We perform three-dimensional numerical simulations to investigate the sedimentation of a single sphere in the absence and presence of a simple cross-shear flow in a yield stress fluid with weak inertia. In our simulations, the settling flow is considered to be the primary flow, whereas the linear cross-shear flow is a secondary flow with amplitude 10 % of the primary flow. To study the effects of elasticity and plasticity of the carrying fluid on the sphere drag as well as the flow dynamics, the fluid is modelled using the elastoviscoplastic constitutive laws proposed by Saramito (J. Non-Newtonian Fluid Mech., vol. 158 (1–3), 2009, pp. 154–161). The extra non-Newtonian stress tensor is fully coupled with the flow equation and the solid particle is represented by an immersed boundary method. Our results show that the fore–aft asymmetry in the velocity is less pronounced and the negative wake disappears when a linear cross-shear flow is applied. We find that the drag on a sphere settling in a sheared yield stress fluid is reduced significantly compared to an otherwise quiescent fluid. More importantly, the sphere drag in the presence of a secondary cross-shear flow cannot be derived from the pure sedimentation drag law owing to the nonlinear coupling between the simple shear flow and the uniform flow. Finally, we show that the drag on the sphere settling in a sheared yield stress fluid is reduced at higher material elasticity mainly due to the form and viscous drag reduction.

Direct numerical simulations are performed to study the flow regimes at the wake behind a strongly streamwise rotating sphere, covering the range of rotation parameters $0\leqslant \unicode[STIX]{x1D6FA}\leqslant 3$ and laminar and transitional Reynolds numbers $Re=250$, 500 and 1000. The wake dynamics is investigated in terms of flow topology, dominant modes and force coefficients. A higher wake complexity is found for growing values of the rotation parameter $\unicode[STIX]{x1D6FA}$ for all the Reynolds numbers investigated. In particular, at low and intermediate $Re$, successive bifurcations entail the development of periodic, quasi-periodic and irregular regimes, constituting a classical scenario of route to chaos, through the destabilization of different structures associated to incommensurate frequencies, which have been analysed by means of flow decomposition techniques. At low $Re$ and high rotation rates, the flow is governed by double-threaded structures due to the destabilization of helical symmetries of azimuthal wavenumber $m=2$, which are not dominant at larger $Re$. Interestingly, at intermediate values of $\unicode[STIX]{x1D6FA}$ and $Re=500$, a bistable dynamics is observed whereby the wake undergoes a random switching between a modulated quasi-periodic regime and an irregular regime, which is associated to a sudden increase of the drag coefficient, on account of the development of a double-celled recirculating bubble. Finally, for $Re=1000$, the flow is already chaotic at $\unicode[STIX]{x1D6FA}=0$, and the evolution with the rotation rate of the flow dynamics is simpler, with wake regimes being characterized by the rotation and massive shedding of vortex loops, that are a continuous deformation through axial rotation of the irregular wake behind the static sphere.

We investigate the turbulence statistics in a multiphase plume made of heavy particles (particle Reynolds number at terminal velocity is 450). Using refractive-index-matched stereoscopic particle image velocimetry, we measure the locations of particles whose buoyancy drives the formation of a multiphase plume, together with the local velocity of the induced flow in the ambient salt–water. Measurements of the mean axial flow in the plume centreplane follow Gaussian profiles and that of the mean radial flow is consistent with integral plume theory. The turbulence characteristics resemble those measured in a bubble plume, including strong anisotropy in the normal Reynolds stresses. However, we observe structural differences between the two multiphase plumes. First, the skewness of the probability density function of the axial velocity fluctuations is not that which would be predicted by simply reversing the direction of a bubble plume. Second, in contrast to a bubble plume, the particle plume has a non-negligible fluid-shear production term in the turbulent kinetic energy (TKE) budget. Third, the radial decay of all measured terms in the TKE budget is slower than those in a bubble plume. Despite these dissimilarities, a bigger picture emerges that applies to both flows. The TKE production by particles (or bubbles) roughly balances the viscous dissipation, except near the plume centreline. The one-dimensional power spectra of the velocity fluctuations show a $-3$ power law that puts both the particle and bubble plume in a category different from single-phase shear-flow turbulence.

