In air-entraining flows, there is often strong turbulence beneath the free surface. We consider the entrainment of bubbles at the free surface by this strong free-surface turbulence (FST). Our interest is the entrainment size distribution (per unit free surface area) $I(a)/A_{\textit{FS}}$, for bubbles with radius $a$ greater than the capillary scale ($\approx 1.3\ \mathrm{mm}$ for air–water on Earth), where gravity dominates surface tension. We develop a mechanistic model based on entrained bubble size being proportional to the minimum radius of curvature of the initial surface deformation. Using direct numerical simulation of a flow that isolates entrainment by FST, we show that, consistent with our mechanism, $I(a)/A_{\textit{FS}} = C_I \, g^{-3} \varepsilon ^{7/3} (2 a)^{-14/3}$, where $g$ is gravity, and $\varepsilon$ is the turbulence dissipation rate. In the limit of negligible surface tension, $C_I\approx 3.62$, and we describe how $C_I$ decreases with increasing surface tension. This scaling holds for sufficiently strong FST such that near-surface turbulence is nearly isotropic, which we show is true for turbulent Froude number ${\textit{Fr}}^2_T = \varepsilon /u_{\textit{rms}} g \gt 0.1$. While we study FST entrainment in isolation, our model corroborates previous numerical results from shear-driven flow, and experimental results from open-channel flow, showing that the FST entrainment mechanism that we elucidate can be important in broad classes of air-entraining flows.

The propagation of linear waves in non-ideal compressible fluids plays a crucial role in numerous physical and engineering applications, particularly in the study of instabilities, aeroacoustics and turbulence modelling. This work investigates linear waves in viscous and heat-conducting non-ideal compressible fluids, modelled by the Navier–Stokes–Fourier equations and a fully arbitrary equation of state (EOS). The linearised governing equations are derived to analyse the dispersion relations when the EOS differs from that of an ideal gas. Special attention is given to the influence of non-ideal effects and various dimensionless numbers on wave propagation speed and attenuation. By extending classical results from Kovásznay (1953 J. Aeronaut. Sci. vol. 20, no. 10, pp. 657–674) and Chu (1965 Acta Mech. vol. 1, no. 3, pp. 215–234) obtained under the ideal gas assumption, this study highlights the modifications introduced by arbitrary EOSs to the linear wave dynamics in non-ideal compressible flows. This work paves the path for an improved understanding and modelling of wave propagation, turbulence and linear stability in arbitrary viscous and heat-conducting fluids.

The complex behaviour of air–liquid interfaces driven into Hele-Shaw channels at high speeds could arise from oscillatory dynamics; yet both the physical and dynamical mechanisms that lead to interfacial oscillations remain unclear. We extend the experiments by Couder et al. (1986, Phys. Rev. A, vol. 34, 5175) to present a systematic investigation of the dynamics that results when a small air bubble is placed at the tip of a steadily propagating air finger in a horizontal Hele-Shaw channel. The system can exhibit steady and oscillatory behaviours, and we show that these different behaviours each occur in well-defined regions of the phase space defined by flow rate and bubble size. For sufficiently large flow rates, periodic finger oscillations give way to disordered dynamics characterised by an irregular meandering of the finger’s tip. At fixed flow rate, the oscillations commence when the bubble size is increased sufficiently that the decreased curvature of the bubble tip in the horizontal plane matches that of the finger tip. This causes the axial pressure gradient along the bubble to vanish, thus rendering the bubble susceptible to lateral perturbations. Differing time scales for finger and bubble restoral allow sustained oscillations to develop in the finger–bubble system. The oscillations cease when the bubble is sufficiently large that it can act as the tip of a single finger. The disordered dynamics at high flow rates are consistent with the transient exploration of unstable periodic states, which suggests that similar dynamics may underlie disorder in viscous fingering.

Small-scale topography can significantly influence large-scale motions in geophysical flows, but the dominant mechanisms underlying this complicated process are poorly understood. Here, we present a systematic experimental study of the effect of small-scale topography on zonal jets. The jet flows form under the conditions of fast rotation, a uniform background $\beta$-effect, and sink–source forcing. The small-scale topography is produced by attaching numerous small cones on the curved bottom plate, and the height of the cones is much smaller than the water depth. It is found that for all tested cases, the energy fraction in the zonal mean flow consistently follows a scaling $E_{uZ}/E_{uT}=C_1 l_f^2\epsilon _{\textit{up}}^{-2/5}\beta _{\textit{eff}}^{6/5}$, where $l_f$ is the forcing scale, $\epsilon _{\textit{up}}$ is the upscale energy transfer rate, and $\beta _{\textit{eff}}$ measures the effective $\beta$-effect in the presence of topography. The presence of the small-scale topography weakens the jet strength notably. Moreover, the effect of topography on energy transfers depends on the topography magnitude $\beta _\eta$, and there exist three regimes. At small $\beta _\eta$, the inverse energy transfers are remarkably diminished while the jet pattern remains unchanged. When $\beta _\eta$ increases, a blocked flow pattern forms, and the jet width reaches saturation, becoming independent of the forcing magnitude and $\beta$. At moderate $\beta _\eta$, the inverse energy fluxes are surprisingly enhanced. A further increase of $\beta _\eta$ leads to a greater reduction of the energy fluxes. We finally examine the effect of topography from the perspective of turbulence–topography interaction.

