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.

The well-known quadratic temperature–velocity (TV) relation is significant for physical understanding and modelling of compressible wall-bounded turbulence. Meanwhile, there is an increasing interest in employing the TV relation for laminar modelling. In this work, we revisit the TV relation for both laminar and turbulent flows, aiming to explain the success of the TV relation where it works, improve its accuracy where it deviates and relax its limitation as a wall model for accurate temperature prediction. We show that the general recovery factor defined by Zhang et al. (J. Fluid. Mech., vol. 739, 2014, pp. 392–440) is not a wall-normal constant in most laminar and turbulent cases. The effective Prandtl number $Pr_e$ is more critical in determining the shape of temperature profiles. The quadratic TV relation systematically deviates for laminar boundary layers irrespective of Mach number and wall boundary conditions. We find a universal distribution of $Pr_e$, based on which the TV relation can be notably improved, especially for cold-wall cases. For turbulent flows, the TV relation as the wall model can effectively improve the near-wall temperature prediction for cold-wall boundary layer cases, but it involves boundary-layer-edge quantities used in the Reynolds analogy scaling, which hinders the application of the wall model in complex flows. We propose a transformation-based temperature wall model by solving inversely the newly developed temperature transformation of Cheng and Fu (Phy. Rev. Fluids, vol. 9, 2024, no. 054610). The dependence on edge quantities is thus removed in the new model and the high accuracy in turbulent temperature prediction is maintained for boundary layer flows.