The motion and deformation of a neutrally buoyant drop in a rectangular channel experiencing a pressure-driven flow at a low Reynolds number has been investigated both experimentally and numerically. A moving-frame boundary-integral algorithm was used to simulate the drop dynamics, with a focus on steady-state drop velocity and deformation. Results are presented for drops of varying undeformed diameters relative to channel height ($D/H$), drop-to-bulk viscosity ratio ($\lambda$), capillary number ($Ca$, ratio of deforming viscous forces to shape-preserving interfacial tension) and initial position in the channel in a parameter space larger than considered previously. The general trend shows that the drop steady-state velocity decreases with increasing drop diameter and viscosity ratio but increases with increasing $Ca$. An opposite trend is seen for drops with small viscosity ratio, however, where the steady-state velocity increases with increasing $D/H$ and can exceed the maximum background flow velocity. Experimental results verify theoretical predictions. A deformable drop with a size comparable to the channel height when placed off centre migrates towards the centreline and attains a steady state there. In general, a drop with a low viscosity ratio and high capillary number experiences faster cross-stream migration. With increasing aspect ratio, there is a competition between the effect of reduced wall interactions and lower maximum channel centreline velocity at fixed average velocity, with the former helping drops attain higher steady-state velocities at low aspect ratios, but the latter takes over at aspect ratios above approximately 1.5.

The acoustic receptivity of Tollmien–Schlichting (TS) waves due to two-dimensional sharp-edged rectangular bumps and gaps in a compressible boundary layer was investigated by direct numerical simulations. The conclusions were based on a new procedure proposed for obtaining the receptivity amplitudes which appeared to be more robust and accurate than previous ones. The procedure is particularly important for the correct evaluation of the receptivity of gaps. The receptivity amplitudes for gaps were smaller than those for bumps, except for the nominally zero height/depth roughness element, where, consistent with a linear behaviour, they had the same absolute value. The procedure also revealed in detail the behaviour in the region downstream of the roughness element where the TS wave is formed (the formation region). This region extends for approximately $50\delta ^*_{b}$, regardless of bump height or gap depth. For bumps, the receptivity scaled superlinearly with bump height while for the gaps it scaled sublinear with depth. This behaviour is associated with the different velocity profiles caused by bumps and gaps in the formation region. We also discussed issues regarding comparison with experiments. Investigation of the effect of compressibility confirmed that, in the subsonic regime, the receptivity reduces with Mach number. Finally, we addressed the receptivity scaling with the acoustic wave amplitude. It was found that the receptivity scales linearly with the acoustic wave amplitude in a range for which experiments available in the literature indicated a superlinearly scaling. Reasons for these conflicting results are discussed.

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.

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.

The on-body flow and near-to-intermediate wake of a 6:1 prolate spheroid at a pitch angle of $\alpha = 10^{\circ }$ and a length-based Reynolds number, ${Re}_L = U_\infty L / \nu = 3 \times 10^4$, are investigated using large eddy simulation (LES) across four stratification levels: ${\textit {Fr}} = U_{\infty }/ND = \infty , 6, 1.9$ and $1$. A streamwise vortex pair, characteristic of non-zero $\alpha$ in unstratified flow over both slender and blunt bodies, is observed. At ${\textit {Fr}} = \infty$ (unstratified) and $6$, the vortex pair has a lateral left–right asymmetry as has been reported in several previous studies of unstratified flow. However, at higher stratification levels of ${\textit {Fr}} = 1.9$ and $1$, this asymmetry disappears and there is a complex combination of body-shed vorticity that is affected by baroclinicity and vorticity associated with internal gravity waves. Even at the relatively weak stratification of ${\textit {Fr}} = 6$, the wake is strongly influenced by buoyancy from the outset: (a) the vertical drift of the wake is more constrained at ${\textit {Fr}} = 6$ than at ${\textit {Fr}} = \infty$ throughout the domain; and (b) the streamwise vortex pair loses coherence by $x/D = 10$ in the ${\textit {Fr}} = 6$ wake, unlike the ${\textit {Fr}} = \infty$ case. For the ${\textit {Fr}} = 1$ wake, flow separation characteristics differ significantly from those at ${\textit {Fr}} = \infty$ and $6$, resulting in a double-lobed wake topology that persists throughout the domain.

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.

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.

Recent studies focusing on the response of turbulent boundary layers (TBLs) to a step change in roughness have provided insight into the scaling and characterisation of TBLs and the development of the internal layer. Although various step-change combinations have been investigated, ranging from smooth-to-rough to rough-to-smooth, the minimum required roughness fetch length over which the TBL returns to its homogeneously rough behaviour remains unclear. Moreover, the relationship between a finite- and infinite-fetch roughness function (and the equivalent sand-grain roughness) is also unknown. In this study, we determine the minimum ‘equilibrium fetch length’ for a TBL developing over a smooth-to-rough step change as well as the expected error in local skin friction if the fetch length is under this minimum threshold. An experimental study is carried out where the flow is initially developed over a smooth wall, and then a step change is introduced using patches of P24 sandpaper. Twelve roughness fetch lengths are tested in this study, systematically increasing from $L = 1\delta _2$ up to $L = 39\delta _2$ (where L is the roughness fetch length and $\delta _2$ is the TBL thickness of the longest fetch case), measured over a range of Reynolds numbers ($4\times 10^3 \leqslant Re_\tau \leqslant 2\times 10^4$). Results show that the minimum fetch length needed to achieve full equilibrium recovery is around $20\delta _2$. Furthermore, we observe that the local friction coefficient, $C_{\! f}$, recovers to within 10 % of its recovered value for fetch lengths $\geqslant 10\delta _2$. This information allows us to incorporate the effects of roughness fetch length on the skin friction and roughness function.

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.

The hydrodynamic forces acting on an undulating swimming fish consist of two components: a drag-based resistive force, and a reactive force originating from the necessary acceleration of an added mass of water. Lighthill’s elongated-body theory, based on potential flow, provides a framework for calculating this reactive force. By leveraging the high aspect ratio of most fish, the theory simplifies the problem into a series of independent two-dimensional slices of fluids along the fish’s body, which exchange momentum with the body and neighbouring slices. Using momentum conservation arguments, Lighthill’s theory predicts the total thrust generated by an undulating fish, based solely on the dimensions and kinematics of its caudal fin. However, the assumption of independent slices has led to the common misconception that the flow produced lacks a longitudinal component. In this paper, we revisit Lighthill’s theory, offering a modern reinterpretation using essential singularities of potential flows. We then extend it to predict the full three-dimensional flow field induced by the fish’s body motion. Our results compare favourably with numerical simulations of realistic fish geometries.