Pressure fluctuations are an important ingredient in turbulence, e.g. in the pressure strain terms which redistribute turbulence among the different fluctuating velocity components. The variation of the pressure fluctuations inside a turbulent boundary layer has hitherto been out of reach of experimental determination. The mechanisms of non-local pressure-related coupling between the different regions of the boundary layer have therefore remained poorly understood. One reason for this is the difficulty inherent in measuring the fluctuating pressure. We have developed a new technique to measure pressure fluctuations. In the present study, both mean and fluctuating pressure, wall pressure, and streamwise velocity have been measured simultaneously in turbulent boundary layers up to Reynolds numbers based on the momentum thickness Rθ ≃ 20000. Results on mean and fluctuation distributions, spectra, Reynolds number dependence, and correlation functions are reported. Also, an attempt is made to test, for the first time, the existence of Kolmogorov's -7/3 power-law scaling of the pressure spectrum in the limit of high Reynolds numbers in a turbulent boundary layer.

The present wind-tunnel experiment demonstrates that a reactive control system is able to decrease the amplitude of random disturbances in a flat-plate boundary layer. The disturbances were induced in a laminar boundary layer by a turbulent free stream. The control system consisted of upstream wall-shear-stress sensors (wall wires) and downstream actuators (suction through holes). An ad hoc threshold-and-delay control algorithm is evaluated and parameter variations were performed in order to find a suitable working point of the control system. Detailed measurements of the flow field show how the control influences the disturbances in the boundary layer, whereas the effect on the mean flow owing to the control is minute. The control system manages to inhibit the growth of the fluctuations of the streamwise velocity component for a considerable distance downstream of the two actuator positions. Further downstream, however, the amplitudes of the fluctuations grow again. The flow rate used to obtain the control effect is one sixth of that necessary if continuous distributed suction is used to reach the same control objective. Finally, correlations and spectra show that the elongation of the structures in the streamwise direction is eliminated in the regions where the control has the largest effect. The spanwise scale of the disturbances is not affected by the control.

A combination of molecular dynamics simulations, theoretical predictions and previous experiments are used in a two-part study to determine the role of the Knudsen layer in rapid granular flows. First, a robust criterion for the identification of the thickness of the Knudsen layer is established: a rapid deterioration in Navier–Stokes order prediction of the heat flux is found to occur in the Knudsen layer. For (experimental) systems in which heat flux measurements are not easily obtained, a rule-of-thumb for estimating the Knudsen layer thickness follows, namely that such effects are evident within 2.5 (local) mean free paths of a given boundary. Secondly, comparisons of simulation and experimental data with Navier–Stokes order theory are used to provide a measure as to when Knudsen-layer effects become non-negligible. Specifically, predictions that do not account for the presence of a Knudsen layer appear reliable for Knudsen layers collectively composing up to 20% of the domain, whereas deterioration of such predictions becomes apparent when the domain is fully comprised of the Knudsen layer.

The numerical study of J. Song & M. L. Banner (J. Phys. Oceanogr. vol. 32, 2002, p. 254) proposed a generic threshold parameter for predicting the onset of breaking within two-dimensional groups of deep-water gravity waves. Their parameter provides a non-dimensional measure of the wave energy convergence rate and geometrical steepening at the maximum of an evolving nonlinear wave group. They also suggested that this parameter might control the strength of breaking events. The present paper presents the results of a detailed laboratory observational study aimed at validating their proposals.

For the breaking onset phase of this study, wave potential energy was measured at successive local envelope maxima of nonlinear deep-water wave groups propagating along a laboratory wave tank. These local maxima correspond alternately to wave group geometries with the group maximum occurring at an extreme carrier wave crest elevation, followed by an extreme carrier wave trough depression. As the nonlinearity increases, these crest and trough maxima can have markedly different local energy densities owing to the strong crest–trough asymmetry. The local total energy density was reconstituted from the potential energy measurements, and made dimensionless using the square of the local (carrier wave) wavenumber. A mean non-dimensional growth rate reflecting the rate of focusing of wave energy at the envelope maximum was obtained by smoothing the local fluctuations.

