We develop a weakly nonlinear theory to revisit the water hammer phenomenon resulting from slow valve manoeuvres. The hydraulic head at the valve is known to be nonlinearly coupled with the flow velocity via a relation derived from Bernoulli’s principle, so that solutions have been so far obtained only via numerical models. The governing equations and boundary conditions indeed yield a nonlinear boundary-value problem, which is here solved using a perturbation approach, Laplace transform and complex analysis. We obtain space- and time-dependent analytical solutions in all of the pipe and validate our results by comparison with standard numerical methods (i.e. Allievi’s method) for determining the exact behaviour of the hydraulic head at the valve. Additionally, we derive algebraic practically relevant closed form expressions for predicting the maximum and minimum hydraulic head values during both valve closure and opening manoeuvres.

This study quantitatively investigates the two-dimensional pseudosteady shock refraction at an inclined air–water interface, referred to as the water wedge, in the weak and strong incident shock strength groups. Numerical simulations are employed to validate the predicted refraction sequences from a previous study (Anbu Serene Raj et al. 2024 J. Fluid Mech. 998, A49). A distinctive irregular refraction pattern, referred to as the bound precursor refraction with a Mach reflection, is numerically validated in the weak shock group. Based on the numerical simulations, an enhanced formulation is proposed to determine the sonic line of the incident flow Mach number ($M_b$) in water, thereby providing an appropriate transition condition for an irregular refraction with a Mach reflection to a free precursor refraction with a Mach reflection transition. Furthermore, comparative studies on solid and water wedges of wedge angle $20^\circ$ reveal discernible differences in the shock reflection patterns. The interplay of the energy dissipation due to the transmitted shock wave and the Richtmyer–Meshkov instability at the air–water interface results in the variation of the triple-point trajectory and transition angles between single Mach reflection (SMR) to transitional Mach reflection (TMR) occurring in air.

The spatio-temporal evolution of very large-scale coherent structures, also known as superstructures, is investigated in both smooth- and rough-wall boundary layers by means of direct numerical simulations up to a frictional Reynolds number of ${\textit{Re}}_\tau = 3\,150$. One smooth-wall and four rough-wall cases are considered, all developing over a region as long as $\sim$60 times the incoming boundary-layer thickness in the streamwise direction. Bio-inspired, biofouling-type topographies are employed for the rough-wall cases, following the previous work of Womack et al. (2022 J. Fluid Mech. vol. 933, p. A38) and Kaminaris et al. (2023 J. Fluid Mech. vol. 961, p. A23). We utilise three-dimensional time series, as well as multiple two-point correlation functions along the boundary layer to capture the detailed length- and time-scale evolution of the superstructures. The results suggest that the presence of roughness significantly amplifies both the strength and the streamwise growth rate of superstructures. Interestingly, however, their ratios relative to the local boundary-layer thickness, $\mathscr{L}_{\!x}/\delta$ and $\mathscr{L}_z/\delta$, remain constant and independent of the streamwise coordinate, indicating that such scaled length scales might constitute a possible flow invariant. Volumetric correlations revealed that all cases induce structures inclined with respect to the mean-flow direction, with those over the rough-wall topographies exhibiting steeper inclination angles. Finally, via proper orthogonal decomposition, pairs of counter-rotating roll modes were detected and found to flank the high- and low-speed superstructures, supporting the conjecture in the literature regarding the mechanisms responsible for the lateral momentum redistribution. The latter also suggests that the way momentum organises itself in high Reynolds number wall-bounded flows might be independent of the roughness terrain underneath.

