We consider a layer of an inviscid fluid with a free surface which is subject to vertical high-frequency vibrations. We derive three asymptotic systems of equations that describe slowly evolving (in comparison with the vibration frequency) free-surface waves. The first set of equations is obtained without assuming that the waves are long. These equations are as difficult to solve as the exact equations for irrotational water waves in a non-vibrating fluid. The other two models describe long waves. These models are obtained under two different assumptions about the amplitude of the vibration. Surprisingly, the governing equations have exactly the same form in both cases (up to the interpretation of some constants). These equations reduce to the standard dispersionless shallow-water equations if the vibration is absent, and the vibration manifests itself via an additional term which makes the equations dispersive and, for small-amplitude waves, is similar to the term that would appear if surface tension were taken into account. We show that our dispersive shallow-water equations have both solitary and periodic travelling wave solutions and discuss an analogy between these solutions and travelling capillary–gravity waves in a non-vibrating fluid.

The deformation of a weakly conducting, ‘leaky dielectric’, drop in a density matched, immiscible weakly conducting medium under a uniform direct current (DC) electric field is quantified computationally. We exclusively consider prolate drops, for which the drop elongates in the direction of the applied field. Furthermore, for the majority of this study, we assume the drop and medium to have equal viscosities. Using axisymmetric boundary integral computations, we delineate drop deformation and breakup regimes in the $Ca_{E}-Re_{E}$ parameter space, where $Ca_{E}$ is the electric capillary number (ratio of the electric to capillary stresses); and $Re_{E}$ is the electric Reynolds number (ratio of charge relaxation to flow time scales), which characterizes the strength of surface charge convection along the interface. For so-called ‘prolate A’ drops, where the surface charge is convected towards the ‘poles’ of the drop, we demonstrate that increasing $Re_{E}$ reduces the critical capillary number for breakup. Moreover, surface charge convection is the cause of an abrupt transition in the breakup mode of a drop from end pinching, where the drop elongates and develops bulbs at its ends that eventually detach, to a breakup mode characterized by the formation of conical ends. On the contrary, the deformation of ‘prolate B’ drops, where the surface charge is convected away from the poles, is essentially unaffected by the magnitude of $Re_{E}$.

The dissolution of a multicomponent nanodrop in a sparingly miscible liquid is studied by molecular dynamics (MD) simulations. We studied both binary and ternary systems, in which nanodroplets are formed from one and two components, respectively. Whereas for a single-component droplet the dissolution can easily be calculated, the situation is more complicated for a multicomponent drop, as the interface concentrations of the drop constituents depend on the drop composition, which changes with time. In this study, the variation of the interface concentration with the drop composition is determined from independent ‘numerical experiments’, which are then used in the theoretical model for the dissolution dynamics of a multicomponent drop. The MD simulations reveal that when the interaction strengths between the drop constituents and the surrounding bulk liquid are significantly different, the concentration of the more soluble component near the drop interface may become larger than in the drop bulk. This effect is the larger the smaller the drop radius. While the present study is limited to binary and ternary systems, the same method can be easily extended to a larger number of components.

Gravity currents often occur on complex topographies and are therefore subject to spatial development. We present experimental results on continuously supplied gravity currents moving from a horizontal to a sloping boundary, which is either concave or straight. The change in boundary slope and the consequent acceleration give rise to a transition from a stable subcritical current with a large Richardson number to a Kelvin–Helmholtz (KH) unstable current. It is shown here that depending on the overall acceleration parameter $\overline{T_{a}}$, expressing the rate of velocity increase, the currents can adjust gradually to the slope conditions (small $\overline{T_{a}}$) or go through acceleration–deceleration cycles (large $\overline{T_{a}}$). In the latter case, the KH billows at the interface have a strong effect on the flow dynamics, and are observed to cause boundary layer separation. Comparison of currents on concave and straight slopes reveals that the downhill deceleration on concave slopes has no qualitative influence, i.e. the dynamics is entirely dominated by the initial acceleration and ensuing KH billows. Following the similarity theory of Turner 1973 (Buoyancy Effects in Fluids. Cambridge University Press), we derive a general equation for the depth-integrated velocity that exhibits all driving and retarding forces. Comparison of this equation with the experimental velocity data shows that when $\overline{T_{a}}$ is large, bottom friction and entrainment are large in the region of appearance of KH billows. The large bottom friction is confirmed by the measured high Reynolds stresses in these regions. The head velocity does not exhibit the same behaviour as the layer velocity. It gradually approaches an equilibrium state even when the acceleration parameter of the layer is large.

