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.

As shown by Wenzel et al. (J. Fluid Mech., vol. 930, 2022, A1), the Eckert number $Ec$ defined using the difference between recovery temperature $\bar{T}_r$ and wall temperature $\bar{T}_w$ can be understood as a meaningful quantity to compare heat-transfer effects inside compressible turbulent boundary layers (for a calorically perfect gas), no matter whether these are caused by different Mach-number or wall-temperature conditions. While the named study deduced this comparative behaviour of $Ec$ from an integral perspective in a strict sense, Cogo et al. (J. Fluid Mech., vol. 974, 2023, A10) performed a systematic parameter study based on the previous findings to look at wall-normal profiles. They have shown that the diabatic parameter $\varTheta$, being equivalent to $Ec$, is capable of categorizing heat-transfer effects for cases at different Mach numbers, even to some extent for some of the wall-normal profiles. Building on this progress, the present paper provides a comprehensive classification of both existing and newly computed super- and hypersonic direct numerical simulation data at various wall temperature conditions into heated cases, adiabatic cases or weakly/moderately/strongly/quasi-incompressibly cooled cases. Hereby, the classification is largely based on the wall-normal position of the temperature peak occurring in cooled boundary-layer cases, which is one of the determining factors for the topological characteristics of diabatic boundary-layer profiles. Integrating high-enthalpy data into the analysis allowed us to confirm the reliability of the proposed classification also in more complex scenarios, where the calorically perfect gas assumption no longer applies and additional heat-transfer mechanisms come into play. While the Eckert number is shown to well characterize heat-transfer effects on most important temperature-related quantities for a wide range of Mach numbers and $\bar {T}_w/\bar {T}_r$ conditions, also the local Reynolds number $Re_{\tau }$ is shown to notably affect the strength of heat-transfer effects. Since both $Ec$ and $Re_{\tau }$ can be determined in advance – or estimated to a reasonable extent – a key advantage of the classification scheme is to allow for an effective a priori estimation of the extent to which heat-transfer effects are to be expected for a given compressible turbulent boundary-layer configuration.

We explored the instability dynamics of the viscous fingering interaction in dual displacement fronts by varying the viscosity configuration. Four regimes of rear-dominated fingering, front-dominated fingering, dual fingering and stable were identified. By using the breakthrough time, which refers to the breakup of the dual displacement fronts, the instability dynamics were modelled, and a regime map was developed. These serve as a tool for effectively harnessing the dual displacement fronts for various applications, such as hydrogeology, petroleum, chemical processes and microfluidics.

We revisit Brenner's seminal work on the Stokes resistance of a slightly deformed sphere (Chem. Engng Sci., vol. 19, 1964, p. 519), evaluate its range of validity and extend its applicability to higher deformations for axisymmetric particles, using hydrodynamic radius as the measure of Stokes resistance. Brenner's method solves the flow around a slightly deformed sphere through two mapping steps: the first mapping translates the surface velocity on the deformed sphere to that over a reference sphere of arbitrary radius using an asymptotic expansion of the flow field in terms of deformation amplitude and a Taylor expansion of the velocity field around the surface of the reference sphere. Subsequently, the second mapping extrapolates the velocity field from the surface of the reference sphere to any point in the fluid using Lamb's general solution for Stokes flow. While the original work addresses slightly deformed spheres to a linear order in deformation amplitude, we demonstrate that the first mapping, in combination with axisymmetric spectral modes (J. Fluid Mech., vol. 936, 2022, R1), can accommodate significant deformations to arbitrary orders of perturbation, and thus is not limited to slightly deformed spheres. Also, while first-order analysis is suitable for nearly spherical particles, second-order terms can provide a reasonable range for significantly higher deformations.

We investigate air-entraining flows where degassing, rather than fragmentation, plays a significant role. Of interest is the power-law slope $\beta$ of the bulk bubble size distribution $N(a)$ during the air-generating period, when the total volume of bubbles is increasing. We study a canonical air-entraining flow created by strong underlying free-surface turbulence. We perform analysis using the population balance equation (PBE) and computations using direct numerical simulations (DNS) with bubble tracking. We quantify the importance of degassing by the ratio of degassing flux ($Q_D$) to entrainment flux ($Q_I$), $\mathcal {D}=Q_D/Q_I$, and the ratio of degassing rate ($\varLambda (a)$) to fragmentation rate ($\varOmega (a)$) for a bubble of radius $a$, $\varLambda (a)/\varOmega (a)$. For a broad range of large Froude numbers ${{Fr}}=U/\sqrt {L g}$, DNS give $\mathcal {D}=\operatorname {O}(1)$ (independent of ${{Fr}}$), showing that degassing is relevant, and $\varLambda (a) \gg \varOmega (a)$, showing that the bubble population is degassing-dominated. In contrast to fragmentation-dominated populations, such as those due to wave breaking, where $\beta =-10/3$, degassing-dominated populations have qualitatively different $N(a)$ during air entrainment. Analysis using the PBE shows that degassing-dominated $\beta$ is a function of $\varLambda (a)$, which has a turbulence-driven regime ($a< a_\varLambda$) and a buoyancy-driven regime ($a>a_\varLambda$). Here, $a_\varLambda$ is the bubble radius where terminal buoyant rise velocity equals $u_{rms}$. Consequently, $N(a)$ exhibits a split power with $\beta (a< a_\varLambda )=-4.\bar {3}$ and $\beta (a>a_\varLambda )=-5.8\bar {3}$ for moderate bubble Reynolds numbers ${{Re}}_b$. For large ${{Re}}_b$, $\beta (a>a_\varLambda )=-4.8\bar {3}$. The DNS strongly confirm these findings for moderate ${{Re}}_b$. By identifying and describing degassing-dominated bubble populations, this work contributes to the understanding and interpretation of broad types of air-entraining problems where degassing plays a relevant role.

We use three-dimensional direct numerical simulations of homogeneous isotropic turbulence in a cubic domain to investigate the dynamics of heavy, chiral, finite-size inertial particles and their effects on the flow. Using an immersed-boundary method and a complex collision model, four-way coupled simulations have been performed, and the effects of particle-to-fluid density ratio, turbulence strength and particle volume fraction have been analysed. We find that freely falling particles on the one hand add energy to the turbulent flow but, on the other hand, they also enhance the flow dissipation: depending on the combination of flow parameters, the former or the latter mechanism prevails, thus yielding enhanced or weakened turbulence. Furthermore, particle chirality entails a preferential angular velocity which induces a net vorticity in the fluid phase. As turbulence strengthens, the energy introduced by the falling particles becomes less relevant and stronger velocity fluctuations alter the solid phase dynamics, making the effect of chirality irrelevant for the large-scale features of the flow. Moreover, comparing the time history of collision events for chiral particles and spheres (at the same volume fraction) suggests that the former tend to entangle, in contrast to the latter which rebound impulsively.