We recently lost a valuable colleague and friend, Paul Steen, and the text below is a brief overview of his life.

In the asymptotic theory of interaction noise from contra-rotating propellers with many blades, previous work in the frequency domain has shown that, for each combination mode/tone, the critical points that dominate the acoustic radiation can be linked to two criteria (sonic and normal-edge) and to a radial event line that spins around the annulus at the same speed as the mode. Thus the speed is different for each combination tone. In real time, however, the interactions precess around the annulus at a single speed, governed by the ‘firing order’ of the interactions, with that speed being (in general) considerably different from that of either the front or rear blade rows. The precession of the interaction is described here in detail and then demonstrated by application to three relevant architectures, including a rotor–stator configuration. The paper then considers how the blade leading-edge design affects the radial variation in the interaction location and analyses the interactions from the viewpoint of a far-field observer. The paper also connects previous time- and frequency-domain results by showing that the sonic and normal-edge criteria can be derived in the time domain using the precession speed. The case of equal blade numbers is also included, for which the precession speed becomes infinite but for which, nonetheless, the sonic and normal-edge criteria still apply.

Laboratory experiments are conducted to investigate the mechanism controlling the formation of stable and unstable acoustic fountains at the free surface of a quiescent body of water. Fountains are induced by focused ultrasonic, a new modality that allows for better spatiotemporal control of water flow. Particle image velocimetry was used to characterize the induced flow field in the vicinity of the ultrasonic focal spot. We used two types of ultrasonic transducers with distinct wave frequencies. We examined three fountain formation regimes by varying the pressure level of the transducers, namely weak, intermediate (stable) and highly forced fountains (explosive). Between different regimes, the fountain height underwent a step-change in response to the increase in acoustic pressure. A force estimation obtained from the flow field shows that the magnitude of axial momentum flux is orders of magnitude lower than that of gravity and surface tension, indicating that the dominant driving force for the fountain generation is the acoustic radiation force (Nightingale et al., Ultrasound Med. Biol., vol. 28, 2002, pp. 227–235). We propose a simple model to estimate the shape of a stable fountain; it accounts for the applied acoustic pressure, gravity, surface tension and axial momentum. The model neglects viscous force, which precludes capturing the intermediate fountain surface curvature. However, the model successfully predicts the geometry of the weak and intermediate fountains.

We report for the first time on triadic resonances in a rotating convection system. Using direct numerical simulations, we find that convective modes in a rotating spherical fluid can excite a pair of inertial modes whose frequencies and wavenumbers match the triadic resonance conditions. Depending on the structures of the convective modes, triadic resonances can lead to the growth of either a pair of modes with lower frequencies and wavenumbers, or a pair of modes with higher frequencies and wavenumbers, providing a possible mechanism for the bi-directional energy cascade. Increased thermal forcing leads to fully developed turbulence, which also exhibits wave-like motions, and is reminiscent of the energy spectrum of inertial wave turbulence. Our results suggest that the interaction of inertial waves plays an important role in rotating convection, which is of great importance in understanding the dynamics of planetary and stellar interiors.

A new lattice Boltzmann model for multicomponent ideal gas mixtures is presented. The model development consists of two parts. First, a new kinetic model for Stefan–Maxwell diffusion amongst the species is proposed and realized as a lattice Boltzmann equation on the standard discrete velocity set. Second, a compressible lattice Boltzmann model for the momentum and energy of the mixture is established. Both parts are consistently coupled through mixture composition, momentum, pressure, energy and enthalpy whereby a passive scalar advection–diffusion coupling is obviated, unlike in previous approaches. The proposed model is realized on the standard three-dimensional lattices and is validated with a set of benchmarks highlighting various physical aspects of compressible mixtures. Stefan–Maxwell diffusion is tested against experiment and theory of uphill diffusion of argon and methane in a ternary mixture with hydrogen. The speed of sound is measured in various binary and ternary compositions. We further validate the Stefan–Maxwell diffusion coupling with hydrodynamics by simulating diffusion in opposed jets and the three-dimensional Kelvin–Helmholtz instability of shear layers in a two-component mixture. Apart from the multicomponent compressible mixture, the proposed lattice Boltzmann model also provides an extension of the lattice Boltzmann equation to the compressible flow regime on the standard three-dimensional lattice.

