We conducted experiments in a laboratory to study turbulent flow over wind generated water waves. The experiments were performed in a wind-wave-current flume with three free stream wind speeds of Uref = 6.0, 8.0 and 10.0 m s−1, corresponding to 10 m equivalent wind speed of U10 = 10.2, 12.2 and 14.1 m s−1 and the root-mean-square wave height of 0.7, 1.1 and 1.7 cm, respectively, at a fetch of 6.2 m. The instantaneous velocity fields above the waves were obtained by using a particle image velocimetry (PIV) technique. The velocity fields were decomposed into the mean, wave-induced and turbulent velocity components. The tested wind waves were primarily dissipated by capillaries and microscale breaking waves. The Bond number and the shear velocity-fetch based Reynolds number were found to correlate with the wind wave regimes well. The turbulent dissipation rates above the water surface were determined based on resolved spatial gradient of instantaneous velocities, where the time-averaged dissipation rate values were calibrated using those estimated from the one-dimensional velocity spectrum in the temporal space. Subsequently, the turbulent kinetic energy (TKE) budget including its production, dissipation, advection and turbulent transport was presented. In addition, conditional averaging analysis of the TKE budgets over leeward, windward sides and all phases was performed. The results showed a strong dependency with the wave phase in the TKE budget terms except for the dissipation. The production-dissipation ratio increased significantly as the wind speed increased, likely attributed to the increased roughness over the substantial coverage of micro-breaking waves.

This paper investigates the transport of drugs delivered by direct injection into the cerebrospinal fluid (CSF) that fills the intrathecal space surrounding the spinal cord. Because of the small drug diffusivity, the dispersion of neutrally buoyant drugs has been shown in previous work to rely mainly on the mean Lagrangian flow associated with the CSF oscillatory motion. Attention is given here to effects of buoyancy, arising when the drug density differs from the CSF density. For the typical density differences found in applications, the associated Richardson number is shown to be of order unity, so that the Lagrangian drift includes a buoyancy-induced component that depends on the spatial distribution of the drug, resulting in a slowly evolving cycle-averaged flow problem that can be analysed with two-time scale methods. The asymptotic analysis leads to a nonlinear integro-differential equation for the spatiotemporal solute evolution that describes accurately drug dispersion at a fraction of the cost involved in direct numerical simulations of the oscillatory flow. The model equation is used to predict drug dispersion of positively and negatively buoyant drugs in an anatomically correct spinal canal, with separate attention given to drug delivery via bolus injection and constant infusion.

Equations relating the pressure at a horizontal seabed, the free-surface profile and the surface pressure are derived for two-dimensional irrotational steady water waves with arbitrary pressure at the free surface. Special cases include gravity, capillary, flexural and wind waves. Without approximations, we show that the free-surface recovery from the bottom pressure requires the resolution of only one first-order ordinary differential equation independent of the surface pressure, thus providing a new general recovery method valid for a broad class of water waves. Another equation provides an explicit expression for the surface pressure as a function of the bottom pressure and of the free surface. Thus, if unknown, the surface pressure can also be recovered if one extra measurement is available. This new recovery procedure is illustrated analytically for the linear approximation of a flexural–capillary–gravity wave, and numerically for fully nonlinear capillary–gravity waves.

We study coherent structures in subsonic turbulent jets subject to a flight stream. A thorough characterisation of the effects of a flight stream on the turbulent field was recently performed by Maia et al. (Phys. Rev. Fluids, vol. 8, 2023, 063902) and fluctuation energy attenuations were observed over a broad range of frequencies and azimuthal wavenumbers. The Kelvin–Helmholtz, Orr and lift-up mechanisms were all shown to be weakened by the flight stream. Here we expand upon that study and model the changes in the dynamics of jets in flight using global resolvent analysis. The resolvent model is found to correctly capture the main effects of the flight stream on the dynamics of coherent structures, which are educed from a large-eddy simulation database using spectral proper orthogonal decomposition. Three modifications of note are: the damping of low-frequency streaky/Orr structures that carry most of the fluctuation energy; a degradation of the low-rank behaviour of the jet in frequencies where modal instability mechanisms are dominant; and a rank decrease at very low Strouhal numbers. The latter effect is underpinned by larger gain separations predicted by the resolvent analysis, due to a reduction in the wavelength of associated flow structures. This leads to a clearer relative dominance of streaky structures generated by the lift-up mechanism, despite the fact that the lift-up mechanism has been weakened with respect to the static jet.

The two-dimensional gravity-driven motion of a relatively dense viscous liquid at the base of a granular mush is investigated using a model that exploits the relative shallowness of the flow. The granular mush obeys a $\mu (I)$-rheology, and we assume that the two phases are segregated throughout the motion. The viscous liquid spreads under gravity, carrying the granular mush above and transporting it outwards as levees at either end of the flow. The accumulation of granular material away from the centre of the deposit produces hydrostatic pressure gradients that retard the viscous gravity current. At later times, the granular mush is quasi-static relative to the moving liquid owing to the balance of outward granular transfer by the liquid and inward hydrostatic pressure gradients associated with the granular free surface. The viscous liquid exhibits a Poiseuille-like flow structure with negligible velocity at both the base and the granular interface. The flow of a fixed volume of viscous liquid becomes self-similar with the effective viscosity quadrupled relative to a classical viscous gravity current owing to the retarding effects of the granular mush. The case of constant input flux of viscous liquid is also analysed. The qualitative features are akin to the fixed volume case with the granular mush forming levees and slowing the viscous spreading. The case in which the upper medium is a Bingham material rather than a granular mush is also discussed, and the same features are observed, demonstrating the importance of the yield criterion in the upper medium.

