We derive a scaling law for the characteristic frequencies of wall pressure fluctuations in swept shock wave/turbulent boundary layer interactions in the presence of cylindrical symmetry, based on analysis of a direct numerical simulations database. Direct numerical simulations in large domains show evidence of spanwise rippling of the separation line, with typical wavelength proportional to separation bubble size. Pressure disturbances around the separation line are shown to be convected at a phase speed proportional to the cross-flow velocity. This information is leveraged to derive a simple model for low-frequency unsteadiness, which extends previous two-dimensional models (Piponniau et al., J. Fluid Mech., vol. 629, 2009, pp. 87–108), and which correctly predicts growth of the typical frequency with the sweep angle. Inferences regarding the typical frequencies in more general swept shock wave/turbulent boundary layer interactions are also discussed.

‘Freeze-out’ of amplitude growth, i.e. the amplitude growth stagnation of a shocked helium–air interface, is realized through a reflected shock, which produces baroclinic vorticity of the opposite sign to that deposited by the first shock. Theoretically, a model is constructed to calculate the relations among the initial parameters for achieving freeze-out. In particular, if the amplitude growth is within the linear regime at the arrival of the reflected shock, the time interval between the impacts of two shock waves is linearly related to the initial perturbation wavelength, and is independent of the initial perturbation amplitude. Experimentally, an air–SF$_6$ (or air–argon) plane interface is adopted to produce a weak reflected shock. Seven experimental runs with specific initial conditions are examined. For all cases, freeze-out is achieved after the reflected shock impact under the designed conditions.

At high surface Péclet numbers, it is common to associate the presence of surfactants with surface immobilization, where a free surface becomes indistinguishable from a no-slip surface. A different mechanism has recently been proposed for longitudinal shear flow along a unidirectional trench (Baier & Hardt, J. Fluid Mech., vol. 949, 2022, A34) wherein, at high Marangoni numbers, the meniscus spanning the finite-length trench becomes a constant-shear-stress surface due to contamination by incompressible surfactant. That model predicts recirculating interfacial flows on the meniscus, a phenomenon that has been observed experimentally (Song et al., Phys. Rev. Fluids, vol. 3, issue 3, 2018, 033303). By finding an explicit solution to the constant-shear-stress model at all protrusion angles and calculating the effective slip length for a dilute mattress of such surfactant-laden trenches, we show that those effective slip lengths are almost indistinguishable from those for a surface whose menisci have the same deflection but have been completely immobilized (i.e. they are no-slip surfaces). This means that, despite the presence of non-trivial recirculating vortical flows on the menisci, the aggregate slip characteristics of such surfaces are that they have been effectively immobilized. This surprising result underscores the need for caution in comparing theory with experiments based on effective slip properties alone.

A bio-inspired, passively deployable flap attached to an airfoil by a torsional spring of fixed stiffness can provide significant lift improvements at post-stall angles of attack. In this work, we describe a hybrid active–passive variant to this purely passive flow control paradigm, where the stiffness of the hinge is actively varied in time to yield passive fluid–structure interaction of greater aerodynamic benefit than the fixed-stiffness case. This hybrid active–passive flow control strategy could potentially be implemented using variable-stiffness actuators with less expense compared with actively prescribing the flap motion. The hinge stiffness is varied via a reinforcement-learning-trained closed-loop feedback controller. A physics-based penalty and a long–short-term training strategy for enabling fast training of the hybrid controller are introduced. The hybrid controller is shown to provide lift improvements as high as 136 % and 85 % with respect to the flapless airfoil and the best fixed-stiffness case, respectively. These lift improvements are achieved due to large-amplitude flap oscillations as the stiffness varies over four orders of magnitude, whose interplay with the flow is analysed in detail.

