The initial stages of high-velocity droplet impact on a shallow water layer are described, with special emphasis given to the spray jet mechanics. Four stages of impact are delineated, with appropriate scalings, and the successively more important influence of the base is analysed. In particular, there is a finite time before which part of the water in the layer remains under the droplet and after which all of the layer is ejected in the splash jet.

We describe new experiments to measure the penetrative entrainment by a turbulent fountain in a steady two-layer stratification. A theoretical model was established by assuming that the stratification consists of two uniform layers, and the penetrative entrainment rate is estimated quantitatively by three independent formulae. Two quasi-uniform layers were observed in the steady state in the laboratory experiments. Experimental results gave a nearly constant dimensionless penetrative entrainment rate (0.65±0.17) across a density interface when the local Richardson number is smaller than 1.2.

Convective dynamos in a rotating spherical shell feature steady zonal flows. This process is studied numerically for Prandtl numbers of 0.1 and 1, Ekman numbers in the range $E=\te{-4}$–$\te{-5}$, magnetic Prandtl number from 0.5 to 10 and Rayleigh numbers up to 100 times supercritical. The zonal flow is mainly of thermal wind origin, and minimizes the shear of the axisymmetric poloidal magnetic field lines, according to Ferraro's law of corotation. The dissipation in the interior of the fluid is mainly ohmic, while the introduction of rigid velocity boundary conditions confines viscous dissipation in the Ekman boundary layers. The root-mean-square amplitude $U_\varphi$ of the zonal flow in the spherical shell scales as $U_\varphi=(F/\Omega)^{0.5}$, F being the buoyancy flux through the shell and $\Omega$ the rotation rate. As a consequence of the corotation law, this scaling relationship is remarkably independent of the magnetic field amplitude. It does not depend on thermal, kinematic and magnetic diffusivities, owing to the large-scale and steady nature of forcing and dissipative processes. The scaling law is in agreement with the zonal-flow amplitude at the external boundary of the Earth's liquid core.

A large-axial-wavenumber asymptotic analysis of inviscid normal modes in an axisymmetric vortex with a weak axial flow is performed in this work. Using a WKBJ approach, general conditions for the existence of regular neutral modes are obtained. Dispersion relations are derived for neutral modes confined in the vortex core (‘core modes’) or in a ring (‘ring modes’). Results are applied to a vortex with Gaussian vorticity and axial velocity profiles, and a good agreement with numerical results is observed for almost all values of k. The theory is also extended to deal with singular modes possessing a critical point singularity. We demonstrate that the characteristics for vanishing viscosity of viscous damped normal modes can also be obtained. Known viscous damped eigenfrequencies for the Gaussian vortex without axial flow are, in particular, shown to be predicted well by our estimates. The theory is also shown to provide explanations for a few of their peculiar properties.

The multifractal theory of turbulence uses a saddle-point evaluation in determining the power-law behaviour of structure functions. Without suitable precautions, this could lead to the presence of logarithmic corrections, thereby violating known exact relations such as the four-fifths law. Using the theory of large deviations applied to the random multiplicative model of turbulence and calculating subdominant terms, we explain here why such corrections cannot be present.

For weak shock waves and small wedge angles the application of three-shock (von Neumann) theory gives no physically realistic solutions and yet experiments clearly show a pattern of reflection of three shocks meeting at a triple point. This disagreement is referred to as the von Neumann paradox, and the reflection pattern as von Neumann reflection (vNR). Some recent numerical computations have indicated the existence of an expansion wave immediately behind the reflected wave as originally suggested by Guderley over fifty years ago. Furthermore, a recent solution of the inviscid transonic equations has indicated the possible existence of a very small, multi-wave structure immediately behind the three-shock confluence. A special shock tube has been constructed which allows Mach stem lengths to be obtained which are more than an order of magnitude larger than those obtainable in conventional shock tubes. Schlieren photographs do indeed show a structure consisting of an expansion wave followed by a small shock situated behind the confluence point, with some indication of smaller scale structures in some tests. This indicates that some of the earlier models of vNR, in the parameter space tested, are incorrect. The size of the region influenced by this small wave system is about 2% of the Mach stem length and it is therefore not surprising that it has not been detected before in conventional shock tube facilities.

An experimental investigation into the structure of a supersonic jet impinging on a large plate is presented. Digital particle image velocimetry (DPIV), shadowgraph photography and acoustic measurements are used to understand the relationship between the unsteady jet structure and the production of tones for nozzle-to-plate spacings between 1 and 5 nozzle exit diameters at a nozzle–pressure ratio equal to 4. Results indicate that the instability of the jet depends on the location of the plate in the shock cell structure of the corresponding free jet and the strength of the standoff shock wave, rather than on the occurrence of recirculation zones in the impingement region. Phase-locked studies show streamwise displacements of the stand-off shock wave, a moving recirculation zone in the subsonic flow in front of the plate, and significant oscillations of both the compression and expansion regions in the peripheral supersonic flow when tones are produced. Sound is shown to be generated by periodic pulsing of the wall jet boundary resulting from periodic motion of the flow in the impingement and near-wall regions of the flow.

