Shallow-water equations with bottom drag and viscosity are used to study the dynamics of roll waves. First, we explore the effect of bottom topography on linear stability of turbulent flow over uneven surfaces. Low-amplitude topography is found to destabilize turbulent roll waves and lower the critical Froude number required for instability. At higher amplitude, the trend reverses and topography stabilizes roll waves. At intermediate topographic amplitude, instability can be created at much lower Froude numbers due to the development of hydraulic jumps in the equilibrium flow. Second, the nonlinear dynamics of the roll waves is explored, with numerical solutions of the shallow-water equations complementing an asymptotic theory relevant near onset. We find that trains of roll waves undergo coarsening due to waves overtaking one another and merging, lengthening the scale of the pattern. Unlike previous investigations, we find that coarsening does not always continue to its ultimate conclusion (a single roll wave with the largest spatial scale). Instead, coarsening becomes interrupted at intermediate scales, creating patterns with preferred wavelengths. We quantify the coarsening dynamics in terms of linear stability of steady roll-wave trains.

High-resolution shock-capturing finite-volume numerical methods are applied to investigate nonlinear geostrophic adjustment of rectilinear fronts and jets in the rotating shallow-water model. Numerical experiments for various jet/front configurations show that for localized initial conditions in the open domain an adjusted state is always attained. This is the case even when the initial potential vorticity (PV) is not positive-definite, the situation where no proof of existence of the adjusted state is available. Adjustment of the vortex, PV-bearing, part of the flow is rapid and is achieved within a couple of inertial periods. However, the PV-less low-energy quasi-inertial oscillations remain for a long time in the vicinity of the jet core. It is demonstrated that they represent a long-wave part of the initial perturbation and decay according to the standard dispersion law ${\sim}t^{-1/2}$. For geostrophic adjustment in a periodic domain, an exact periodic nonlinear wave solution is found to emerge spontaneously during the evolution of wave perturbations allowing us to conjecture that this solution is an attractor. In both cases of adjustment in open and periodic domains, it is shown that shock-formation is ubiquitous. It takes place immediately in the jet core and, thus, plays an important role in fully nonlinear adjustment. Although shocks dissipate energy effectively, the PV distribution is not changed owing to the passage of shocks in the case of strictly rectilinear flows.

Measurements were made of fully developed turbulent pipe flows subjected to temporal pressure gradients larger than those considered previously. Integral profile parameters exhibited similar qualitative trends when time was scaled with the flow rise time and this differed from corresponding spatial development of flows subjected to steady streamwise pressure gradients. Velocity profiles were initially characterized by significant reduction of their wake component. This was accompanied by turbulence intensity freezing in the core region of the pipe and a rapid turbulence increase in the near-wall region that propagated towards the pipe centre. The final phase of the acceleration was characterized by reconstitution of the wake, producing a velocity profile inflection and the generation of turbulence in that vicinity. These features were not observed in previous investigations, possibly due to the present use of higher ramp rates and initial Reynolds numbers.

Three-dimensional advection of passive tracers in non-inertial flows is studied in a finite cylinder confined by two parallel endwalls by means of numerical simulations and laboratory experiments. The fluid is set in motion through steady or time-periodic forcing by in-plane motion of the endwalls via a given forcing protocol. The numerical analysis centres on a dynamical-systems approach and concerns symmetry-based identification of coherent structures in the web of tracer paths (collectively defining the flow topology) for a number of archetypal flow configurations. The role of the flow topology in the process of tracer transport is investigated by numerical tracking of finite-size material objects released at strategic locations in the flow. Experimental validation of key aspects of the numerical results has been carried out in laboratory experiments by flow visualization with dye and flow measurement via three-dimensional particle tracking velocimetry.

While acoustic scatter from oceanic turbulence is sensitive to temperature–salinity covariations, there are unfortunately no published measurements of the turbulent temperature–salinity co-spectrum. Several models have been proposed for the form of the co-spectrum of two scalars in turbulence, but they all produce unsatisfactory results when applied to the turbulent scattering equations (either predicting negative scattering cross-sections in some regimes or predicting implausible levels of correlation between temperature and salinity at some scales). A new model is proposed and shown to give physically plausible scattering predictions in all density regimes. High-frequency acoustic data illustrate the importance of the co-spectrum for acoustic scattering, but were collected in a density regime where there is little difference between the co-spectrum models.