In the context of transition analysis, linear input–output analysis determines the worst-case disturbances to a laminar base flow based on a generic right-hand-side volumetric/boundary forcing term. The worst-case forcing is not physically realizable, and, to our knowledge, a generic framework for posing physically realizable worst-case disturbance problems is lacking. In natural receptivity analysis, disturbances are forced by matching (typically local) solutions within the boundary layer to outer solutions consisting of free-stream vortical, entropic and acoustic disturbances. We pose a scattering formalism to restrict the input forcing to a set of realizable disturbances associated with plane-wave solutions of the outer problem. The formulation is validated by comparing with direct numerical simulations of a Mach 4.5 flat-plate boundary layer. We show that the method provides insight into transition mechanisms by identifying those linear combinations of plane-wave disturbances that maximize energy amplification over a range of frequencies. We also discuss how the framework can be extended to accommodate scattering from shocks and in shock layers for supersonic flow.

The dynamics of charge-free dielectric droplets under electric fields in weakly conducting fluids is relevant to a large range of technological and environmental applications. In Sorgentone & Vlahovska (J. Fluid Mech., vol. 951, 2022, R2), the authors investigate by asymptotic theory and boundary integral simulations the motility states of a pair of dissimilar droplets at an arbitrary orientation to the electric field and find a state of steady motility that is carefully balanced between repulsion and contact.

We consider the stability of laminar high-Reynolds-number flow through a planar channel formed by a rigid wall and a heavy compliant wall under longitudinal tension with motion resisted by structural damping. Numerical simulations indicate that the baseline state (with Poiseuille flow and a flat wall) exhibits two unstable normal modes: the Tollmien–Schlichting (TS) mode and a surface-based mode which manifests as one of two flow-induced surface instabilities (FISI), known as travelling wave flutter (TWF) and static divergence (SD), respectively. In the absence of wall damping the system is unstable to TWF, where the neutrally stable wavelength becomes shorter as the wall mass increases. With wall damping, TWF is restricted to long wavelengths through interaction with the most unstable centre mode, while for wall damping greater than a critical value the system exhibits an SD mode with a two branch neutral stability curve; the critical conditions along the upper and lower branches are constructed in the limit of large wall damping. We compute the Reynolds–Orr and activation energy descriptions of these neutrally stable FISI by continuing the linear stability analysis to the following order in perturbation amplitude. We find that both FISI are primarily driven by the working of normal stress on the flexible wall, lower-branch SD has negative activation energy, while upper-branch SD approaches zero activation energy in the limit of large wall damping. Finally, we elucidate interaction between TS and TWF modes for large wall mass, resulting in stable islands within unstable regions of parameter space.

In this paper, the acoustic resonance mechanism for different axisymmetric screech modes of the underexpanded jets that impinge on an inclined plate is investigated experimentally. The ideally expanded Mach number of jets ($M_j$) ranges from 1.05 to 1.56. The nozzle-to-plate distance at the jet axis and the impingement angle are respectively set as 5.0$D$ and $30^{\circ }$, where $D$ is the nozzle exit diameter. The acoustic results show that the $M_j$ range for the A2 screech mode of impinging jets is broader than that of underexpanded free jets, and a new axisymmetric screech mode A3 appears. With the increase of $M_j$, the effect of the impinging plate on the shock cell structures of jets becomes obvious gradually, and the second suboptimal peaks are evident in the axial wavenumber spectra of mean shock structures. The coherent flow structures at screech frequencies are extracted from time-resolved schlieren images via the spectral proper orthogonal decomposition (SPOD). The axial wavenumber spectra of the selected SPOD modes suggest that the A1, A2 and A3 screech modes are respectively closed by the guided jet modes that are energized by the interactions between the Kelvin–Helmholtz wavepacket and the first three shock wavenumber peaks. The upstream- and downstream-propagating waves that constitute the screech feedback loop are analysed by applying wavenumber filters to the wavenumber spectra of SPOD modes. The frequencies of these three screech modes can be predicted by the phase constraints between the nozzle exit and the rear edge of the third shock cell. For the A3 mode, the inclined plate invades the third shock cell with the increase of $M_j$, and the phase constraint cannot be satisfied at the lower side of the jets, which leads the A3 mode to fade away. The present results suggest that external boundaries can modulate the frequency and mode of jet screech by changing the axial spacings of shock cells.