A numerical study of the flow-induced vibration of two elastically mounted cylinders in tandem and staggered arrangements at Reynolds number $Re=200$ is presented. The cylinder centres are set at a streamwise distance of 1.5 cylinder diameters, placing the rear cylinder in the near-wake region of the front cylinder for the tandem arrangement. The cross-stream or lateral offset is varied between 0 and 5 cylinder diameters. The two cylinders are identical, with the same elastic mounting, and constrained to oscillate only in the cross-flow direction. The variation of flow behaviours is examined for static cylinders and for elastic mountings of a range of spring stiffnesses, or reduced velocity. At least seven major modes of flow response are identified, delineated by whether the oscillation is effectively symmetric, and the strength of the influence of the flow through the gap between the two cylinders. Submodes of these are also identified based on whether or not the flow remains periodic. More subtle temporal behaviours, such as period doubling, quasi-periodicity and chaos, are also identified and mapped. Across all of these regimes, the amplitudes of vibration and the magnitude of the fluid forces are quantified. The modes identified span the parameter space between two important limiting cases: two static bodies at varying lateral offset; and two elastically mounted bodies in a tandem configuration at varying spring stiffnesses. Some similarity in the response of extremely stiff or static bodies and extremely slack bodies is shown. This is explained by the fact that the slack bodies are free to move to an equilibrium position and stop, effectively becoming a static system. However, the most complex behaviour appears between these limits, when the bodies are in reasonably close proximity, and the natural structural frequency is close to the vortex shedding frequency of a single cylinder. This appears to be driven by the interplay between a series of time scales, including the vortex formation time, the advection time across the gap between the cylinders and the oscillation period of both bodies. This points out an important difference between this multi-body system and the classic single-cylinder vortex-induced vibration: two bodies in close proximity will not oscillate in a synchronised, periodic manner when their natural structural frequencies are close to the nominal vortex shedding frequency of a single cylinder.

The incompressible flow of a uniform fluid, which fills a rigid spheroid rotating about an arbitrary axis fixed in an inertial frame, is dominated at small Rossby and Ekman numbers by the rotation through the Coriolis force. The effects of rotation on the flow can be found by treating the Coriolis force modified by a pressure gradient as a skew-symmetric bounded linear operator $\boldsymbol{{\mathcal{C}}}$ acting on smooth inviscid incompressible flows in the spheroid. It is shown that the space of incompressible polynomial flows of degree $N$ or less in the spheroid is invariant under $\boldsymbol{{\mathcal{C}}}$ for any $N$. The skew symmetry of $\boldsymbol{{\mathcal{C}}}$ implies the Coriolis operator $\boldsymbol{{\mathcal{C}}}$ is non-defective for such flows with an orthogonal set of eigenmodes (inertial and geostrophic modes) which form a basis for the finite-dimensional space of spheroidal polynomial flows. The eigenmodes are tilted if the rotation axis is not aligned with the symmetry axis of the spheroid. The non-defective property of $\boldsymbol{{\mathcal{C}}}$ enables enumeration of the modes and proof of their completeness using the Weierstrass polynomial approximation theorem. The fundamental tool, which is required to establish invariance of spheroidal polynomial flows under $\boldsymbol{{\mathcal{C}}}$ and completeness of the Coriolis modes, is that the solution of the polynomial Poisson–Neumann problem, i.e. Poisson’s equation with Neumann boundary condition and polynomial data, in a spheroid is a polynomial. The Coriolis modes of degree one and all geostrophic modes are explicitly constructed. Only the modes of degree one have non-zero angular momentum in the boundary frame.

