Direct numerical simulations of a temporally developing, low-speed, variable-density, turbulent, plane mixing layer are performed. The Navier–Stokes equations in the low-Mach-number approximation are solved using a novel algorithm based on an extended version of the velocity–vorticity formulation used by Kim et al. (J. Fluid Mech., vol 177, 1987, 133–166) for incompressible flows. Four cases with density ratios $s=1,2,4$ and 8 are considered. The simulations are run with a Prandtl number of 0.7, and achieve a $Re_{\unicode[STIX]{x1D706}}$ up to 150 during the self-similar evolution of the mixing layer. It is found that the growth rate of the mixing layer decreases with increasing density ratio, in agreement with theoretical models of this phenomenon. Comparison with high-speed data shows that the reduction of the growth rates with increasing density ratio has a weak dependence with the Mach number. In addition, the shifting of the mixing layer to the low-density stream has been characterized by analysing one-point statistics within the self-similar interval. This shifting has been quantified, and related to the growth rate of the mixing layer under the assumption that the shape of the mean velocity and density profiles do not change with the density ratio. This leads to a predictive model for the reduction of the growth rate of the momentum thickness, which agrees reasonably well with the available data. Finally, the effect of the density ratio on the turbulent structure has been analysed using flow visualizations and spectra. It is found that with increasing density ratio the longest scales in the high-density side are gradually inhibited. A gradual reduction of the energy in small scales with increasing density ratio is also observed.

We present results from experiments designed to measure near-surface turbulence generated by rainfall. Laboratory experiments were performed using artificial rain falling at near-terminal velocity in a wind–wave channel filled with synthetic seawater. In this first series of experiments, no wind was generated and the receiving seawater was initially at rest. Rainfall rates from 40 to $190~\text{mm}~\text{h}^{-1}$ were investigated. Subsurface turbulent velocities of the order of $O(10^{-2})~\text{m}~\text{s}^{-1}$ are generated near the interface below the depth of the cavities generated by the rain drop impacts. The turbulence appears independent of rainfall rates. At depth larger than the size of the cavities, the turbulent velocity fluctuations decay as $z^{-3/2}$. Turbulent length scales also appear to scale with the size of the impact cavities. In these seawater experiments, a freshwater lens is established at the water surface due to the rain. At the highest rain rate studied, the resulting buoyancy flux appears to lead to a shallower subsurface mixed layer and a slight decrease of the turbulent kinetic energy dissipation. Finally, direct measurements and inertial estimates of the turbulent kinetic energy dissipation show that approximately 0.1–0.3 % of the kinetic energy flux from the rain is dissipated in the form of turbulence. This is consistent with existing freshwater measurements and suggests that high levels of dissipation occur at depths and scales smaller than those resolved here and/or that other phenomena dissipate a considerable amount of the total kinetic energy flux provided by rainfall.

A numerical investigation of normal-mode perturbations of a two-dimensional periodic finite-amplitude gravity wave propagating on a vertically sheared current of constant vorticity is considered. For this purpose, an extension of the method developed by Rienecker & Fenton (J. Fluid Mech., vol. 104, 1981, pp. 119–137) is used for the numerical computations of the finite-amplitude waves on a linear shear current. This method enables to compute accurately waves with or without critical layers and pressure anomalies. The numerical results of the linear stability analysis extend the weakly nonlinear analytical results of Thomas et al. (Phys. Fluids, vol. 24, 2012, 127102) to fully nonlinear waves. In particular, the restabilization of the Benjamin–Feir modulational instability, whatever the depth, for an opposite shear current is confirmed. For these sideband instabilities, the numerical results show some deviations with the weakly nonlinear theory as the wave steepness of the basic wave and vorticity are increased. Besides the modulational instabilities, new instability bands corresponding to quartet and quintet instabilities, which are not sideband disturbances, are discovered. The present numerical results show that with opposite shear currents, increasing the shear reduces the growth rate of the most unstable sideband instabilities but enhances the growth rate of these quartet instabilities, which eventually dominate the Benjamin–Feir modulational instabilities.

