Helmholtz decomposition of velocity perturbations is performed in conjunction with linear stability analysis to examine the effects of flow-thermodynamics interactions on the stability of high-speed boundary layers. A corresponding decomposition of the pressure field is also proposed. The contributions of perturbation solenoidal kinetic, dilatational kinetic and internal energy to the various instability modes are examined as a function of Mach number ($M$). As expected, dilatational and pressure field effects play an insignificant part in the first-mode behaviour at all Mach numbers. The second (Mack) mode, however, is dominantly dilatational in nature, and perturbation internal energy is significant compared to perturbation kinetic energy. The observed behaviour is explicated by examining the key processes of production and pressure-dilatation. Production of the second-mode dilatational kinetic energy is mostly due to the solenoidal-dilatational covariance stress tensor interacting with the mean (background) velocity gradient. This cross production component also inhibits the first mode. The dilatational pressure facilitates energy transfer from the kinetic to the internal field in the near‐wall region, whereas the energy transfer away from the wall is mostly due to the solenoidal pressure work. The dilatational characters of the fast and slow modes are also examined. The fast mode is dominantly dilatational at both $M=4$ and $M=6$, while the nature of the slow mode is dependent on $M$. Finally, Helmholtz decomposition of the perturbation momentum vector is performed. Interestingly, both first and second modes are dominated by solenoidal components of momentum.

Numerical simulations of the interaction of internal solitary waves (ISWs) of opposite polarity are conducted by solving the incompressible Euler equations under the Boussinesq approximation. A double-pycnocline stratification is used. A method to determine when ISWs of both polarities exist is also presented. The simulations confirm previous work that the interaction of waves of the same polarity are soliton-like; however, here it is shown that when a fast ISW with the same polarity as a Korteweg–de Vries (KdV) solitary wave catches up and interacts with a slower ISW of opposite polarity, the interaction can be far from soliton-like. The energy in the fast KdV-polarity wave can increase by more than a factor of 5 while the energy in the slower negative-KdV-polarity wave can decrease by 50 %. Large trailing wave trains may be generated and in some cases multiple ISWs with KdV polarity may be formed by the interaction.

The wake flow past an axisymmetric body of revolution at a diameter-based Reynolds number $Re=u_{\infty }D/\nu =5000$ is investigated via a direct numerical simulation. The study is focused on identification of coherent vortical motions and the dominant frequencies in this flow. Three dominant coherent motions are identified in the wake: the vortex shedding motion with the frequency of $St=fD/u_{\infty }=0.27$, the bubble pumping motion with $St=0.02$, and the very-low-frequency (VLF) motion originated in the very near wake of the body with the frequency $St=0.002$–$0.005$. The vortex shedding pattern is demonstrated to follow a reflectional symmetry breaking mode, whereas the vortex loops are shed alternatingly from each side of the vortex shedding plane, but are subsequently twisted and tangled, giving the resulting wake structure a helical spiraling pattern. The bubble pumping motion is confined to the recirculation region and is a result of a Görtler instability. The VLF motion is related to a stochastic destabilisation of a steady symmetric mode in the near wake and manifests itself as a slow, precessional motion of the wake barycentre. The VLF mode with $St=0.005$ is also detectable in the intermediate wake and may be associated with a low-frequency radial flapping of the shear layer.

Drop impact experiments allow the modelling of a wide variety of natural processes, from raindrop impacts to planetary impact craters. In particular, interpreting the consequences of planetary impacts requires an accurate description of the flow associated with the cratering process. In our experiments, we release a liquid drop above a deep liquid pool to investigate simultaneously the dynamics of the cavity and the velocity field produced around the air–liquid interface. Using particle image velocimetry, we analyse quantitatively the velocity field using a shifted Legendre polynomial decomposition. We show that the velocity field is more complex than considered in previous models, in relation to the non-hemispherical shape of the crater. In particular, the velocity field is dominated by degrees 0 and 1, with contributions from degree 2, and is independent of the Froude and the Weber numbers when these numbers are large enough. We then derive a semi-analytical model based on the Legendre polynomial expansion of an unsteady Bernoulli equation coupled with a kinematic boundary condition at the crater boundary. This model explains the experimental observations and can predict the time evolution of both the velocity field and the shape of the crater, including the initiation of the central jet.

