The statistics of unsteady turbulence with uniform stratification $N$ (Brunt–Väisälä frequency) and shear $\alpha ({=}\,{\rm d}U_{1}/{\rm d}x_{3})$ are analysed over the entire time range ($0 \,{<}\, \alpha t \,{<}\, \infty$) using rapid distortion theory (RDT) over a wide range of Richardson number ${\hbox{\it Ri}} ({=}\,N^{2}/\alpha^{2})$, and initial conditions. The solutions are found to be described by the Legendre functions of complex degree with pure-imaginary argument and are compared with previously published results of both direct numerical simulations (DNS) and experiments. In the initial stage of development many of the characteristics are similar to those in stratified flow with no shear, since the turbulence is determined by $\hbox{\it Nt}$ at the leading order, and the effects of vertical shear $\alpha$ generally appear at higher order. It is shown how in developing turbulence for ${\hbox{\it Ri}} \,{>}\, 0$ and ${\hbox{\it Ri}} \,{>}\, 0.25$ respectively, oscillatory momentum and positive and negative density fluxes develop. Above a critical value of ${\hbox{\it Ri}}_{\hbox{\scriptsize\it crit}} ({\sim}\,0.3)$, their average values are persistently countergradient. This structural change in the turbulence is the primary mechanism whereby stable stratification reduces the fluxes and the production of variances. It is quite universal and differs from the energy and stability mechanisms of Richardson (1926) and Taylor (1931). The long-time asymptotics of the energy ratio ER$({=}\,\hbox{\it PE/VKE}$) of the potential energy to the vertical kinetic energy generally decreases with ${\hbox{\it Ri}}({\geq}\,0.25)$, reaching the smallest value of $3/2$ when there is no shear (${\hbox{\it Ri}} \,{\rightarrow}\, \infty$). For strong mean shear (${\hbox{\it Ri}}\,{<}\,0.25$), RDT significantly overestimates ER since (as in unstratified shear flow) it underestimates the vertical kinetic energy VKE. The RDT results show that the asymptotic values of the energy ratio ER and the normalized vertical density flux are independent of the initial value of ER, in agreement with DNS. This independence of the initial condition occurs because the ratios of the contributions from the initial values $P\!E_{0}$ and $K\!E_{0}$ are the same for PE and VKE and can be explained by the linear processes. Stable stratification generates buoyancy oscillations in the direction of the energy propagation of the internal gravity wave and suppresses the generation of turbulence by mean shear. Because the shear distorts the wavenumber fluctuations, the low-wavenumber spectrum of the vertical kinetic energy has the general form $E_{33}(k)\,{\propto}\, (\alpha tk)^{-1}$, where $(L_{X} \alpha t)^{-1} \,{\ll}\,k \,{\ll}\,L_{X}^{-1}$ ($L_{X}$: integral scale). The viscous decay is controlled by the shear, so that the components of larger streamwise wavenumber $k_{1}$ decay faster. Then, combined with the spectrum distortion by the shear, the energy and the flux are increasingly dominated by the small-$k_{1}$ components as time elapses. They oscillate at the buoyancy period $\pi/N$ because even in a shear flow the components as $k_{1} \,{\rightarrow}\, 0$ are weakly affected by the shear. The effects of stratification $N$ and shear $\alpha$ at small scales are to reduce both VKE and PE. Even for the same ${\hbox{\it Ri}}$, larger $N$ and $\alpha$ reduce the high-wavenumber components of VKE and PE. This supports the applicability of the linear assumption for large $N$ and $\alpha$. At large scales, the stratification and shear effects oppose each other, i.e. both VKE and PE decrease due to the stratification but they increase due to the shear. We conclude that certain of these unsteady results can be applied directly to estimate the properties of sheared turbulence in a statistically steady state, but others can only be applied qualitatively.