In the current study, selective forcing of cross-flow instability modes evolving on a $45^{\circ }$ swept wing at $Re=2.17\times 10^{6}$ is achieved by means of spanwise-modulated plasma actuators, positioned near the leading edge. In the perspective of laminar flow control, the followed methodology holds on the discrete roughness elements/upstream flow deformation (DRE/UFD) approach, thoroughly investigated by e.g. Saric et al. (AIAA Paper 1998-781, 1998), Malik et al. (J. Fluid Mech., vol. 399, 1999, pp. 85–115) and Wassermann & Kloker (J. Fluid Mech., vol. 456, 2002, pp. 49–84). The possibility of using active devices for UFD provides several advantages over passive means, allowing for a wider range of operating $Re$ numbers and pressure distributions. In the present work, customised alternating current dielectric barrier discharge plasma actuators have been designed, manufactured and characterised. The authority of the actuators in forcing monochromatic stationary cross-flow modes at different spanwise wavelengths is assessed by means of infrared thermography. Moreover, quantitative spatio-temporal measurements of the boundary layer velocity field are performed using time-resolved particle image velocimetry. The results reveal distinct steady and unsteady forcing contributions of the plasma actuator on the boundary layer. It is shown that the actuators introduce unsteady fluctuations in the boundary layer, amplifying at frequencies significantly lower than the actuation frequency. In line with the DRE/UFD strategy, forcing a sub-critical stationary mode, with a shorter wavelength compared to the naturally selected mode, results in less amplified primary vortices and related fluctuations, compared to the critical forcing case. The effect of the forcing on the flow stability is further inspected by combining the measured actuators body force with the numerical solution of the laminar boundary layer and linear stability theory. The simplified methodology yields fast and computationally cheap estimates on the effect of steady forcing (magnitude and direction) on the boundary layer stability.

The hydrodynamics and rheology of a sheared dilute gas–solid suspension, consisting of inelastic hard spheres suspended in a gas, are analysed using an anisotropic Maxwellian as the single particle distribution function. For the simple shear flow, the closed-form solutions for granular temperature and three invariants of the second-moment tensor are obtained as functions of the Stokes number ($St$), the mean density ($\unicode[STIX]{x1D708}$) and the restitution coefficient ($e$). Multiple states of high and low temperatures are found when the Stokes number is small, thus recovering the ‘ignited’ and ‘quenched’ states, respectively, of Tsao & Koch (J. Fluid Mech., vol. 296, 1995, pp. 211–246). The phase diagram is constructed in the three-dimensional ($\unicode[STIX]{x1D708},St,e$)-space that delineates the regions of ignited and quenched states and their coexistence. The particle-phase shear viscosity and the normal-stress differences are analysed, along with related scaling relations on the quenched and ignited states. At any $e$, the shear viscosity undergoes a discontinuous jump with increasing shear rate at the ‘quenched–ignited’ transition. The first (${\mathcal{N}}_{1}$) and second (${\mathcal{N}}_{2}$) normal-stress differences also undergo similar first-order transitions: (i) ${\mathcal{N}}_{1}$ jumps from large to small positive values and (ii) ${\mathcal{N}}_{2}$ from positive to negative values with increasing $St$, with the sign change of ${\mathcal{N}}_{2}$ identified with the system making a transition from the quenched to ignited states. The superior prediction of the present theory over the standard Grad’s method and the Burnett-order Chapman–Enskog solution is demonstrated via comparisons of transport coefficients with simulation data for a range of Stokes number and restitution coefficient.