A prediction framework for the mean quantities in a compressible turbulent boundary layer (TBL) with given Reynolds number, free-stream Mach number and wall-to-recovery ratio as inputs is proposed based on the established scaling laws regarding the velocity transformations, skin-friction coefficient and temperature–velocity (TV) relations. The established velocity transformations that perform well for collapsing the compressible mean profiles onto incompressible ones in the inner layer are used for the scaling of such inner-layer components of mean velocity, while the wake velocity scaling is determined such that self-consistency is achieved under the scaling law for the skin-friction coefficient. A total of 44 compressible TBLs from six direct numerical simulations databases are used to validate the proposed framework, with free-stream Mach numbers ranging from 0.5 to 14, friction Reynolds numbers ranging from 100 to 2400, and wall-to-recovery ratios ranging from 0.15 to 1.9. When incorporated with the scaling laws for velocity transformation from Griffin et al. (2021, Proc. Natl Acad. Sci., vol. 118, e2111144118), the skin-friction coefficient from Zhao & Fu (2025, J. Fluid Mech., vol. 1012, R3) and the TV relation from Duan & Martín (2011, J. Fluid Mech., vol. 684, pp. 25–59), the prediction errors in the mean velocity and temperature profiles remain within $4.0\,\%$ and $6.0\,\%$, respectively, across all tested cases. Correspondingly, the skin-friction and wall-heat-transfer coefficients are also accurately predicted, with root mean square prediction errors of approximately $3 \,\%$. When adopting different velocity transformation methods that are valid for inner-layer scaling, the root mean square prediction errors in the mean velocity and temperature profiles remain below $2.3\,\%$ and $3.6\,\%$, respectively, which further highlights the universality of the proposed framework.

Solid atmospheric particles, such as ice crystals, pollen, dust, ash and microplastics, strongly influence Earth’s climate, ecosystems and air quality. Previous studies have typically relied on analytical models valid only for very small particles or experiments in liquids, where the particle-to-fluid density ratio $R$ is much lower than values encountered in the atmosphere. We combine a novel experimental set-up with particle-resolved direct numerical simulations to study the settling of sub-millimetre ellipsoids in still air. Particle shapes span elongation and flatness values $ 0.2 \leqslant {\textit{EL}}, {\textit{FL}} \leqslant 1.0$ at a density ratio $ R = 1000$ and particle Reynolds numbers $ 2.1 \lt {\textit{Re}}_{\!p} \lt 4.5$, a regime well below the onset of wake-induced instabilities. Nonetheless, we observe unexpectedly rich dynamics: all non-spherical particles exhibit damped oscillatory motion, and some triaxial ellipsoids follow fully three-dimensional, non-planar trajectories due to rotation about all three axes. Simulations at lower density ratios ($ R = 10, 100$) confirm that these behaviours are driven by strong lateral forces happening only at $R=1000$. Key settling characteristics exhibit nonlinear and non-trivial dependencies on shape. In the two-dimensional phase space of elongation and flatness, settling velocity is symmetric about the principal diagonal ($ {\textit{EL}} = {\textit{FL}}$), while oscillation frequency and damping rate show symmetry about the anti-diagonal. Flatness strongly influences pressure drag, while elongation governs lateral drift and swept volume, which can reach up to ten times the particle diameter and four times the volume-equivalent sphere, respectively.

We investigate the energetics of mixing induced by a continuously supplied dense current (density $\rho _0$) propagating beneath a lighter ambient fluid (density $\rho _a$) along a horizontal rigid boundary within a rectangular domain. The flow fields are computed using direct numerical simulations (DNS) performed with the Nek5000 spectral element solver. Mixing is quantified through the temporal evolution of the background potential energy, which exhibits a linear increase over time. This linear trend enables the definition of a dimensionless mixing parameter $\gamma$, representing the rate of background potential energy growth. The value of $\gamma$ depends on the initial density contrast for a fixed volumetric discharge at the source, characterised by the dimensionless source Froude number. The results reveal a non-monotonic dependence of $\gamma$ on the source Froude number, highlighting a complex interaction between flow forcing and mixing efficiency. We find that, under the assumption of uniform mixing along the current’s length, a fraction $\gamma /2$ of the total supplied energy is invested in mixing along a horizontal distance equal to the height of the inlet.

