We define ‘surface layer’ (SL) as an inertia-dominated turbulence region outside a viscous or roughness surface-adjacent sub-layer (SAS) that is characterised by linear scaling of specific coherence length scales on wall-normal distance, $z$. We generalise the mechanisms that underlie the formation of the classical inertial SL in the shear-dominated turbulent boundary layer (TBL) to wall-bounded turbulent flows with zero mean shear. Using particle image velocimetry data from two wind tunnel facilities, we contrast the classical TBL SL with a non-classical shear-free SL generated within grid turbulence advected over an impermeable plate using two grids with different turbulence length scales. Integral-scale variations with $z$ and other statistics are quantified. In both shear-dominated and shear-free SLs we observe well-defined linear increases in $z$ of the streamwise integral scale of vertical velocity fluctuations. In grid turbulence the shear-free SL initiates just above the SAS that confines friction-generated motions. By contrast, the TBL SL forms with non-zero mean shear rate that extends streamwise coherence lengths of streamwise fluctuations. In both flow classes only the integral scales of vertical fluctuating velocity increase linearly with $z$, indicating that the SL is generated by the blockage of vertical fluctuations in the vertical. Whereas the SAS in the TBL is much thinner than in the grid-turbulence flows, the generation of a shear-free SL by the interaction of turbulence eddies and a surface depends on the relative thinness of the SAS. We conclude that the common generalisable SL mechanism is direct blockage of vertical fluctuations by the impermeable surface.

Significant progress has been made in understanding planetary core dynamics using numerical models of rotating convection (RC) in spherical shell geometry. However, the behaviour of forces in these models within various dynamic regimes of RC remains largely unknown. Directional anisotropy, scale dependence and the role of dynamically irrelevant gradient contributions in incompressible flows complicate the representation of dynamical balances in spherical shell RC. In this study, we systematically compare integrated and scale-dependent representations of mean and fluctuation forces and curled forces (which contain no gradient contributions) separately for the three components ($\hat {r},\hat {\theta },\hat {\phi }$). The analysis is performed with simulations in a range of convective supercriticality $Ra_T/Ra_T^{c}=1.2{-}297$ where $Ra_T$ and $Ra^{c}_T$ are the Rayleigh and critical Rayleigh numbers, respectively and Ekman number $E=10^{-3}{-}10^{-6}$, with fixed Prandtl number $Pr=1$, along with no-slip and fixed flux boundaries. We have excluded regions from each boundary of the spherical shell, with a thickness equivalent to ten velocity boundary layers, which provides a consistent representation of the bulk dynamics between the volume-averaged force and curled force balance in the parameter space studied. Radial, azimuthal and co-latitudinal components exhibit distinct force and curled force balances. The total magnitudes of the mean forces and mean curled forces exhibit a primary thermal wind balance; the corresponding fluctuating forces are in a quasi-geostrophic primary balance, while the fluctuating curled forces transition from a Viscous–Archimedean–Coriolis balance to an Inertia–Viscous–Archimedean–Coriolis balance with increasing $Ra_T/Ra_T^{c}$. The curled force balances are more weakly scale-dependent compared to the forces, and do not show clear cross-over length scales. The fluctuating force and curled force balances are broadly consistent with three regimes of RC (weakly nonlinear, rapidly rotating and weakly rotating), but do not exhibit sharp changes with $Ra_{T}/Ra_{T}^{c}$, which inhibits the identification of precise regime boundaries from these balances.

This study investigates the energy exchange between coherent structures in flows over four low-aspect-ratio (low-) plates using the tomographic particle image velocimetry dataset originally obtained by Zhu et al. (2024. J. Fluid Mech. 983, A35). The chord-based Reynolds number is $5400$, with fixed angle of attack $6 ^\circ$. In this study, multiscale proper orthogonal decomposition is applied to extract the coherent structures, including those associated with the vortex-shedding frequency $St_1$ and its subharmonic counterpart. Subsequently, the coherent kinetic energy budget is analysed with a focus on inter-scale energy transfer. This study demonstrates that the energy transfer between the scales centred at $St_1$ and $0.5\,St_1$ can exhibit a reverse or forward direction, depending on the transformation pattern of the leading-edge vortices (LEVs). Specifically, different triadic interactions are excited during the LEV transformation, and manifest themselves during the formation of hairpin vortices downstream. Understanding this nonlinear energy transfer is essential for elucidating mechanisms underlying the development of turbulence in three-dimensional flows over low- plates.