This paper investigates the linear and nonlinear evolution of radiating modes in supersonic boundary layers under the influence of impinging sound waves. It is found that the ensuing boundary-layer response is extraordinarily large for a subset of the sound frequency and incident angle, and the resonant over-reflection, corresponding to the reflection coefficient becoming infinite, occurs at a particular pairing of frequency and incident angle. At this point, the reflected wave coincides with a locally neutral radiating mode, which emits spontaneously sound in the form of Mach waves. A fundamental resonance takes place between the incident wave and the radiating mode. Viewed in a developing boundary layer, the response is rendered finite by introducing non-parallelism and nonlinear effects near the neutral location of the radiating mode, where the sound wave directly excites the radiating mode and/or acts on the pre-existing radiating mode. Inhomogeneous amplitude equations are derived to describe the excitation as well as the nonlinear development of the radiating mode in the two regimes where non-equilibrium and non-parallelism play a leading-order role, respectively. A composite amplitude equation is then constructed to take into account both non-parallelism and non-equilibrium effects. This amplitude equation is, with an appropriate initial condition, solved to quantify the impact of the impinging sound wave on the linear and nonlinear instability characteristics of the radiating mode. The far-field analysis shows that the Mach wave field of the radiating mode is changed significantly due to the incident sound.

This paper investigates the linear and nonlinear evolution of radiating modes under the influence of the spontaneously emitted Mach waves in a simple set-up of the supersonic boundary layers that develop in the entry region of a channel formed by two parallel semi-infinite flat plates. Two scenarios are considered. The first occurs in the boundary layers having identical wall conditions, where the Mach wave emitted by a radiating mode in one boundary layer influences the instability in the other. The second scenario takes place when the wall temperatures are different, in which case the spontaneously radiated Mach wave is reflected by the other boundary layer back to act on the radiating mode. Appropriate amplitude equations with the acoustic feedback effect being accounted for are derived. In each case, the effect of the spontaneously emitted sound contributes a linear term of delay type to the respective amplitude equation. For the first scenario, analytical and numerical studies of the amplitude equations show that due to the back action of the spontaneously radiated Mach wave, the amplitude exhibits rapid oscillations, and in the case of enhanced feedback effects, its envelope experiences near extinction followed by resurrection. The study of the coupled equations shows that the two modes with different initial amplitudes either undergo oscillations before attenuating, or terminate a finite-distance singularity at different locations. For the second scenario, the acoustic feedback produces similar effects in a broad range of wall temperature. The effects become pronounced, and the dependence on the wall temperature becomes more sensitive when the latter approaches the value corresponding to the resonance. Estimates suggest that such acoustic feedback is likely to be present in typical wind tunnel experiments and models for scramjet combustors.

Advective–diffusive transport in Poiseuille flow through a channel with partially absorbing walls is a classical problem with applications to a broad range of natural and engineered scenarios, ranging from solute and heat transport in porous and fractured media to absorption in biological systems and chromatography. We study this problem from the perspective of transverse distributions of surviving mass and velocity, which are a central ingredient of recent stochastic models of transport based on the sampling of local flow velocities along trajectories. We show that these distributions tend to asymptotic equilibria for large times and travel distances, and derive rigorous explicit expressions for arbitrary reaction rate. We find that the equality of flux-weighted and breakthrough distributions that holds for conservative transport breaks in the presence of reaction, and that the average velocity of the scalar plume is no longer fully characterized by the transverse distribution of flow velocities sampled at a given time.

The classical Cox–Voinov theory of contact line motion provides a relation between the macroscopically observable contact angle, and the microscopic wetting angle as a function of contact-line velocity. Here, we investigate how viscoelasticity, specifically the normal stress effect, modifies the wetting dynamics. Using the thin film equation for the second-order fluid, it is found that the normal stress effect is dominant at small scales yet can significantly affect macroscopic motion. We show that the effect can be incorporated in the Cox–Voinov theory through an apparent microscopic angle, which differs from the true microscopic angle. The theory is applied to the classical problems of drop spreading and dip coating, which shows how normal stress facilitates (inhibits) the motion of advancing (receding) contact lines. For rapid advancing motion, the apparent microscopic angle can tend to zero, in which case the dynamics is described by a regime that was already anticipated in Boudaoud (Eur. Phys. J. E, vol. 22, 2007, pp. 107–109).

