Scaling laws for the thrust production and power consumption of a purely pitching hydrofoil in ground effect are presented. For the first time, ground-effect scaling laws based on physical insights capture the propulsive performance over a wide range of biologically relevant Strouhal numbers, dimensionless amplitudes and dimensionless ground distances. This is achieved by advancing previous scaling laws (Moored & Quinn (AIAA J., 2018, pp. 1–15)) with physics-driven modifications to the added mass and circulatory forces to account for ground distance variations. The key physics introduced are the increase in the added mass of a foil near the ground and the reduction in the influence of a wake-vortex system due to the influence of its image system. The scaling laws are found to be in good agreement with new inviscid simulations and viscous experiments, and can be used to accelerate the design of bio-inspired hydrofoils that oscillate near a ground plane or two out-of-phase foils in a side-by-side arrangement.

The effect of a counter-current gas flow on the linear stability of an inclined falling liquid film switches from destabilizing to stabilizing, as the flow confinement is increased. We confront this linear effect with the response of nonlinear surface waves resulting from long-wave interfacial instability. For the strongest confinement studied, the gas flow damps both the linear growth rate and the amplitude of nonlinear travelling waves, and this holds for waves of the most-amplified frequency and for low-frequency solitary waves. In the latter case, waves are shaped into elongated humps with a flat top that resist secondary instabilities. For intermediate confinement, the linear and nonlinear responses are opposed and can be non-monotonic. The linear growth rate of the most-amplified waves first decreases and then increases as the gas velocity is increased, whereas their nonlinear amplitude is first amplified and then damped. Conversely, solitary waves are amplified linearly but damped nonlinearly. For the weakest confinement, solitary waves are prone to two secondary instability modes that are not observed in unconfined falling films. The first involves waves of diminishing amplitude slipstreaming towards their growing leading neighbours. The second causes wave splitting events that lead to a train of smaller, shorter waves.

Turbulent convection plays a crucial role in many natural environments and engineering applications. One of the most fundamental questions is how the heat flux depends on the thermal driving and fluid property. It has been proposed that when the fluid layer experiences extremely strong thermal driving, the boundary layers will become fully turbulent and the so-called ultimate regime will emerge. In this work we proposed a numerical experiment in which the thermal boundary layer can be spatially decoupled with the viscous one. We demonstrate that, once the thermal boundary layer is fully decoupled from the viscous boundary layer and locates entirely inside the turbulent bulk, the scaling laws corresponding to the ultimate regime can be obtained, namely $Nu \sim Ra^{1/2}Pr^{1/2}$ and $Re \sim Ra^{1/2} Pr^{-1/2}$ with $Nu$ being the Nusselt number, $Re$ the Reynolds number, $Ra$ the Rayleigh number and $Pr$ the Prandtl number, respectively. Therefore, our results support the physical conjecture of the ultimate regime for the turbulent convection.

An analytical solution of the impulsive impact of a cylindrical body submerged below a calm water surface is obtained by solving a free boundary problem. The shape of the cross-section of the body is arbitrary. The integral hodograph method is applied to derive the complex velocity potential defined in a parameter plane. The boundary-value problem is reduced to a Fredholm integral equation of the first kind in the velocity magnitude on the free surface. The velocity field, the impulsive pressure on the body surface and the added mass are determined in a wide range of depths of submergence for various cross-sectional shapes, such as a flat plate, a circular cylinder and a rectangle.

Abrupt depth transitions (ADTs) have recently been identified as potential causes of ‘rogue’ ocean waves. When stationary and (close-to-) normally distributed waves travel into shallower water over an ADT, distinct spatially localized peaks in the probability of extreme waves occur. These peaks have been predicted numerically, observed experimentally, but not explained theoretically. Providing this theoretical explanation using a leading-order-physics-based statistical model, we show, by comparing to new experiments and numerical simulations, that the peaks arise from the interaction between linear free and second-order bound waves, also present in the absence of the ADT, and new second-order free waves generated due to the ADT.