Similarity laws of mean velocity profiles and turbulence characteristics of Couette–Poiseuille turbulent flow (C-P flow) have been studied experimentally. The global parameters of C-P flow are the Reynolds number $\hbox{\it Re}^{\ast}$ and the dimensionless shear stress gradient $\mu$ and flow parameter $\beta $. The effects of these parameters on the turbulence structure have also been considered in the wall region and turbulent core region, respectively. In the wall region, the wall law varies greatly with $\mu $ but slightly with $\hbox{\it Re}^{\ast}$. Typically, the additive constant $B$ of the logarithmic law (or Van Driest damping factor $A^{+}$) is shown to depend only on $\mu $. Turbulence characteristics are also strongly influenced by $\mu $, but not much by $\hbox{\it Re}^{\ast}$. Because the relation $\mu \,{=}\, {-}\hbox{\it Re}^{\ast}$ holds in plane Poiseuille flow and $\hbox{\it Re}^{\ast}$ has little effect on the similarity laws for C-P flows, the low-Reynolds-number effect on the additive constant and turbulence quantities for plane Poiseuille flow can be attributed to the $\mu $ effect. In the turbulent core region, however, there is a great difference in the defect law of the velocity profile and the distribution of turbulence intensity between Poiseuille (P)- and Couette (C)-types flows. For P-type flow, an effective friction velocity $u^{\ast}_{e}$ and a new coordinate $\eta \,{=}\,y- h_{s}$ are recommended for the universal profile, where $y\,{=}\,h_{s}\,{=}\,\delta _{p}$ is the position of $\tau \,{=}\,0$ and $\delta _{p}$ is considered to be appropriate as a characteristic length scale of turbulence. For C-type flow, a different effective friction velocity $u^{\ast}_{c,}$ the characteristic length scale 2$h$ and the wall coordinate $y$ are preferred. The turbulence activity away from the wall is extremely high for $\mu \,{>}\,0$ and low for $\mu \,{<}\,0$. A strong sweep event plays a dominant role in the Reynolds shear stress when $0\,{<}\,\mu \,{<}\,50$, whereas strong ejection from the near-wall region prevails in the case of negative $\mu$ with a small absolute value.

We examine the properties of stationary, barotropic flows over isolated topographic features such as seamounts. According to a general variational principle, flows that maximize or minimize the energy in a set of isovortical flows are stationary and stable. Using this, it is shown that a large class of stable and stationary attached anticyclones exists at a seamount. Those with positive potential vorticity (PV) are maximum energy states, while those with negative PV are minimum energy states. If the seamount is circular, there are also stable attached cyclones, but these are destabilized by irregularities in the topographic shape, unlike the anticyclones. Numerical simulations broadly support these theoretical predictions, but also highlight the importance of time-dependent processes, particularly in cases in which the vortex collides with the seamount.

This article examines the structure and stability of detonations in mixtures of gases and solid particles via direct numerical simulation. Cases with both reactive and inert particles are considered. First, the two-phase flow model is presented and the assumptions that it is based upon are discussed. Steady-wave structures admitted by the model are subsequently analysed. Next, the algorithm employed for the numerical simulations is described. It is a recently developed high-order shock-capturing algorithm for compressible two-phase flows. The accuracy of the algorithm has been verified through a series of code validation and numerical convergence tests, some of which are included in this article. Subsequently, numerical results for both one-dimensional and two-dimensional detonations are presented and discussed. These results show that the mass, momentum and energy transfers between the two phases result in detonation structures that are substantially different from those observed in the corresponding purely gaseous flows. The effect of certain important parameters, such as particle reactivity, particle volume fraction, and particle diameter, are examined in detail. The numerical results predict that increased particle reactivity suppresses the flow instabilities and increases the detonation velocities. It is further predicted that sufficiently high particle volume fractions can cause detonation quenching regardless of particle reactivity.

We devise a stochastic model for the effects of breaking waves and fit its distribution functions to laboratory and field data. This is used to represent the space–time structure of momentum and energy forcing of the oceanic boundary layer in turbulence-resolving simulations. The aptness of this breaker model is evaluated in a direct numerical simulation (DNS) of an otherwise quiescent fluid driven by an isolated breaking wave, and the results are in good agreement with laboratory measurements. The breaker model faithfully reproduces the bulk features of a breaking event: the mean kinetic energy decays at a rate approaching $t^{-1}$, and a long-lived vortex (eddy) is generated close to the water surface. The long lifetime of this vortex (more than 50 wave periods) makes it effective in energizing the surface region of oceanic boundary layers. Next, a comparison of several different DNS of idealized oceanic boundary layers driven by different surface forcing (i.e. constant current (as in Couette flow), constant stress, or a mixture of constant stress plus stochastic breakers) elucidates the importance of intermittent stress transmission to the underlying currents. A small amount of active breaking, about 1.6% of the total water surface area at any instant in time, significantly alters the instantaneous flow patterns as well as the ensemble statistics. Near the water surface a vigorous downwelling–upwelling pattern develops at the head and tail of each three-dimensional breaker. This enhances the vertical velocity variance and generates both negative- and positive-signed vertical momentum flux. Analysis of the mean velocity and scalar profiles shows that breaking effectively increases the surface roughness $z_o$ by more than a factor of 30; for our simulations $z_o/\lambda \,{\approx}\, 0.04$ to 0.06, where $\lambda$ is the wavelength of the breaking wave. Compared to a flow driven by a constant current, the extra mixing from breakers increases the mean eddy viscosity by more than a factor of 10 near the water surface. Breaking waves alter the usual balance of production and dissipation in the turbulent kinetic energy (TKE) budget; turbulent and pressure transports and breaker work are important sources and sinks in the budget. We also show that turbulent boundary layers driven by constant current and constant stress (i.e. with no breaking) differ in fundamental ways. The additional freedom provided by a constant-stress boundary condition permits finite velocity variances at the water surface, so that flows driven by constant stress mimic flows with weakly and statistically homogeneous breaking waves.