Crowdy et al. (2023 Phys. Rev. Fluids, vol. 8, 094201), recently showed that liquid suspended in the Cassie state over an asymmetrically spaced periodic array of alternating cold and hot ridges such that the menisci spanning the ridges are of unequal length will be pumped in the direction of the thermocapillary stress along the longer menisci. Their solution, applicable in the Stokes flow limit for a vanishingly small thermal Péclet number, provides the steady-state temperature and velocity fields in a semi-infinite domain above the superhydrophobic surface, including the uniform far-field velocity, i.e. pumping speed, the key engineering parameter. Here, a related problem in a finite domain is considered where, opposing the superhydrophobic surface, a flow of liquid through a microchannel is bounded by a horizontally mobile smooth wall of finite mass subjected to an external load. A key assumption underlying the analysis is that, on a unit area basis, the mass of the liquid is small compared with that of the wall. Thus, as shown, rather than the heat equation and the transient Stokes equations governing the temperature and flow fields, respectively, they are quasi-steady and, as a result, governed by the Laplace and Stokes equations, respectively. Under the further assumption that the ridge period is small compared with the height of the microchannel, these equations are resolved using matched asymptotic expansions which yield solutions with exponentially small asymptotic errors. Consequently, the transient problem of determining the velocity of the smooth wall is reduced to an ordinary differential equation. This approach is used to provide a theoretical demonstration of the conversion of thermal energy to mechanical work via the thermocapillary stresses along the menisci.

We derive a depth-averaged model consistent with the $\mu (I)$ rheology for an incompressible granular flow down an inclined plane. The first two variables of the model are the depth and the depth-averaged velocity. The shear is also taken into account via a third variable called enstrophy. The obtained system is a hyperbolic system of conservation laws, with an additional equation for the energy. The system is derived from an asymptotic expansion of the flow variables in powers of the shallow-water parameter. This method ensures that the model is fully consistent with the rheology. The velocity profile is a Bagnold profile at leading order and the first-order correction to this profile can be calculated for flows that are not steady uniform. The first-order correction to the classical granular friction law is also consistently written. As a consequence, the instability threshold of the steady uniform flow is the same for the depth-averaged model and for the governing equations. In addition, a higher-order version that contains diffusive terms is also presented. The spatial growth rate, the phase velocity and the cutoff frequency of the version with diffusion are in good agreement with the experimental data and with the theoretical predictions for the rheology. The mathematical structure of the equations enables us to use well-known and stable numerical solvers. Numerical simulations of granular roll waves are presented. The model has the same limitations as the $\mu (I)$ rheology, in particular for the solid/ liquid and liquid/gas transitions, and needs therefore a regularisation for these transitions.

The formation of Kelvin–Helmholtz-like rollers (referred to as K–H rollers) over riblet surfaces has been linked to the drag-increasing behaviour seen in certain riblet geometries, such as sawtooth and blade riblets, when the riblet size reaches sufficiently large viscous scales (Endrikat et al. (2021a), J. Fluid Mech. 913, A37). In this study, we focus on the sawtooth geometry of fixed physical size, and experimentally examine the response of these K–H rollers to further increases in viscous scaled riblet sizes, by adopting the conventional approach of increasing freestream speeds (and consequently, the friction Reynolds number). Rather than continual strengthening, the present study shows a gradual weakening of these K–H rollers with increasing sawtooth riblet size. This is achieved by an analysis of the roller geometric characteristics using both direct numerical simulations and hot-wire anemometry databases at matched viscous scaled riblet spacings, with the former used to develop a novel methodology for detecting these rollers via streamwise velocity signatures (e.g. as acquired by hot wires). Spectral analysis of the streamwise velocity time series, acquired within riblet grooves, reveals that the frequencies (and the streamwise wavelengths) of the K–H rollers increase with increasing riblet size. Cross-correlation spectra, estimated from unique two-point hot-wire measurements in the cross-plane, show a weakening of the K–H rollers and a reduction in their wall-normal coherence with increasing riblet size. Besides contributing to our understanding of the riblet drag-increasing mechanisms, the present findings also have implications for the heat transfer enhancing capabilities of sawtooth riblets, which have been associated previously with the formation of K–H rollers. The present study also suggests conducting future investigations by decoupling the effects of viscous scaled riblet spacing and friction Reynolds numbers, to characterise their influence on the K–H rollers independently.