Earlier investigations of steady two-dimensional marginally separated laminar boundary layers have shown that the non-dimensional wall shear (or equivalently the negative non-dimensional perturbation displacement thickness) is governed by a nonlinear integro-differential equation. This equation contains a single controlling parameter $\Gamma$ characterizing, for example, the angle of attack of a slender airfoil and has the important property that (real) solutions exist up to a critical value $\Gamma_c$ of $\Gamma$ only. Here we investigate three-dimensional unsteady perturbations of an incompressible steady two-dimensional marginally separated laminar boundary layer with special emphasis on the flow behaviour near $\Gamma_c$. Specifically, it is shown that the integro–differential equation which governs these disturbances if $\Gamma_c\,{-}\,\Gamma\,{=}\,O(1)$ reduces to a nonlinear partial differential equation – known as the Fisher equation – as $\Gamma$ approaches the critical value $\Gamma_c$. This in turn leads to a significant simplification of the problem allowing, among other things, a systematic study of devices used in boundary-layer control and an analytical investigation of the conditions leading to the formation of finite-time singularities which have been observed in earlier numerical studies of unsteady two-dimensional and three-dimensional flows in the vicinity of a line of symmetry. Also, it is found that it is possible to construct exact solutions which describe waves of constant form travelling in the spanwise direction. These waves may contain singularities which can be interpreted as vortex sheets. The existence of these solutions strongly suggests that solutions of the Fisher equation which lead to finite-time blow-up may be extended beyond the blow-up time, thereby generating moving singularities which can be interpreted as vortical structures qualitatively similar to those emerging in direct numerical simulations of near critical (i.e. transitional) laminar separation bubbles. This is supported by asymptotic analysis.

High resolution direct numerical simulation (DNS) (512×1024×512) and large-eddy simulation (LES) of a shear-free mixing layer are presented. The geometry of the flow consists of two layers with different turbulence intensities that are in contact and interact through a fairly thin mixing layer. This geometry is used to explore the influence of inhomogeneities in the characteristic length scales, times scales and energy scales on the turbulence properties. Comparison of DNS results is made with the Veeravalli & Warhaft (J. Fluid Mech.207, 191–229, 1989) experiment. The LES is performed on a 32×64×32 grid using an eddy-viscosity model. The use of such a model appears to be justified by the very weak departures from isotropy that are observed in the shear-free mixing layer. The LES predictions are compared with the filtered DNS data and show that the eddy viscosity model performs very well in predicting the energy profile as well as the deviation from Gaussianity in the turbulent velocity field statistics.

Experiments were carried out in an axisymmetric turbulent jet from 2 to 6 diameters downstream at exit Reynolds numbers of 78 400, 117 600, and 156 800. Data were collected using the 138 hot-wire probe constructed by Citriniti & George (2000). The proper orthogonal decomposition (POD) was then applied to a double Fourier transform in time and azimuthal direction of the two-point velocity correlation tensor. Azimuthal mode-0, which dominated the dynamics at $x/D\,{=}\,3$ in the previous experiments, dies off rapidly downstream towards a non-zero value. For the higher azimuthal modes, the peak shifts from mode-6 towards lower azimuthal mode numbers with increasing downstream distance, until the peak is at mode-2 by the end of the potential core. The POD eigenspectra collapse in similarity variables for the mixing layer at all downstream positions, and are nearly independent of Reynolds number. Reconstruction of the full-field streamwise velocity component using the dominant POD modes shows clearly the evolution of the flow with downstream position, from ‘volcano-type’ eruptions at 2 to 3 diameters downstream to a ‘propeller-like’ pattern where the number of blades diminishes downstream.

The evolution of the most energetic features in the far-field region of a turbulent axisymmetric jet are investigated by applying the proper orthogonal decomposition (POD) to the double Fourier transform in time and azimuthal direction of the two-point velocity correlations of the streamwise velocity component. Measurements were obtained from 15 to 69 diameters downstream at jet exit Reynolds numbers ranging from 40 000 to 84 700 using two different experimental techniques, an array of 139 long hot wires and an articulated array of 15 hot wires.