The generation of internal gravity waves by a vertically oscillating cylinder that is tilted to the horizontal in a stratified Boussinesq fluid of constant buoyancy frequency, $N$, is investigated. This variant of the widely studied horizontal configuration – where a cylinder aligned with a plane of constant gravitational potential induces four wave beams that emanate from the cylinder, forming a cross pattern known as the ‘St. Andrew’s Cross’ – brings out certain unique features of radiated internal waves from a line source tilted to the horizontal. Specifically, simple kinematic considerations reveal that for a cylinder inclined by a given angle $\unicode[STIX]{x1D719}$ to the horizontal, there is a cutoff frequency, $N\sin \unicode[STIX]{x1D719}$, below which there is no longer a radiated wave field. Furthermore, three-dimensional effects due to the finite length of the cylinder, which are minor in the horizontal configuration, become a significant factor and eventually dominate the wave field as the cutoff frequency is approached; these results are confirmed by supporting laboratory experiments. The kinematic analysis, moreover, suggests a resonance phenomenon near the cutoff frequency as the group-velocity component perpendicular to the cylinder direction vanishes at cutoff; as a result, energy cannot be easily radiated away from the source, and nonlinear and viscous effects are likely to come into play. This scenario is examined by adapting the model for three-dimensional wave beams developed in Kataoka & Akylas (J. Fluid Mech., vol. 769, 2015, pp. 621–634) to the near-resonant wave field due to a tilted line source of large but finite length. According to this model, the combination of three-dimensional, nonlinear and viscous effects near cutoff triggers transfer of energy, through the action of Reynolds stresses, to a circulating horizontal mean flow. Experimental evidence of such an induced mean flow near cutoff is also presented.

Premixed low swirling flames (methane–air and hydrogen–methane–air) are experimentally investigated for three different regimes. Stable, local transient to instability and limit cycle regimes corresponding to three distinct equivalence ratios are considered. Dynamic mode decomposition is applied to the hydrogen–air–methane flame to retrieve the modes frequencies, growth rates and spatial distributions for each regime. The results indicate that a vortical wave propagating along the flame front is associated with the transition from stability to instability. In addition, it is shown that a key effect on stability is the location of the non-oscillating (0 Hz) flame component. The phase-averaged unsteady motion of the flames over one cycle of oscillation shows the vortical wave rolling up the flame front. The Rayleigh index maps are formed to identify the region of driving and damping of the self-sustained oscillation, while the flame transfer function phase leads to the propagation mode of the perturbations along the flame front. The second mechanism identified concerns the swirl number fluctuation induced by the mode conversion. By utilizing hypotheses for the flow field and the flame structure, it is pointed out that those mechanisms are at work for both flames (methane–air and hydrogen–methane–air) and their effects on the unsteady heat release are determined. Both unsteady heat release contributions, the vortical wave induces flame surface fluctuations and swirl number oscillation induces unsteady turbulent burning velocity, are in phase opposition and of similar amplitudes.

We perform numerical simulations of a turbulent channel flow over an hyper-elastic wall. In the fluid region the flow is governed by the incompressible Navier–Stokes (NS) equations, while the solid is a neo-Hookean material satisfying the incompressible Mooney–Rivlin law. The multiphase flow is solved with a one-continuum formulation, using a monolithic velocity field for both the fluid and solid phase, which allows the use of a fully Eulerian formulation. The simulations are carried out at Reynolds bulk $Re=2800$ and examine the effect of different elasticity and viscosity of the deformable wall. We show that the skin friction increases monotonically with the material elastic modulus. The turbulent flow in the channel is affected by the moving wall even at low values of elasticity since non-zero fluctuations of vertical velocity at the interface influence the flow dynamics. The near-wall streaks and the associated quasi-streamwise vortices are strongly reduced near a highly elastic wall while the flow becomes more correlated in the spanwise direction, similarly to what happens for flows over rough and porous walls. As a consequence, the mean velocity profile in wall units is shifted downwards when shown in logarithmic scale, and the slope of the inertial range increases in comparison to that for the flow over a rigid wall. We propose a correlation between the downward shift of the inertial range, its slope and the wall-normal velocity fluctuations at the wall, extending results for the flow over rough walls. We finally show that the interface deformation is determined by the fluid fluctuations when the viscosity of the elastic layer is low, while when this is high the deformation is limited by the solid properties.