Direct numerical simulations of three-dimensional homogeneous turbulence under rapid rigid rotation are conducted for a fixed large reynolds number and a sequence of decreasing rossby numbers to examine the predictions of resonant wave theory. the theory states that ‘slow modes’ of the velocity, with zero wavenumber parallel to the rotation axis ($k_z{=}0$), will decouple at first order from the remaining ‘fast modes’ and solve an autonomous system of two-dimensional navier–stokes equations for the horizontal velocity components, normal to the rotation axis, and a two-dimensional passive scalar equation for the vertical velocity component, parallel to the rotation axis. The navier–stokes equation for three-dimensional rotating turbulence is solved in a $128^3$ mesh after being diagonalized via ‘helical decomposition’ into normal modes of the coriolis term. A force supplies constant energy input at intermediate scales. to verify the theory, we set up a corresponding simulation for the two-dimensional navier–stokes equation and two-dimensional passive scalar equation to compare them with the slow-mode dynamics of the three-dimensional rotating turbulence. the simulation results reveal that there is a clear inverse energy cascade to the large scales, as predicted by two-dimensional navier–stokes equations for resonant interactions of slow modes. as the rotation rate increases, the vertically averaged horizontal velocity field from three-dimensional navier–stokes converges to the velocity field from two-dimensional navier–stokes, as measured by the energy in their difference field. likewise, the vertically averaged vertical velocity from three-dimensional navier–stokes converges to a solution of the two-dimensional passive scalar equation. the slow-mode energy spectrum approaches $k_h^{-5/3},$ where $k_h$ is the horizontal wavenumber, and, as in two dimensions, energy flux becomes closer to constant the greater the rotation rate. furthermore, the energy flux directly into small wavenumbers in the $k_z{=}0$ plane from non-resonant interactions decreases, while fast-mode energy concentrates closer to that plane. the simulations are consistent with an increasingly dominant role of resonant triads for more rapid rotation.

High-precision measurements of the Nusselt number Nu as a function of the Rayleigh number Ra have been made in water-filled 1m diameter cylindrical cells of aspect ratio $\Gamma {=} $0.67, 1, 2, 5, 10 and 20. The measurements were conducted at the Prandtl number $Pr {\approx} 4$ with Ra varying from $1{\times} 10^7$ to $5{\times} 10^{12}$. When corrections for the finite conductivity of the top and bottom plates are made, the estimates obtained of $Nu_{\infty}$ for perfectly conducting plates may be described by a combination of two power laws $Nu_{\infty} {=} C_{1}(\Gamma)Ra^{\beta_1}+C_{2}(\Gamma)Ra^{\beta_2}$ for all the aspect ratios. The fitted exponents $\beta_1 {=}0.211$ and $\beta_2 {=} 0.332$ are very close to $1/5$ and $1/3$ respectively, which have been predicted by Grossmann & Lohse for the II$_u$ and IV$_u$ regimes in their model. It is also found that $Nu_{\infty}$ is generally smaller for larger $\Gamma$ but the difference is only a few percent and for $\Gamma{\gtrsim} 10$ the asymptotic large-$\Gamma$ behaviour may have been reached.

An experimental investigation of the stability of longitudinal rolls in a horizontal layer heated from below in the presence of a Poiseuille flow is carried out. This study follows on from the theoretical work of Clever & Busse (J. Fluid Mech., vol. 229, 1991, p. 517) who detected a wavy instability for a range of relatively low Rayleigh and Reynolds numbers depending on the Prandtl number. In the present study, an air flow is circulating in a rectangular channel of transverse aspect ratio 10 for Rayleigh numbers of 6300 and 9000 and Reynolds numbers from 100 to 174. The system exhibits a wavy pattern only if the flow is continuously excited. The amplitude of the waves grows as they propagate downstream and the frequency of the oscillations is equal to the frequency of the imposed disturbance. The bifurcation from steady longitudinal rolls to unsteady wavy rolls is thus a convective instability. A mode by mode study is performed by measuring the wave velocity and the spatial growth of the instability along the channel for a large range of the imposed frequency. The phase velocity is found to depend only on the Reynolds number, and is nearly equal to the bulk velocity of the flow for all the modes in the range of parameters under study. The maximum spatial growth rate corresponding to the most unstable mode as well as the corresponding frequency decrease with decreasing Reynolds number or Rayleigh number, providing a decrease in the wavelength. This feature is in agreement with the theoretical results of Clever & Busse (1991).

