Whilst surface-stress integration remains the standard approach for fluid force evaluation, control-volume integral methods provide deeper physical insights through functional relationships between the flow field and the resultant force. In this work, by introducing a second-order tensor weight function into the Navier–Stokes equations, we develop a novel weighted-integral framework that offers greater flexibility and enhanced capability for fluid force diagnostics in incompressible flows. Firstly, in addition to the total force and moment, the weighted integral methods establish, for the first time, rigorous quantitative connections between the surface-stress distribution and the flow field, providing potential advantages for flexible body analyses. Secondly, the weighted integral methods offer alternative perspectives on force mechanisms, through vorticity dynamics or pressure view, when the weight function is set as divergence-free or curl-free, respectively. Thirdly, the derivative moment transformation (DMT)-based integral methods (Wu et al., J. Fluid Mech. vol. 576, 2007, 265–286) are generalised to weighted formulations, by which the interconnections among the three DMT methods are clarified. In the canonical problem of uniform flow past a circular cylinder, weighted integral methods demonstrate advantages in yielding new force expressions, improving numerical accuracy over original DMT methods, and enhancing surface-stress analysis. Finally, a force expression is derived that relies solely on velocity and acceleration at discrete points, without spatial derivatives, offering significant value for experimental force estimation. This weighted integral framework holds significant promise for flow diagnostics in fundamentals and applications.

The merging of two turbulent fronts without mean shear is investigated by direct numerical simulations. The turbulent streams are created by prescribing instantaneous velocity fields from precursor simulations of homogeneous isotropic turbulence (HIT) as inlet conditions for spatially evolving turbulent merging. The fronts are initially separated by a distance $H$ and convected with a uniform free stream velocity $U_{\infty }$. The inlet turbulence intensity varies in the range of $0.24 \leqslant u^{\prime}/U_{\infty } \leqslant 0.47$, while the inlet Taylor-scale Reynolds number is in the range of $151 \leqslant \textit{Re}_{\lambda } \leqslant 317$. As the flow develops in the streamwise direction, two distinct regions are identified: (i) an initial linear decay region, where the two turbulent fronts gradually approach each other without noticeable interaction; and (ii) a rapid decay region, where the opposing turbulent fronts influence one another and eventually merge. The flow statistics collapse once the streamwise coordinate is rescaled as $x^{+} = (x/H) (u^{\prime}/U_{\infty })$, suggesting that the merging location is imposed by large scales. An analysis conditioned to the developing turbulent/non-turbulent interfaces (TNTIs) reveals that, within the merging region, conditional mean enstrophy profiles deviate from those observed in ‘classical’ TNTIs, indicating a locally more homogenous flow. Within this region of interaction, the surface area of the TNTI increases while the volume of irrotational fluid steadily decreases, resulting in the generation of fine-scale structures. These findings support that turbulent merging is a multiscale process, where both the largest and smallest scales of motion intervene.

Hydrodynamic instability can occur when a viscous fluid is driven rapidly through a flexible-walled channel, including a multiplicity of steady states and distinct families of self-excited oscillations. In this study we use a computational method to predict the stability of flow through a planar finite-length rigid channel with a segment of one wall replaced by a thin pre-tensioned elastic beam of negligible mass. For large external pressures, this system exhibits a collapsed steady state that is unstable to low-frequency self-excited oscillations, where the criticality conditions are well approximated by a long-wavelength one-dimensional (1-D) model. This oscillation growing from a collapsed state exhibits a reduced inlet driving pressure compared with the corresponding steady flow, so the oscillating state is energetically more favourable. In some parameter regimes this collapsed steady state is also unstable to distinct high-frequency normal modes, again predicted by the 1-D model. Conversely, for lower external pressures, the system exhibits an inflated steady state that is unstable to another two modes of self-excited oscillation, neither of which are predicted by the lower-order model. One of these modes becomes unstable close to the transition between the upper and lower steady states, while the other involves small-amplitude oscillations about a highly inflated wall profile with large recirculation vortices within the cavity. These oscillatory modes growing from an inflated steady state exhibit a net increase in driving pressure compared with the steady flow, suggesting a different mechanism of instability to those growing from a collapsed state.