Plumes occur in many natural and industrial settings, such as chimney smoke, volcanic eruptions and deep water oil spills. A plume function, $\unicode[STIX]{x1D6E4}$, is used to characterize plumes and jets. The far-field behaviour of these flows has been studied in great detail while the near-field behaviour has not quite received the same attention. We examine near-field phenomena such as radial constriction, termed necking, and vortex structure formations with new high resolution direct numerical simulations. Four lazy plumes with increasing values of the source plume parameter, $\unicode[STIX]{x1D6E4}_{0}$, are simulated. We study the evolution of entrainment and the plume function. The original assumptions, that Reynolds stresses dominate viscous shear stresses, do not hold for lazy plumes in the near field. Due to this, a deviation from self-similarity occurs initially and is corrected by a large entrainment coefficient caused by vortex stretching and compression. After correcting for the virtual origin, comparison between theory and simulations shows a monotonic decay of $\unicode[STIX]{x1D6E4}$ towards pure plume behaviour. The entrainment coefficient asymptotes to a widely accepted constant value for plumes.

The unsteady flow estimation problem of wall-bounded turbulence, numerically benchmarked by Chevalier et al. (J. Fluid Mech., vol. 552, 2006, pp. 167–187), is re-tackled with simple approaches. A turbulent channel flow at $Re_{\unicode[STIX]{x1D70F}}=100$ with periodic boundary conditions is reconstructed with linear stochastic estimation only based on the wall measurement, i.e. the wall shear stress in the streamwise and spanwise directions as well as the wall pressure over the entire wavenumber space. The results reveal that instantaneous information on the wall governs the success of the estimation in the vicinity of the wall ($y^{+}\lesssim 20$). The degrees of agreement are equivalent to those reported by Chevalier et al. using the extended Kalman filter as well as the ensemble Kalman filter performed in this study. This suggests that the instantaneous information on the wall dictates the reconstruction rather than the prediction step in these state observers solving the dynamical system. Subsequently, we feed the velocity components given by the linear stochastic estimation via the body-force term into the Navier–Stokes system: such an observer slightly improves the estimation in the log layer, indicating a small benefit of involving a dynamical system but over-suppression of turbulent motions beyond the viscous sublayer due to their low correlation with the wall measurement. Errors in the estimation grow in the buffer layer and prevent further reconstruction toward the centreline even if we relax the feedback forcing and let the flow evolve nonlinearly through the observer. We also discuss the flow components truly reconstructible from the wall measurement, which has limited degrees of freedom and poor correlation across wavenumbers.

Considering structure functions of the streamwise velocity component in a framework akin to the extended self-similarity hypothesis (ESS), de Silva et al. (J. Fluid Mech., vol. 823, 2017, pp. 498–510) observed that remarkably the large-scale (energy-containing range) statistics in canonical wall-bounded flows exhibit universal behaviour. In the present study, we extend this universality, which was seen to encompass also flows at moderate Reynolds number, to Taylor–Couette flow. In doing so, we find that also the transversal structure function of the spanwise velocity component exhibits the same universal behaviour across all flow types considered. We further demonstrate that these observations are consistent with predictions developed based on an attached-eddy hypothesis. These considerations also yield a possible explanation for the efficacy of the ESS framework by showing that it relaxes the self-similarity assumption for the attached-eddy contributions. By taking the effect of streamwise alignment into account, the attached-eddy model predicts different behaviour for structure functions in the streamwise and in the spanwise directions and that this effect cancels in the ESS framework – both consistent with the data. Moreover, it is demonstrated here that also the additive constants, which were previously believed to be flow dependent, are indeed universal at least in turbulent boundary layers and pipe flow where high Reynolds number data are currently available.