Direct numerical simulations were performed to characterize fully developed supersonic turbulent channel flows over isothermal rough walls. The effect of roughness was incorporated using a level-set/volume-of-fluid immersed boundary method. Turbulence statistics of five channel flows are compared, including one reference case with both walls smooth and four cases with smooth top walls and bottom walls with two-dimensional (2-D) and three-dimensional (3-D) sinusoidal roughnesses. Results reveal a strong dependence of the turbulence on the roughness topography and the associated shock patterns. Specifically, the 2-D geometries generate strong oblique shock waves that propagate across the channel and are reflected back to the rough-wall side. These strong shocks are absent in the smooth-wall channel and are significantly weaker in cases with 3-D roughness geometries, replaced by weak shocklets. At the impingement locations of the shocks on the top wall in the 2-D roughness cases, localized augmentations of turbulence shear production are observed. Such regions of augmented production also exist for the 3-D cases, at a much weaker level. The oblique shock waves are thought to be responsible for a more significant entropy generation for cases with 2-D surfaces than those with 3-D ones, leading to a higher irreversible heat generation and consequently higher temperature values in 2-D roughness cases. In the present supersonic channels, the effects of roughness extend beyond the near-wall layer due to the shocks. This suggests that outer layer similarity may not fully apply to a rough-wall supersonic turbulent flow.

We present the analytical solution for the two-dimensional velocity and density fields within an approximation for laminar stratified inclined duct (SID) flows where diffusion dominates over inertia in the along-channel momentum equation but is negligible in the density transport equation. We refer to this approximation as the hydrostatic/gravitational/viscous in momentum and advective in density (HGV-A) approximation due to the leading balances in the governing equations. The analytical solution is valid for laminar flows in a two-layer configuration in the limit of long ducts. The non-dimensional volume flux within the HGV-A approximation is given by $Fr^* ={{Re}}_g/(AK)$, which is a control parameter with ${{Re}}_g$ the gravitational Reynolds number, $A$ the aspect ratio of the duct and $K$ a geometrical parameter that depends on the tilt of the duct and is obtained from the analytical solution. This analytical solution was validated against results from laboratory experiments, and allows us to gain new insight into the dynamics and properties of SID flows. Most importantly, constant values of $Fr^*$ describe, in both horizontal and inclined ducts, the transitions between increasingly turbulent flow regimes: from laminar flow, to interfacial waves, to intermittent turbulence and sustained turbulence.

Nonlinearity of density stratification modulates buoyancy effects. We report results from a body-inclusive large eddy simulation of a wake in nonlinear stratification, specifically for a circular disk at diameter-based Reynolds number (${\it Re} $) of $5000$. Five density profiles are considered; the benchmark has linear stratification and the other four have hyperbolic tangent profiles of the same thickness to model a pycnocline. The disk moves inside the central core of the pycnocline in two of those four cases and, in the other two cases with a shifted density profile, the disk moves partially/completely outside the pycnocline. The maximum buoyancy frequency ($N_{max}$) for all the profiles is the same. The first part of the study investigates the centred cases. Non-uniform stratification results in increasing wake turbulence relative to the benchmark owing to reduced suppression of turbulence production as well as wave trapping in the pycnocline. Steady lee waves are also quantified to understand the limitations of linear theory. The second part pays attention to the effect of a relative shift between the pycnocline and the disk. The wake defect velocity decays substantially faster in the cases with a shift and the wake has higher turbulence level. The effect of disk location on the Kelvin wake waves (a family of steady waves within the pycnocline) and its modal form is obtained and explained by solving the Taylor–Goldstein equation. The family of unsteady internal gravity waves that are generated by the wake is also studied and the effect of disk shift is quantified.

