We use the Korteweg–de Vries (KdV) equation, supplemented with several forcing/friction terms, to describe the evolution of wind-driven water wave packets in shallow water. The forcing/friction terms describe wind-wave growth due to critical level instability in the air, wave decay due to laminar friction in the water at the air–water interface, wave stress in the air near the interface induced by a turbulent wind and wave decay due to a turbulent bottom boundary layer. The outcome is a modified KdV–Burgers equation that can be a stable or unstable model depending on the forcing/friction parameters. To analyse the evolution of water wave packets, we adapt the Whitham modulation theory for a slowly varying periodic wave train with an emphasis on the solitary wave train limit. The main outcome is the predicted growth and decay rates due to the forcing/friction terms. Numerical simulations using a Fourier spectral method are performed to validate the theory for various cases of initial wave amplitudes and growth and/or decay parameter ranges. The results from the modulation theory agree well with these simulations. In most cases we examined, many solitary waves are generated, suggesting the formation of a soliton gas.

The present experimental study shows that a nozzle with optimal flexibility can enhance the impulse and entrainment of a pulsed jet. Near the nozzle exit, vortex rings emanating from the flexible nozzle move faster because of the timely release of the elastic energy (stored during the expansion) to the jet, which is maximized at the structural stiffness that needs to be optimally tuned to the jet acceleration. The total circulation, hydrodynamic impulse and entrained fluid volume are enhanced substantially. Interestingly, we find that the same condition for optimal flexibility to maximize the hydrodynamic impulse and circulation of the primary vortex ring of the continuous jet (Choi & Park, J. Fluid Mech., vol. 949, 2022, A39) holds universally for the pulsed jet, indicating that abrupt jet termination is irrelevant to the impulse augmentation mechanism. Compared to the rigid counterpart, increments of the impulse (${\sim }400\,\%$) and entrainment (${\sim }220\,\%$) of a pulsed jet in the present study are considerably larger than those ($200\,\%$ and $50\,\%$, respectively) in a continuous jet from previous studies, which is attributed to the significant suppression of negative pressure at the nozzle exit by the collapsing motion of the flexible nozzle in the phase with the jet-driven upstream propagation of the surface wave on the nozzle. This universal mechanism provides a guideline for a novel jet propulsor using a flexible nozzle, for example, for small-scale underwater robots.

Buoyancy-driven bubbly flows play pivotal roles in various scenarios, such as the oxygenation and mixing in the upper ocean and the reaction kinetics in chemical and bio-reactors. This work focuses on the convective flow induced by the localised release of large air bubbles ($D_b=3.7$ mm, ${Re}_b=950$) in a water tank, exploring the resulting flow and the transition from laminar to disturbed states as a function of the Rayleigh number ranging from $3\times 10^3$ to $2\times 10^5$. At low ${Ra}$ the flow is smooth and laminar with weak temporal oscillations, while a highly disturbed state appears above a critical value ${Ra}_c$. A theoretical analysis is presented that links the mean flow circulation to the Rayleigh number. Through an experimental investigation, utilising three-dimensional particle tracking velocimetry and flow visualisation, we confirm the theory presented, and characterise the laminar to disturbed transition in the system. These findings not only enhance our fundamental understanding of buoyancy-driven convective flows but also hold significant implications for practical applications, particularly in the optimisation of bio-reactor design and other industrial processes reliant on controlled convective dynamics.

Laboratory experiments and particle-resolved simulations are employed to investigate the settling dynamics of a pair of rigidly connected spherical particles of unequal size. They yield a detailed picture of the transient evolution and the terminal values of the aggregate's orientation angle and its settling and drift velocities as functions of the aspect ratio and the Galileo number $Ga$, which denotes the ratio of buoyancy and viscous forces acting on the aggregate. At low to moderate values of $Ga$, the aggregate's orientation and velocity converge to their terminal values monotonically, whereas for higher $Ga$-values the aggregate tends to undergo a more complex motion. If the aggregate assumes an asymmetric terminal orientation, it displays a non-zero terminal drift velocity. For diameter ratios much larger than one and small $Ga$, the terminal orientation of the aggregate becomes approximately vertical, whereas when $Ga$ is sufficiently large for flow separation to occur, the aggregate orients itself such that the smaller sphere is located at the separation line. Empirical scaling laws are obtained for the terminal settling velocity and orientation angle as functions of the aspect ratio and $Ga$ for diameter ratios from 1 to 4 and particle-to-fluid density ratios from 1.3 to 5. An analysis of the accompanying flow field shows the formation of vortical structures exhibiting complex topologies in the aggregate's wake, and indicates the formation of a horizontal pressure gradient across the larger sphere, which represents the main reason for the emergence of the drift velocity.