We investigate interactions between two like-signed vortices over either an isolated seamount or a basin (a depression in the bathymetry), using a quasi-geostrophic, two-layer model on the $f$-plane. When the vortex pair is centred over the seamount, the vortices are pushed together by the secondary flow generated in the bottom layer, facilitating their merger. Over a basin, the deep anomalies are much stronger and their interaction strains out the surface vortices. The results are supported by an analytical estimation of the initial potential vorticity anomalies in the lower layer and by analysis of the linear stability of a single vortex over the bathymetry. Similar phenomena are observed when the vortex pair is displaced from the bathymetric centre and when the initial vortices are initially compensated. Sub-deformation-scale vortices are less influenced by bathymetry than larger vortices. The results help explain asymmetries noted previously in turbulence simulations over bathymetry.

If a body of inviscid fluid is disturbed, it will typically eject a jet of fluid. If the effects of gravity and surface tension are negligible, these jets travel in straight lines, with the tips approaching a constant velocity. Earlier works have concentrated upon jets which result from the occurrence of shocks or singularities in the fluid flow. In this paper, by contrast, we describe the simplest case, in two dimensions: an infinitely deep body of inviscid fluid, with no surface tension or gravitational forces acting, responds to a generic impulsive disturbance. We find that, contrary to some earlier suggestions, the jet has a hyperbolic profile (away from its tip and its base).

The statistical relation of residual stress between averaged and filtered compressible flow, known as Reynolds stress in the Reynolds-averaged Navier–Stokes equation (RANS) and subgrid-scale (SGS) stress in a large eddy simulation (LES), serves a significant role in high-Reynolds-numbers wall-bounded turbulence modelling. However, existing residual stress relations are not universally applicable due to additional assumptions or variables not directly derived from compressible turbulence modelling. To establish an effective and accurate residual stress relation, a theoretical study accompanied by numerical verification has been carried out. By introducing a novel pair of average and filter operators with commutative properties, the statistical relations of residual stress for compressible flows are derived. Then, a realisation and verification of the stress relation is carried out within the finite volume method framework to facilitate the application of the proposed stress relation in engineering turbulence modelling. The reliability of the residual stress relation is confirmed using the compressible channel turbulence at various Mach numbers and compressible boundary layer flow. The stress relation formula effectively establishes the decomposition between Reynolds stress and subgrid-scale stress of the compressible flows. The proposed residual stress relation and filter operators may contribute to the compressible turbulence modelling, including the development of the wall model, SGS model and RANS/LES hybrid strategy for high-Reynolds-number turbulence modelling.

The Reynolds analogy is revisited and the van Driest equation is established for fully developed particle-laden compressible turbulent channel flow (CTCF). A correction function is introduced into the classical approximate solution of the van Driest equation based on numerical observations. The refined Reynolds analogy is validated in both single-phase and particle-laden CTCFs. The newly proposed mean temperature–velocity relation agrees very well with numerical results. The turbulence modulation caused by inertial particles in CTCF is also studied through two-way coupling point-particle direct numerical simulation. Similar to its incompressible counterpart, the mean velocity of background flow is unchanged in the presence of inertial particles. However, it is discovered that the mean temperature of background flow is attenuated due to the interplay between carrier flow and adiabatic particles. The temperature attenuation rate (TAR) is employed to describe this phenomenon, which is defined as the integral of mean temperature profile with respect to mean velocity normalized by the product of wall temperature and central mean velocity. The numerical results manifest that the inertial particles can cause considerable temperature attenuation across the channel. It is further found that the Reynolds analogy and recovery factors are reduced by inertial particles. The refined Reynolds analogy can reproduce the TAR obtained from numerical simulations. In addition, the energy transfer analysis reveals that the temperature attenuation caused by the motion of adiabatic particles is mainly attributed to the suppression of turbulent dissipation.

When a water wave group encounters a floating body, it forces the body into motion; this motion radiates waves that modify the wave group. This study considers a floating body in the form of a two-dimensional (2-D) rectangular block constrained to heaving motion. The focus is on how the 2-D block modifies infragravity (IG) waves, a type of nonlinear low-frequency wave in the wave group. The IG waves transmitted beyond the block comprise two types: (i) bound IG waves generated by nonlinear interactions of first-order carrier waves, and (ii) free IG waves released due to discontinuities in flow potential created by the block. A systematic parameter sweep reveals that, when heaving motion is allowed, the transmitted IG waves differ significantly from those of stationary blocks. In some cases, heaving motion enables attenuation of the total transmitted IG waves, while stationary blocks cannot achieve similar effects. Only small-sized blocks are considered; they are ‘small’ compared with the IG wavelengths. The findings are relevant to dual-purpose wave energy converters designed for energy generation and coastal protection, floating breakwaters and other small-sized floating structures such as ships and some icebergs: the heaving motion of these objects may modify IG waves, thereby influencing harbour resonance, near-shore currents, beach erosion, wave forcing on ice shelves and coastal inundation.