We conduct direct numerical simulations to investigate the synchronisation of Kolmogorov flows in a periodic box, with a focus on the mechanisms underlying the asymptotic evolution of infinitesimal velocity perturbations, also known as conditional leading Lyapunov vectors. This study advances previous work with a spectral analysis of the perturbation, which clarifies the behaviours of the production and dissipation spectra at different coupling wavenumbers. We show that, in simulations with moderate Reynolds numbers, the conditional leading Lyapunov exponent can be smaller than a lower bound proposed previously based on a viscous estimate. A quantitative analysis of the self-similar evolution of the perturbation energy spectrum is presented, extending the existing qualitative discussion. The prerequisites for obtaining self-similar solutions are established, which include an interesting relationship between the integral length scale of the perturbation velocity and the local Lyapunov exponent. By examining the governing equation for the dissipation rate of the velocity perturbation, we reveal the previously neglected roles of the strain rate and vorticity perturbations, and uncover their unique geometrical characteristics.

The flow of an incompressible fluid in a rapidly rotating cubic cavity librating at a low frequency around an axis through the midpoints of opposite edges features synchronous waves with a foliation pattern that is quasi-invariant in the axial direction. These waves are emitted from the equatorial edges (the edges furthest away from the axis) and travel into the interior in a retrograde fashion about the eastern equatorial vertices. These waves are interpreted as topographic Rossby waves, consistent with the lack of closed geostrophic contours for the rotating container. They are analysed in detail at small Ekman numbers, both in the linear regime, corresponding to the limit of zero libration amplitude (Rossby number $ Ro \to 0$), and in the weakly nonlinear regime with small but finite $ Ro$. The waves subsist in the linear regime and coexist with a network of shear layers that are predicted by linear inviscid analysis to focus towards the equatorial edges. However, viscous effects stop the focusing at a distance from the edges that scales with $E^{1/2}$. The large inclination of the oblique walls with the rotation axis, together with the vanishing depth at the equatorial edges, provide the conditions for singular behaviour in the Rossby waves as $E\to 0$. Within a distance of the eastern equatorial vertices also scaling with $E^{1/2}$, the nonlinear contributions have a self-similar structure whose enstrophy density scales as $E^{-16/3} Ro ^2$. This means that $ Ro$ must be reduced considerably faster than $E$ for nonlinear contributions to be negligible as $E\to 0$.

Direct numerical simulations are performed to study turbulence generated by the interaction of multiple temporally evolving circular jets with jet Mach numbers $M_J=0.6$ and $1.6$, and a jet Reynolds number of 3000. The jet interaction produces decaying, nearly homogeneous isotropic turbulence, where the root-mean-squared (r.m.s.) fluctuation ratio between the streamwise and transverse velocities is approximately 1.1, consistent with values observed in grid turbulence. In the supersonic case, shock waves are generated and propagate for a long time, even after the turbulent Mach number decreases. A comparison between the two Mach number cases reveals compressibility effects, such as reductions in the velocity derivative skewness magnitude and the non-dimensional energy dissipation rate. For the r.m.s. velocity fluctuations, $u_{rms}$, and the integral scale of the streamwise velocity, $L_u$, the Batchelor turbulence invariant, $u_{rms}^2 L_u^5$, becomes nearly constant after the turbulence has decayed for a certain time. In contrast, the Saffman turbulence invariant, $u_{rms}^2 L_u^3$, continuously decreases. Furthermore, temporal variations of $u_{rms}^2$ and $L_u$ follow power laws, with exponents closely matching the theoretical values for Batchelor turbulence. The three-dimensional energy spectrum $E(k)$, where $k$ is the wavenumber, exhibits $E(k) \sim k^4$ for small wavenumbers. This behaviour is consistently observed for both Mach number cases, indicating that the modulation of small-scale turbulence by compressibility effects does not affect the decay characteristics of large scales. These results demonstrate that jet interaction generates Batchelor turbulence, providing a new direction for experimental investigations into Batchelor turbulence using jet arrays.