Tsunami waves induced by landslides are a threat to human activities and safety along coastal areas. In this paper, we characterize experimentally the waves generated by the gravity-driven collapse of a dry granular column into water. Three nonlinear wave regimes are identified depending on the Froude number ${Fr}_f$ based on the ratio of the velocity of the advancing granular front and the velocity of linear gravity waves in shallow water: transient bores for large ${Fr}_f$, solitary waves for intermediate values of ${Fr}_f$, and nonlinear transition waves at small ${Fr}_f$. The wave amplitude relative to the water depth increases with ${Fr}_f$ in the three regimes but with different nonlinear scalings, and the relative wavelength is an increasing or decreasing function of ${Fr}_f$ depending on the wave regime. Two of these wave regimes are rationalized by considering that the advancing granular front acts as a vertical piston pushing the water, while the last one is found to be a transition from shallow- to deep-water conditions. The present modelling contributes to a better understanding of the rich hydrodynamics of the generated waves, with coastal risk assessment as practical applications.

The classical theory by Jeffery predicts that, in the absence of Brownian fluctuations, a thin rigid platelet rotates continuously in a shear flow, performing periodic orbits. However, a stable orientation is possible if the surface of the platelet displays a hydrodynamic slip length $\lambda$ comparable to or larger than the thickness of the platelet. In this article, by solving the Fokker–Plank equation for the orientation distribution function and corroborating the analysis with boundary integral simulations, we quantify a threshold Péclet number, ${Pe}_{c}$, above which such alignment occurs. We found that for ${Pe}$ smaller than ${Pe}_{c}$, but larger than a second threshold, a regime emerges where Brownian fluctuations are strong enough to break the platelet's alignment and induce rotations, but with a period of rotation that depends on the value of $\lambda$. For ${Pe}$ below this second threshold, slip has a negligible effect on the orientational dynamics. We use these thresholds to classify the dynamics of graphene-like nanoplatelets for realistic values of $\lambda$ and apply our results to the quantification of the orientational contribution to the effective viscosity of a dilute suspension of nanoplatelets with slip. We find a non-monotonic variation of this term, with a minimum occurring when the slip length is comparable to the thickness of the particle.

We study flow-induced vibrations of a square prism free to oscillate in two degrees of freedom (cross-flow (CF) and inline (IL)), and placed in flow at varying angles of attack, by measuring the prism's displacement and flow-induced forces in both degrees of freedom experimentally, and conducting hydrogen bubble visualizations, as well as bubble image velocimetry. At large angles of attack (where $\alpha = 45^{\circ }$ corresponds to the case where one of the edges of the prism sees the flow first), we observe a two degree of freedom vortex-induced vibration (VIV) response with a figure-eight trajectory, similar to what has been observed for a cylinder with two degrees of freedom. As the angle of attack is decreased, the figure-eight trajectory transitions to a teardrop trajectory, suggesting a $1:1$ ratio between the oscillation frequencies in the CF and IL directions. The VIV response remains to be the dominant response down to an angle of attack of $\alpha = 20^{\circ }$. At angles of attack of $\alpha = 10^{\circ }$ and $\alpha = 15^{\circ }$, the VIV response becomes negligible and elliptical trajectories are observed at higher reduced velocities. These elliptical trajectories then become four-lobe trajectories with amplitudes mainly in the CF direction at the lowest angles of attack (where a side of the square sees the flow first) and galloping-type response is observed, where the amplitude of oscillations is increased with increasing reduced velocity. Deviations from a typical galloping response are observed due to synchronizations between the shedding frequency and oscillation frequency at ranges of higher reduced velocities.

Resolvent analysis of the linearized Navier–Stokes equations provides useful insight into the dynamics of transitional and turbulent flows and can provide a model for the dominant coherent structures within the flow, particularly for flows where the linear operator selectively amplifies one particular force component, known as the optimal force mode. Force and response modes are typically obtained from a singular-value decomposition of the resolvent operator. Despite recent progress, the cost of resolvent analysis for complex flows remains considerable, and explicit construction of the resolvent operator is feasible only for simplified problems with a small number of degrees of freedom. In this paper we propose two new matrix-free methods for computing resolvent modes based on the integration of the linearized equations and the corresponding adjoint system in the time domain. Our approach achieves an order of magnitude speedup when compared with previous matrix-free time-stepping methods by enabling all frequencies of interest to be computed simultaneously. Two different methods are presented: one based on analysis of the transient response, providing leading modes with fine frequency discretization; and another based on the steady-state response to periodic forcing, providing optimal and suboptimal modes for a discrete set of frequencies. The methods are validated using a linearized Ginzburg–Landau equation and applied to the three-dimensional flow around a parabolic body.