The effectiveness of polymer drag reduction by targeted injection is studied in comparison with that of a uniform concentration (or polymer ocean) in a turbulent channel flow. Direct numerical simulations are performed using a pseudo-spectral code to solve the coupled equations of a viscoelastic fluid using the finitely extensible nonlinear elastic dumbbell model with the Peterlin approximation. Light and heavy particles are used to carry the polymer in some cases, and polymer is selectively injected into specific flow regions in the other cases. Drag reduction is computed for a polymer ocean at a viscosity ratio of $\beta = 0.9$ for simulation validation, and then various methods of polymer addition at $\beta = 0.95$ are compared for their drag-reduction performance and general effect on the flow. It was found that injecting polymer directly into regions of high axial strain inside and around coherent vortical structures was the most effective at reducing drag, while injecting polymer very close to the walls was the least effective. The targeting methods achieved up to 2.5 % higher drag reduction than an equivalent polymer ocean, offering a moderate performance boost in the low drag-reduction regime.

This work investigates the Richtmyer–Meshkov instability (RMI) at gas/viscoelastic interfaces with an initial single-mode perturbation both experimentally and theoretically. By systematically varying the compositions and concentrations of hydrogels, a series of viscoelastic materials with tuneable mechanical properties is created, spanning from highly viscous to predominantly elastic. Following shock impact, the interface exhibits two distinct types of perturbations: small-amplitude, short-wavelength perturbations inherited from initial single-mode condition, and large-amplitude, long-wavelength perturbations arising from viscous effects. For hydrogels with high loss factors, viscosity dominates the interface dynamics, leading to pronounced V-shaped deformation of the entire interface accompanied by a rapid decay of the initial single-mode perturbation. In contrast, for hydrogels with low loss factors, elasticity plays a prominent role, leading to sustained oscillations of the single-mode perturbation. By employing the Maxwell model to simultaneously incorporate both viscous and elastic effects, a comprehensive linear theory for RMI at gas/viscoelastic interfaces is developed, which shows good agreement with experimental results in the early stages. Although deviations arise at later times due to factors such as the shear-thickening feature of hydrogels and three-dimensional effects, the model well reproduces the oscillation behaviour of single-mode perturbations. In particular, it effectively captures the trend that increasing elasticity reduces both oscillation period and amplitude, providing key insights into the role of material properties in interface dynamics.

This article follows on from Scott & Cambon (J. Fluid Mech., vol. 979, 2024, A17) and Scott (Phys. Rev. E, vol. 111, 2025, 035101). Like those articles, it concerns weak, decaying homogeneous turbulence in a rotating, stably stratified fluid with constant Brunt–Väisälä frequency, $N$. The difference is that here we consider the case in which $\beta =2{\varOmega} /N$ is close to $1$, where ${\varOmega}$ is the rotation rate. Because this renders inertial-gravity waves only weakly dispersive, wave-turbulence theory, which played a prominent role in the earlier studies, no longer applies. Indeed, wave-turbulence analysis does not appear here. Nonetheless, much of the analytical framework, based on modal decomposition, carries over, as do many of the conclusions. The flow is expressed as a sum of wave and non-propagating (NP) modes and their weak-turbulence mode-amplitude evolution equations are derived for small $\beta -1$. The NP component is found to evolve independently of the wave one, following an amplitude equation which is precisely that of the previous studies in the limit $\beta \rightarrow 1$. The NP component induces coupling between wave modes and, without it, the wave component has purely linear decay. The mode-amplitude equations are integrated numerically using a scheme similar to that of classical direct numerical simulation and results given. We find an inverse energy cascade of the NP component, whereas the presence of that component induces a forward cascade, hence significant dissipation, of the wave component. Detailed results are given for the energy, energy spectra and energy fluxes of the two components.