We develop the time-dependent regularised 13-moment equations for general elastic collision models under the linear regime. Detailed derivation shows the proposed equations have super-Burnett order for small Knudsen numbers, and the moment equations enjoy a symmetric structure. A new modification of Onsager boundary conditions is proposed to ensure stability as well as the removal of undesired boundary layers. Numerical examples of one-dimensional channel flows is conducted to verified our model.

The Reynolds number dependence of the normalised energy dissipation rate $C_{\epsilon }=\epsilon L/u^3$ is studied, where $\epsilon$ is the energy dissipation rate, $L$ is the integral length scale and $u$ is the root-mean-square velocity. We present the derivation of the exact relationship between the normalised energy dissipation rate and integrated form of the Kármán–Howarth equation in homogeneous isotropic turbulence. The present mathematical formulation is valid for both forced and decaying turbulence. The discussion of $C_{\epsilon }$ is developed under the assumption that the term resulting from the nonlinear energy transfer appearing in $C_{\epsilon }$ is constant at sufficiently high-Reynolds-number turbulence. The fact that the integrated term originating from nonlinear energy transfer is constant plays the role of a lower bound in $C_{\epsilon }$, implying that the energy dissipation rate is finite in high-Reynolds-number turbulence. Furthermore, the origin of the non-equilibrium dissipation law could be the imbalance between $u$ and ${\rm d}L/{\rm d}t$, the influence of external forces, or both. In decaying turbulence with forced turbulence as the initial condition, the imbalance between $u$ and ${\rm d}L/{\rm d}t$ causes the non-equilibrium dissipation law. The validity of the theoretical analysis is investigated using direct numerical simulations of the forced and decaying turbulence.

Many fluid flow configurations nominally contain symmetries, which are always imperfect in real systems. In this study, we reduce the degree of rotational symmetry and break the mirror symmetry of an annular combustor’s thermoacoustic model by using non-uniform flame response distributions. It is known that, in the linear regime, asymmetries lift the degeneracy of some azimuthal thermoacoustic eigenvalues, which are nominally degenerate in the symmetric case. In this work, we prove that a second asymmetric perturbation, which does not restore any trivial symmetry, can be exploited to create an exceptional point (EP). If the only source of asymmetry is the non-uniform distribution of flame responses, at this symmetry-breaking induced EP the single remaining eigenvector is a perfectly spinning mode. We demonstrate that symmetry-breaking induced EPs may be linearly unstable. For an EP obtained for vanishingly small asymmetric perturbations, the linearly stable/unstable nature of the EP follows that of the degenerate eigenvalue of the perfectly symmetric system. Our results are derived theoretically with a low-order model, and validated on a state-space model extracted from experimental data.

A complete three-dimensional long-wave polar–Cartesian equation is developed in the frequency domain. This development employs an auxiliary axis system oriented locally in the bottom gradient direction. The long-wave limit of the two-dimensional polar–Cartesian steep-slope equation is also derived. An approximate explicit expression of the coefficients is developed without restrictions on bed steepness. This is achieved by utilising a rational function approximation of the $\arctan$ function, which arises from the formulation of the vertical profile of the flow parameters. Additionally, long-wave equations in both two and three dimensions are developed in the time domain. Simulations of the long-wave equations are compared with those of the extended shallow-water equation for two-dimensional test cases, as well as for the quasi-three-dimensional scenario of oblique incidence. Our equations exhibit better agreement with the exact solutions in the majority of the test cases.

History effects play a significant role in determining the velocity in boundary layers with pressure gradients, complicating the identification of a velocity scaling. This work pivots away from traditional velocity analysis to focus on fluid acceleration in boundary layers with strong adverse pressure gradients. We draw parallels between the transport equation of the velocity in an equilibrium spatially evolving boundary layer and the transport equation of the fluid acceleration in temporally evolving boundary layers with pressure gradients, establishing an analogy between the two. To validate our analogy, we show that the laminar Stokes solution, which describes the flow immediately after the application of a pressure gradient force, is consistent with the present analogy. Furthermore, fluid acceleration exhibits a linear scaling in the wall layer and transitions to logarithmic scaling away from the wall after the initial period, mirroring the velocity in an equilibrium boundary layer, lending further support to the analogy. Finally, by integrating fluid acceleration, a velocity scaling is derived, which compares favourably with data as well.