Since supersonic test facilities have tunnel noise that strongly influences boundary layer transition experiments, the determination of tunnel noise is of great significance to properly evaluate and interpret experimental results. The composition of tunnel noise, which consists of acoustic, entropy and vorticity modes, highly influences the boundary layer receptivity. The measurement of the separate modes is a major goal of ongoing research. In this study, the properties of stagnation point probes for a newly developed modal decomposition method for tunnel noise are investigated by direct numerical simulation. Pressure and heat flux responses of a stagnation point probe to various entropy and acoustic mode input functions are analysed to investigate how tunnel noise is perceived by corresponding sensor types. The interaction of the incident mode and the shock wave upstream of the probe is analysed and the resulting wave pattern in the subsonic region between shock wave and probe is evidenced. It is found that pure incident acoustic or entropy modes cause acoustic and entropy waves downstream of the shock wave whose strengths differ depending on the incident mode. The resulting wave pattern downstream of the shock wave is determined by postshock acoustic waves propagating bidirectionally between shock wave and probe. Formulating a model equation linking pressure and heat flux fluctuations to the initially caused postshock acoustic and entropy wave, a criterion for the applicability of stagnation point probes measuring pressure and heat flux fluctuations in the new modal decomposition method can be deduced: to distinguish between the incident mode types based on their pressure and heat flux signal the perception of initially generated entropy waves downstream of the shock wave by the heat flux sensor is crucial. The transfer function between entropy waves and heat flux is shown to have low pass filter characteristics and the cutoff Strouhal number could be estimated by control theory. The analysis of the frequency response to continuous incident waves corroborated this cutoff Strouhal number.

The chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this is also the case for the infinite-dimensional dynamics of Navier–Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics – specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier–Stokes equations as they do in lower-dimensional systems – is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell at $Re=2500$, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.

This article deals with the Hadamard instability of the so-called $\unicode[STIX]{x1D707}(I)$ model of dense rapidly sheared granular flow, as reported recently by Barker et al. (J. Fluid Mech., vol. 779, 2015, pp. 794–818). The present paper presents a more comprehensive study of the linear stability of planar simple shearing and pure shearing flows, with account taken of convective Kelvin wavevector stretching by the base flow. We provide a closed-form solution for the linear-stability problem and show that wavevector stretching leads to asymptotic stabilization of the non-convective instability found by Barker et al. (J. Fluid Mech., vol. 779, 2015, pp. 794–818). We also explore the stabilizing effects of higher velocity gradients achieved by an enhanced-continuum model based on a dissipative analogue of the van der Waals–Cahn–Hilliard equation of equilibrium thermodynamics. This model involves a dissipative hyperstress, as the analogue of a special Korteweg stress, with surface viscosity representing the counterpart of elastic surface tension. Based on the enhanced-continuum model, we also present a model of steady shear bands and their nonlinear stability against parallel shearing. Finally, we propose a theoretical connection between the non-convective instability of Barker et al. (J. Fluid Mech., vol. 779, 2015, pp. 794–818) and the loss of generalized ellipticity in the quasi-static field equations. Apart from the theoretical interest, the present work may suggest stratagems for the numerical simulation of continuum field equations involving the $\unicode[STIX]{x1D707}(I)$ rheology and variants thereof.

We computationally investigate the Richtmyer–Meshkov instability of a density interface with a single-mode perturbation in a two-fluid, ion–electron plasma with no initial magnetic field. Self-generated magnetic fields arise subsequently. We study the case where the density jump across the initial interface is due to a thermal discontinuity, and select plasma parameters for which two-fluid plasma effects are expected to be significant in order to elucidate how they alter the instability. The instability is driven via a Riemann problem generated precursor electron shock that impacts the density interface ahead of the ion shock. The resultant charge separation and motion generates electromagnetic fields that cause the electron shock to degenerate and periodically accelerate the electron and ion interfaces, driving Rayleigh–Taylor instability. This generates small-scale structures and substantially increases interfacial growth over the hydrodynamic case.