This paper characterizes the geometry of converging near-elliptic shock waves at a Mach number of 6. The converging shocks are produced by elliptic conical surfaces with shapes made up from adjacent straight generators, each deflected a constant angle from the free-stream direction. Combined shock tunnel experiments and numerical analyses are conducted to depict the evolution of the converging shock waves for several elliptic entry aspect ratios $AR$s (i.e. the ratio of the major axis to the minor axis). It is revealed that the deviation from axial symmetry is amplified as the shock front approaches the centreline, which results in different shock interaction types compared with the axisymmetric case. Three typical shock interaction types are classified depending on various ARs. For a small AR, faster shock strengthening in the major plane dominates, although a Mach reflection (type A) that resembles the axisymmetric flow field is formed. However, for a sufficiently large AR, the shock strengthening is eventually terminated by the intersection of the weaker shocks in the minor plane owing to their smaller off-centre distances, which results in a regular reflection (type B). Between these two interaction patterns, there is a critical AR for which both the shock fronts in the major and minor planes intersect at the centreline coincidentally, and this critical intersection (type C) exhibits an extreme case of a shock front converging to a singular point. This study indicates that deviation from axial symmetry affects the evolution of the shock structures in converging flow.

We present a theoretical and computational study of the mechanics of a jet formed from a large amplitude, axisymmetric, capillary-gravity wave on the surface of a liquid pool in a cylindrical container. Jetting can cause a pronounced overshoot of the interface at the axis of symmetry. A linear theory presented earlier in Farsoiya et al. (J. Fluid Mech., vol. 826, 2017, pp. 797–818) was shown to be incapable of describing this jet. To understand its mechanics, we present here the inviscid, weakly nonlinear solution to the initial value problem where the initial surface perturbation is a single Fourier–Bessel mode on quiescent fluid. The theory predicts that energy injected into a primary (Bessel) mode initially is transferred nonlinearly to a spectrum of modes. The extent of the theoretically predicted energy transfer is found to be very accurate for modes up to the second harmonic. We show using numerical simulations that the jet originates as a small dimple formed at the trough of the wave, analogous to similar observations in bubble cavity collapse (Duchemin et al., Phys. Fluids, vol. 14, 2002, pp. 3000–3008; Lai et al., Phys. Rev. Lett. vol. 121, 2018, 144501). The theory is able to describe the jet overshoot and the velocity and pressure fields in the liquid qualitatively, but does not capture the temporal evolution of the dimple or the thinning of the jet neck leading to pinchoff. Modal analysis shows that the latter phenomenon requires higher-order approximations, beyond the second order presented here. The nonlinear theory yields explicit analytical expressions without any fitting parameters which are systematically tested against numerical simulations of the incompressible Euler equation. The theory contains cylindrical analogues of the singularities corresponding to second harmonic resonance (Wilton, Lond. Edinb. Dubl. Phil. Mag. J. Sci., vol. 29, 1915, pp. 688–700). The connection of these to triadic resonant interactions among capillary-gravity waves in a cylindrical confined geometry is discussed.