Hypersonic flow over a two-dimensional compression corner with a Mach number of 7.7 and unit Reynolds number of 4.2 × 106 m−1 is numerically investigated. Special emphasis is given to the onset of global instability with respect to three-dimensional perturbations. Global stability analysis is performed for various ramp angles and wall temperatures. It is found that the shock-induced separated flow system becomes unstable when the ramp angle is beyond a certain value. The critical ramp angle increases slightly with the wall temperature, although the length of the separation region is significantly enlarged. The global instability is shown to be closely linked with the occurrence of secondary separation beneath the primary separation bubble. A criterion is established based on a scaled ramp angle defined in the triple-deck theory to predict the global stability boundary, which depends on the free-stream conditions and geometries only.

The primary object of this study is to derive new scaling laws for passive scalar statistical quantities in temporally and spatially evolving plane turbulent jet flows. We apply Lie symmetry analysis to the equations governing the evolution of the first three statistical moments for the passive scalar quantities. The analysis is based on novel forms of the two-point velocity–scalar and scalar–scalar correlation equations, which are naturally based on the statistical moments derived from the instantaneous velocities and not on those of the fluctuation velocities from the Reynolds decomposition. The newly derived invariant solutions recover the gaps in the classical self-similarity analysis from three major perspectives. First, the scaling laws are constructed as the direct consequence of the symmetry approach, while an a priori set of similarity scales is not required. Second, unlike in the classical laws, which show self-similarity primarily for the first moments, here self-similarity is theoretically shown up to the third moments. Third, there is a symmetry breaking induced by the integral invariant of the mean temperature that connects the scaling symmetries of momentum and passive scalar equations, which results in a close coupling of the scaling exponents of the first two moments and, further, determines the scaling exponents of all higher moments. To verify the new theoretical findings, we employ data from two direct numerical simulations of the mixing of a passive scalar driven by a temporally evolving turbulent jet. The direct numerical simulation data very clearly validate the new scaling laws up to the third moments.

Hill's vortex is a classical solution of the incompressible Euler equations which consists of an axisymmetric spherical region of constant vorticity matched to an irrotational external flow. This solution has been shown to be a member of a one-parameter family of steady vortex rings and as such is commonly used as a simple analytic model for a vortex ring. Here, we model the decay of a Hill's vortex in a weakly rotating flow due to the radiation of inertial waves. We derive analytic results for the modification of the vortex structure by rotational effects and the generated wave field using an asymptotic approach where the rotation rate, or inverse Rossby number, is taken to be small. Using this model, we predict the decay of the vortex speed and radius by combining the flux of vortex energy to the wave field with the conservation of peak vorticity. We test our results against numerical simulations of the full axisymmetric Navier–Stokes equations.

Coherent structures in the far field of a round turbulent jet are investigated experimentally and modelled by local linear stability analysis (LSA) and local resolvent analysis (RA). The study aims to determine the potential and limitations of mean flow-based linear models predicting the far field dynamics. Particular emphasis is placed on the high wavenumber and frequency range. The study is based on time-resolved stereoscopic particle image velocimetry (PIV) data acquired in the self-similar region of the jet. Spectral proper orthogonal decomposition (SPOD) is applied to the dataset to identify empirical coherent structures with azimuthal wavenumbers ranging from $m=0$ to $m=\pm 5$. The leading SPOD mode features low-rank behaviour over a wide frequency range and is found to account for the major part of total turbulent production. Thus, the leading SPOD mode captures the anisotropic part of turbulence, which is still significant even at the highest resolved frequencies reaching into the inertial subrange. The LSA determines stable but discrete eigenmodes that are excellently in line with the SPOD modes. This applies especially to modes at mid-range to high frequencies and higher azimuthal wavenumbers where the LSA predicts strongly decaying modes. Moreover, the RA modes are in very good agreement with LSA and SPOD modes, indicating a predominantly resonant mechanism. The present study shows that an unexpectedly wide range of turbulent scales in the self-similar region of the jet can be reproduced based on linearized mean-field models.