For the cases of idealized nonlinear wave groups investigated, the observations confirmed the evolutionary trends of the modelling results of Song & Banner (2002) with regard to predicting breaking onset. The measurements confirmed the proposed common breaking threshold growth rate of 0.0014±0.0001, as well as the predicted key evolution times: the time taken to reach the energy maximum for recurrence cases; and the time to reach the breaking threshold and then breaking onset, for breaking cases.

After the initiation and subsequent cessation of breaking, the measured wave packet mean energy losses and loss rates associated with breaking produced an unexpected finding: the post-breaking mean wave energy did not decrease to the mean energy level corresponding to maximum recurrence, but remained significantly higher. Therefore, pre-breaking absolute wave energy or mean steepness do not appear to be the most fundamental determinants of post-breaking wave packet energy density.

However, the dependence of the fractional breaking energy loss of wave packets on the parametric growth rate just before breaking onset proposed by Song & Banner (2002) was found to provide a plausible collapse to our laboratory data sets, within the experimental uncertainties. Further, when the results for the energy loss rate per unit width of breaking front were expressed in terms of a breaker strength parameter b multiplying the fifth power of the wave speed, it is found that b was also strongly correlated with the parametric growth rate just before breaking. Measured values of b obtained in this investigation ranged systematically from 8 × 10−5 to 1.2 × 10−3. These are comparable with open ocean estimates reported in recent field studies.

For simple parallel shear flows on the f-plane and the equatorial β-plane we derive an energy norm for zonally invariant perturbations. It is used to derive the linear stability boundary for when these flows are inertially stable in the classical sense but may be destabilized due to unequal rates of diffusion of momentum and heat. The analysis is valid when there are arbitrary, zonally invariant, no-slip boundaries which are perfect thermal conductors.

The creeping flow around a sphere settling along the axis of a closed axisymmetric container is obtained both theoretically and experimentally. The numerical technique for solving the Stokes equations uses the classical Sampson expansion; the boundary conditions on the sphere are satisfied exactly and those on the container walls are applied in the sense of least squares. This is an extension to the axisymmetric case of the technique for solving various two-dimensional flow problems. Two types of axisymmetric container are considered here as examples: circular cylinders closed by planes at both ends, and cones closed by a base plane. Calculated streamlines patterns show various sets of eddies, depending upon the geometry and the sphere position. Results are in agreement with earlier Stokes flow calculations of eddies in corners and in closed containers. Experiments use laser interferometry to measure the vertical displacement of a steel bead a few millimetres in diameter settling in a container filled with a very viscous silicone oil. The Reynolds number based on the sphere radius is typically of the order of 10−5. The accuracy on the vertical displacement is 50nm. Experiments show that the motion towards the apex of a cone is much slower than that towards a plane; the bead takes hours to reach the micrometre size roughness asperities on a conical wall, as compared with minutes to reach those on a plane wall. The numerical results for the drag force are in excellent agreement with experiments both for the cylindrical and the conical containers. With standard computer accuracy, the present numerical technique applies when the gap between the sphere and the nearby wall is larger than about one radius. For a sphere in the vicinity of any plane horizontal wall, these results also match with a previous analytical solution. That solution is in excellent agreement with our experimental results at small distances from the wall (typically less than a diameter, depending on the sphere size).

This paper presents a large-Reynolds-number asymptotic analysis of viscous centre modes on an arbitrary axisymmetrical vortex with an axial jet. For any azimuthal wavenumber m and axial wavenumber k, the frequency of these modes is given at leading order by ω0 = mΩ0 + kW0 where Ω0 and W0 are the angular and axial velocities of the vortex at its centre. These modes possess a multi-layer structure localized in an O(Re−1/6) neighbourhood of the vortex. By a multiple-scale matching analysis, we demonstrate the existence of three different families of viscous centre modes whose frequency expands as ω(n) ∼ ω0 + Re−1/3ω1 + Re−1/2ω(n)2. One of these families is shown to have unstable eigenmodes when H0 = 2Ω0k(2kΩ0 − mW2) < 0 where W2 is the second radial derivative of the axial flow in the centre. The growth rate of these modes is given at leading order by σ ∼ (3/2)(H0/4)1/3Re−1/3. Our results prove that any vortex with a jet (or jet with swirl) such that Ω0W2 ≠ 0 is unstable if the Reynolds number is sufficiently large. The spatial structure of the viscous centre modes is obtained and simple approximations which capture the main feature of the eigenmodes are also provided.