Supercritical ${\rm CO}_2$ injection and dissolution into deep brine aquifers allow its sequestration within geological formations. After injection, ${\rm CO}_{2}$ gas phase is buoyancy-driven over the denser aqueous brine, reaching an apparent gravitational stable distribution. However, ${\rm CO}_2$ dissolution in brine propels convection since the mixture is even denser than the underlying brine. This process still needs to be characterised comprehensively. Here, we investigate the irreversible mixing of dissolved ${\rm CO}_2$ in brine through laboratory-scale numerical experiments utilising the Hele-Shaw model (Letelier et al., J. Fluid Mech., vol. 864, 2019, pp. 746–767) and a fully miscible two-fluid system. In this scenario, mixing the less dense fluid – mimicking ${\rm CO}_{2}$ gas phase – with the heavier fluid – representing aqueous brine – catalyses cabbeling-powered convection. Our numerical simulations recover the laboratory results in porous media by Neufeld et al. (Geophys. Res. Lett., vol. 37, issue 22, 2010, L22404) and may explain the scaling law obtained by Backhaus et al. (Phys. Rev. Lett., vol. 106, issue 10, 2011, 104501) in Hele-Shaw cells. More remarkably, we show that the mass flux between the two analogue fluids, characterised by the Sherwood number $ {{Sh}}$, obeys the universal scaling law $ {{Sh}}\sim {{Ra}}\, \vartheta _{scalar}$, with $ {{Ra}}$ the Rayleigh number and $\vartheta _{scalar}$ the mean scalar dissipation rate. This paper sheds light on the fluid dynamics and solubility trapping in geological carbon sequestration.

In this work, the linear responses of turbulent mean flow to both harmonic and stochastic forcing are investigated for supersonic channel flow. Well-established universal relations are utilized to obtain efficiently the mean profiles with a large parameter space, with the bulk Mach number up to 5 and the friction Reynolds number up to $10^4$, so a systematic parameter study is feasible. The most amplified structure takes the form of streamwise velocity and temperature streaks forced optimally by the streamwise vortices. The outer peak of the pre-multiplied energy amplification corresponds to the large-scale motion, whose spanwise wavelength ($\lambda _z^+$) is very insensitive to compressibility effects. In contrast, the classic inner peak representing small-scale near-wall motions disappears for the stochastic response with increasing Mach number. Meanwhile, the small-scale motions become much less coherent. A decomposition of the forcing identifies different effects of the incompressible counterpart and the thermodynamic components. Wall-cooling effects, arising with high Mach number, increase the spacing of the most amplified near-wall streaks; the spacing becomes nearly invariant with Mach number if expressed in semi-local units. Meanwhile, the coherence of stochastic response with $\lambda _z^+>90$ is enhanced, but on the other hand, with $\lambda _z^+<90$ it is decreased. The geometrical self-similarity of the response in the mid-$\lambda _z$ range is still roughly satisfied, insensitive to Mach number. Finally, theoretical analyses of the perturbation equations are presented to help with understanding the scaling of energy amplification.

This study investigates the effect of structural damping on vortex-induced vibration (VIV) of a circular cylinder when the mass ratio is below its critical value. It is confirmed by water-channel experiments and a reduced-order model (ROM) that the previously identified phenomenon of VIV forever, i.e. resonance oscillations at any reduced velocity, persists even with high structural damping. Of interest, the ROM results reveal that the wake mode for VIV forever is unstable with a constant positive growth rate with increasing reduced velocity, while the experimental results suggest that VIV forever is associated with a synchronisation between the non-stationary cylinder vibration frequency and the vortex-shedding frequency.

The collapse of a gas or vapour bubble near a non-porous boundary is directed at the boundary due to the asymmetry induced by the nearby boundary. High surface pressure and shear stress from this collapse can damage, or clean, the surface. A porous boundary, such as a filter, would act similarly to a non-porous boundary but with reduced asymmetry and thus reduced effect. Prior research has measured the cleaning effect of bubbles on filters using ultrasonic cleaning, but it is not known how the bubble dynamics are fundamentally affected by the porosity of the surface. We address this question experimentally by investigating how the standoff distance, porosity, pore size and pore shape affect two collapse properties: bubble displacement and bubble rebound size. We show that these properties depend primarily on the standoff distance and porosity of the boundary and extend a previously developed numerical model that approximates this behaviour. Using the numerical model in combination with experimental data, we show that bubble displacement and bubble rebound size each collapse onto respective single curves.