The stability and transition in the bottom boundary layer under a solitary wave are analysed in the presence of finite-amplitude disturbances. First, the receptivity of the boundary layer is investigated using a linear input-output analysis, in which the environment noise is modelled as distributed body forces. The most ‘dangerous’ perturbations in a time frame until flow reversal are found to be arranged as counter-rotating streamwise-constant vortices. One of these vortex configurations is then selected and deployed to nonlinear equations, and streaks of various amplitudes are generated via the lift-up mechanism. By means of secondary stability analysis and direct numerical simulations, the dual role of streaks in the boundary-layer transition is shown. When the amplitude of streaks remains moderate, these elongated features remain stable until the adverse-pressure-gradient stage and have a dampening effect on the instabilities developing thereafter. In contrast, when the low-speed streaks reach high amplitudes exceeding 15 % of the free stream velocity at the respective phase, they become highly unstable to secondary sinuous modes in the outer shear layers. Consequently, a subcritical transition to turbulence, i.e. bypass transition, can be initiated already in the favourable-pressure-gradient region ahead of the wave crest.

Transition to turbulence in high-speed flows is determined by multiple parameters, many of which are not fully understood, leading to problems in developing physics-based prediction methods. In this contribution, we compare transition mechanisms in configurations with unswept and swept leading edges that are exposed to free-stream acoustic disturbances. Direct numerical simulations are run at a Mach number of six with the same free-stream noise, consisting of either fast or slow acoustic disturbances, with two different amplitudes to explore the linear and nonlinear aspects of receptivity and transition. For the unswept configuration, receptivity follows an established mechanism involving synchronisation of fast acoustic disturbances with boundary-layer modes. At high forcing amplitudes, transition proceeds via the formation of streaks and their eventual breakdown. In the swept case, the process of streak-induced transition is modified by the presence of a cross-flow instability in the leading-edge region. Linear stability analysis confirms the presence of a cross-flow mode as well as weaker first and second mode waves. Both fast and slow types of forcing independently stimulate an unusual transition mechanism involving significantly narrower streaks than those arising from the cross-flow instability behind the swept leading edge or those induced nonlinearly in the unswept case. In the observed transition process, the cross-flow mode leads to a thin layer of streamwise vorticity that breaks up under the influence of high spanwise wavenumber disturbances. These disturbances first appear in the leading-edge region.

This paper presents a computational study on the free forward flight and sideslip manoeuvre of an insect-like flapping-wing flyer modelled after the hummingbird hawkmoth (Macroglossum stellatarum), with Reynolds number ${\sim}3000$. The numerical model integrated a Navier–Stokes fluid solver with the Newtonian free-body dynamics of the flyer. A generic proportional–integral–derivative (PID)-based wing kinematics controller was used to achieve stable controlled flight. State-equation analyses of flight dynamics were helpful in identifying the roles of kinematic wing actions and for establishing control coefficients for stable flight. Forward flights up to a speed of $4.3~\text{m}~\text{s}^{-1}$ were simulated, which show that the wingbeat frequency decreased below the hovering frequency for cruising flight in the low- and medium-speed range, and higher frequency was only needed for high-speed flight. Similarly, the aerodynamic power consumption was also lower than that for hovering flight over the simulated speed range, due to the contribution of wing drag to overall lift. In addition, flight with higher speed tends to be more efficient in terms of energy consumption for the same distance travelled. In a complete sideslip manoeuvre, the model hawkmoth took approximately 20 wing cycles to translate laterally 4.5 wing lengths to its right and another 30 wing cycles to stabilize hovering at the new location. Slightly higher wingbeat frequency and power were required during the sideslipping phase to adjust for drop in lift due to body roll. The evolution of the vortical wakes reflects quite well the major mechanisms of force generation that were at play at key stages in these flights.