Despite the widely acknowledged significance of turbulence-driven clustering, a clear topological definition of particle cluster in turbulent dispersed multiphase flows has been lacking. Here we introduce a definition of coherent cluster based on self-similarity, and apply it to distributions of heavy particles in direct numerical simulations of homogeneous isotropic turbulence, with and without gravitational acceleration. Clusters show self-similarity already at length scales larger than twice the Kolmogorov length, as indicated by the fractal nature of their surface and by the power-law decay of their size distribution. The size of the identified clusters extends to the integral scale, with average concentrations that depend on the Stokes number but not on the cluster dimension. Compared to non-clustered particles, coherent clusters show a stronger tendency to sample regions of high strain and low vorticity. Moreover, we find that the clusters align themselves with the local vorticity vector. In the presence of gravity, they tend to align themselves vertically and their fall speed is significantly different from the average settling velocity: for moderate fall speeds they experience stronger settling enhancement than non-clustered particles, while for large fall speeds they exhibit weakly reduced settling. The proposed approach for cluster identification leverages the Voronoï diagram method, but is also compatible with other tessellation techniques such as the classic box-counting method.

The reconstruction of water wave elevation from bottom pressure measurements is an important issue for coastal applications, but corresponds to a difficult mathematical problem. In this paper we present the derivation of a method which allows the elevation reconstruction of water waves in intermediate and shallow waters. From comparisons with numerical Euler solutions and wave-tank experiments we show that our nonlinear method provides much better results for the surface elevation reconstruction compared to the linear transfer function approach commonly used in coastal applications. More specifically, our method accurately reproduces the peaked and skewed shape of nonlinear wave fields. Therefore, it is particularly relevant for applications on extreme waves and wave-induced sediment transport.

We explore the effect of an applied homogeneous magnetic field on the electrospraying characteristics of a ferrofluid in the cone-jet mode. A sulfolane-based ferrofluid mixed with the ionic liquid ethyl ammonium nitrate has been synthesized. These mixtures have negligible volatility under ambient conditions and remain stable under a very wide range of electrical conductivities $K$. Magnetized Taylor cones spray with the same current emission characteristics as their non-magnetized counterparts in the shared voltage and flow rate parameter space. However, the magnetized Taylor cones studied remained stable at voltages 23 % lower than the non-magnetized spray; they also access flow rates 30 % and 40 % lower in ferrofluids with $K=0.3$ and $0.01~\text{S}~\text{m}^{-1}$. In the lower voltage ranges available only to magnetized tips, unusually long stable cones are observed. The magnetic stabilization mechanism responsible for these two effects remains unclear. It is noteworthy that these strong effects arise even when the tip curvature of the strictly magnetized liquid is orders of magnitude smaller than that for the strictly electrified liquid.