This paper studies the sustainability of plastrons on superhydrophobic (SHPo) surfaces made of longitudinal micro-trenches covered by nano-grass with the main interest on hydrodynamic friction drag reduction in high-speed flows of open water, which represent the operating conditions of common watercraft. After revising the shear-driven drainage model to address the air diffusion for SHPo surfaces, the existing theories are combined to reveal the trends of how the immersion depth, air saturation level and shear stress affect the maximum attainable plastron length. Deviations from the theories by the dynamic effect at the two ends of the trench, the interfacial contaminations and turbulent fluctuation are also discussed. A combinatorial series of well-defined SHPo trench surfaces (4 cm × 7 cm in size with varying trench widths, depths, lengths and roughnesses) is microfabricated and attached underneath a 4 m long motorboat on seawater in turbulent flows up to 7.2 m s−1 (shear rate ∼83 000 s−1 and friction Reynolds number ∼5500). Because the plastron can provide a substantial slip only while its air–water interfaces are pinned (or only slightly depinned) at the trench top, two underwater cameras are employed to differentiate the pinned (and slightly depinned) interfaces from the depinned (and no) interfaces. In addition to achieving pinned plastrons on 6 cm long trenches aligned to high-speed flows in open water, the experimental results corroborate the theoretical estimations, supporting the design of SHPo surfaces for field applications.

The interaction of a hollow droplet impacting a solid surface occurs in several applications, including controllable biomedicine and thermal spray coating. Understanding the physics of the hollow droplet spreading is the key to maintaining the mass transfer process in all relevant applications. In this work, a comprehensive experimental, numerical and theoretical study is performed on water hollow droplets impacting a rigid surface to better understand the flattening process of a hollow droplet. In the numerical part, compressible Navier–Stokes equations are solved using the volume of fluid (VOF) method in a two-dimensional (2-D)-axisymmetric model. The comparison of simulation results with the experimental photographs shows that the numerical solution can correctly predict the hollow droplet shape evolution. The results show that the spreading diameter and height of the counter-jet formed after the hollow droplet impact grow with impact velocity. Investigating the size and location of the entrapped bubble shows an optimum bubble size that facilitates the hollow droplet flattening. It is also shown that the ripples on splats produced by the hollow droplets with a larger bubble size are higher than those of small bubbles. In the end, a theoretical model is developed to analyse the maximum spreading diameter of the hollow droplet impact analytically. Its prediction is in good agreement with the experimental and numerical results.

Understanding the hysteretic behaviour in fluid–fluid displacement processes in porous media is critical in many engineering applications. In this work, we study the quasi-static immiscible displacement process in two-dimensional porous media during cyclic injections in the context of carbon geosequestration. The role of wettability on the residual trapping of CO$_2$ is investigated numerically using an extended interface tracking algorithm. Despite that higher CO$_2$ saturation can be achieved in CO$_2$-wet porous media after the first CO$_2$ injection, the majority of CO$_2$ is found to be unstable and can be mobilised during subsequent water injection processes. An improvement in the residual trapping of CO$_2$ is observed as the number of injection cycles increases, which is associated with the dispersion of continuous CO$_2$ ganglia into numerous smaller blobs. Compared with either water-wet or CO$_2$-wet porous media, it is found that less CO$_2$ is trapped within the neutral-wet ones at equilibrium state after a sufficient number of injection cycles. The hysteretic behaviour of saturation between water/CO$_2$ injection cycles is found to follow an exponential decay, which eventually reaches a finite value. This process corresponds to the shift of the mobile region during displacement from typical capillary fingering to a less ramified regime, which ultimately converges towards main flow channels. This work highlights the hysteretic behaviour during cyclic injections, providing insights on the wettability impacts on multiphase flow in porous media, which is of great importance in applications such as carbon geosequestration and geological hydrogen storage.

When surface waves interact with ambient turbulence, the two affect each other mutually. Turbulent eddies get redirected, intensified and periodically stretched and compressed, while the waves suffer directional scattering. We study these mutual interactions experimentally in the water channel laboratory at the Norwegian University of Science and Technology (NTNU) Trondheim. Long groups of waves were propagated upstream on currents with identical mean flow but different turbulence properties, created by an active grid at the current inlet. The subsurface flow in the spanwise–vertical plane was measured with stereo particle-image velocimetry. Comparing the subsurface velocity fields before and after the passage of a wave group, a strong enhancement of streamwise vorticity is observed which increases rapidly towards the surface for $k_0z\gtrsim -0.3$ ($z$, vertical distance from still surface; $k_0$, carrier wavenumber) in qualitative agreement with theory. Next, we measure the broadening of the directional wave spectrum at increasing propagation distance. The rate of directional diffusion is greatest for the turbulent case with the highest energy at the longest length scales whereas the highest total turbulent kinetic energy overall did not produce the most scattering. The variance of directional spectra is found to increase linearly as a function of propagation time.