The present paper investigates the receptivity of inviscid first and second modes in super/hypersonic boundary layers due to the interaction between a weak free-stream acoustic wave and a small isolated surface roughness element. The large-Reynolds-number asymptotic analysis reveals the detailed processes of the excitation. The distortion of the acoustic signature by the curved wall contributes to the leading-order receptivity, producing an eigenmode of $O({\mathcal{E}}h)$ amplitude, where ${\mathcal{E}}\ll 1$ is the magnitude of the acoustic wave and $h\ll 1$ the roughness height normalised by the local boundary-layer thickness $\unicode[STIX]{x1D6FF}$. The interactions between the roughness-induced mean-flow distortion and the acoustic signature contribute to the second-order receptivity, which is of $O({\mathcal{E}}hR^{-1/3})$ with $R\gg 1$ being the Reynolds number based on $\unicode[STIX]{x1D6FF}$. Interestingly, the leading-order receptivity is equivalent to a canonic receptivity problem, the excitation by time-periodic blowing and suction through a local slot on the wall, and the effective periodic outflux velocity forced from the underneath Stokes layer can be determined explicitly in terms of the roughness shape function. This equivalence holds when $h=O(R^{-1/3})$, for which the roughness-induced mean-flow distortion becomes nonlinear. A systematic parametric study is carried out for the excitation of the first and second modes by both fast and slow free-stream acoustic waves, and the dependence of the receptivity efficiency on the relevant parameters is provided. Interestingly, the second-order receptivity could become dominant (e.g. in the case of slow acoustic waves with low frequencies and small incident angles), but the present mathematical theory remains valid. In order to check the accuracy of the asymptotic predictions, we have carried out direct numerical simulations (DNS) and also extended the existing finite-Reynolds-number theory to the supersonic regime. The asymptotic solutions agree with the results given by the finite-Reynolds-number calculations and DNS when $R$ is sufficiently large (typically $R=O(10^{5})$). An improved large-Reynolds-number approach is developed by replacing the non-penetration boundary condition by an unsteady outflux, which accounts for the $O(R^{-1/2})$ viscous correction by the Stokes layer. With this modification, the accuracy of the receptivity calculation at moderate Reynolds numbers (approximately a few thousands) is improved remarkably.

The dynamics of a hyperelastic splitter plate interacting with the laminar wake flow of a circular cylinder is investigated numerically at a Reynolds number of 80. By decreasing the plate’s stiffness, four regimes of flow-induced vibrations are identified: two regimes of periodic oscillation about a symmetric position, separated by a regime of periodic oscillation about asymmetric positions, and finally a regime of quasi-periodic oscillation occurring at very low stiffness and characterized by two fundamental (high and low) frequencies. A linear fully coupled fluid–solid analysis is then performed and reveals the destabilization of a steady symmetry-breaking mode, two high-frequency unsteady modes and one low-frequency unsteady mode, when varying the plate’s stiffness. These unstable eigenmodes explain the emergence of the nonlinear self-sustained oscillating states and provide a good prediction of the oscillation frequencies. A comparison with nonlinear calculations is provided to show the limits of the linear approach. Finally, two simplified analyses, based on the quiescent-fluid or quasi-static assumption, are proposed to further identify the linear mechanisms at play in the destabilization of the fully coupled modes. The quasi-static static analysis allows an understanding of the behaviour of the symmetry-breaking and low-frequency modes. The quiescent-fluid stability analysis provides a good prediction of the high-frequency vibrations, unlike the bending modes of the splitter plate in vacuum, as a result of the fluid added-mass correction. The emergence of the high-frequency periodic oscillations can thus be predicted based on a resonance condition between the frequencies of the hydrodynamic vortex-shedding mode and of the quiescent-fluid solid modes.

In this paper we re-examine the flow produced by the normal impact of a laminar liquid jet onto an infinite plane when the flow is dominated by surface tension. Over the range of parameters we consider, which are typical of water from a tap over a kitchen sink, it is observed experimentally that after impact the liquid spreads radially over the plane away from the point of impact in a thin film. It is also observed that, at a finite radius, there is an abrupt increase in thickness of the film which has been identified as a hydraulic jump. Once the jump is formed this radius remains constant in time and, further, is independent of the orientation of the surface showing that gravity is unimportant (Bhagat et al., J. Fluid Mech., vol. 851, 2018, R5). We show that the application of conservation of momentum in the film, subject only to viscosity and surface tension and ignoring gravity completely, predicts a singularity in the curvature of the liquid film and consequently a jump in the depth of the film at a finite radius. This location is almost identical to the radius of the jump predicted by conservation of energy and agrees with experimental observations. We also provide the correct boundary condition to be applied at an interface, where there is a change in interfacial area as a result of the fluid flow, that accounts for the energy change associated with fluid molecules’ exchange between the interface and the bulk.