The shear layer instability in the flow between two counter-rotating disks enclosed by a cylinder is investigated experimentally and numerically, for radius-to-height ratio $\Gamma \,{=}\, R/h$ between 2 and 21. For sufficiently large rotation ratio, the internal shear layer that separates two regions of opposite azimuthal velocities is prone to an azimuthal symmetry breaking, which is investigated experimentally by means of visualization and particle image velocimetry. The associated pattern is a combination of a sharp-cornered polygonal pattern, as observed by Lopez et al. (2002) for low aspect ratio, surrounded by a set of spiral arms, first described by Gauthier et al. (2002) for high aspect ratio. The spiral arms result from the interaction of the shear layer instability with the Ekman boundary layer over the faster rotating disk. Stability curves and critical modes are experimentally measured for the whole range of aspect ratios, and are found to compare well with numerical simulations of the three-dimensional time-dependent Navier–Stokes equations over an extensive range of parameters. Measurements of a local Reynolds number based on the shear layer thickness confirm that a shear layer instability, with only weak curvature effect, is responsible for the observed patterns. This scenario is supported by the observed onset modes, which scale as the shear layer radius, and by the measured phase velocities.

Direct numerical simulations and a self-similar analysis of the single-fluid Boussinesq Rayleigh–Taylor instability and transition to turbulence are used to investigate Rayleigh–Taylor turbulence. The Schmidt, Atwood and bulk Reynolds numbers are $Sc\,{=}\,1$, $A\,{=}\,0.01$, $Re \,{\le}\, 3000$. High-Reynolds-number moment self-similarity, consistent with the the energy cascade interpretation of dissipation, is used to analyse the DNS results. The mixing layer width obeys a differential equation with solution $h(t;C_o,h_0)\,{=}\,\fourth C_o Agt^2+ \sqrt{AgC_o}h^{1/2}_0 t+h_0$; the result for $h(t;C_o,h_0)$ is a rigorous consequence of only one ansatz, self-similarity. It indicates an intermediate time regime in which the growth is linear and the importance of a virtual origin. At long time the well-known $h \sim \fourth C_o Agt^2$ scaling dominates. The self-similar analysis indicates that the asymptotic growth rate is not universal. The scalings of the second-order moments, their dissipations, and production–dissipation ratios, are obtained and compared to the DNS. The flow is not self-similar in a conventional sense – there is no single length scale that scales the flow. The moment similarity method produces three different scalings for the turbulence energy-containing length scale, $\ell$, the Taylor microscale, $\la$, and the Kolmogorov dissipation scale, $\eta$. The DNS and the self-similar analysis are in accord showing $\ell \,{\sim}\, Agt^2$, $\la \,{\sim}\, t^{1/2}$ and $\eta \,{\sim}\, (({A^2g^2}/{\nu^3})t)^{-1/4}$ achieving self-similar behaviour within three initial eddy turnovers of the inception of the turbulence growth phase at bulk Reynolds numbers in the range of ${\it Re}\,=\,800$–1000 depending on initial conditions. A picture of a turbulence in which the largest scales grow, asymptotically, as $t^2$ and the smallest scales decrease as $t^{-1/4}$, emerges. As a consequence the bandwidth of the turbulence spectrum grows as $t^{9/4}$ and is consistent with the $R_t^{3/4}$ Kolmogorov scaling law of fully developed stationary turbulent flows. While not all moments are consistent, especially the dissipations and higher-order moments in the edge regions, with the self-similar results it appears possible to conclude that: (i) the turbulence length scales evolve as a power of $h(t;C_o,h_0)$; (ii) $\al$, as demonstrated mathematically for self-similar Rayleigh–Taylor turbulence and numerically by the DNS, is not a universal constant; (iii) there is statistically significant correlation between decreasing $\alpha$ and lower low-wavenumber loading of the initial spectrum.

The classical problem of nonlinear capillary waves on two-dimensional fluid sheets is reconsidered. The problem is formulated in terms of a complex potential, and solutions are sought using Fourier series expansions. A collocation technique combined with Newton's method is used to compute the Fourier coefficients numerically. Using this procedure, the exact solutions of Kinnersley (1976) are recomputed and various symmetric and antisymmetric wave profiles are presented, including the limiting configurations which exhibit trapped bubbles of air. Most important, three new solution branches which bifurcate nonlinearly from the symmetric Kinnersley solution branch are identified. The wave profiles along these new branches do not possess the symmetry or antisymmetry of the Kinnersley solutions, although their limiting configurations also display trapped air bubbles. No bifurcations are found along the antisymmetric Kinnersley solution branch.