Direct numerical simulation of the three-dimensional (3-D) wake transition of a heated square cylinder subjected to horizontal cross-flow is performed in the presence of buoyancy. In order to capture the effects of large-scale heating, a non-Oberbeck–Boussinesq model is utilized, which includes the governing equations for compressible gas flow. All computations are performed at low free stream Mach number $M=0.1$ using air (free stream Prandtl number, $Pr=0.71$) as the working fluid. The 3-D instability modes A and B, which correspond to free stream Reynolds numbers of 180 and 250, are observed with longer and shorter spanwise wavelengths, respectively, and the onset of three-dimensionality is triggered at a Reynolds number of 173. In the presence of buoyancy, baroclinic vorticity production in the near-wake plays an important role for streamwise vorticity generation. The chaotic wake of the Mode-A instability bifurcates into periodic and quasiperiodic wakes at various heating levels, expressed by the overheat ratio, $\varepsilon =(T_w-T_\infty )/T_\infty$, where $T_w$ and $T_\infty$ are the temperature of the cylinder surface and the ambient air, respectively. At low heating ($\varepsilon =0.2$), the 3-D Mode-A instability is suppressed leading to a two-dimensional wake flow. Further increase in heating, again brings back the three-dimensionality in the wake through Mode-E instability. The variation of thermophysical properties and the effective Reynolds number with increase in heating level around the cylinder is examined. It is shown that the effect of thermophysical properties competes with the baroclinic streamwise vorticity generation at higher levels of heating ($\varepsilon \geqslant 0.4$) to control the 3-D modes and wake dynamics.

Predicting the magnitude of the slip velocity of non-tracer particles with respect to the surrounding fluid is crucial to address both fundamental and practical questions involving dispersed turbulent flows. Here we derive an analytical model to predict the slip velocity of spherical particles in homogeneous isotropic turbulence. We modulate the particle equation of motion according to the inertial filtering framework, and obtain closed-form expressions for the mean slip velocity magnitude as a function of the governing parameters. These are compared against laboratory measurements and direct numerical simulations, demonstrating close agreement for both light and heavy particles, both smaller and larger than the Kolmogorov scales. The predictive value of the model and its implications are discussed, as well as the range of validity of the underlying assumptions.