A hypersonic shock wave/laminar boundary-layer interaction over a canonical $25{-}55^{\circ }$ double-cone configuration is numerically investigated. A moderate-enthalpy flow of $5~\text{MJ}~\text{kg}^{-1}$ with a Mach number of 9.87 and a unit Reynolds number of $1.5\times 10^{5}~\text{m}^{-1}$ is considered. Special emphasis is given to the influence of leading-edge bluntness. The results indicate that the double-cone flow is insensitive to small bluntness in terms of shock structures, separation region sizes and surface pressure and heat flux distributions. A critical nose radius is observed, beyond which the separation bubble grows dramatically. The numerical data are analysed and interpreted based on a triple-deck formulation. It is shown that the sudden change in flow features is mainly caused by pressure overexpansion on the first cone due to leading-edge bluntness, such that the skin friction upstream of the separation is significantly reduced and the upstream pressure can no longer resist the large adverse pressure gradient induced by shock impingement. An estimation of the critical radius is established based on the pressure correlations of Blick & Francis (AIAA J., vol. 4 (3), 1966, pp. 547–549) for spherically blunted cones. Simulations at a higher enthalpy with the presence of both vibrational relaxation and air chemistry show a similar trend with increasing nose radius. The proposed criterion agrees well with the experimental observations.

Our experimental study focuses on the density and velocity field in two layers of fluid separated by a sharp density interface. Turbulence is generated by a non-invasive stirrer, a Taylor–Couette tank, and the interface is stabilized with a source of saline fluid and a source of fresh water at the bottom and top of the tank, respectively. The same volume fluxes are withdrawn by two sinks to maintain a constant volume of fluid in the tank. Our results confirm past experiments and show that a strong vertical exchange of fluid occurs close to the inner cylinder and across the interface, where the vertical turbulent length scales appear to be suppressed. For low values of kinetic energy supplied to the system, the interface may act as a rigid boundary for the turbulent eddies, with a reduction of the vertical length scales although it seems not to affect the horizontal length scales. The vertical buoyancy flux extracted at the top of the tank is fairly well reproduced by the measured correlation $\overline{\unicode[STIX]{x1D70C}^{\prime }w^{\prime }}$ between density and vertical velocity fluctuations across the interface. Quadrant analysis of the correlation terms reveals that the greatest contribution to salt flux is given by eddies that carry the lighter fluid from top to bottom across the interface. The mixing process is accompanied by a single wake-like disturbance, with a radial front advancing in the azimuthal direction across the interface, acting as a blade, and with a period that decreases with rotation rate. The wake favours the smoothing of the density step and, in a simplified model, we assume that the turbulent diffusion is active during a fraction of the cycle in the wake-mixing region, with diffusivity proportional to the transverse length scale and the speed of the wake. The mixing region is the domain between the nose of the wave-like perturbation and the section where the interface becomes ‘darker’ again after being mixed by the vortexes. The results of this model are in a fair agreement with the experiments. The potential energy of the interfacial perturbations is only a small part of the missing turbulent kinetic energy, defined as the difference in the turbulent kinetic energy between a well-mixed fluid and a two-layer fluid. Further analysis is needed to explain the mechanism of generating these perturbations and the factors that control their periodicity.

We study the dynamics of cavitation bubbles and induced jets in a thin liquid gap bounded by two rigid walls. The bubbles are generated experimentally with a focused laser pulse and are compared to simulations. The gap height $H$ and the distance of the position of bubble nucleation $h$ with respect to the nearest wall are varied. The bubble dynamics is recorded at 500 000 frames per second and is compared to simulation results from the compressible volume of fluid solver based on OpenFOAM that takes into account viscosity and surface tension. Good agreement of the spatio-temporal bubble dynamics between experiments and simulations is obtained. The findings are that the parameter space consists of three regions with distinct jetting dynamics that are characterized by two dimensionless parameters: the normalized gap height, $\unicode[STIX]{x1D702}=H/R_{max}$, where $R_{max}$ is the spherical equivalent radius of the bubble at maximum expansion, and the normalized stand-off distance of the bubble measured from the centre of the gap, $\unicode[STIX]{x1D701}=(H/2-h)/R_{max}$. The three qualitatively distinct jetting behaviours are the transferred jet impacting on the distant wall, the double jet as a result of a bubble splitting and impacting on both walls and the directed jet from a conically shaped bubble impacting on the closest wall. The impact velocity of the liquid jets onto the walls can reach more than $200~\text{m}~\text{s}^{-1}$ and strongly depends on the gap height and bubble position. The simulations reveal that the viscous boundary layers affect the bubble splitting and therefore the directions of jetting. Additionally, we found that with increasing length of the thin gap $L$ the bubble oscillation period increases and converges for sufficiently large gaps.