In the present paper, we investigate the path instabilities and the transition scenario for oblate homogeneous spheroids falling (or ascending) freely in a quiescent and unconfined Newtonian fluid under the action of gravity, buoyancy and hydrodynamic forces. The problem depends on three independent external parameters: the aspect ratio $\unicode[STIX]{x1D712}=d/a$ , where $d$ is the diameter and $a$ the length of the axisymmetry axis of the spheroid; the non-dimensionalized mass $m^{\ast }=m/(\unicode[STIX]{x1D70C}d^{3})$ , $m$ being the mass of the spheroid and $\unicode[STIX]{x1D70C}$ the fluid density; and the Galileo number, defined as $G=\sqrt{(m^{\ast }-V^{\ast })gd^{3}}/\unicode[STIX]{x1D708}$ . In the definition of the Galileo number, $V^{\ast }=V/d^{3}$ is the non-dimensionalized volume, $g$ the gravitational acceleration and $\unicode[STIX]{x1D708}$ is the kinematic viscosity. Asymptotic solutions (regimes) are investigated in seven $\unicode[STIX]{x1D712}=\text{const}.$ parameter planes going from $\unicode[STIX]{x1D712}=10$ (very flat spheroid) to $\unicode[STIX]{x1D712}=1.1$ (an almost spherical shape), for $m^{\ast }$ going from 0 to 5 and Galileo numbers up to 300 (i.e. Reynolds numbers roughly up to 500). The obtained results provide a link between the known scenario of a homogeneous disk and that, well known, of a sphere. The scenario of the flat spheroid of aspect ratio 10 has many common features with that of an infinitely thin disk, but the finite thickness brings about significant quantitative differences. At the opposite side of the investigated aspect ratio interval, the dynamics of the spheroid of aspect ratio 1.1 is found very close to that of a perfect sphere except for small density ratios (smaller than approximately 0.5). Very light spheroids of aspect ratio 1.1 move along vertical zig-zagging trajectories. At intermediate aspect ratios, the strong subcritical effects and the characteristic zig-zagging and fluttering motion, typical for flat bodies, progressively disappear. The tumbling regime remains remarkably stable and is shown to be present down to $\unicode[STIX]{x1D712}=2$ . An interesting result consists in the evidence of the first two bifurcations typical for the sphere scenario (leading to steady oblique and oblique oscillating trajectories) present both for very flat and thick spheroids but absent at intermediate aspect ratios. The results pertaining to spheroids might be more useful in practical applications than those obtained for too idealized thin disks and perfect spheres.

We consider the hydrodynamic stability of homogeneous, incompressible and rotating ellipsoidal fluid masses. The latter are the simplest models of fluid celestial bodies with internal rotation and subjected to tidal forces. The classical problem is the stability of Roche–Riemann ellipsoids moving on circular Kepler orbits. However, previous stability studies have to be reassessed. Indeed, they only consider global perturbations of large wavelength or local perturbations of short wavelength. Moreover many planets and stars undergo orbital motions on eccentric Kepler orbits, implying time-dependent ellipsoidal semi-axes. This time dependence has never been taken into account in hydrodynamic stability studies. In this work we overcome these stringent assumptions. We extend the hydrodynamic stability analysis of rotating ellipsoids to the case of eccentric orbits. We have developed two open-source and versatile numerical codes to perform global and local inviscid stability analyses. They give sufficient conditions for instability. The global method, based on an exact and closed Galerkin basis, handles rigorously global ellipsoidal perturbations of unprecedented complexity. Tidally driven and libration-driven elliptical instabilities are first recovered and unified within a single framework. Then we show that new global fluid instabilities can be triggered in ellipsoids by tidal effects due to eccentric Kepler orbits. Their existence is confirmed by a local analysis and direct numerical simulations of the fully nonlinear and viscous problem. Thus a non-zero orbital eccentricity may have a strong destabilising effect in celestial fluid bodies, which may lead to space-filling turbulence in most of the parameters range.

We study so-called wave boundary layers (BLs) arising in a stably stratified fluid at large times. The BL is a narrow domain near the surface and/or bottom of the fluid; with increasing time, gradients of buoyancy and horizontal velocity in the BL grow sharply and the BL thickness tends to zero. The non-stationary BL can arise both as a result of linear evolution of the initial perturbation and under the action of an external force (tangential stress exerted on the fluid surface in our case). We analyse both the variants and find that the ‘forced’ BLs are much more intense than the ‘free’ ones. In the ‘free’ BLs all fields are bounded and the gradients of buoyancy and horizontal velocity grow linearly in time, whereas in the ‘forced’ BL only the vertical velocity is bounded and the buoyancy and horizontal velocity grow linearly in time. As to the gradients in the ‘forced’ BL, the vertical velocity gradient grows in time linearly and the gradients of buoyancy and horizontal velocity grow quadratically. In both of the cases we determine exact solutions in the form of expansions in the vertical wave modes and find asymptotic solutions valid at large times. The comparison between them shows that the asymptotic solutions approximate the exact ones fairly well even for moderate times.