Data-driven discovery of governing equations is of great significance for helping us understand intrinsic mechanisms and build physical models. Recently, numerous highly innovative algorithms have emerged, aimed at inversely discovering the underlying governing equations from data, such as sparse regression-based methods and symbolic regression-based methods. Along this direction, a novel dimensional homogeneity constrained gene expression programming (DHC-GEP) method is proposed in this work. The DHC-GEP simultaneously discovers the forms and coefficients of functions using basic mathematical operators and physical variables, without requiring preassumed candidate functions. The constraint of dimensional homogeneity is capable of filtering out the overfitting equations effectively. The key advantages of DHC-GEP compared with the original gene expression programming, including being more robust to hyperparameters, the noise level and the size of datasets, are demonstrated on two benchmark studies. Furthermore, DHC-GEP is employed to discover the unknown constitutive relations of two representative non-equilibrium flows. Galilean invariance and the second law of thermodynamics are imposed as constraints to enhance the reliability of the discovered constitutive relations. Comparisons, both quantitative and qualitative, indicate that the derived constitutive relations are more accurate than the conventional Burnett equations in a wide range of Knudsen numbers and Mach numbers, and are also applicable to the cases beyond the parameter space of the training data.

The interaction between a uniform current with a circular cylinder submerged in a fluid covered by a semi-infinite ice sheet is considered analytically. The ice sheet is modelled as an elastic thin plate, and the fluid flow is described by the linearised velocity potential theory. The Green function or the velocity potential due to a source is first obtained. As the water surface is divided into two semi-infinite parts with different boundary conditions, the Wiener–Hopf method (WHM) offers significant advantages over alternative approaches and is consequently adopted. To do that, the distribution of the roots of the dispersion equation for fluid fully covered by an ice sheet in the complex plane is first analysed systematically, which does not seem to have been done before. The variations of these roots with the Froude number are investigated, especially their effects or factorisation and decomposition required in the WHM. The result is verified by comparing with that obtained from the matched eigenfunction expansion method. Through differentiating the Green function with respect to the source position, the potentials due to multipoles are obtained, which are employed to construct the velocity potential for the circular cylinder. Extensive results are provided for hydrodynamic forces on the cylinder and wave profiles, and some unique features are discussed. In particular, it is found that the forces can be highly oscillatory with the Froude number when the body is below the ice sheet, whereas such an oscillation does not exist when the body is below the free surface.

Using experiments and a depth-averaged numerical model, we study instabilities of two-phase flows in a Hele-Shaw channel with an elastic upper boundary and a non-uniform cross-section prescribed by initial collapse. Experimentally, we find increasingly complex and unsteady modes of air-finger propagation as the dimensionless bubble speed $Ca$ and level of collapse are increased, including pointed fingers, indented fingers and the feathered modes first identified by Cuttle et al. (J. Fluid Mech., vol. 886, 2020, A20). By introducing a measure of the viscous contribution to finger propagation, we identify a $Ca$ threshold beyond which viscous forces are superseded by elastic effects. Quantitative prediction of this transition between ‘viscous’ and ‘elastic’ reopening regimes across levels of collapse establishes the fidelity of the numerical model. In the viscous regime, we recover the non-monotonic dependence on $Ca$ of the finger pressure, which is characteristic of benchtop models of airway reopening. To explore the elastic regime numerically, we extend the depth-averaged model introduced by Fontana et al. (J. Fluid Mech., vol. 916, 2021, A27) to include an artificial disjoining pressure that prevents the unphysical self-intersection of the interface. Using time simulations, we capture for the first time the majority of experimental finger dynamics, including feathered modes. We show that these disordered states evolve continually, with no evidence of convergence to steady or periodic states. We find that the steady bifurcation structure satisfactorily predicts the bubble pressure as a function of $Ca$, but that it does not provide sufficient information to predict the transition to unsteady dynamics that appears strongly nonlinear.

This study demonstrates an experimental realization of turbulence control strategies previously explored by Choi et al. (J. Fluid Mech., vol. 262, 1994, pp. 75–110) through numerical simulations. To conduct the experiments, a deformable surface with a streamwise array of 16 independently controlled actuators was developed. A real-time particle image velocimetry (RT-PIV) system was also created for flow measurements. The objective of the control strategy was to target the sweep and ejection motions of the vortex shedding from a spherical cap placed in a laminar boundary layer. Reactive control strategies consisted of wall-normal surface deformations that opposed or complied with the wall-normal (v) or streamwise (u) velocity fluctuations obtained from the RT-PIV. The results showed two primary outcomes of the control approach. Firstly, it effectively hindered the advancement of sweep motions towards the wall. Secondly, it disrupted the periodic shedding of vortices. The v-control with opposing wall motions and u-control with compliant wall motions exhibited strong inhibition of sweep motions, while the v-control with compliant and u-control with opposing wall motions showed weaker inhibition. All reactive control cases resulted in the disruption of vortex shedding. In some instances, this disruption was accompanied by increased turbulent kinetic energy due to the generation of additional flow motions. However, the v-control with opposing wall motions significantly reduced the vortex-shedding energy while maintaining total turbulent kinetic energy close to or below that of the unforced flow. Overall, the experiments show the effectiveness of reactive control strategies in mitigating sweep motions and disrupting vortical structures, offering insights for developing reactive control strategies.