The propagation of waves from a vertical uplift of a slender rectangular fault in a sea of constant depth is discussed, accounting for water compressibility, gravity and seabed elasticity. The compressed water column results in the generation of acoustic–gravity waves that travel at the speed of sound in water. Acoustic–gravity waves are found to terminate after a finite time, with the decay time most influenced by seabed rigidity, which is in contrast to the rigid stationary-phase model where signals persist indefinitely. At certain frequencies acoustic–gravity waves couple with the elastic seabed and travel at the shear velocity (speed of sound in an elastic solid). Improved estimates of the critical frequencies are derived. Moreover, besides the usual tsunami, a second – very small amplitude – surface wave mode travelling at the speed of sound arises under certain frequencies. We derive the cut-off frequency for this mode. The acoustic modes possess a frequency spectrum which depends on the time evolution and spatial properties of the rupture. We find that appropriate filtering of the acoustic–gravity wave signal can reveal characteristic peaks that encode information on the fault's geometry and dynamics.

We extend the scaling relations of strongly (stably) stratified turbulence from the geophysical regime of unity Prandtl number to the astrophysical regime of extremely small Prandtl number applicable to stably stratified regions of stars and gas giants. A transition to a new turbulent regime is found to occur when the Prandtl number drops below the inverse of the buoyancy Reynolds number, i.e. $Pr\,Rb<1$, which signals a shift of the dominant balance in the buoyancy equation. Application of critical balance arguments then derives new predictions for the anisotropic energy spectrum and dominant balance of the Boussinesq equations in the $Pr\,Rb\ll 1$ regime. We find that all the standard scaling relations from the unity $Pr$ limit of strongly stratified turbulence simply carry over if the Froude number, $Fr$, is replaced by a modified Froude number, $Fr_M\equiv Fr/(Pr\,Rb)^{1/4}$. The geophysical and astrophysical regimes are thus smoothly connected across the $Pr\,Rb=1$ transition. Applications to vertical transport in stellar radiative zones and modification to the instability criterion for the small-scale dynamo are discussed.

Density segregation of multi-component granular mixtures in a dense, gravity-driven flow over a rough and bumpy periodic chute surface is studied using theory and simulations. An existing theoretical model for predicting the steady-state concentration field of each species in a binary mixture using the forces acting on the particles is generalised for multi-component mixtures in this work. In addition, the rheological model for binary mixtures is also extended to multi-component mixtures. In contrast to the percolation velocity-based empirical segregation models that do not account for the rheology and need prior knowledge of the velocity field, the present approach accounts for the inter-coupling of rheology with segregation. The momentum balance equations are solved, along with the mixture rheological model as well as the multi-component density segregation model, to obtain concentration fields using an iterative numerical method. The theoretical predictions are compared with discrete element method (DEM) simulations for ternary, quaternary and quinary granular mixtures differing in density. The steady-state profiles for the concentration of different species as well as other flow properties predicted from the theory are in excellent agreement with the DEM simulation results for a variety of compositions over a range of inclination angles for different density ratios. Through this work, we take the first essential step towards proposing a generalised particle force-based segregation theory for multi-component mixtures differing in size and/or density.

The present study investigates the profiles of statistically axisymmetric turbulent jets with arbitrary buoyancy. Analytical expressions for the shape of the radial velocity, Reynolds stress and radial scalar flux profiles are derived from the governing equations by assuming self-similar Gaussian mean velocity and scalar profiles. Previously these have only been derived for the special cases of pure jets and plumes, whereas the present study generalises them to arbitrary buoyancies. These are then used to derive analytical expressions for the turbulent Schmidt/Prandtl numbers, which, along with the mean profiles, are shown to give predictions in agreement with existing literature.