Here we show that monochromatic long-crested corrugations on an otherwise flat seafloor can coherently scatter the energy of an oblique incident internal wave to multiple multi-directional higher-mode internal waves via a series of resonant interactions. We demonstrate that a resonance between seabed corrugations and a normally or slightly oblique incident internal wave results in a series of follow-up resonant interactions, which take place between the same corrugations and successively resonated shorter waves. A chain resonance of internal waves that carries energy to small scales is thus obtained, and we find that the Richardson number decreases by several orders of magnitude over the corrugated patch. If the incidence angle is large, and the incident wave perfectly satisfies a resonance condition with the topography, it turns out that not many higher-mode resonance or near-resonance conditions can be satisfied, such that energy stays confined within the first few modes. Nevertheless, if the incident waves are sufficiently detuned from satisfying a perfect resonance condition with the seabed corrugations, then we show that this frequency detuning may balance off the large detuning due to oblique incidence, leading to a chain resonance that again carries energy to small scales. The evolution of the incident and resonated wave amplitudes is predicted from the envelope equation for internal waves over resonant seabed topography in a three-dimensional rotating fluid, which we derive considering the Boussinesq and $f$-plane approximations with $f$ the Coriolis frequency, linear density stratification and small-amplitude corrugations. Our results suggest that topographic features on the ocean floor with a well-defined dominant wavenumber vector, through the chain resonance mechanism elucidated here, may play a more important role than previously thought in the enhancement of diapycnal mixing and energy dissipation.

Direct numerical simulations are used to examine the behaviour of wall-pressure fluctuations $p_{w}$ in a flat-plate turbulent boundary layer with large adverse and favourable pressure gradients, involving separation and reattachment. The Reynolds number $Re_{\unicode[STIX]{x1D703}}$ based on momentum thickness is equal to 300, 600 and 900. Particular attention is given to effects of Reynolds number on root-mean-square (r.m.s.) values, frequency/power spectra and instantaneous fields. The possible scaling laws are also examined as compared with the existing direct numerical simulation and experimental data. The r.m.s. value of $p_{w}$ normalized by the local maximum Reynolds shear stress $-\unicode[STIX]{x1D70C}\overline{uv}_{max}$ (Simpson et al. J. Fluid Mech. vol. 177, 1987, pp. 167–186; Na & Moin J. Fluid Mech. vol. 377, 1998b, pp. 347–373) leads to near plateau (i.e. $p_{w\,rms}/-\unicode[STIX]{x1D70C}\overline{uv}_{max}=2.5\sim 3$) in the adverse pressure gradient and separated regions in which the frequency spectra exhibit good collapse at low frequencies. The magnitude of $p_{w\,rms}/-\unicode[STIX]{x1D70C}\overline{uv}_{max}$ is however reduced down to 1.8 near reattachment where good collapse is also obtained with normalization by the local maximum wall-normal Reynolds stress $\unicode[STIX]{x1D70C}\overline{vv}_{max}$. Near reattachment, $p_{w\,rms}/-\unicode[STIX]{x1D70C}\overline{vv}_{max}=1.2$ is attained unambiguously independently of the Reynolds number and pressure gradient. The present magnitude (1.2) is smaller than (1.35) obtained for step-induced separation by Ji & Wang (J. Fluid Mech. vol. 712, 2012, pp. 471–504). The reason for this difference is intrinsically associated with convective nature of a pressure-induced separation bubble near reattachment where the magnitude of $p_{w\,rms}$ depends essentially on the favourable pressure gradient. The resulting mean flow acceleration leads to delay of the r.m.s. peak after reattachment. Attention is also given to structures of $p_{w}$. It is shown that large-scale spanwise rollers of low pressure fluctuations are formed above the bubble, whilst changing to large-scale streamwise elongated structures after reattachment. These large-scale structures become more prominent with increasing $Re_{\unicode[STIX]{x1D703}}$ and affect $p_{w}$ significantly.