The theoretical predictions are compared with numerical results for the q-vortex model (or Batchelor vortex) for Re ≥ 105. For all modes, a good agreement is demonstrated for both the frequency and the spatial structure.

We investigate the two-dimensional flow of a liquid foam around a circular obstacle by measuring all the local fields necessary to describe this flow: velocity, pressure, and bubble deformations and rearrangements. We show how our experimental set-up, a quasi-two-dimensional ‘liquid pool’ system, is adapted to the determination of these fields: the velocity and bubble deformations are easy to measure from two-dimensional movies, and the pressure can be measured by exploiting a specific feature of this system, a two-dimensional effective compressibility. To describe accurately neighbour swapping (so-called ‘T1’ processes), we propose a new, tensorial descriptor. All these quantities are evaluated via an averaging procedure that we justify by showing that the fluctuations of the fields are essentially Gaussian. The flow is extensively studied in a reference experimental case; the velocity presents an overshoot in the wake of the obstacle, and the pressure is maximum at the leading side and minimal at the trailing side. The study of the elastic deformations and of the velocity gradients shows that the transition between plug flow and yielded regions is smooth. Our tensorial description of T1s highlights their correlation both with the bubble deformations and the velocity gradients. A salient feature of the flow, notably for the velocity and T1 distribution, is a marked fore–aft asymmetry, the signature of the elastic behaviour of the foam. We show that the results do not change qualitatively when various control parameters (flow rate, bubble area, fluid fraction, bulk viscosity, obstacle size and boundary conditions) vary, identifying a robust quasi-static regime. These results are discussed in the framework of the foam rheology literature. A movie is available with the online version of the paper.

This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg–de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modelled by the Chezy law, but also, importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media.

We consider the dynamical processes by which a dense, porous object floating in a body of liquid becomes waterlogged and sinks. We first generalize the classic model of capillary rise in a porous medium to present an analytically tractable model of the process, which is valid for objects that are very shallow compared to their horizontal extent. We find an analytical expression for the time taken for the object to sink under this approximation. We use a series of boundary integral simulations to show that decreasing the horizontal extent of the object decreases the time taken to sink. We find that the results of these numerical simulations are in good quantitative agreement with a series of laboratory experiments. Finally, we discuss the implications of our work for pumice fragments, often found floating in open water after a volcanic eruption, occasionally even supporting plant, animal and human remains.

We address the sensitivity of large-eddy simulations (LES) to parametric uncertainty in the subgrid-scale model. More specifically, we investigate the sensitivity of the LES statistical moments of decaying homogeneous isotropic turbulence to the uncertainty in the Smagorinsky model free parameter Cs (i.e. the Smagorinsky constant). Our sensitivity methodology relies on the non-intrusive approach of the generalized Polynomial Chaos (gPC) method; the gPC is a spectral non-statistical numerical method well-suited to representing random processes not restricted to Gaussian fields. The analysis is carried out at Reλ, =, 100 and for different grid resolutions and Cs distributions. Numerical predictions are also compared to direct numerical simulation evidence. We have shown that the different turbulent scales of the LES solution respond differently to the variability in Cs. In particular, the study of the relative turbulent kinetic energy distributions for different Cs distributions indicates that small scales are mainly affected by changes in the subgrid-model parametric uncertainty.

The response of choked nozzles and supersonic diffusers to one-dimensional flow perturbations is investigated. Following previous arguments in the literature, small flow perturbations in a duct of spatially linear steady velocity distribution are determined by solution of a hyper-geometric differential equation. A set of boundary conditions is then developed that extends the existing work to a nozzle of arbitrary geometry. This analysis accommodates the motion of a plane shock wave and makes no assumption about the nozzle compactness. Numerical simulations of the unsteady, quasi-one-dimensional Euler equations are performed to validate this analysis and also to indicate the conditions under which the perturbations remain approximately linear.

The nonlinear response of compact choked nozzles and supersonic diffusers is also investigated. Simple analyses are performed to determine the reflected and transmitted waveforms, as well as conditions for unchoke, ‘over-choke’ and unstart. This analysis is also supported with results from numerical simulations of the Euler equations.

