In an open channel flow, deviations to the lower topography can induce abrupt changes in the wave height, known as hydraulic jumps. This phenomenon occurs when the flow switches from subcritical to supercritical (or vice versa), and is commonly observed in rivers, flumes and weirs. Theoretical insight is typically sought through the study of reduced models such as the forced Korteweg–de Vries equation, in which previous work has predominantly focused on either stationary formulations or the initial transient behaviour caused by perturbations. In a joint theoretical and numerical study of the free-surface Euler equations, Keeler & Blyth (J. Fluid Mech., vol. 993, 2024, A9) have detected a new class of unsteady solutions to this problem. These emerge from an unstable steady solution, and feature large-amplitude time-periodic ripples emitted from a sudden decrease in the water depth forced by topography, known as a hydraulic fall.

The complex behaviours of supersonic turbulent boundary-layer flows interacting with combustion is explored through the use of direct numerical simulations. The chosen flow model is a non-premixed hydrogen–air flame ignited within a three-dimensional supersonic turbulent flat-plate boundary layer operating at a Mach number of 2.33 and a friction Reynolds number of approximately 1000. The simulation involves a finite-rate model for the hydrogen–air reaction including 9 species and 19 steps with adiabatic non-catalytic wall conditions. The inlet flow consists of preheated air in the main stream and a hydrogen film injected in proximity to the cold wall, inducing mixing and ignition in the outer layer. The specific inlet configuration leads to two successive transition stages of distinct mechanisms, with the first stage related to the mixing-layer instability and the second one to the boundary-layer instability. The excess Reynolds stresses resulting from the transition exhibit a downstream decrease, with values being lower than canonical adiabatic profiles, particularly in the outer layer. This difference is attributed to the reduced local friction Reynolds numbers, associated with the non-classical wall-normal distributions of density and viscosity. The effect of combustion on the recovery process and skin friction is analysed in comparison with a supplemented chemistry-frozen setting. Velocity–temperature and velocity–species correlations are further examined. The wall-normal profiles of turbulent Prandtl number tend to classical non-reacting values and the turbulent Schmidt number is only slightly affected by the reaction. The strong Reynolds analogy linking velocity and temperature fluctuations is found to be invalid in the outer layer due to the presence of large-scale temperature fluctuations and the inhomogeneity of hydrogen gas distribution, as revealed by a spectrum analysis. A statistical analysis of elementary reactions at varying wall-normal distances is provided, highlighting the dominance of hydrogen atom depletion in the inner region and the prevalence of water vapour production in the outer region, together contributing to the chemical heat release across the boundary layer. Turbulence–chemistry interaction is assessed through a comparison of mean turbulent and laminar chemical heat release rates, showing significant differences of up to an order of magnitude. Using a simple diagnostic tool, the modelling of fluctuations associated with hydrogen gas and two intermediate products is found to be crucial for turbulent chemistry closure.

In this note, we study the effect of viscosity gradients on the energy dissipated by the motion of microswimmers and the associated efficiency of that motion. Using spheroidal squirmer model swimmers in weak linearly varying viscosity fields, we find that efficiency depends on whether they generate propulsion from the back (pushers) or the front (pullers). Pushers are faster and more efficient when moving down gradients, but slower and less efficient moving up viscosity gradients, and the opposite is true for pullers. However, both pushers and pullers display negative viscotaxis, therefore pushers dynamically tend to the most efficient orientation, while pullers tend to the least. We also evaluate the effect of shape on power expenditure and efficiency when swimming in viscosity gradients, and find that in general, the change in both due to gradients decreases monotonically with increasing slenderness. This work shows how shape and gait play an important role in determining dynamics and efficiency in inhomogeneous environments, and demonstrating that both efficiency minimizing and maximizing stable dynamical states are possible.