The present article investigates the stability of Rayleigh–Bénard convection in a composite system consisting of a horizontal fluid layer overlying a fluid-saturated Darcy porous layer subjected to a time-periodic temperature distribution. The bottom surface is heated periodically with time, whereas a Biot number-dependent thermal boundary condition represents the heat transfer at the upper surface. The Beavers–Joseph–Saffman–Jones condition describes the ‘slip’ at the interface of the domains, and the Lions interface condition governs the normal force balance, incorporating a dynamic pressure term. The Chebyshev tau method and Fourier analysis are utilised to obtain linear instability bounds, which are compared with strong global and asymptotic limits derived from the nonlinear analysis using the energy method. Four deliberately chosen configurations of superposed fluid- and porous-layer systems are investigated. Two configurations validate the analysis through the limiting cases of the classical Darcy–Bénard and Rayleigh–Bénard systems obtained by setting the fluid-to-porous depth ratio $(\hat {d})$ to zero and infinity, respectively. The other two configurations involve layers with equal depths $(\hat {d} =1)$ and a shallow fluid layer overlying a porous layer $(\hat {d} \sim 0.1)$. For these cases, modulation substantially influences the onset of convection. In the last case, the linear theory points out that modulation parameters can control the dominant convective mode (fluid/porous). Furthermore, unlike the previously reported studies, the nonlinear stability bounds are found to be significantly lower than the linear instability bounds, indicating the possibility of subcritical instabilities in the presence of modulation. The region of subcritical instabilities increases with modulation amplitude.

Supersonic jets impinging on a ground plane produce a highly unsteady jet shear layer, often resulting in extremely high noise level. The widely accepted mechanism for this jet resonance involves a feedback loop consisting of downstream-travelling coherent structures and upstream-propagating acoustic waves. Despite the importance of coherent structures, often referred to as disturbances, that travel downstream, a comprehensive discussion on the disturbance convection velocity has been limited due to the challenges posed by non-intrusive measurement requirements. To determine the convection velocity of disturbances in the jet shear layer, a high-speed schlieren flow visualisation is carried out, and phase-averaged wave diagrams are constructed from the image sets. The experiments are conducted using a Mach 1.5 jet under various nozzle pressure ratios and across a range of impingement distances. A parametric analysis is performed to examine the influence of nozzle pressure ratio on the convection velocity and phase lead/lag at specific impingement distances. The results reveal that impingement tonal frequency is nearly independent of the disturbance convection velocity, except in cases of staging behaviour. They also demonstrated that slower downstream convection velocity of the disturbance corresponds to larger coherent structures, resulting in increased noise levels. Based on the observation of acoustic standing waves, an acoustic speed-based frequency model has been proposed. With the help of the allowable frequency range calculated from the vortex-sheet model, this model can provide a good approximation for the majority of axisymmetric impingement tonal frequencies.

One-degree-of-freedom flow-induced vibration (FIV) and energy harvesting through FIV of an elastically mounted circular cylinder with mechanically coupled rotation were investigated numerically for low Reynolds number 100, mass ratio 8 and a wide range of reduced velocities. The aims of this study are to investigate the effect of the flow direction angle $\beta$ on the vibration and energy harvesting through FIV. Two types of lock-in are found: vortex-induced vibration (VIV) and galloping. The response amplitude increases with the increase of $\beta$ in both regimes. Both VIV response and galloping regimes are found for $\beta$ = 45° to $\beta$ = 90°. For $\beta$ = −90° to $\beta$ = 0°, only VIV response regimes are found. The fluid force and fluid torque play different roles in exciting/damping the vibration. In the high-amplitude gallop regime, the fluid force excites the vibration, and the torque damps the vibration. Energy harvesting at flow direction angle 90° is investigated as this flow direction has the maximum galloping amplitude. The energy harvesting is achieved by a linear electric damping coefficient in the numerical model. The maximum harvestable power in the galloping regime is significantly greater than that in the VIV regime, and it increases with the increase of the reduced velocity. When the reduced velocity is 20, the harvested power is over 20 times that in the VIV regime, and can further increase if reduced velocity further increases. The maximum efficiency over all simulated parameters is 0.424, occurring when the reduced velocity is 20, and electric damping factor is 0.04.