Starting from the coupled Boltzmann–Enskog (BE) kinetic equations for a two-particle system consisting of hard spheres, a hyperbolic two-fluid model for binary, hard-sphere mixtures was derived in Fox (2019, J. Fluid Mech. 877, 282). In addition to spatial transport, the BE kinetic equations account for particle–particle collisions, using an elastic hard-sphere collision model, and the Archimedes (buoyancy) force due to spatial gradients of the pressure in each phase, as well as other forces involving spatial gradients. The ideal-fluid–particle limit of this model is found by letting one of the particle diameters go to zero while the other remains finite. The resulting two-fluid model has closed terms for the spatial fluxes and momentum exchange due to the excluded volume occupied by the particles, e.g. a momentum-exchange term $\boldsymbol {F}_{\!\!fp}$ that depends on gradients of the fluid density $\rho _f$, fluid velocity $\boldsymbol{u}_{f}$ and fluid pressure $p_f$. In Zhang et al. (2006, Phy. Rev. Lett. 97, 048301), the corresponding unclosed momentum-exchange term depends on the divergence of an unknown particle–fluid–particle (pfp) stress (or pressure) tensor. Here, it is shown that the pfp-pressure tensor ${\unicode{x1D64B}}_{\!pfp}$ can be found in closed form from the expression for $\boldsymbol {F}_{\!\!fp}$ derived in Fox (2019, J. Fluid Mech. 877, 282). Remarkably, using this expression for ${\unicode{x1D64B}}_{\!pfp}$ ensures that the two-fluid model for ideal-fluid–particle flow is well posed for all fluid-to-particle material-density ratios $Z = \rho _f / \rho _p$.

We present a mathematical model to investigate heat transfer and mass transport dynamics in the wave-driven free-surface boundary layer of the ocean under the influence of long-crested progressive surface gravity waves. The continuity, momentum and convection–diffusion equations for fluid temperature are solved within a Lagrangian framework. We assume that eddy viscosity and thermometric conductivity are dependent on Lagrangian coordinates, and derive a new form of the second-order Lagrangian mass transport velocity, applicable across the entire finite water depth. We then analyse the convective heat dynamics influenced by the free-surface boundary layer. Rectangular distributions of free-surface temperature (i.e. a Dirichlet boundary condition) are considered, and analytical solutions for thermal boundary layer temperature fields are provided to offer insights into free-surface heat transfer mechanisms affected by ocean waves. Our results suggest the need to improve existing models that neglect the effects of free-surface waves and the free-surface boundary layer on ocean mass transport and heat transfer.

The propulsive efficiency of flying and swimming animals propelled by oscillatory appendages typically peaks within a narrow Strouhal number range of $0.20 \lt St \lt 0.40$. Motivated by the ubiquitous presence of stratification in natural environments, we numerically investigate the optimal Strouhal numbers $S{t_m}$ for an oscillating foil in density stratified fluids. Our results reveal that $S{t_m}$ increases with the strength of stratification characterised by the internal Froude number $Fr$, giving rise to markedly higher values under strong stratifications compared with those observed in homogeneous fluids. The propulsive efficiency tends to maximise when there is a resonance between the oscillations of the foil and the fluid, as inferred from a fitted line in the ($St$, $Fr$) parameter space, which shows that $S{t_m}$ is proportional to $Fr^{-1}$. We further uncover that the significant increase in $S{t_m}$ in strongly stratified regimes is fundamentally driven by fluid entrainment. During this process, the oscillating foil induces perturbations in the density field, resulting in buoyancy-driven restoring forces which alter the pressure distribution on the foil and thus the hydrodynamic forces. Notably, only under strongly stratified conditions, where dominant buoyancy effects confine the density transport to the vicinity of the oscillating foil, the intensified density perturbation due to the increase in $St$ can be effectively harnessed to enhance thrust production, thereby contributing to the elevated $S{t_m}$. These insights suggest that oscillatory propulsors should adopt new kinematic strategies involving relatively large Strouhal numbers to achieve efficient cruising in strongly stratified environments.