In this paper, a phase-change model based on a geometric volume-of-fluid (VOF) framework is extended to simulate nucleate boiling with a resolved microlayer and conjugate heat transfer. Heat conduction in both the fluid and solid domains is simultaneously solved, with interfacial heat-transfer resistance (IHTR) imposed. The present model is implemented in the open-source software Basilisk with adaptive mesh refinement (AMR), which significantly improves computational efficiency. However, the approximate projection method required for AMR introduces strong oscillations within the microlayer due to intense heat and mass transfer. This issue is addressed using a ghost fluid method, allowing nucleate boiling experiments to be successfully replicated. Compared with previous literature studies, the computational cost is reduced by three orders of magnitude. We investigated the impact of contact angle on nucleate boiling through direct numerical simulation (DNS). The results show that the contact angle primarily influences the bubble growth by altering the hydrodynamic behaviour within the microlayer, rather than the thermal effect. An increase in contact angle enhances contact line mobility, resulting in a slower bubble growth, while maintaining an approximately constant total average mass flux. Furthermore, the sensitivity of bubble dynamics to the contact angle diminishes as the angle decreases. Finally, a complete bubble cycle from nucleation to detachment is simulated, which, to our knowledge, has not been reported in the open literature. Reasonable agreement with experimental data is achieved, enabling key factors affecting nucleate boiling simulations in the microlayer regime to be identified, which were previously obscured by limited simulation time.

In this work, we derive higher-order transport equations starting from the Boltzmann equation using a second-order accurate distribution function within the 13-moment framework. The equations are shown to be unconditionally linearly stable and consistent with Onsager’s symmetry principle. We also show that the equations comply with the second law of thermodynamics by establishing the non-negativity of the bulk entropy generation rate using the linearised form of the proposed equations. The force-driven Poiseuille flow problem, a standard benchmark problem, is selected to establish the validity of the equations. A complete analytical solution for this problem is proposed and compared against the Navier–Stokes, regularised 13, Grad 13 solutions and direct simulation Monte Carlo data. The proposed solution captures key rarefaction effects, including the Knudsen layer, non-uniform bimodal pressure profile, non-Fourier heat flux and the characteristic temperature dip at the centre. The analytical solution for the field variables indicates that the equations outperform the existing models in the slip- and transition-flow regimes for the problem considered. These satisfactory results point to the accuracy and applicability of the proposed equations, and the equations hold significant promise for rarefied gas dynamics at large Knudsen numbers.

This study considers the global instability of unidirectional flows through single, and double, bifurcation models using linear stability and direct numerical simulation (DNS). The motivation is respiratory flows, so we consider flow in both directions, through two geometries. We identify conditions (quantified by the Reynolds number, ${Re}=U^*D/\nu$, where $U^*$ is the peak centreline velocity, $D$ is the primary pipe diameter and $\nu$ is the kinematic viscosity) where temporal fluctuations occur using DNS. We calculate the linear stability of the steady flows, identifying the critical Reynolds number and leading unstable modes. For flows from single to double pipe, the critical Reynolds number is dependent on the number of bifurcations in the domain, but the mode structures are similar, with growth observed in regions dominated by longitudinal vortices formed by the centrifugal imbalance of flows passing through curved bifurcations. Flows in the opposite direction, from double to single pipe, also depend on the number of bifurcations in the domain. The flow through the double-bifurcation case undergoes two spatial symmetry-breaking bifurcations, altering the mode structure and critical Reynolds number. In all cases, the critical Reynolds number closely matches with temporal fluctuations observed from DNS, suggesting transition is the result of a linear instability, similar to other curved geometries like toroidal and helical pipes. We compare the frequencies of the modes with the frequencies observed from DNS, finding a close match during both initial and saturated flows. These results are important for understanding respiratory flows where turbulent mixing and streaming contribute to gas transport.

We developed a numerical method to investigate the effects of flow properties and phase transition between a gas and a liquid on sloshing-induced impact pressures acting on the walls of a partially filled tank. The conservation equations of mass, momentum and energy, as well as a transport equation for the volume fraction, were solved by considering flow compressibility, surface tension and phase transition. We modelled the phase transition by employing a mass transfer model, and validated our numerical method against experimental data. We investigated the effects of flow compressibility and density ratio between gas and liquid, representing a range similar to that of natural gas and hydrogen. We examined the effects of phase transition on sloshing-induced impact loads caused by a single-impact wave with gas pockets. Compressibility, density ratio and phase transition significantly affected the flow of the liquid–gas interface in the tank and, consequently, the impact pressure. The gas compressibility, caused by a single-wave impact with gas pockets, reduced the impact pressures significantly. Although the influence of density ratio on impact pressures is often emphasised, we demonstrated that, for impacts with gas pockets, the gas density was decisive and not the density ratio. With increasing gas density, the shape of the liquid–gas interface changed, and the pressure peak decreased. For the cases investigated, the viscosity of the liquid phase hardly influenced the impact pressures. Furthermore, the phase change during condensation considerably reduced the impact pressure peak. The pressure fluctuations after the first impact were strongly damped due to the vaporisation process.