The study investigates the flow structure, dynamics of the large-scale coherent structures, drag forces and sediment entrainment mechanisms generated by a circular array of diameter D containing rigid emerged circular cylinders of diameter d placed in a smooth-bed channel of depth h under strong shallow flow conditions (D/h = 20, d/D = 0.0125 and 0.025). Eddy resolving simulations are conducted with different values of the solid volume fraction 0.025 ≤ SVF ≤ 0.1 and non-dimensional frontal area per unit volume (2.56 ≤ aD ≤ 10.2, where for the present configuration, a = SVF(1/d)4/${\rm \pi}$) and a fixed channel Reynolds number (Reh = 10 000). The flow conditions are such that a vortex street (VS)-type of shallow wake is expected to form for a solid cylinder of diameter D. Findings are compared with previous results obtained for cases with moderately shallow conditions 2.5 ≤ D/h ≤ 3.5 and with the limiting case of flow past a solid cylinder with D/h = 20. For moderately shallow conditions, the core of the main horseshoe vortex (HV) forming around the array occupies a small fraction of the water column and its coherence is the largest in front of the array. By contrast, for very shallow conditions, multiple HVs form around the array and their cores occupy a large fraction of the water column. Moreover, the coherence of most of these HVs peaks close to the sides of the array. Only for relatively large aD values, the main HV extends over the whole upstream face of the array, similar to the limiting case of a solid cylinder. For sufficiently high aD, a secondary instability is present inside the near wake that leads to the formation of parallel horizontal vortices in the vicinity of the wake roller vortices. The changes in the wake structure with decreasing SVF and aD are qualitatively similar to those observed for cases with moderately shallow flow conditions, with the antisymmetric wake shedding mode being suppressed for aD ≤ 2.5. However, the Strouhal number associated with the shedding of wake rollers, which is still close to 0.2 for a solid cylinder with D/h = 20, can be as high as 0.4 for aD = 2.5. The paper also discusses how the steady wake and total wake lengths and the strength of the bleeding flow vary with the SVF. Simulation results show that the capacity of the flow to entrain sediment inside and around the array peaks for SVF = 0.05 and aD = 5.2, with sediment entrainment monotonically increasing with the SVF outside the array and monotonically decreasing inside the array. Despite the differences in the flow structure next to and inside the array, the variation of the mean, time-averaged streamwise drag coefficient of the solid cylinders ${\bar{C}_d}$ with aD is close to that observed for arrays with moderately shallow flow conditions. The combined drag coefficient for the array decreases with increasing flow shallowness, with the decay being stronger for relatively large values of aD.

We study the dynamics of hydraulic fracturing of an elastic solid in a Hele-Shaw cell. Compared with hydraulic fractures in an infinite elastic bulk, the viscous resistance comes mainly from the drag by the two parallel plates that forms the Hele-Shaw cell rather than by the fluid–solid interface. Such a feature leads to a different nonlinear differential–integral system that describes the coupled evolution of the fracture shape and pressure field. Our theory leads to hydraulic fractures of cusp shapes in the neighbourhood of the fracture tip, which is consistent with recent experimental observations. Accordingly, there exists no pressure singularity at the location of the fracture tip, which is also fundamentally different from our previous understandings of hydraulic fracturing of elastic solids.

Communities of swimming microorganisms often thrive near liquid–air interfaces. We study how such ‘active carpets’ shape their aquatic environment by driving biogenic transport in the water column beneath them. The hydrodynamic stirring that active carpets generate leads to diffusive upward fluxes of nutrients from deeper water layers, and downward fluxes of oxygen and carbon. Combining analytical theory and simulations, we examine the biogenic transport by studying fundamental metrics, including the single and pair diffusivity, the first passage time for particle pair encounters and the rate of particle aggregation. Our findings reveal that the hydrodynamic fluctuations driven by active carpets have a region of influence that reaches orders of magnitude further in distance than the size of the organisms. These non-equilibrium fluctuations lead to a strongly enhanced diffusion of particles, which is anisotropic and space dependent. Fluctuations also facilitate encounters of particle pairs, which we quantify by analysing their velocity pair correlation functions as a function of distance between the particles. We found that the size of the particles plays a crucial role in their encounter rates, with larger particles situated near the active carpet being more favourable for aggregation. Overall, this research broadens our comprehension of aquatic systems out of equilibrium and how biologically driven fluctuations contribute to the transport of fundamental elements in biogeochemical cycles.

We perform direct numerical simulations of actively controlled laminar separated wakes around low-aspect-ratio wings with two primary goals: (i) reducing the size of the separation bubble and (ii) attenuating the wing tip vortex. Instead of preventing separation, we modify the three-dimensional (3-D) dynamics to exploit wake vortices for aerodynamic enhancements. A direct wake modification is considered using optimal harmonic forcing modes from triglobal resolvent analysis. For this study, we consider wings at angles of attack of $14^\circ$ and $22^\circ$, taper ratios $0.27$ and $1$, and leading edge sweep angles of $0^\circ$ and $30^\circ$, at a mean-chord-based Reynolds number of $600$. The wakes behind these wings exhibit 3-D reversed-flow bubble and large-scale vortical structures. For tapered swept wings, the diversity of wake vortices increases substantially, posing a challenge for flow control. To achieve the first control objective for an untapered unswept wing, root-based actuation at the shedding frequency is introduced to reduce the reversed-flow bubble size by taking advantage of the wake vortices to significantly enhance the aerodynamic performance of the wing. For both untapered and tapered swept wings, root-based actuation modifies the stalled flow, reduces the reversed-flow region and enhances aerodynamic performance by increasing the root contribution to lift. For the goal of controlling the tip vortex, we demonstrate the effectiveness of actuation with high-frequency perturbations near the tip. This study shows how insights from resolvent analysis for unsteady actuation can enable global modification of 3-D separated wakes and achieve improved aerodynamics of wings.