Simulations have been performed with a fairly narrow band numerical gravity wave model (higher-order nls type) and a computational domain of dimensions $128\times 128$ typical wavelengths. The simulations are initiated with $\thicksim6\times10^{4}$ fourier modes corresponding to truncated jonswap spectra and different angular distributions giving both short- and long-crested waves. A development of the spectra on the so-called benjamin–feir timescale is seen, similar to the one reported by dysthe et al. (J. Fluid Mech. vol. 478, 2003, p.1). The probability distributions of surface elevation and crest height are found to fit theoretical distributions found by tayfun (J. Geophys. Res. Vol. 85, 1980, p. 1548) very well for elevations up to four standard deviations (for realistic angular spectral distributions). moreover, in this range of the distributions, the influence of the spectral evolution seems insignificant. for the extreme parts of the distributions a significant correlation with the spectral change can be seen for very long-crested waves. For this case we find that the density of large waves increases during spectral change, in agreement with a recent experimental study by onorato et al. (J. Fluid Mech. 2004 submitted).

Near-wall structures in turbulent natural convection at Rayleigh numbers of $10^{10}$ to $10^{11}$ at A Schmidt number of 602 are visualized by a new method of driving the convection across a fine membrane using concentration differences of sodium chloride. The visualizations show the near-wall flow to consist of sheet plumes. A wide variety of large-scale flow cells, scaling with the cross-section dimension, are observed. Multiple large-scale flow cells are seen at aspect ratio (AR)= 0.65, while only a single circulation cell is detected at AR= 0.435. The cells (or the mean wind) are driven by plumes coming together to form columns of rising lighter fluid. The wind in turn aligns the sheet plumes along the direction of shear. the mean wind direction is seen to change with time. The near-wall dynamics show plumes initiated at points, which elongate to form sheets and then merge. Increase in rayleigh number results in a larger number of closely and regularly spaced plumes. The plume spacings show a common log–normal probability distribution function, independent of the rayleigh number and the aspect ratio. We propose that the near-wall structure is made of laminar natural-convection boundary layers, which become unstable to give rise to sheet plumes, and show that the predictions of a model constructed on this hypothesis match the experiments. Based on these findings, we conclude that in the presence of a mean wind, the local near-wall boundary layers associated with each sheet plume in high-rayleigh-number turbulent natural convection are likely to be laminar mixed convection type.

The Rayleigh–Taylor instability of a system of three fluids separated by one unstable and one stable interface has been investigated experimentally. The experiments were gravitationally driven and conducted with miscible liquids consisting of salt solutions and fresh water. The lower two layers are initially gravitationally stable and are formed by depositing the lighter fluid on top of a thicker layer of the heavier one. The relatively thick top layer is initially separated from the two lower layers by a rigid barrier that is removed at the start of an experiment. In situations where the density of the bottom-layer fluid equals that of the top-layer fluid, the resulting turbulent flow is found to be self-similar as demonstrated by the collapse of the mean concentration distributions as well as the behaviour of the decay of the peak of the mean concentration profiles. In this configuration, the erosion of the bottom layer by the turbulence generated by the upper unstable interface is found to be small. When the density of the bottom-layer fluid is increased above that of the top-layer fluid, the degree of erosion is further decreased. In the cases where the lower interface is stably stratified at late-time, the entrainment rate E at the lower (statically stable) interface is found to follow a power law of the Richardson number, i.e. $E \propto Ri^{-n}$, with $n {\approx} 1.3$, a result in agreement with studies of mixing induced by oscillating grids. When the density of the bottom-layer fluid is decreased below that of the top-layer fluid, the erosion increases as expected. However, in this case, the overall density distribution is such that it is globally Rayleigh–Taylor unstable at late time. In this situation, the turbulent mixing region at late times grows similarly to that of single-interface Rayleigh–Taylor instability with approximately the same value of the growth constant. In these late-time unstable experiments the density profile approaches that of an equivalent two-layer Rayleigh–Taylor unstable system.