In this study, the resolvent-based estimation (RBE) is further generalised to cases with arbitrarily sampled measurements in time, where the generalised RBE is denoted as GRBE in this study. Different from the RBE that constructs the transfer function at each frequency, the GRBE minimises the estimation error energy in the physical temporal domain by considering the forcing and noise statistics. The GRBE is validated by estimating the complex Ginzburg–Landau equation and turbulent channel flows with the friction Reynolds number $Re_{\tau }=186$, 547 and 934, where the results from the RBE are also included. When the measurements are temporally resolved, the estimation results of the two approaches are equivalent to each other, and both match well with the reference numerical results. For the temporally unresolved cases, the estimation errors from the GRBE are obviously lower than those from the RBE. After validation, the GRBE is applied to investigate the impacts of the abundance of the measured information, including the temporal information and sensor types, on the estimation accuracy. Compared with the mean square error (MSE) in the estimation with temporally resolved measurements, that with measurements at only one snapshot, i.e. without any temporal information, increases by approximately $15\,\%$. On the other hand, it can effectively improve the estimation accuracy by increasing the number of sensor types. With temporally resolved measurements, the relative MSE decreases by $12.3\,\%$ when the sensor types increase from $\lbrace \tau _u \rbrace$ to $\lbrace \tau _u,\tau _w,p \rbrace$, where $\tau_u$, $\tau_w$ and p are the streamwise shear stress, spanwise shear stress and pressure at the wall. Finally, several existing forcing models are incorporated into the GRBE to investigate their performance in the linear estimation of flow state. The wall-distance-dependent model (W-model) results match well with the optimal linear estimations when the measurements are temporally unresolved. Meanwhile, with the increase of temporal information of the measurement, the estimation errors from the tested W-model and the scale-dependent model ($\lambda$-model) both increase, which contradicts the tendency observed in the optimal linear GRBE estimation results. Such a phenomenon highlights the importance of proper modelling of the forcing in the temporal domain for the accuracy of flow state estimation.

We perform simulations of an impulsively started, axisymmetric viscoelastic jet exiting a nozzle and entering a stagnant gas phase using the open-source code Basilisk. This code allows for efficient computations through an adaptively refined volume-of-fluid technique that can accurately capture the deformation of the liquid–gas interface. We use the FENE-P constitutive equation to describe the viscoelasticity of the liquid, and employ the log-conformation transformation, which provides stable solutions for the evolution of the conformation tensor as the jet thins down under the action of interfacial tension. For the first time, the entire jetting and breakup process of a viscoelastic fluid is simulated, including the pre-shearing flow through the nozzle, which results in an inhomogeneous initial radial stress distribution in the fluid thread that affects the subsequent breakup dynamics. The evolution of the velocity field and the elastic stresses in the nozzle are validated against analytical solutions where possible, and the early-stage dynamics of the jet evolution are compared favourably to the predictions of linear stability theory. We study the effect of the flow inside the nozzle on the thinning dynamics of the viscoelastic jet (which develops distinctive ‘beads-on-a-string’ structures) and on the spatio-temporal evolution of the polymeric stresses in order to systematically explore the dependence of the filament thinning and breakup characteristics on the initial axial momentum of the jet and the extensibility of the dissolved polymer chains.

Arrays of heaving buoy type wave energy converters (WECs) are a promising contender to harness the renewable power of ocean waves on a commercial scale but require strategies to achieve efficient capture of wave energy over broad frequency bands for economic viability. A WEC-array design is proposed for absorption over a target frequency range in the two-dimensional water wave context by spatially grading the resonant properties of WECs via linear spring–damper power take-off mechanisms. The design is based on theories for rainbow reflection and rainbow absorption, which incorporate analyses based on Bloch wave modes and pole–zero pairs in complex frequency space. In contrast to previous applications of these theories, the influence of a higher-order passband and associated pole–zero pairs are shown to influence absorption at the high-frequency end of the target interval. The theories are used to inform initialisations for optimisation algorithms, and an optimised array of only five WECs is shown to give near-perfect absorption ($\geq$99 %) over the target interval. Broadband absorption is demonstrated when surge and pitch motions are released, for irregular sea states, and for incident wave packets in the time domain, where the time-domain responses are decomposed into Bloch modes to connect with the underlying theory.