Zonal jet formation in $\beta$-plane turbulence is investigated with the focus on whether an accurate closure can be developed for the eddy momentum fluxes due to small-scale random forcing. The approach of Srinivasan and Young (J. Atmos. Sci., vol. 71, 2014, pp. 2169–2185) is developed to give a relatively simple expression for the local Reynolds stress, due to a white-in-time random forcing with characteristic length scale much less than the jet spacing. In typical jet flows, however, it is demonstrated that the Srinivasan–Young flux is not the full story because momentum fluxes due to jet-scale waves, present as a result of distinct barotropic instabilities of the eastward and westward jets, respectively, also play a key role in the momentum balance. Numerical simulations that explicitly include the random forcing are then compared with those in which the Srinivasan–Young closure is applied. For typical jet flows, good agreement of the equilibrium zonal flow is found provided that the closure simulation is not truncated to be purely zonal, i.e. jet-scale secondary barotropic instabilities are allowed to develop. Flows in which the geometry or external forcing acts to suppress the development of secondary instabilities are also simulated, and for these flows the Srinivasan–Young closure is shown to be successful as a purely zonal closure. It is argued that vortex condensates in isotropic forced-dissipative 2D turbulence are an example of this latter situation.

We experimentally investigate the settling of millimetre-sized thin disks in quiescent air. The range of physical parameters is chosen to be relevant to plate crystals settling in the atmosphere: the diameter-to-thickness aspect ratio is $\chi =25\unicode{x2013}60$, the Reynolds numbers based on the disk diameter and fall speed are $Re=O(10^2)$ and the inertia ratio is $I^*=O(1)$. Thousands of trajectories are reconstructed for each disk type by planar high-speed imaging, using the method developed by Baker & Coletti (J. Fluid Mech., vol. 943, 2022, A27). Most disks either fall straight vertically with their maximum projected area normal to gravity or tumble while drifting laterally at an angle $<20^\circ$. Two of the three disk sizes considered exhibit bimodal behaviour, with both non-tumbling and tumbling modes occurring with significant probabilities, which stresses the need for a statistical characterization of the process. The smaller disks (1 mm in diameter, $Re=96$) have a stronger tendency to tumble than the larger disks (3 mm in diameter, $Re=360$), at odds with the diffused notion that $Re=100$ is a threshold below which falling disks remain horizontal. Larger fall speeds (and, thus, smaller drag coefficients) are found with respect to existing correlations based on experiments in liquids, demonstrating the role of the density ratio in setting the vertical velocity. The data supports a simple scaling of the rotational frequency based on the equilibrium between drag and gravity, which remains to be tested in further studies where disk thickness and density ratio are varied.

We investigate the equilibrium of an axisymmetric system consisting of sessile and pendent drops on pre-stretched nonlinear elastic membranes. The membrane experiences large deformations due to the drop's weight and interfacial interactions. We first show that force balance alone leads to non-unique equilibrium solutions. Identifying the system's equilibrium with the minimum of its free energy, we then demonstrate that the equilibrium solution is made unique by requiring the continuity of meridional stretches across the three-phase contact circle. For a special class of nonlinear elastic materials – $I_2$ materials – we then compute the equilibrium configurations of the drop–membrane system for a range of drop volumes and membrane pre-tensions. Finally, the present work facilitates two important applications: (a) the membrane's pre-tension and current tension are related exactly to help in utilizing the system as an elastocapillary probe for membrane pre-tension and (b) we suggest an experimental protocol for measuring the membrane's surface properties.

Experimental measurements in a wind tunnel of the unsteady force and moment that a fluid exerts on flexible flapping aerofoils are not trivial because the forces and moments caused by the aerofoil's inertia and others structural tensions at the pivot axis have to be obtained separately and subtracted from the direct measurements with a force/torque sensor. Here we derive from the nonlinear beam equation general relations for the force and torque reactions at the leading edge of a pitching aerofoil in terms of the fluid force and moment on the aerofoil and its kinematics, involving geometric and structural parameters of the flexible aerofoil. These relations are validated by comparing high-resolution numerical simulations of the flow–structure interaction of a two-dimensional flexible aerofoil pitching about its leading edge with direct force and torque measurements in a wind tunnel.