We perform numerical simulations of two-dimensional strongly stratified flows in a square periodic domain $(y,z)$ forced by a steady mode with vorticity of the form $\sin (k_{\textit{y f}}y)\sin (k_{\textit{z f}}z)$, where $(k_{\textit{y f}},k_{\textit{z f}})$ are fixed wavenumbers. It is shown that such deterministic forcing can lead to a transition to turbulence and the emergence of horizontal layers (so-called vertically sheared horizontal flows, VSHFs) similarly as for random stochastic forcing. The flow characteristics are studied depending on the Froude and Reynolds numbers. Furthermore, the mechanisms of layers formation are disentangled. Triadic instabilities first lead to the growth of pairs of wavevectors that resonate with each of the four forced wavevectors. Quadratic interactions between these resonant modes and the forcing also drive the growth of several non-resonant modes at the same growth rate. Since the forcing comprises the wavevectors $\pm (k_{\textit{y f}},k_{\textit{z f}})$ and their mirror symmetric with respect to the horizontal $\pm (k_{\textit{y f}},-k_{\textit{z f}})$, there exist enslaved/bound modes with the same horizontal wavenumber and different vertical wavenumbers. Hence, the quadratic interactions between the latter modes force a second generation of modes among which some are VSHFs. Their growth rate is twice the growth rate of the primary resonant modes. Such a mechanism is similar to resonant quartets (Newell, J. Fluid Mech., 1969, vol. 35, no 2, pp. 255–271; Smith & Waleffe, Phys. Fluids, 1999, vol. 11, no 6, pp. 1608–1622). When the forcing is restricted to only the two wavevectors $\pm (k_{\textit{y f}},k_{\textit{z f}})$, the second generation of enslaved/bound modes all have a non-zero horizontal wavenumber. However, further quadratic interactions can force VSHF. Thus, horizontal layers also emerge, but with a growth rate equal to the number of quadratic interactions times the growth rate of the primary instability.

We present a back-in-time analysis for the origin of vorticity in viscous separated flows over immersed bodies, using the adjoint-vorticity framework recently introduced by Xiang et al. (2025 J. Fluid Mech. vol. 1011, A33. The solution of the adjoint-vorticity equations yields the volume density of mean deformation, which captures the stretching and tilting of the earlier vorticity that leads to the terminal value. The analysis also takes into account the boundary contributions of vorticity and its flux. Three examples are considered. Steady, axisymmetric separation in the flow over a sphere at Reynolds number $Re=200$ is shown to be established due to wall flux from both upstream and downstream of separation, the latter contribution being absent from the classical description by Lighthill. For unsteady separation at higher $Re=300$, the streamwise vorticity within the wake hairpin vortex is traced back, quantitatively, to the azimuthal vorticity on the sphere. The third configuration is the flow over a prolate spheroid at $Re=3000$. The null vorticity at three-dimensional separation originates from the cancellation of opposite interior contributions adjacent to the separation surface. The contribution from the downstream side migrates across the separation surface into the upstream region due to a tilting effect – a fundamental distinction between two- and three-dimensional separation. We also examine the detached vortical structures. The streamwise vorticity in the primary vortex originates from tilting of near-wall azimuthal vorticity, differing from Lighthill’s conjecture that the origin is streamwise near-wall vorticity that arises due to the reduced Coriolis force. Finally, a necklace vortex in the turbulent wake is traced back in time, and is shown to have contributions from the spheroid trailing-edge shed shear layer and the large-scale counter-rotating primary vortices.

Shock trains compress incoming supersonic flow through a series of shock wave/turbulent boundary layer interactions (STBLIs) that occur in rapid streamwise succession. In this work, the global flow changes across the entire turbulent shock train are analysed as the confluence of local changes imparted by individual STBLIs. For this purpose, wall-resolved large eddy simulations are used on a constant area, back-pressured channel configuration with an entry Mach number of 2.0. Local changes due to individual STBLIs are evaluated in terms of deviations from incoming, near-equilibrium boundary layers, by systematically examining properties of the mean flow structure and turbulent statistics. The first STBLI in the train induces a strongly separated region, which interrupts inner layer dynamics and incites wall-normal deflection of turbulent structures, leading to prominent outer layer Reynolds stress amplification and related transport phenomena. Downstream of the first STBLI, the thickened, turbulent wall layer repeatedly interacts with subsequent shock waves in the train, resulting in cyclic attenuation and amplification of turbulent stresses, localised incipient separations and variations in mean momentum flux gradients. Decreases in mean Mach number along the shock train result in downstream shocks weakening to the point that interactions with the turbulent boundary layer impart negligible changes on the local flow. Consequently, after a sufficient streamwise extent, the boundary layers asymptote towards a new equilibrium state, thus recovering certain classical properties of near-wall turbulence. Among the features that reappear are a self-similar, adverse pressure gradient velocity profile and the restoration of the autonomous roller-streak cycle.