The linear stability of a periodic array of vortices in stratified fluid is studied by modal stability analysis. Two base flows are considered: the two-dimensional Taylor–Green vortices and the Stuart vortices. In the case of the two-dimensional Taylor–Green vortices, four types of instability are identified: the elliptic instability, the pure hyperbolic instability, the strato-hyperbolic instability and the mixed hyperbolic instability, which is a mixture of the pure hyperbolic and the strato-hyperbolic instabilities. Although the pure hyperbolic instability is most unstable for the non-stratified case, it is surpassed by the strato-hyperbolic instability and the mixed hyperbolic instability for the stratified case. The strato-hyperbolic instability is dominant at large wavenumbers. Its growth rate tends to a constant along each branch in the large-wavenumber and inviscid limit, implying that the strato-hyperbolic instability is not stabilized by strong stratification. Good agreement between the structure of the strato-hyperbolic instability mode and the corresponding local solution is observed. In the case of the Stuart vortices, the unstable modes are classified into three types: the pure hyperbolic instability, the elliptic instability and the mixed-type instability, which is a mixture of the pure hyperbolic and the elliptic instabilities. Stratification decreases the growth rate of the elliptic instability, which is expected to be stabilized by stronger stratification, although it is not completely stabilized within the range of Froude numbers considered. The present results imply that both the pure hyperbolic instability and the strato-hyperbolic instability are important in stably stratified flows of geophysical or planetary scale.

The collapse under surface tension of a long axisymmetric capillary, held at both ends and softened by a travelling heater, is used to determine the viscosity or surface tension of silica glasses. Capillary collapse is also used in the manufacture of some optical fibre preforms. Typically, a one-dimensional (1-D) model of the closure of a concentric fluid annulus is used to relate a measure of the change in the cross-sectional geometry, for example the external radius, to the desired information. We here show that a two-dimensional (2-D) asymptotic model developed for drawing of optical fibres, but with a unit draw ratio, may be used and yields analytic formulae involving a single dimensionless parameter, the scaled heater speed $V$, equivalently a capillary number. For a capillary fixed at both ends, this 2-D model agrees with the 1-D model and offers the significant benefit that it enables determination of both the surface tension and viscosity from a single capillary-collapse experiment, provided the pulling tension in the capillary during collapse is measured. The 2-D model also enables our investigation of the situation where both ends of the capillary are not fixed, so that the capillary cannot sustain a pulling tension. Then the collapse of the capillary is markedly different from that predicted by the 1-D model and the ability to determine both surface tension and viscosity is lost.

The unsteady behaviour of a rarefied gas caused by a sudden change of the angular velocity of a sphere, placed in an otherwise quiescent gas, is investigated based on the linearized Bhatnagar–Gross–Krook model of the Boltzmann equation and the diffuse reflection boundary condition. The initial and boundary value problem is solved numerically by the method of characteristics, which is capable of tracking the discontinuity of the velocity distribution function moving in the phase space. The transient behaviour of the macroscopic quantities, such as the circumferential flow velocity and shear stress as well as the heat flow around the sphere, is clarified for a wide range of the Knudsen number. Furthermore, the long-time behaviour of the macroscopic quantities is elucidated, that is, they approach terminal values with a rate $t^{-3/2}$ for $t\gg 1$, with $t$ being a time variable. The analytical expression for the free molecular gas as well as for the slip flow is obtained. It is found that the circumferential heat flow reverses its direction as time proceeds when the Knudsen number is finite. More precisely, the direction is the same as that of the circumferential velocity of the sphere in the initial stage and opposite in the final stage, being reversed at some point of time depending on the distance from the sphere. This makes a clear contrast with the case of a free molecular gas, for which the heat flow is always in the direction of the sphere rotation in finite time and vanishes in the long-time limit.

In the present study, high-speed droplet impingement on typical curved surfaces is numerically investigated to analyse the inherent complex wave structures and cavitation. A three-component compressible multi-phase flow model is utilised considering fluid phase transitions, but the calculation of coupling with the solid structure is neglected. A detailed comparative analysis is presented of the dynamic processes, including the evolution of confined water-hammer shock waves, occurrence and collapse of cavities and spatiotemporal pressure distribution on concave, convex and flat surfaces. The synclastic curvature of a concave surface can increase a shock wave's strength, but an incongruous curvature can decrease its strength and a flat surface has moderate intensity. Both homogenous and near-surface heterogeneous cavitation can occur in three cases; the cavitation is the strongest in the concave case and, hence, the collapse waves are strongest running toward the surface. The pressure wave distributions and their evolutions are more complex in curved surface impacts than in flat surfaces. Both the confined shock wave inside the impacted droplet and near-surface lateral jet are weakest, and the near-surface cavitation level is also lowest in the convex case. Therefore, it can be inferred that a convex surface can reduce the possible surface damage during high-speed impingement. The two-dimensional axisymmetric numerical results show that both the converging and diverging motions of waves intensify, which further increases the curvature influence on concave surface damage.