The lift force acting on a slender body of revolution that separates from a thin wing in supersonic flow is analysed using Prandtl–Glauert linearized theory, scattering theory and asymptotic methods. It is shown that this lift is associated with multi-scattering of the wing-induced shock wave by the body surface. The local and global lift coefficients are obtained in simple analytical forms. It is shown that the total lift is mainly induced by the first scattering. Contributions from second, third and higher scatterings are zero in the leading-order approximation. This greatly simplifies calculations of the lift force. The theoretical solution for the flow field is compared with numerical solutions of three-dimensional Euler equations and experimental data at free-stream Mach number 2. There is agreement between the theory and the computations for a wide range of shock-wave strength, demonstrating high elasticity of the leading-order asymptotic approximation. Theoretical and experimental distributions of the cross-sectional normal force coefficient agree satisfactorily, showing robustness of the analytical solution. This solution can be applied to the moderate supersonic (Mach numbers from 1.2 to 3) multi-body interaction problem for crosschecking with other computational or engineering methods.

The effects of the Coriolis force on the elliptical instability are studied experimentally in cylindrical and spherical rotating containers placed on a table rotating at a fixed rate . For a given set-up, changing the ratio ΩG of global rotation to flow rotation leads to the selection of various unstable modes due to the presence of resonance bands, in close agreement with the normal-mode theory. No instability occurs when ΩG varies between −3/2 and −1/2 typically. On decreasing ΩG toward −1/2, resonance bands are first discretized for ΩG<0 and progressively overlap for −1/2 ≪ ΩG < 0. Simultaneously, the growth rates and wavenumbers of the prevalent stationary unstable mode significantly increase, in quantitative agreement with the viscous short-wavelength analysis. New complex resonances have been observed for the first time for the sphere, in addition to the standard spin-over. We argue that these results have significant implications in geo- and astrophysical contexts.

Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re ≫ 1 and horizontal Froude number Fh ≪ 1 are presented. The results are interpreted on the basis of a scaling analysis of the governing equations. The analysis suggests that there are two different strongly stratified regimes according to the parameter . When , viscous forces are unimportant and lv scales as lv ∼ U/N (U is a characteristic horizontal velocity and N is the Brunt–Väisälä frequency) so that the dynamics of the flow is inherently three-dimensional but strongly anisotropic. When , vertical viscous shearing is important so that (lh is a characteristic horizontal length scale). The parameter is further shown to be related to the buoyancy Reynolds number and proportional to (lO/η)4/3, where lO is the Ozmidov length scale and η the Kolmogorov length scale. This implies that there are simultaneously two distinct ranges in strongly stratified turbulence when : the scales larger than lO are strongly influenced by the stratification while those between lO and η are weakly affected by stratification. The direct numerical simulations with forced large-scale horizontal two-dimensional motions and uniform stratification cover a wide Re and Fh range and support the main parameter controlling strongly stratified turbulence being . The numerical results are in good agreement with the scaling laws for the vertical length scale. Thin horizontal layers are observed independently of the value of but they tend to be smooth for < 1, while for > 1 small-scale three-dimensional turbulent disturbances are increasingly superimposed. The dissipation of kinetic energy is mostly due to vertical shearing for < 1 but tends to isotropy as increases above unity. When < 1, the horizontal and vertical energy spectra are very steep while, when > 1, the horizontal spectra of kinetic and potential energy exhibit an approximate k−5/3h-power-law range and a clear forward energy cascade is observed.

The relationship between the upstream boundary layer and the low-frequency, large-scale unsteadiness of the separated flow in a Mach 2 compression ramp interaction is investigated by performing wide-field particle image velocimetry (PIV) and planar laser scattering (PLS) measurements in streamwise–spanwise planes. Planar laser scattering measurements in the upstream boundary layer indicate the presence of spanwise strips of elongated regions of uniform momentum with lengths greater than 40δ. These long coherent structures have been observed in a Mach 2 supersonic boundary layer (Ganapathisubramani, Clemens & Dolling 2006) and they exhibit strong similarities to those that have been found in incompressible boundary layers (Tomkins & Adrian 2003; Ganapathisubramani, Longmire & Marusic 2003). At a wall-normal location of y/δ=0.2, the inferred instantaneous separation line of the separation region is found to oscillate between x/δ=−3 and −1 (where x/δ=0 is the ramp corner). The instantaneous spanwise separation line is found to respond to the elongated regions of uniform momentum. It is shown that high- and low-momentum regions are correlated with smaller and larger size of the separation region, respectively. Furthermore, the instantaneous separation line exhibits large-scale undulations that conform to the low- and high-speed regions in the upstream boundary layer. The low-frequency unsteadiness of the separation region/shock foot observed in numerous previous studies can be explained by a turbulent mechanism that includes these elongated regions of uniform momentum.