A new experimental and theoretical approach is presented to examine the dynamic lift forces that are generated in the compression of both fresh powder snow and wind-packed snow. At typical skiing velocities of 10 to 30ms$^{-1}$ the duration of contact of a ski or snowboard with the snow will vary from 0.05 to 0.2s depending on the length of the planing surface and its speed. No one, to our knowledge, has previously measured the dynamic behaviour of snow on such a short time scale and, thus, there are no existing measurements of the excess pore pressure that can build-up in snow on this time scale. Using a novel porous cylinder–piston apparatus, we have measured the excess pore pressure that would build-up beneath the piston surface and have also measured its subsequent decay due to the venting of the air from the snow at the porous wall of the cylinder. In further experiments, in which the air is slowly and deliberately drained to avoid a build-up in pore pressure, we have been able to separate out the force exerted by the ice crystal phase as a function of its instantaneous deformation. A theoretical model for the pore pressure relaxation in the porous cylinder is then developed using consolidation theory. Dramatically different dynamic behaviour is observed for two different snow types, one (wind-packed) giving a steady continuous relaxation of the excess pore pressure and the other (fresh powder) leading to a piston rebound with negative pore pressure. A feature of the rebound is the apparent debonding of sintered ice crystals after maximum compression. This behaviour is described well by introducing a debonding coefficient where the debonding force is proportional to the expansion velocity of the medium. The experimental and theoretical approach presented herein and the previous generalized lubrication theory for compressible porous media, have laid the foundation for understanding the detailed dynamic response of soft porous layers to rapid deformation.

The effect of rotation on a homogeneous turbulent shear flow has been studied by means of a series of direct numerical simulations with different rotation numbers. The evolution of passive scalar fields with mean gradients in each of the three orthogonal directions in the flow was investigated in order to elucidate the effect of rotation on turbulent scalar transport. Conditions of the near-wall region of a boundary layer were approached by using a rapid shear and therefore, comparisons could be made with rapid distortion theory based on the linearized equations of the flow and scalar transport. Reynolds stresses, pressure–strain correlations and two-point velocity correlations were computed and turbulent structures were visualized. It is shown that rotation has a strong influence on the time development of the turbulent kinetic energy, the anisotropy of the flow and on the turbulent structures. Furthermore, rotation significantly affects turbulent scalar transport. The transport rate of the scalar and the direction of the scalar flux vector show large variations with different rotation numbers, and a strong alignment was observed between the scalar flux and the principal axes of the Reynolds stress tensor. The ratio of the turbulent and scalar time scales is influenced by rotation as well. The predictions of the linear theory of the turbulent one-point statistics and the scalar flux agreed fairly well with direct numerical simulation (DNS) results based on the full nonlinear governing equations. Nonetheless, some clear and strong nonlinear effects are observed in a couple of cases which significantly influence the development of the turbulence and scalar transport.

The methods that have come to be known as the Malkus–Howard–Busse (MHB) and the Constantin–Doering–Hopf (CDH) techniques have, over the past few decades, produced the few rigorous statements available about average properties (e.g. momentum and heat transport) of turbulent flows governed by the Navier–Stokes equation and the heat equation. In this, the first of two papers investigating upper bounds on the heat transport in infinite-Prandtl-number convection, we show that the methods of MHB and CDH yield equivalent optimal bounds: as at a saddle–one from above, and one from below.

We also demonstrate that here, in contrast to earlier applications of the CDH method, the simplest possible, one-parameter, ‘test function’ does not capture the leading-order scaling associated with the fully optimal solution. We explore the consequences of a two-parameter test function in modifying the scaling of the upper bound. In the case of no-slip, the suggestion is that a hierarchy of test functions of increasing complexity is required to yield the correct limiting behaviour.

The well-known Rayleigh criterion is a necessary and sufficient condition for inviscid centrifugal instability of axisymmetric perturbations. We have generalized this criterion to disturbances of any azimuthal wavenumber m by means of large-axial-wavenumber WKB asymptotics. A sufficient condition for a free axisymmetric vortex with angular velocity $\Omega(r)$ to be unstable to a three-dimensional perturbation of azimuthal wavenumber m is that the real part of the growth rate \[\sigma (r) =-{\rm i}m\Omega(r)+\sqrt{-\phi(r)}\] is positive at the complex radius $r{=}r_0$ where $\partial \sigma (r)/\partial r{=}0$, i.e. \[\phi'(r_0) =-2{\rm i}m\Omega'(r_0)\sqrt{-\phi(r_0)},\] where $\phi{=}(1/r^3)\partial{r^4\Omega^2}/\partial {r}$ is the Rayleigh discriminant, provided that some a posteriori checks are satisfied. The application of this new criterion to various classes of vortex profiles shows that the growth rate of non-axisymmetric disturbances decreases as m increases until a cutoff is reached. The criterion is in excellent agreement with numerical stability analyses of the Carton & McWilliams (1989) vortices and allows one to analyse the competition between the centrifugal instability and the shear instability. The generalized criterion is also valid for a vertical vortex in a stably stratified and rotating fluid, except that ϕ becomes $\phi{=}(1/r^3)\partial{r^4(\Omega+\Omega_b)^2/\partial r$, where $\Omega_b$ is the background rotation about the vertical axis. The stratification is found to have no effect. For the Taylor–Couette flow between two coaxial cylinders, the same criterion applies except that $r_0$ is real and equal to the inner cylinder radius. In sharp contrast, the maximum growth rate of non-axisymmetric disturbances is then independent of m.