The interaction between an incident shock wave and a Mach-6 undisturbed hypersonic laminar boundary layer over a cold wall is addressed using direct numerical simulations (DNS) and wall-modelled large-eddy simulations (WMLES) at different angles of incidence. At sufficiently high shock-incidence angles, the boundary layer transitions to turbulence via breakdown of near-wall streaks shortly downstream of the shock impingement, without the need of any inflow free-stream disturbances. The transition causes a localized significant increase in the Stanton number and skin-friction coefficient, with high incidence angles augmenting the peak thermomechanical loads in an approximately linear way. Statistical analyses of the boundary layer downstream of the interaction for each case are provided that quantify streamwise spatial variations of the Reynolds analogy factors and indicate a breakdown of the Morkovin's hypothesis near the wall, where velocity and temperature become correlated. A modified strong Reynolds analogy with a fixed turbulent Prandtl number is observed to perform best. Conventional transformations fail at collapsing the mean velocity profiles on the incompressible log law. The WMLES prompts transition and peak heating, delays separation and advances reattachment, thereby shortening the separation bubble. When the shock leads to transition, WMLES provides predictions of DNS peak thermomechanical loads within $\pm 10\,\%$ at a computational cost lower than DNS by two orders of magnitude. Downstream of the interaction, in the turbulent boundary layer, the WMLES agrees well with DNS results for the Reynolds analogy factor, the mean profiles of velocity and temperature, including the temperature peak, and the temperature/velocity correlation.

We present a new data reconstruction method with supervised machine learning techniques inspired by super resolution and inbetweening to recover high-resolution turbulent flows from grossly coarse flow data in space and time. For the present machine-learning-based data reconstruction, we use the downsampled skip-connection/multiscale model based on a convolutional neural network, incorporating the multiscale nature of fluid flows into its network structure. As an initial example, the model is applied to the two-dimensional cylinder wake at $Re_D = 100$. The reconstructed flow fields by the present method show great agreement with the reference data obtained by direct numerical simulation. Next, we apply the current model to a two-dimensional decaying homogeneous isotropic turbulence. The machine-learned model is able to track the decaying evolution from spatial and temporal coarse input data. The proposed concept is further applied to a complex turbulent channel flow over a three-dimensional domain at $Re_{\tau }=180$. The present model reconstructs high-resolved turbulent flows from very coarse input data in space, and also reproduces the temporal evolution for appropriately chosen time interval. The dependence on the number of training snapshots and duration between the first and last frames based on a temporal two-point correlation coefficient are also assessed to reveal the capability and robustness of spatio-temporal super resolution reconstruction. These results suggest that the present method can perform a range of flow reconstructions in support of computational and experimental efforts.

We analyse both numerically and experimentally the stability of the steady jetting tip streaming produced by focusing a liquid stream with another liquid current when they coflow through the orifice of an axisymmetric nozzle. We calculate the global eigenmodes characterizing the response of this configuration to small-amplitude perturbations. In this way, the critical conditions leading to the instability of the steady jetting tip streaming are determined. The unstable perturbations are classified according to their oscillatory character and to the region where they originate (convective and absolute instability). We derive and explain in terms of the velocity field a simple scaling law to predict the diameter of the emitted jet. The numerical stability limits are compared with experimental results, finding reasonable agreement. The experiments confirm the existence of the two instability mechanisms predicted by the global stability analysis.