We discuss the modal, linear stability analysis of generalized Couette–Poiseuille (GCP) flow between two parallel plates moving with relative speed in the presence of an applied pressure gradient vector inclined at an angle $0\leqslant \phi \leqslant 90^\circ$ to the plate relative velocity vector. All possible GCP flows can be described by a global Reynolds number $Re$, $\phi$ and an angle $0\leqslant \theta \leqslant 90^\circ$, where $\cos \theta$ is a measure of the relative weighting of Couette flow to the composite GCP flow. This provides a novel and uncommon group of generally three-dimensional base velocity fields with wall-normal twist, for which Squire's theorem does not generally apply, requiring study of oblique perturbations with wavenumbers $(\alpha,\beta )$. With $(\theta,\phi )$ fixed, the neutral surface $f(\theta,\phi ;Re,\alpha,\beta )=0$ in $(Re, \alpha,\beta )$ space is discussed. A mapping from GCP to plane Couette–Poiseuille flow stability is found that suggests a scaling relation $Re^*\alpha /k = H(\theta ^*)$ that collapses all critical parameters, where ${Re}^*= Re\,({\alpha _1}/{\alpha })\,({\sin \theta }/{\sin {\theta }^*})$ and $\tan \theta ^*=({\alpha _{1}}/\alpha )\tan \theta$, with $\alpha _1=\alpha \cos \phi +\beta \sin \phi$. This analysis does not, however, directly reveal global critical properties for GCP flow. The global $Re_{cr}(\theta,\phi )$ shows continuous variation, while $\alpha _{cr}(\theta,\phi )$ and $\beta _{cr}(\theta,\phi )$ show complex behaviour, including discontinuities owing to jumping of critical states across neighbouring local valleys (in $Re$) or lobes of the neutral surface. The discontinuity behaviour exists for all low $\phi$. For $\phi \gtrsim 21^\circ$, variations of $\alpha _{cr}(\theta )$ and $\beta _{cr}(\theta )$ are generally smooth and monotonic.

The entrainment of ambient fluid into a variable-density jet is typically quantified using an entrainment coefficient $\alpha$. Here, we investigate the dependence of $\alpha$ on the ratio of the jet's density $\rho _m$ and that of the ambient fluid $\rho _0$. Current parametrisations of $\alpha$ rely on a scaling inferred from early laboratory experiments (Ricou & Spalding, J. Fluid Mech., vol. 11, 1961, pp. 21–32). We demonstrate analytically that the experiments preclude definitive conclusions regarding the dependence of $\alpha$ on $\rho _m / \rho _0$ and that the underlying physical processes therefore warrant closer attention. To investigate the physics behind the dependence of entrainment on the density ratio we use a Favre-averaged entrainment decomposition. The decomposition is applied to data from large-eddy simulations of jets characterised by density ratios $\rho _m / \rho _0$ spanning over two orders of magnitude that have been verified against experimental data. Changes in the shape of the velocity profile are a significant contributor to entrainment in the near field due to the breakdown of the potential core, and persist over larger streamwise distances in heavy releases than in light releases. Therefore, to focus exclusively on the effects of density ratio, we study the region where the shape changes have become small but the density ratio is still significant. We show that the dimensionless turbulent kinetic energy production and mean kinetic energy flux depend strongly on the density ratio, both for our large-eddy simulation data and for recent experiments. Despite this, the entrainment coefficient is practically constant in this region and has value $\alpha \approx 0.07$ for all simulations.