The mechanisms of unsteadiness in nominally two-dimensional (2-D) shock/turbulent-boundary-layer interactions (STBLIs) cannot be directly extended to three-dimensional (3-D) STBLIs, because of differences in interaction structure; swept 3-D interactions, including the sharp-fin and swept-compression-ramp configurations, are of particular interest in this work. Complications arise from the observation that the separation length employed to scale low-frequency unsteadiness in 2-D (spanwise homogeneous) interactions is not a global property of 3-D (swept) interactions, due to the quasi-conical symmetry of the latter. Also, flow separation in 3-D interactions is topologically different, in that closure of the primary separation cannot occur without breaking the quasi-conical symmetry of the interaction – consequently, the unsteady properties of the separation are different. To address these points, large-eddy simulations are performed to assess unsteadiness in 3-D interactions, with the aim of understanding key differences relative to analogous 2-D interactions, the former of which have received less attention in the literature. The mechanism underlying the prominent band of low-frequency unsteadiness (two decades below the characteristic boundary-layer frequency) is shown to be significantly muted in swept interactions. An interesting scaling for the band of mid-frequency unsteadiness is uncovered (at least one decade below the characteristic boundary-layer frequency). This is a consequence of the observed connection between coherent fluctuations in the separated shear layer and local mean-flow gradients, indicating a mix between competing 2-D and 3-D free-interaction scaling laws. In contrast, high-frequency fluctuations largely retain the 2-D scaling introduced by the incoming turbulent boundary layer. The spatial structure of the mid-frequency coherence in 3-D STBLIs is isolated, revealing the significant influence of these convective coherent structures on shock rippling/corrugation, as well as a spanwise dependence of coherence size consistent with the 3-D mean-flow similarity scaling. Finally, the dynamic linear response of a representative 3-D interaction is compared to that of a representative 2-D interaction; the absolute instability present in the 2-D interaction is not present in the 3-D interaction. The coincident absence of both the absolute instability and associated band of low-frequency unsteadiness in 3-D STBLIs underscores the significance of this absolute instability in facilitating low-frequency unsteadiness in 2-D interactions.

The development of a wing-tip vortex of a rectangular, square-tipped wing having a NACA 0012 airfoil at a chord Reynolds number $Re_{c_{w}}=2\times 10^{5}$, under the effect of synthetic jet actuation, was experimentally studied. Five control configurations were considered: case C1 with momentum coefficient $C_{\unicode[STIX]{x1D707}}=0.001$ and actuation frequency $F^{+}=0.075$; case C2 with $C_{\unicode[STIX]{x1D707}}=0.001$ and $F^{+}=0.15$; case C3 with $C_{\unicode[STIX]{x1D707}}=0.001$ and $F^{+}=0.3$; case C4 with $C_{\unicode[STIX]{x1D707}}=0.001$ and $F^{+}=0.6$; and case C5 with $C_{\unicode[STIX]{x1D707}}=0.001$ and $F^{+}=1.2$. Under the most effective configuration, case C3, the vortex was stretched and appeared to be diffuse with a nearly 40 % decrease in the peak circumferential velocity and 50 % decrease in the core axial vorticity. The vortex core radius largely broadened suggesting that the lower-frequency control configuration allowed the synthetic jet to travel larger distances into the vortex bringing turbulent structures within its core resulting in increased mixing and subsequently a more diffuse vortex.

We identify an incorrect term in the expression for the rate of change of circulation of a material volume including an interface between two fluids which appears in Brøns et al. (J. Fluid Mech., vol. 758, 2014, pp. 63–93).