The POD eigenspectra for the far jet were found to follow equilibrium similarity scaling sooner than the single-point statistics. The first POD eigenspectrum accounts for more than 50% of the resolved streamwise energy, and has three major peaks: a dominant one at azimuthal mode-2 for near-zero frequency, another at mode-1 at a constant local Strouhal number, $f \delta_{1/2}/U_c$, and a third smaller one for azimuthal mode-0 at near-zero frequency. The eigenspectra were independent of downstream position, but similar to those obtained as early as $x/D\,{=}\,6$ in Part 1 of this paper. All the eigenspectra integrated over frequency peaked at azimuthal mode-2.

A three-dimensional direct numerical simulation (DNS) study of a spatially evolving planar turbulent reacting jet is reported. Combustion of methane with air is modelled using a four-step reduced mechanism in the non-premixed regime. A total of eight chemical species are integrated in time along with the fluid mechanical fields. The solution of the compressible Navier–Stokes equations is obtained numerically for moderately low Mach number. A large computational grid, with 100 million grid points, is required in order to resolve the flame. The cold flow Reynolds number is 3000. The focus of the study is to investigate the dynamics of extinction fronts in three-dimensional turbulent flows. A novel data reduction and identification algorithm was developed to postprocess the large DNS database and extract the shape of the evolving flame surface including its edges and their propagation velocity. The joint probability density function (p.d.f.) of edge velocity and scalar dissipation was obtained and the results indicate that the three-dimensional flame edges propagate with a velocity that is largely controlled by the local rate of scalar dissipation, or equivalently in terms of the local Damköhler number at the flame edge, as predicted by theory. Naturally, the effects of unsteadiness in this flow produce a broad joint p.d.f. The statistics collected also suggest that the mean value of the hydrogen radical reaction rate are very small in the turbulent regions of the flow owing to the functional form of the hydrogen radical reaction rate itself. The consequence of these results in the context of turbulent combustion modelling is discussed. Additional statistical and morphological information of the flame is provided.

Numerical simulations of turbulent polymer solutions using the FENE-P model are used to characterize the action of polymers on turbulence in drag-reduced flows. The energetics of turbulence is investigated by correlating the work done by polymers on the flow with turbulent structures. Polymers are found to store and to release energy to the flow in a well-organized manner. The storage of energy occurs around near-wall vortices as has been anticipated for a long time. Quite unexpectedly, coherent release of energy is observed in the very near-wall region. Large fluctuations of polymer work are shown to re-energize decaying streamwise velocity fluctuations in high-speed streaks just above the viscous sublayer. These distinct behaviours are used to propose an autonomous regeneration cycle of polymer wall turbulence, in the spirit of Jiménez & Pinelli (1999).

Sonic-boom noise penetrating under a deep ocean is affected by its time-dependent interaction with the surface waves, which can significantly influence the perceived sound pressure level and tonal content of the disturbances at depth far greater than expected from the flat-ocean (Sawyers) model. The present theory assumes a small surface slope and a high water-to-air density ratio; the ocean surface in the analysis is modelled by a sinusoidal surface-wave train. The analysis shows that a distinct acoustic wave mode in the form of a packet of wavelets emerges in the sound field far below the surface and attenuates with increasing distance in a manner similar to the cylindrical spreading of monochromatic waves. The latter feature renders the surface waviness influence an effect of first-order importance, overwhelming the primary noise field at large depth. Detailed properties of the deep-water wave fields are examined and illustrated for the case of an incident N-wave, for which an explicit, analytic solution is obtained. The result reveals a similarity structure of the wave field with two distinct time scales and the invariance characteristics of the cylindrically spreading waves, in accord with the group-velocity concept of dispersive waves. An example is given of the interaction, illustrating the underwater waveform, sound-pressure and frequency levels.