Drop impact onto a thin liquid film of another liquid is observed and characterized using a high-speed video system. A new mode of splash – a complete, simultaneous corona detachment – has been observed, which is the result of the lamella breakup near the wall film. The abrupt outward and upward displacement of the lamella leads to an extreme stretching of the corona wall, resulting in its rapid thinning and a rupture. This rupture triggers propagating Taylor–Culick rims, which rapidly spread, meet and, thus, undercut simultaneously the entire corona, resulting in its detachment. Special experiments with the spreading corona impingement onto a fixed needle, supplement the physical evidence of the above-mentioned mechanism. A self-consistent theory of the observed phenomena is proposed and compared with experiments, exhibiting good agreement.

A mathematical model is developed to investigate seabed heat transfer processes under long-crested ocean waves. The unsteady convection–diffusion equation for water temperature includes terms depending on the velocity field in the laminar boundary layer, analogous to mass transfer near the seabed. Here we consider regular progressive waves and standing waves reflected from a vertical structure, which complicate the convective term in the governing equation. Rectangular and Gaussian distributions of seabed temperature and heat flux are considered. Approximate analytical solutions are derived for uniform and trapezoidal currents, and compared against predictions from a numerical solver of the full equations. The effects of heat source profile, location and strength on heat transfer dynamics in the thermal boundary layer are explained, providing insights into seabed temperature forced convection mechanisms enhanced by free-surface waves.

In two-dimensional decaying homogeneous isotropic turbulence, kinetic energy and enstrophy are respectively transferred to larger and smaller scales. In such spatiotemporally complex dynamics, it is challenging to identify the important flow structures that govern this behaviour. We propose and employ numerically two flow-modification strategies that leverage the inviscid global conservation of energy and enstrophy to design external forcing inputs that change these quantities selectively and simultaneously, and drive the system towards steady-state or other late-stage behaviour. One strategy employs only local flow field information, while the other is global. We observe various flow structures excited by these inputs and compare them with recent literature. Energy modification is characterized by the excitation of smaller wavenumber structures in the flow than enstrophy modification.

After decades of research efforts, wind–wave interaction mechanisms have been recognized as extremely elusive. The reason is the complex nature of the problem, which combines complex coupling mechanisms between turbulent wind and water waves with the presence of multiple governing parameters, such as the friction Reynolds number of the wind, the water depth and the wind fetch. As shown unequivocally here, the use of suitable flow settings allows us to reduce the complex problem of wind–wave interaction to its essential features, mainly as a function of the sole friction Reynolds number of the wind. The resulting numerical solution allows us to study the interactions between water and air layers with their own fluid properties, and to unveil very interesting features, such as an oblique wave pattern travelling upstream and a wave-induced Stokes sublayer. The latter is responsible for a drag reduction mechanism in the turbulent wind. Despite the simulated flow conditions being far from the intense events occurring at the ocean–atmosphere interface, the basic flow phenomena unveiled here may explain some experimental evidence in wind–wave problems. Among other things, the wave-induced Stokes sublayer may shed light on the large scatter of the drag coefficient data in field measurements where swell waves of arbitrary directions are often present. Hence the present results and the developed approach pave the way for the understanding and modelling of the surface fluxes at the ocean–atmosphere interface, which are of overwhelming importance for climate science.

This paper presents an analytic model for the analysis of co-planar turbine fences that partially span the width of a channel in which the flow is driven by a sinusoidally oscillating driving head. The thrust presented by the turbines reduces the flow rate through the channel leading to a solution for overall power that is dependent upon turbine resistance and flow blockage as well as on channel characteristics. We introduce a return parameter, in terms of power per turbine area, to assess optimum turbine fence deployment for a given channel. We find that the optimal deployment rests on a universal curve independent of the channel characteristics, and that these characteristics – namely the integrated channel bed friction and a modified channel Froude number – move the optimum along this curve. We find that blockage considerations play a large role in the performance of a tidal farm – its achievable power, optimal return, channel flow rate reduction and device thrust – and that the scales of blockage must be considered even when designing relatively unblocked farms. The impact of the channel characteristics on the optimal arrangement, alongside environmental constraints that may limit permissible flow blockage, are quantified and discussed.