Free surface flows driven by boundary undulations are observed in many biological phenomena, including the feeding and locomotion of water snails. To simulate the feeding strategy of apple snails, we develop a centimetric robotic undulator that drives a thin viscous film of liquid with the wave speed $V_w$. Our experimental results demonstrate that the behaviour of the net fluid flux $Q$ strongly depends on the Reynolds number $Re$. Specifically, in the limit of vanishing $Re$, we observe that $Q$ varies non-monotonically with $V_w$, which has been successfully rationalised by Pandey et al. (Nat. Commun., vol. 14, no. 1, 2023, p. 7735) with the lubrication model. By contrast, in the regime of finite inertia (${Re} \sim O(1)$), the fluid flux continues to increase with $V_w$ and completely deviates from the prediction of lubrication theory. To explain the inertia-enhanced pumping rate, we build a thin-film, two-dimensional model via the asymptotic expansion in which we linearise the effects of inertia. Our model results match the experimental data with no fitting parameters and also show the connection to the corresponding free surface shapes $h_2$. Going beyond the experimental data, we derive analytical expressions of $Q$ and $h_2$, which allow us to decouple the effects of inertia, gravity, viscosity and surface tension on free surface pumping over a wide range of parameter space.

The development of thermal boundary layers and plume near a section-triangular roof under different isothermal heating conditions has been the focus of numerous numerical studies. However, flow transition in this type of flow has never been observed experimentally. Here, phase-shifting interferometry and thermistor measurements are employed to experimentally observe and quantify the flow transitions in a buoyancy-driven flow over an isothermal section-triangular roof. Visualisation of temperature contours is conducted across a wide range of Rayleigh numbers from laminar at 103 to chaotic state at 4 × 106. Power spectral density of the temperature measurements reveals the type of bifurcations developing as the Rayleigh number is increased. This flow transition is characterised as a complex bifurcation route with the presence of two fundamental frequencies, a low and a high frequency. We found that the thermal stratification in the environment plays a significant role in the flow transition. The spatial development of flow is also quantitatively and qualitatively described. In addition to clarifying flow transition in experiments, the work demonstrates the implementation of phase-shifting interferometry and punctual temperature measurements for characterisation of near-field flow over a heated surface.

Rayleigh–Bénard convection in a rotating spherical shell provides a simplified model for convective dynamics of planetary and stellar interiors. Over the past decades, the problem has been studied extensively via numerical simulations, but most previous simulations set the Prandtl number $Pr$ to unity. In this study we build more than 200 numerical models of rotating convection in a spherical shell over a wide range of $Pr$ ($10^{-2}\le Pr \le 10^2$). By increasing the Rayleigh number $Ra$, we characterise four different flow regimes, starting from the linear onset to multiple modes, then transitioning to the geostrophic turbulence and eventually approaching the weakly rotating regime. In the multiple modes regime, we show evidence of triadic resonances in numerical models with different $Pr$, which may provide a generic mechanism for the transition from laminar to turbulence in rotating convection. We analyse scaling behaviours of the heat transfer and convective flow speeds in numerical simulations, paying particular attention to the $Pr$ dependence. We find that the so-called diffusion-free scaling for the heat transfer cannot reconcile all numerical models with different $Pr$ in the geostrophic turbulence regime. However, the characteristic flow speeds at different $Pr$ roughly follow a unified scaling that can be described by visco-Archimedean–Coriolis force balances, though the scaling tends to approach the Coriolis-inertial-Archimedean force balance at low $Pr$. We also show that transition behaviours from rotating to non-rotating convection depend on $Pr$. The transition criteria based on heat transfer and flow morphology would be rather different when $Pr>1$, but the two criteria are consistent for cases with $Pr\le 1$. Both scaling behaviours and transition behaviours suggest that the heat transfer is controlled by the boundary layers while the convective flow speeds are mainly determined by the force balance in the bulk for cases with $Pr>1$, which is in line with recent experimental results with moderate to high $Pr$. For cases with $Pr \le 1$, both the heat transfer and convective velocities are approaching the inviscid dynamics in the bulk. We also briefly analysed the magnitude and scaling of zonal flows at different $Pr$, showing that the zonal flow amplitude rapidly increases as $Pr$ decreases.