When body of fluid with a salinity gradient is heated from an isolated vertical wall instabilities can form. These have been observed in many experiments. The evolving temperature boundary layer causes the fluid to rise and generates horizontal salinity gradients and vertical shear. These background temperature and salinity gradients and the shear can all drive (or inhibit) the instabilities. The time-dependent nature of the background gradients has previously restricted linear stability analysis to some limits where a quasi-static assumption could be made. However, in many of the experiments this assumption is not valid. We investigate the instabilities over the full range of salinity gradients and applied heating, from what is essentially the thermal problem to the strong salinity gradient limit, with two different heating rates. The approach taken is to find the optimal evolution of a quadratic energy-like measure of the amplitude of the instabilities as the background state evolves. This involves a matrix optimization problem. The choice of quadratic measure is not predetermined, but selected to minimize this optimal growth. This approach has been developed previously for the purely thermal case of heating from isolated horizontal and vertical boundaries, and to the double-diffusive problem of heating a salinity gradient from a horizontal lower boundary. We show that there are three regimes of instability when a salinity gradient is heated from a sidewall. There are the small and large Prandtl number regimes observed previously in the purely thermal problem. As the salinity gradient increases the shear-driven small Prandtl number mode is suppressed and only the large Prandtl number mode is observed. For larger salinity gradients a double-diffusive mode of instability appears, which initially has an order-one aspect ratio. As the salinity gradient further increases it evolves into the thin almost horizontal intrusions of the quasi-static analysis. These findings are in line with experimental observations.

We investigate the Marangoni instability in a thin polymeric liquid film heated from the free surface. The polymeric solutions are usually a binary mixture of a Newtonian solvent with a polymeric solute, and exhibit viscoelastic behaviour. In the presence of a temperature gradient, stratification of these solutes can take place via the Soret effect, giving rise to the solutocapillary effect at the free surface. Considering this cross-diffusive effect and incorporating the effects of gravity, here we analyse the stability characteristics of this polymeric film when bounded between its deformable free surface and a poorly conductive rigid substrate. Linear stability analysis around the quiescent base state reveals that, under the combined influences of thermosolutocapillarity and the elasticity of the liquid, apart from the monotonic disturbances, two different oscillatory instabilities can emerge in this system. The characteristics of each instability mode are discussed, and a complete stability picture is perceived in terms of the phase diagrams, identifying the model parameter regimes for which a particular instability mode becomes dominant.

In a channel flow with a sudden expansion, whether for three-dimensional (3-D) pipe and channel flows, or for two-dimensional (2-D) channel flow, it is known that increasing the Reynolds number beyond a critical value $Re_c$ induces a symmetry breaking Pitchfork bifurcation. The linear stability analysis of the symmetric steady solution enables the $Re_c$ to be determined efficiently and thus the influence of the expansion ratio ($ER$), defined as the ratio between upstream and downstream diameter regarding the expansion, to be explored. In this study, we investigate the behaviour of the flow after 2-D sudden expansions while varying the $ER$ and the inlet flow profile, e.g. corresponding to a transition profile between a plug and a Poiseuille flow that could be reached for a flow after a sudden constriction upstream. Results demonstrate that imposing a plug flow at the inlet gives a higher $Re_c$ than any other profile and that the concomitant recirculation zones are shorter. We show that these results can be rationalized using basic convection–diffusion arguments.