Wetting of a planar solid substrate is investigated in the presence of a macroscopic particle in the complete wetting regime. A drop of silicone oil spreads on the substrate and its macroscopic edge is incident on the particle at the late stage of spreading. The drop–particle interaction is observed in detail by shadowgraph and interferometry. Although the spreading drop edge is pinned by the particle for a short time, a sharp acceleration occurs when the liquid starts wetting the extra surface area offered by the particle and forming a meniscus. This process yields a net gain in spreading speed. A theoretical model based on the classical wetting dynamics dictated by Cox’s law is developed. It predicts that the capillary energy of the meniscus gives rise to a rapid motion of the liquid edge, showing good agreement with the dynamics observed in the experiments.

Wavepackets obtained as solutions of the flow equations linearised around the mean flow have been shown in recent work to yield good agreement, in terms of amplitude and phase, with those educed from turbulent jets. Compelling agreement has been demonstrated, for the axisymmetric and first helical mode, up to Strouhal numbers close to unity. We here extend the range of validity of wavepacket models to Strouhal number $St=4.0$ and azimuthal wavenumber $m=4$ by comparing solutions of the parabolised stability equations with a well-validated large-eddy simulation of a Mach 0.9 turbulent jet. The results show that the near-nozzle dynamics can be correctly described by the homogeneous linear model, the initial growth rates being accurately predicted for the entire range of frequencies and azimuthal wavenumbers considered. Similarly to the lower-frequency wavepackets reported prior to this work, the high-frequency linear waves deviate from the data downstream of their stabilisation locations, which move progressively upstream as the frequency increases.

This experimental study focuses on the effect of horizontal boundaries with pyramid-shaped roughness elements on the heat transfer in rotating Rayleigh–Bénard convection. It is shown that the Ekman pumping mechanism, which is responsible for the heat transfer enhancement under rotation in the case of smooth top and bottom surfaces, is unaffected by the roughness as long as the Ekman layer thickness $\unicode[STIX]{x1D6FF}_{E}$ is significantly larger than the roughness height $k$. As the rotation rate increases, and thus $\unicode[STIX]{x1D6FF}_{E}$ decreases, the roughness elements penetrate the radially inward flow in the interior of the Ekman boundary layer that feeds the columnar Ekman vortices. This perturbation generates additional thermal disturbances which are found to increase the heat transfer efficiency even further. However, when $\unicode[STIX]{x1D6FF}_{E}\approx k$, the Ekman boundary layer is strongly perturbed by the roughness elements and the Ekman pumping mechanism is suppressed. The results suggest that the Ekman pumping is re-established for $\unicode[STIX]{x1D6FF}_{E}\ll k$ as the faces of the pyramidal roughness elements then act locally as a sloping boundary on which an Ekman layer can be formed.

Accurate prediction of the spatial evolution of turbulent wake flows under pressure gradient conditions is required in some engineering applications such as the design of high-lift devices and wind farms over topography. In this paper, we aim to develop an analytical model to predict the evolution of a turbulent planar wake under an arbitrary pressure gradient condition. The model is based on the cross-stream integration of the streamwise momentum equation and uses the self-similarity of the mean flow. We have also made an experimentally supported assumption that the ratio of the maximum velocity deficit to the wake width is independent of the imposed pressure gradient. The asymptotic response of the wake to the pressure gradient is also investigated. After its derivation, the model is successfully validated against experimental data by comparing the evolution of the wake width and maximum velocity deficit. The inputs of the model are the imposed pressure gradient and the wake width under zero pressure gradient. The model does not require any parameter tuning and is deemed to be practical, computationally fast, accurate enough, and therefore useful for the scientific and engineering communities.