The biharmonic streamfunction is naturally employed in the complex-variable formulation of free-boundary transient problems in two-dimensional Stokes flow. In the event that analytical solutions are not obtainable, biharmonic boundary-integral methods (BBIMs) are frequently used. By using the well-known analytical solution of Hopper (J. Fluid Mech. vol. 213, 1990, pp. 349–375) for the Stokes-flow coalescence of two cylinders, it is demonstrated that the widely used direct BBIM formulation admits hypersingular integrals when solving evolving free-surface problems in the presence of a cusp, irrespective of the degree of piecewise-polynomial shape functions used to represent the curvilinear free surface. It is also shown that the hypersingularity which arises in the dynamic free-surface cusp formation is of the same fundamental form as that arising in both the static-singularity driven-cavity problem and the submergence or withdrawal of a solid plate relative to a free surface (Moffatt, J. Fluid Mech. vol. 18, 1964, pp. 1–18). The hypersingular BBIM integrals do not admit regularization, finite-part integration or Gauss–Chebyshev integration in the normal sense: the natural BBIM is fundamentally ill-posed in the presence of singularities born of boundary motion. In such cases, the Almansi representation should be used in order to guarantee accurate numerical solutions.

We investigate the dynamics of a gravity current that propagates along the interface of a two-layer fluid. The results of the well-studied symmetric case are reproduced in which the upper- and lower-layer depth of the ambient are equal and the density of the intrusion is the average density of the ambient. In addition, we present the first detailed examination of asymmetric circumstances in which the density of the intrusion differs from the mean density of the ambient and in which the upper- and lower-layer fluid depths are unequal. The general equations derived by J. Y. Holyer & H. E. Huppert (J. Fluid Mech.) vol. 100, 1980, pp. 739–767,), which predict the speed and vertical extent of the gravity current head, are re-expressed in a simpler form that employs the Boussinesq approximation. Approximate analytic solutions are determined using perturbation theory. The predictions are compared with the results of laboratory experiments. We find excellent agreement if the density of the gravity current is the average of the upper- and lower-layer densities weighted by the respective depths of the two layers. However, exact theory significantly underpredicts the gravity current speeds if the current density differs from this weighted-mean average. The discrepancy is attributed to the generation of waves that lead and trail the gravity current head. Empirical support for this assertion is provided through an examination of the observed wave characteristics.

The excitation of internal gravity waves by fluid intrusions that propagate along the interface between a uniform upper layer and a uniformly stratified lower layer is examined by way of laboratory experiments. Intrusions are generated using a simple lock-release apparatus. Experiments are conducted in which the density gradient of the uniformly stratified layer, the density jump across the interface and the density difference between the lock fluid and the uniform upper layer are varied.

In all cases, the fluid intrusions travelled at a constant speed. The forcing imparted by the generated internal gravity waves did not deform the intrusion head or significantly retard the intrusion's rate of forward advance. For a limited range of density parameters, good agreement was obtained between the experimental data and the two-layer analytical theory of J. Y. Holyer & H. E. Huppert (J. Fluid Mech. vol. 100 (1980), pp. 739–767) which provides estimates for the intrusion speed and depths of penetration into the upper and lower layers. Internal gravity wave excitation is due to the initial collapse of the lock fluid and the forcing imparted by the head of the intrusion. Waves are visualized and their amplitudes measured using ‘synthetic schlieren’.

The vertical flux of horizontal momentum due to internal gravity wave excitation is related to measurable properties of the fluid intrusion. This analysis suggests that outflows produced by tall convective storms that travel along the tropopause may excite non-hydrostatic internal gravity waves in the stratosphere whose momentum flux, at least during the transient generation time, is comparable to that of waves generated by topographic forcing.

We consider the flow of an incompressible particle-laden fluid through the application of the so-called ‘dusty-gas’ equations, which treat the fluid/particle suspension as two continua. The two phases are described by their individual field equations and interact through a Stokes-drag mechanism. The particular flow we consider is of boundary-layer type, corresponding to the downstream development of a Glauert-type jet adjacent to a horizontal boundary (the inclusion of the particulate phase requires the flow to be non-self-similar). We solve the governing boundary-layer equations through a numerical spatial marching technique in the three distinct cases of (i) weak gravitational influence, (ii) a jet ‘above’ a wall under the action of gravity and (iii) a jet ‘below’ a wall under the action of gravity.

The qualitative and quantitative features of the three cases are quite different and are presented in detail. Of particular interest is the development of a stagnation point in the particle velocity field at a critical downstream location in case (i), the development of fluid/particle flow reversal in case (ii) and the development of ‘shock’ solutions and particle-free regions in case (iii). Asymptotic descriptions are given of the critical phenomena, which support the numerical results. It is found that inclusion of a Saffman force has no substantial effect on either the location or structure of the stagnation-point region.