High-fidelity measurements of velocity and concentration are carried out in a neutral jet (NJ) and a negatively buoyant jet (NBJ) by injecting a jet of fresh water vertically downwards into ambient fresh and saline water, respectively. The Reynolds number ($Re$) based on the pipe inlet diameter ($d$) and the source velocity ($W_o$) is approximately 5900 in all the experiments, while the source Froude number based on density difference is approximately 30 in the NBJ experiments. Velocity and concentration measurements are obtained in the region $17 \leq z/d \leq 40$ ($z$ being the axial coordinate) using particle image velocimetry and planar laser induced fluorescence techniques, respectively. Consistent with the literature on jets, the centreline velocity ($W_c$) decays as $z^{-1}$ in the NJ, but in the NBJ, $W_c$ decays faster along $z$ due to the action of negative buoyancy. Nonetheless, the mean velocity ($W$) and concentration ($C$) profiles in both the flows exhibit self-similar Gaussian form, when scaled by the local centreline parameters ($W_c,C_c$) and the jet half-widths ($r^\ast _{W},r^\ast _{C}$). On the other hand, the turbulence statistics and Reynolds stress in the NBJ do not scale with $W_c$. The results of autocorrelation functions, integral length scales and two-dimensional correlation maps show the similarity of turbulence structure in the NJ and the NBJ when the axial and radial distances are normalised by the local jet half-width. Further, the spectra and probability density functions are similar on the axis and only minor differences are seen near the jet interface. The above findings are fundamentally consistent with our recent analysis (Milton-McGurk et al., J. Fluid Mech., 2020b), where we observed that the mean and turbulence statistics in the NBJ have different development characteristics. Overall, we find that the turbulence structure of the NBJ (when scaled by local velocity and length scales) is very similar to the momentum-driven NJ, and the differences (e.g. spreading rate, scaling of turbulence intensities, etc.) between the NJ and the NBJ seem to be of secondary importance.

The development of free-stream disturbances in flow over a vertically vibrating flat plate with a slender leading edge is investigated. The evolution of the optimal inflow perturbation that results in the maximum amplification is computed to investigate the effect of the plate vibration on the development of free-stream disturbance, secondary instability of streaks and subsequently the bypass transition to turbulence. It is observed that the plate vibration leads to periodic change of the angle of attack, shifting the free-stream disturbance to the upper or lower side of the plate. Therefore, the development of steady inflow perturbations, which receive the largest amplification, is interrupted by the vibration, and the perturbation amplification via the lift-up mechanism is weakened. The vibration brings a second peak of perturbation growth at the vibration frequency, leading to high-frequency free-stream perturbations penetrating into the base boundary layer, which is not observed in flow over a stationary plate owing to the sheltering mechanism. This resonance of the flow perturbation and the vibrating plate is explained by the staggering effect of the leading edge. Further, the direct numerical simulations with the optimal inflow perturbation imposed on the inflow boundary show that the vertical vibration of the plate leads to streamwise periodic vorticity near the edge of the boundary layer. This inhomogeneity of the streamwise vorticity brings about streamwisely localized distortion of the low-speed streaks and, thus, an intermittent secondary instability. Therefore, before the streaks break down to turbulence, they undergo several rounds of secondary instabilities, resulting in an elongated bypass transition process.

Instability to Tollmien–Schlichting waves is one of the primary routes to transition to turbulence for two-dimensional boundary layers in quiet disturbance environments. Cancellation of Tollmien–Schlichting waves using surface heating was first demonstrated in the experiments of Liepmann et al. (J. Fluid Mech., vol. 118, 1982, pp. 187–200) and Liepmann & Nosenchuck (J. Fluid Mech., vol. 118, 1982, pp. 201–204). Here we consider a similar theoretical formulation that includes the effects of localised (unsteady) wall heating/cooling. The resulting problem is closely related to that of Terent'ev (Prikl. Mat. Mekh., vol. 45, 1981, pp. 1049–1055; Prikl. Mat. Mekh., vol. 48, 1984, pp. 264–272) on the generation of Tollmien–Schlichting waves by a vibrating ribbon, but with thermal effects. The nonlinear receptivity problem based on triple-deck scales is formulated and the linearised version solved both analytically as well as numerically. The most significant result is that the wall heating/cooling function can be chosen such that there is no pressure response to the disturbance, meaning there is no generation of Tollmien–Schlichting waves. Numerical calculations substantiate this with an approximation based on the exact analytical result. Previous numerical studies of the unsteady triple-deck equations have shown difficulties in capturing the convective wave packet that develops in the initial-value problem and we show that these arise from the choice of time steps as well as the range of the Fourier modes taken.