Rock–ice avalanches are liquid–granular flows that consist of a mixture of rock, ice and a liquid. The dynamics that distinguishes these types of flows from other geophysical flows is the ice melting. This process is responsible for mass and momentum transfers between the solid and liquid components of the mixture and for the effects of lubrication and fluidization that reduce the mixture strength. In this work, we analyse the problem from a mathematical point of view. Starting from the partial differential equations of a complete three-phase approach, we identify two basic assumptions that can be used to build a framework of classes of simplified models. The implications of these assumptions on the physical description of the flow are carefully analysed for each class, and particular attention is paid to the simplification of the melting process expressed in terms of mass and momentum transfers. Moreover, the derived framework allows us to classify the existing literature models and to identify a new class of models that can be considered a reasonable trade-off between simplicity and completeness. Finally, the mathematical nature of each class is investigated by performing an in-depth analysis of the eigenvalues. Results show that the most simplified models are strictly hyperbolic, while the most complete approaches are affected by a loss of hyperbolicity in given ranges of the model parameters. Further research is necessary to understand the reasons and the numerical implications of this feature.

We study the correlation function and mean linear response function of the velocity Fourier mode of statistically steady-state, homogeneous and isotropic turbulence in Eulerian and Lagrangian coordinates through direct numerical simulation (DNS). As the Lagrangian velocity, we here adopt Kraichnan's Lagrangian-history framework where Lagrangian particles are labelled with current positions and their velocities are measured at some time before. This Lagrangian velocity is numerically calculated with a method known as the passive vector method. Our first goal is to study the relation between the correlation function and the mean linear response function in Eulerian and Lagrangian coordinates. Such a relation is known to be important in analysing the closed set of equations for the two functions, which are obtained by direct-interaction-approximation-type closures. We demonstrate numerically that the fluctuation–dissipation theorem (proportionality between the two functions) does not hold. The relation is further investigated with general analytical expressions of the mean linear response function under stochastic settings, which are known as the fluctuation-response relations in non-equilibrium statistical mechanics. Our second goal is to identify characteristic times associated with the two functions and to compare the times between the Eulerian and Lagrangian coordinates. Our DNS result supports the common view that the Eulerian characteristic times have the sweeping-time scaling ($\propto k^{-1}$, where $k$ is the wavenumber) for both functions and the Lagrangian characteristic times in the inertial range have the Kolmogorov-time scaling ($\propto k^{-2/3}$) for both functions.

We theoretically and numerically investigate the instabilities driven by diffusiophoretic flow, caused by a solutal concentration gradient along a reacting surface. The important control parameters are the Péclet number $Pe$, which quantifies the ratio of the solutal advection rate to the diffusion rate, and the Schmidt number $Sc$, which is the ratio of viscosity and diffusivity. First, we study the diffusiophoretic flow on a catalytic plane in two dimensions. From a linear stability analysis, we obtain that for $Pe$ larger than $8{\rm \pi}$ mass transport by convection overtakes that by diffusion, and a symmetry-breaking mode arises, which is consistent with numerical results. For even larger $Pe$, nonlinear terms become important. For $Pe > 16{\rm \pi}$, multiple concentration plumes are emitted from the catalytic plane, which eventually merge into a single larger one. When $Pe$ is even larger ($Pe \gtrsim 603$ for Schmidt number $Sc=1$), there are continuous emissions and merging events of the concentration plumes. The newly found flow states have different flow structures for different $Sc$: for $Sc\geqslant 1$, we observe the chaotic emission of plumes, but the fluctuations of concentration are only present in the region near the catalytic plane. In contrast, for $Sc<1$, chaotic flow motion occurs also in the bulk. In the second part of the paper, we conduct three-dimensional simulations for spherical catalytic particles, and beyond a critical Péclet number again find continuous plume emission and plume merging, now leading to a chaotic motion of the phoretic particle. Our results thus help us to understand the experimentally observed chaotic motion of catalytic particles in the high $Pe$ regime.

