wave maker. The measurements are analysed by comparing them with numerical simulations of the air flow over waves. In these numerical simulations the sensitivity to the choice of turbulence closure is studied. We considered both first-order turbulence closure schemes based on the eddy viscosity concept, and a second-order Reynolds stress model. The comparison shows that turbulence closure schemes based on the eddy viscosity concept overestimate the modulation of the Reynolds stress in a significant part of the vertical domain. When an eddy viscosity closure is used, the overestimated modulation of the Reynolds stress gives a significant contribution to the wave growth rate. Our results confirm the conclusions Belcher & Hunt reached on the basis of the rapid distortion theory.

The ratio of the wind speed to the phase speed of the paddle wave in the experiment varies between 3 and 6. The observed amplitudes of the velocity and pressure perturbation are in excellent agreement with the simulations. Comparison of the observed phases of the pressure and velocity perturbations shows that the numerical model underpredicts the downwind phase shift of the undulating flow.

The sheltering coefficients for the flow over hills and the growth rates of waves that are slow compared to the wind calculated with the Reynolds stress model are in excellent agreement with the analytical model of Belcher & Hunt. Extending the calculations to fast waves, we find that the energy flux to waves travelling almost as fast as the wind is increased on going from the mixing length turbulence closure to the Reynolds stress model.

Unsteady turbulence in uniformly stratified unsheared flow is analysed using rapid distortion theory (RDT). For inviscid flow with no molecular diffusion the theory shows how the initial conditions, such as the initial turbulent kinetic energy KE0 and potential energy PE0, determine the partition of energy between the potential energy associated with density fluctuation and the kinetic energy associated with each of the velocity components during the subsequent development of the turbulence. One parameter is an exception to this sensitivity to initial conditions, namely the limit at large time of the ratio of potential energy to vertical kinetic energy. In the linear theory, this ratio depends neither on the Reynolds number Re, nor the Prandtl number Pr nor the Froude number Fr. This is consistent with turbulence measurements in the atmosphere, wind tunnel and water tank experiments, and with large-eddy simulations, where similar values of the ratio are found. The RDT results are extended to show the effects of viscosity and diffusion where Re is not very large, explaining the sensitivity of the spectra and the fluxes to the value of the Prandtl number Pr. When Pr is larger than 1, the high-wavenumber components of the three-dimensional spectra induce a vertical flux of temperature (density) that is positive (negative), and therefore ‘countergradient.’ On the other hand, when the thermal diffusivity is stronger and Pr is less than 1, lower-wavenumber components become countergradient sooner since the high-wavenumber components are prevented from becoming countergradient. When all the wavenumber components are integrated to derive the total vertical density flux, it becomes countergradient more quickly and more strongly in high-Pr than in low-Pr turbulence. All these theoretically derived differences between high-Pr and low-Pr turbulence are consistent with the experimental measurements in water tank and wind tunnel experiments and numerical simulations. It is shown that the initial kinetic and potential energy spectrum forms E(k) and S(k) near k = 0 determine the long-time limit values of the variances and the covariances, including their decay rate with time. In the special case of Pr = 1, the oscillation time period of the three-dimensional spectrum function is independent of the wavenumber and is the same as that of an inviscid fluid with the effect of viscosity/diffusion being limited to the damping of all the wavenumber components in-phase with each other. Furthermore, the non-dimensional ratios of the covariances, including the normalized vertical density flux and the anisotropy tensor, agree with the inviscid results if S(k) is proportional to E(k), or if either S(k) or E(k) is identically zero. However, even when Pr = 1, in the ‘one-dimensional spectrum’ in the x-direction, there is a transitory countergradient flux for high wavenumbers; only in this case is there a qualitative difference with the three-dimensioanl spectrum. This paper shows that the characteristic differences in the behaviour of stably stratified turbulence reported in previous DNS experiments at moderate Reynolds numbers can largely be explained by linear oscillations and simple molecular or eddy diffusion rather than by any new kinds of nonlinear mixing processes.

Experimental results for the statistics of turbulent reactive plumes behind line sources are given. The results are obtained with sufficient spatial and time scale resolution and cover the near and far fields. The results for the conserved scalar, which is constructed from the two reactive scalars, are consistent with those measured in passive thermal plumes. The results for the reactive scalars show that within the experimental range, turbulent mixing is the dominant contributor to the spread rate of the plume and the contribution from chemical reaction is small. However it is found that chemical reaction has a large effect on the decay of the plume reactant concentration and this effect depends on the stoichiometric mixture fraction. The gradient model for turbulent diffusion and the conventional model for the dissipation time scale of scalar variance have been tested and it is found that they are satisfactory in the far field of the plume. However large errors can result in the near field. Also it is found that the turbulent diffusitivities derived from the conserved and reactive scalars are about the same. Various models for the mean chemical reaction rate have been checked and it is concluded that an interpolation between the frozen and equilibrium limits for the covariance of the two reactive scalars will model the mean chemical reaction rate reasonably well.

An analytic formula is derived for the Rayleigh growth rate of a fluid cylinder immersed in a fluid of equal viscosity, and an extension is given for concentric fluid threads.

The work suggests a simple qualitative model of the wind wave ‘horse-shoe’ patterns often seen on the sea surface. The model is aimed at explaining the persistent character of the patterns and their specific asymmetric shape. It is based on the idea that the dominant physical processes are quintet resonant interactions, input due to wind and dissipation, which balance each other. These processes are described at the lowest order in nonlinearity. The consideration is confined to the most essential modes: the central (basic) harmonic and two symmetric oblique satellites, the most rapidly growing ones due to the class II instability. The chosen harmonics are phase locked, i.e. all the waves have equal phase velocities in the direction of the basic wave. This fact along with the symmetry of the satellites ensures the quasi-stationary character of the resulting patterns.

Mathematically the model is a set of three coupled ordinary differential equations for the wave amplitudes. It is derived starting with the integro-differential formulation of water wave equations (Zakharov's equation) modified by taking into account small (of order of quartic nonlinearity) non-conservative effects. In the derivation the symmetry properties of the unperturbed Hamiltonian system were used by taking special canonical transformations, which allow one exactly to reduce the Zakharov equation to the model.

The study of system dynamics is focused on its qualitative aspects. It is shown that if the non-conservative effects are neglected one cannot obtain solutions describing persistent asymmetric patterns, but the presence of small non-conservative effects changes drastically the system dynamics at large times. The main new feature is attractive equilibria, which are essentially distinct from the conservative ones. For the existence of the attractors a balance between nonlinearity and non-conservative effects is necessary. A wide class of initial configurations evolves to the attractors of the system, providing a likely scenario for the emergence of the long-lived three-dimensional wind wave patterns. The resulting structures reproduce all the main features of the experimentally observed horse-shoe patterns. In particular, the model provides the characteristic ‘crescent’ shape of the wave fronts oriented forward and the front-back asymmetry of the wave profiles.