Driving oscillatory flow around an obstacle generates, due to inertial rectification, a steady ‘streaming’ flow that is useful in a host of microfluidic applications. While theory has focused largely on two-dimensional flows, streaming in many practical microfluidic devices is three-dimensional due to confinement. We develop a three-dimensional streaming theory around an obstacle in a microchannel with a Hele-Shaw-like geometry, where one dimension (depth) is much shorter than the other two dimensions. Utilizing inertial lubrication theory, we demonstrate that the time-averaged streaming flow has a three-dimensional structure. Notably, the flow reverses direction across the depth of the channel, which is a feature not observed in less confined streaming set-ups. This feature is confirmed by our experiments of streaming around a cylinder sandwiched in a microchannel. Our theory also predicts that the streaming velocity decays as the inverse cube of the distance from the cylinder, faster than that expected from previous two-dimensional approaches. We verify this velocity decay quantitatively using particle tracking measurements from experiments of streaming around cylinders with different aspect ratios at different driving frequencies.

The formation of step-like ‘density staircase’ distributions induced by stratification and turbulence has been widely studied, and can be explained by the ‘instability’ of a sufficiently strongly stably stratified turbulent flow due to the decrease of the turbulent density flux with increasing stratification via the ‘Phillips mechanism’ (Phillips, Deep Sea Research and Oceanographic Abstracts, vol. 19, 1972, pp. 79–81. Elsevier). However, such density staircases are not often observed in ocean interiors, except in regions where double-diffusion processes are important, leading to thermohaline staircases. Using reduced-order models for the evolution of velocity and density gradients, we analyse staircase formation in stratified and sheared turbulent flows. Under the assumption of inertial scaling $\epsilon \sim U^3/L$ for the kinetic energy dissipation rate $\epsilon$, where $U$ and $L$ are characteristic velocity and length scales, we determine ranges of bulk Richardson numbers ${Ri}_{b}$ and turbulent Prandtl numbers ${Pr}_{T}$ for which staircases can potentially form and show that the Phillips mechanism only survives in the limit of sufficiently small turbulent Prandtl numbers. For relevant oceanic parameters, a range of turbulent Prandtl numbers above which the system is not prone to staircases is found to be ${Pr}_T \simeq 0.5\unicode{x2013}0.8$. Since several studies indicate that the turbulent Prandtl number in stably stratified turbulence and in ocean interiors is usually above this threshold, this result supports the empirical observation that staircases are not favoured in ocean interiors in the presence of relatively homogeneous and sustained turbulence. We also show that our analysis is robust to other scalings for $\epsilon$ (such as the more strongly stratified scaling $\epsilon \sim U^{2}N_{c}$, where $N_{c}$ is a characteristic value of the buoyancy frequency), supporting our results in both shear-dominated and buoyancy-dominated turbulent regimes as well as in weakly and strongly stratified regimes.

The evolution of a turbulent boundary layer over truncated cones in staggered and random configurations is investigated using direct numerical simulations. Three random planform densities and one staggered, which closely match the configuration in the recent experiment by Womack et al. (J. Fluid Mech., vol. 933, 2022, A38), are considered. The experiment is also used to validate the present computations which are found to reproduce the flow measurements very accurately. Spanwise heterogeneity in the mean streamwise velocity in the form of high and low momentum pathways was detected in both the staggered and random arrangements, indicating the presence of secondary flows in the cross-stream plane. The momentum pathways found in the staggered arrangements are much weaker and confined near the roughness crest, while in the random arrangements they approach the edge of the boundary layer. We found a clear correlation of the momentum pathways with the leading edge of the roughness, while the correlation to the local topography was very weak for all the cases studied. Vorticity transport analysis is also employed to explore the nature of the secondary flows formed, highlighting the strong presence of vortex stretching near the leading edge of the roughness. The impact of the momentum pathways on the flow statistics, especially in the random arrangements, is demonstrated by the breakdown of outer-layer similarity.

The actuator line method (ALM) is used extensively in wind turbine and rotor simulations. However, its original uncorrected formulation overestimates the forces near the tip of the blades and does not reproduce well forces on translating wings. The recently proposed vortex-based smearing correction for the ALM is a correction based on physical and mathematical properties of the simulation that allows for a more accurate and general ALM. So far, to correct the forces on the blades, the smearing correction depended on an iterative process at every time step, which is usually slower, less stable and less deterministic than direct methods. In this work, a non-iterative process is proposed and validated. First, we propose a formulation of the nonlinear lifting line that is equivalent to the ALM with smearing correction, showing that the results are practically identical for a translating wing. Then, by linearizing the lifting line method, the iterative process of the correction is substituted by the direct solution of a small linear system. No significant difference is observed in the results of the iterative and non-iterative corrections, in both wing and rotor simulations. Additional contributions of the present work include the use of a more accurate approximation for the velocity induced by a smeared vortex segment and the implementation of a free-vortex wake model to define the vortex sheet, which contribute to the accuracy and generality of the method. The results presented here may motivate the adoption of the ALM by other communities, for example, in fixed-wing applications.