This study is inspired by the velvety structures on an owl's upper wing surface. Anechoic wind tunnel experiments were conducted to study the effect of the velvety structures on trailing edge noise as well as the boundary layer flow of a flat plate model. The tests were conducted in The Hong Kong University of Science and Technology low-speed wind tunnel, ultra-quiet noise injection test and evaluation device (UNITED). It was found that the trailing edge noise spectra are significantly modified by the velvety structures. In general, the velvety structures increase the low-frequency noise below a cross-over Strouhal number $St_c$ but reduce the spectral level at higher frequencies. The velvety surface also changes the boundary layer characteristics in terms of the boundary layer thickness, non-dimensional velocity distribution and turbulence distribution. Vortex shedding is suppressed by the velvety coating despite the blunt trailing edge. An analytic model is proposed for the trailing edge noise of a flat plate, including the effect of finite trailing edge thickness and velvety structures on the flat plate surface. The model uses the near wake distribution of the mean and fluctuating velocities in the streamwise direction as the input. The predictions, which require no empirical corrections, match well with the experiments for both the baseline and velvet-coated configurations. With a detailed non-dimensional analysis, this study proposes a potential aeroacoustic function of velvet structures, i.e. noise control through manipulation of boundary layer statistics.

As a stiff polymer tumbles in shear flow, it experiences compressive viscous forces that can cause it to buckle and undergo a sequence of morphological transitions with increasing flow strength. We use numerical simulations to uncover the effects of these transitions on the steady shear rheology of a dilute suspension of stiff polymers. Our results agree with classic scalings for Brownian rods in relatively weak flows but depart from them above the buckling threshold. Signatures of elastoviscous buckling include enhanced shear thinning and an increase in the magnitude of normal stress differences. We discuss our findings in the light of past work on rigid Brownian rods and non-Brownian elastic fibres and highlight the subtle role of thermal fluctuations in triggering instabilities.

A theoretical model is derived to predict the heat fluxes at the upper and lower horizontal surfaces of an internally heated (IH) convection cell by extending the well-known Grossmann & Lohse (Phys. Rev. Lett., vol. 86, 2001, pp. 3316–3319) theory. The approach of Creyssels (J. Fluid Mech., vol. 900, 2020, p. A39) is generalized for a fluid heated internally and uniformly, confined between top and bottom plates of equal temperature. For each plate, a Nusselt number is defined and an analytical formula is given to predict its variations with the Rayleigh and Prandtl numbers. The turbulent flow produced in the upper half of the IH convection cell is very similar to that observed in standard Rayleigh–Bénard convection. On the contrary, the lower plate is swept by the large scale flow that circulates through the entire cell. The corresponding boundary layer is therefore modelled by a laminar boundary layer of the Blasius type. These predictions are consistent with the independent theoretical scalings proposed by Wang et al. (Geophys. Res. Lett., vol. 48, issue 4, 2021, p. e2020GL091198) and they are confirmed by the numerical results obtained by Goluskin & van der Poel (J. Fluid Mech., vol. 791, 2016, p. R6) and Wang et al. (Geophys. Res. Lett., vol. 48, issue 4, 2021, p. e2020GL091198).

We study the behaviour of circular flexible loops sedimenting in a viscous fluid by numerical simulations and linear stability analysis. The numerical model involves a local slender-body theory approximation for the flow coupled to the Euler–Bernoulli elastic forces for an inextensible fibre. Starting from an inclined circle, we simulate the dynamics using truncated Fourier modes to observe three distinct regimes of motion: absolute stability, two- and three-dimensional dynamics, depending on the relative importance of the elastic and gravitational forces. We identify the governing parameter and develop a simple semi-analytic stability criterion, which we verify numerically. In all cases, sedimenting loops converge to stable, planar shape equilibria with one free parameter related to the initial conditions and material properties of the fibre.