The buoyancy-driven motion of a deformable bubble rising near a vertical hydrophilic wall is studied numerically. We focus on moderately inertial regimes in which the bubble undergoes low-to-moderate deformations and would rise in a straight line in the absence of the wall. Three different types of near-wall motion are observed, depending on the buoyancy-to-viscous and buoyancy-to-capillary force ratios defining the Galilei ($Ga$) and Bond ($Bo$) numbers of the system, respectively. For low enough $Ga$ or large enough $Bo$, bubbles consistently migrate away from the wall. Conversely, for large enough $Ga$ and low enough $Bo$, they perform periodic near-wall bounces. At intermediate $Ga$ and $Bo$, they are first attracted to the wall down to a certain critical distance, and then perform bounces with a decreasing amplitude before stabilising at this critical separation. Periodic bounces are accompanied by the shedding of a pair of streamwise vortices in the wake, the formation of which is governed by the near-wall shear resulting from the no-slip condition. These vortices provide a repulsive force that overcomes the viscous resistance of the fluid to the departing motion, making the bubble capable of returning to the region where it is attracted again to the wall. Although periodic, the shedding/regeneration cycle of these vortices is highly asymmetric with respect to the lateral bubble displacements, vortices being shed when the gap left between the bubble and the wall reaches its maximum, and reborn only when this gap comes back to its minimum.

We investigated the effects of two-dimensional sharp-edged rectangular bumps on Tollmien–Schlichting (TS) wave evolution using direct numerical simulation. The bump height, $h$, ranged from 5 % to 40 % of the local displacement thickness, $\delta ^*$. Behind the bump, a recirculating flow region could be formed whose length increased nonlinearly with $h$. The bump height effect on the TS wave, which was the dominant, scaled super-exponentially with $h$. We also showed a substantial effect of the $\delta ^*$-based Reynolds number, ${\textit {Re}} _{\delta ^*}$. Firstly, the bump wake extended with ${\textit {Re}} _{\delta ^*}$, promoting larger TS wave growth rates. The second effect is related to proximity to the upper branch of the instability loop, accounting for the influence of the TS frequency, as well. It dictates the bump impact increases as it gets closer to transition, either by the bump moving downstream or the transition moving upstream. For a 40 % high bump, for example, changing the ${\textit {Re}} _{\delta ^*}$ at the bump location from 1500 to 2000 increased $\Delta N$ by a factor of 2 ($\Delta N$ represents a measure of a surface irregularity effect on the smooth plate N-factor). We also found that $(\Delta N)_{max}$ increases linearly with ${\textit {Re}} _{hh}$. Results in the subsonic regime showed that the bump impact attenuates with Mach number up to 0.7 but above it, stabilisation is surpassed by the destabilising effect caused by the recirculation lengthening. This is mostly associated with the bump wake that extends with the pressure gradient which increases substantially towards the sonic speed. This is enhanced if the surface is adiabatic rather than isothermal.

We experimentally and numerically characterize rapidly rotating radiatively driven thermal convection, beyond the sole heat transport measurements reported by Bouillaut et al. (Proc. Natl Acad. Sci., vol. 118, 2021, e2105015118). Based on a suite of direct numerical simulations (DNS) and additional processing of the experimental data collected by Bouillaut et al. (Proc. Natl Acad. Sci., vol. 118, 2021, e2105015118), we report the simultaneous validation of the scaling predictions of the ‘geostrophic turbulence’ regime – the diffusivity-free or ‘ultimate’ regime of rapidly rotating convection – for the heat transport and the temperature fluctuations. Following such cross-validation between DNS and laboratory experiments, we further process the numerical data to validate the ‘geostrophic turbulence’ scaling predictions for the flow velocity and horizontal scale. Radiatively driven convection thus appears as a versatile set-up for the laboratory observation of the diffusivity-free regimes of various convective flows of geophysical and/or astrophysical interest.