The effects of initial conditions on grid turbulence are investigated for low to moderate Reynolds numbers. Four grid geometries are used to yield variations in initial conditions and a secondary contraction is introduced to improve the isotropy of the turbulence. The hot-wire measurements, believed to be the most detailed to date for this flow, indicate that initial conditions have a persistent impact on the large-scale organization of the flow over the length of the tunnel. The power-law coefficients, determined via an improved method, also depend on the initial conditions. For example, the power-law exponent m is affected by the various levels of large-scale organization and anisotropy generated by the different grids and the shape of the energy spectrum at low wavenumbers. However, the results show that these effects are primarily related to deviations between the turbulence produced in the wind tunnel and true decaying homogenous isotropic turbulence (HIT). Indeed, when isotropy is improved and the intensity of the large-scale periodicity, which is primarily associated with round-rod grids, is decreased, the importance of initial conditions on both the character of the turbulence and m is diminished. However, even in the case where the turbulence is nearly perfectly isotropic, m is not equal to −1, nor does it show an asymptotic trend in x towards this value, as suggested by recent analysis. Furthermore, the evolution of the second- and third-order velocity structure functions satisfies equilibrium similarity only approximately.

The fragmentation of a laminar undulating liquid sheet flowing in quiescent air is investigated. Combining various observations and measurements we propose a sequential atomization scenario describing the overall sheet–drop transition in this configuration. The undulation results from a controlled primary Kelvin–Helmholtz instability. As the liquid travels through the undulating pattern, it experiences transient accelerations perpendicular to the sheet. These accelerations trigger a secondary instability responsible for the amplification of spanwise thickness modulations of the sheet. This mechanism, called the ‘wavy corridor’, is responsible for the sheet free edge indentations from which liquid ligaments emerge and break, forming drops. The final drop size distribution is of a Gamma type characterized by a unique parameter independent of the operating conditions once drop sizes are normalized by their mean.

The details of the start-up transient vortex structure that forms near the junction of an impulsively started plate and a stationary plate where a step jump in velocity occurs at the plate surfaces are investigated. Numerical simulations have been conducted in a geometry representative of recent experiments of this flow. The experiments did not have access to data at very early times following the impulsive start, but they did suggest that the flow undergoes transitions from a viscous-dominated phase to an inertia-dominated phase. The numerical simulations presented here are designed to explore the early viscous-dominated transients. The simulations show that when the non-dimensional time, τ = tU2/ν (t is the time that the plate has been in motion and ν is the kinematic viscosity), is less than 100, the development process is dominated by viscous forces. In this regime similarity scaling is used to collapse the data, which scale as . The simulation results at low τ, when evaluated using entrainment diagrams, show an unsteady transition process consisting of the following stages. Initially, the flow consists of a non-rotating vorticity front with a single critical point for τ < 40. For 40 < τ < 50, the flow has three critical points, two nodes and a saddle. A rotational leading jet head develops for τ > 50 as the outermost node evolves into a spiral focus. The simulations span the viscous range to the inertial range. In the inertial range, for τ > 103, the flow structure scales as t5/6, as was observed in the experiments.

Mean flow profiles, skin friction, and integral parameters for boundary layers developing naturally over a wide variety of fully aerodynamically rough surfaces are presented and discussed. The momentum thickness Reynolds number Reθ extends to values in excess of 47000 and, unlike previous work, a very wide range of the ratio of roughness element height to boundary-layer depth is covered (0.03 < h/δ > 0.5). Comparisons are made with some classical formulations based on the assumption of a universal two-parameter form for the mean velocity profile, and also with other recent measurements. It is shown that appropriately re-written versions of the former can be used to collapse all the data, irrespective of the nature of the roughness, unless the surface is very rough, meaning that the typical roughness element height exceeds some 50% of the boundary-layer momentum thickness, corresponding to about .