The Weissenberg effect, or rod-climbing phenomenon, occurs in non-Newtonian fluids where the fluid interface ascends along a rotating rod. Despite its prominence, theoretical insights into this phenomenon remain limited. In earlier work, Joseph & Fosdick (1973, Arch. Rat. Mech. Anal. vol. 49, pp. 321–380) employed domain perturbation methods for second-order fluids to determine the equilibrium interface height by expanding solutions based on the rotation speed. In this work, we investigate the time-dependent interface height through asymptotic analysis with dimensionless variables and equations using the Giesekus model. We begin by neglecting inertia to focus on the interaction between gravity, viscoelasticity and surface tension. In the small-deformation scenario, the governing equations indicate the presence of a boundary layer in time, where the interface rises rapidly over a short time scale before gradually approaching a steady state. By employing a stretched time variable, we derive the transient velocity field and corresponding interface shape on this short time scale, and recover the steady-state shape on a longer time scale. In contrast to the work of Joseph and Fosdick, which used the method of successive approximations to determine the steady shape of the interface, we explicitly derive the interface shape for both steady and transient cases. Subsequently, we reintroduce small but finite inertial effects to investigate their interaction with viscoelasticity, and propose a criterion for determining the conditions under which rod climbing occurs. Through numerical computations, we obtain the transient interface shapes, highlighting the interplay between time-dependent viscoelastic and inertial effects.

A long-standing conceptual debate regarding the identification and independence of first Mack and cross-flow instabilities is clarified over a Mach 5.9 sharp wing at zero angle of attack and varying sweep angles. Their receptivity of the boundary layers to three-dimensional slow acoustic and vorticity waves is investigated using linear stability theory, direct numerical simulation and momentum potential theory (MPT). Linear stability theory demonstrates that the targeted slow mode appears as the oblique first mode at small sweep angles ($0^\circ$ and $15^\circ$) and transitions to the cross-flow mode at larger sweep angles ($30^\circ$ and $45^\circ$). Direct numerical simulation indicates that both the oblique first mode and cross-flow mode share identical receptivity pathways: for slow acoustic waves, the pathway comprises ‘leading-edge damping–enhanced exponential growth–linear growth’ stages. For vorticity waves, it consists of ‘leading-edge damping–non-modal growth–linear growth’ stages. Momentum potential theory decomposes the fluctuation momentum density into vortical, acoustic and thermal components, revealing unified receptivity mechanisms: for slow acoustic waves, the leading-edge damping is caused by strong acoustic components generated through synchronization. The enhanced exponential growth stage is dominated by steadily growing vortical components, with acoustic and thermal components remaining at small amplitudes. For vorticity waves, leading-edge disturbances primarily consist of vortical components, indicating a distinct mechanism from slow acoustic waves. Non-modal stages originate from adjustments among MPT components. Vortical components dominate the linear growth stage for both instabilities. These uniform behaviours between first Mack and cross-flow modes highlight their consistency.

Investigations into the effects of polymers on small-scale statistics and flow patterns were conducted in a turbulent von Kármán swirling (VKS) flow. We employed the tomographic particle image velocimetry technique to obtain full information on three-dimensional velocity data, allowing us to effectively resolve dissipation scales. Under varying Reynolds numbers ($R_\lambda =168{-}235$) and polymer concentrations ($\phi =0{-}25\ {\textrm{ppm}}$), we measured the velocity gradient tensor (VGT) and related quantities. Our findings reveal that the ensemble average and probability density function (PDF) of VGT invariants, which represent turbulent dissipation and enstrophy along with their generation terms, are suppressed as polymer concentration increases. Notably, the joint PDFs of the invariants of VGT, which characterise local flow patterns, exhibited significant changes. Specifically, the third-order invariants, especially the local vortex stretching, are greatly suppressed, and strong events of dissipation and enstrophy coexist in space. The local flow pattern tends to be two-dimensional, where the eigenvalues of the rate-of-strain tensor satisfy a ratio $1:0:-1$, and the vorticity aligns with the intermediate eigenvector of the rate-of-strain tensor, while it is perpendicular to the other two. We find that these statistics observations can be well described by the vortex sheet model. Moreover, we find that these vortex sheet structures align with the symmetry axis of the VKS system, and orient randomly in the horizontal plane. Further investigation, including flow visualisation and conditional statistics on vorticity, confirms the presence of vortex sheet structures in turbulent flows with polymer additions. Our results establish a link between single-point statistics and small-scale flow topology, shedding light on the previously overlooked small-scale structures in polymeric turbulence.