Flow in a cylinder driven by the co-rotation of the top and bottom endwalls displays distinct instability mechanisms, depending on whether its aspect ratio (length to diameter) is large or small. When the cylinder length is about the same as its diameter, the two mechanisms compete and lead to a stable mixed mode solution. Using numerical computations of the three-dimensional Navier–Stokes equations, together with equivariant bifurcation and normal form theory, we explore the nature of this mode competition, both from a flow physics point of view and from a dynamical systems with symmetry point of view. The results help to clarify the distinct behaviours observed experimentally in some short and long cylinder flows.

In this and the companion paper (Part 2) we examine experimentally, computationally, and analytically the behaviour of breaking-wave-induced macrovortices during startup conditions. Widely separated breakwaters and rip current topographies are chosen as opposite ends of the parameter space. Part 1 examines generation mechanisms using phase-resolving and phase-averaged approximations, and suggests several simple predictive relations for general behaviour. Vortex trajectories and shedding periods for wave breaking on widely spaced breakwaters are also considered in detail. Results show broad agreement with theoretical trajectories. Predictions of vortex shedding periods on breakwater heads show excellent agreement with computations. Part 2 examines startup macrovortices on rip current topographies using computations and laboratory experiments, and changes in behaviour as the system transitions from wide to narrow gap width.

The linear stability of rapid granular flow down a slope under gravity to the longitudinal perturbations is analysed using hydrodynamic equations. It is demonstrated that the steady flow uniform along the flow direction becomes unstable to long-wavelength perturbations longitudinal to the flow direction for certain parameter ranges to form a density wave, in contrast to the finite-wavelength instability for transverse perturbations (Forterre & Pouliquen 2002). It is shown that the instability can be understood as the long-wave instability of kinematic waves in a quasi-one-dimensional system. The results are compared with our previous molecular dynamics simulations (Mitarai & Nakanishi 2001), where spontaneous density wave formation was found.

In this paper we consider the dynamic behaviour of a fluid-loaded elastic plate of dimensionless length $2L$, set in a rigid baffle in the presence of uniform mean flow of dimensionless speed $U$, and including plate pre-tension (dimensionless tension $T$) and a spring foundation (dimensionless spring constant $\lambda$). Our aim is to extend previous analytical work of Crighton & Oswell (1991), who examined an infinite plate with $T\,{=}\,\lambda\,{=}\,0$. This is achieved by considering a plate which is long on the scale of typical plate bending waves, and then by expressing the unsteady motion as a superposition of infinite-plate waves which are continuously rescattered in near-field regions at the plate leading and trailing edges. We find that the finite plate possesses resonant solutions which are temporally unstable, both for parameter values for which the infinite plate is convectively unstable ($T$ and/or $\lambda$ sufficiently small) and parameter values for which it is stable ($T$ and/or $\lambda$ sufficiently large). It is shown that instability is present in the absence of structural damping on the finite plate, in agreement with numerical results of Lucey & Carpenter (1992). Neutral resonant states are also found, for which we derive a generalization of the Landahl (1962) and Benjamin (1963) concept of wave energy. Finally, we replace the linear pre-tension $T$ by the nonlinear tension induced by bending, and analyse the nonlinear evolution of the states of negative wave energy in the presence of weak structural damping. We show that the system possesses points of minimum action at non-zero frequency, which act as attractors, predicting the existence of nonlinear fluttering motion.

We investigate the solutions of the Navier–Stokes equations that describe the steady flow of rivulets down an inclined surface. We find that the shape of the free surface is given by an analytic formula obtained by solving the equation that expresses the condition of static equilibrium under the action of gravity and surface tension, independently of the velocity field and of any assumption concerning the rheology of the liquid. The velocity field is then obtained by solving (in general numerically) a Poisson equation in the domain defined by the cross-section of the rivulet. The isovelocity contours are perpendicular to the free surface. Various properties of the solutions are given as functions of the parameters of the problem. Two special analytic solutions are presented. The exact solutions suggest that the lubrication approximation, frequently employed to investigate problems similar to the present one, predicts reasonably well the global properties of the rivulet provided the static contact angle is not too large.

A systematic dimensional rationale is proposed here to analyse the electrohydro-dynamic equations governing liquid electrospraying phenomena in the well-known steady cone-jet mode with no ambient discharges. As a result, a general, unified description of the complete parametrical space for the emitted current and droplet size is given. Four main distinct subspaces, their relevant boundaries and corresponding scaling laws are identified. Laws already proposed fit in their appropriate region, and previously unknown laws are found. A closed solution for the electric current I when inertia and polarization forces dominate is obtained, in agreement with published experimental results.