Microvortex generators (MVGs) are a promising solution to control shock wave/turbulent boundary layer interactions (SBLIs), especially in supersonic inlets. In this study, we examine the effects of a microramp vortex generator on an SBLI generated by an oblique shock wave and a turbulent boundary layer using direct numerical simulations (DNSs). Two cases, with and without the presence of a microramp, are compared in terms of their mean and unsteady flow features at free-stream Mach number equal to 2 and friction Reynolds number at the inviscid shock impingement equal to 600. The long integration period allows us to assess how microramps affect the typical low-frequency unsteadiness observed in SBLIs, and the data generated may serve as a reference for simulations of lower fidelity or reduced order models. The analysis shows that the three-dimensional microramp wake alters the interaction region dramatically, inducing a significant spanwise modulation and topology change of the separation. For example, tornado-like structures redistribute the flow in both the spanwise and wall-normal directions inside the recirculation region. The increase in momentum close to the wall by the ramp vortices effectively delays the onset of the separation and, thus, the separation length, but at the same time leads to a significant increase in the intensity of the wall-pressure fluctuations. We then characterise the mutual interaction between the arch-like vortices around the ramp wake and the SBLI. The specific spanwise vorticity shows that these vortices follow the edge of the separation and their intensity, apart from mean compressibility effects, is not affected by the shocks. The shocks, instead, are deformed in shape by the periodic impingement of the vortices, although the spectral analysis did not reveal any significant trace of their shedding frequency in the separation region. These Kelvin–Helmholtz vortices, however, may be relevant in the closure of the separation bubble. Fourier analysis also shows a constant increase, in both value and magnitude, in the low-frequency peak all along the span, suggesting that the motion of the separation shock remains coherent while being disturbed by the arch-like vortices and oscillating at a higher frequency in absolute terms.

Melting and solidification in periodically time-modulated thermal convection are relevant for numerous natural and engineering systems, for example, glacial melting under periodic sun radiation and latent thermal energy storage under periodically pulsating heating. It is highly relevant for the estimation of melt rate and melt efficiency management. However, even the dynamics of a solid–liquid interface shape subjected to a simple sinusoidal heating has not yet been investigated in detail. In this paper, we offer a better understanding of the modulation frequency dependence of the melting and solidification front. We numerically investigate periodic melting and solidification in turbulent convective flow with the solid above and the melted liquid below, and sinusoidal heating at the bottom plate with the mean temperature equal to the melting temperature. We investigate how the periodic heating can prevent the full solidification, and the resulting flow structures and the quasi-equilibrium interface height. We further study the dependence on the heating modulation frequency. As the frequency decreases, we found two distinct regimes, which are ‘partially solid’ and ‘fully solid’. In the fully solid regime, the liquid freezes completely, and the effect of the modulation is limited. In the partially solid regime, the solid partially melts, and a steady or unsteady solid–liquid interface forms depending on the frequency. The interface height can be derived based on the energy balance through the interface. In the partially solid regime, the interface height oscillates periodically, following the frequency of modulation. Here, we propose a perturbation approach that can predict the dependency of the oscillation amplitude on the modulation frequency.

The transient shear-induced particle migration of frictional non-Brownian suspensions is studied using particle-resolved simulations. The numerical method – the fictitious domain method – is well suited to heterogeneous flows thanks to a frame-invariant formulation of the subgrid (lubrication) corrections that does not involve the ambient flow (Orsi et al., J. Comput. Phys., vol. 474, 2023, 111823). The paper aims to give an accurate quantitative picture of the mass and momentum balance during the flow. The various assumptions and local constitutive laws that together form the suspension balance model (SBM) are thoroughly examined. To this purpose, the various quantities of interest are locally averaged in space and time, and their profile across the channel is extensively studied, with specific attention to the time evolution of the different contributions, either hydrodynamic in nature or from contact interactions, to the shear and normal stresses. The latter, together with the velocity gradient in the wall-normal direction and the volume fraction profile, yield the local constitutive laws, which are compared with their counterpart obtained in homogeneous shear flow. A fair agreement is observed except in a layering area at the boundaries and at the very centre of the channel. In addition, the main assumption of the SBM, i.e. the local relation between the hydrodynamic force on the particles and the particle flux, is meticulously investigated. The hydrodynamic force is found to be mainly a drag, except in the lower range of the probed volume fractions, where a non-drag contribution is observed.

In this paper, we numerically study the mechanism of the oscillatory flow dynamics associated with the tip vortex cavitation (TVC) over an elliptical hydrofoil section. Using our recently developed three-dimensional variational multiphase flow solver, we investigate the TVC phenomenon at Reynolds number $Re = 8.95 \times 10^5$ via dynamic subgrid-scale modelling and the homogeneous mixture theory. To begin, we examine the grid resolution requirements and introduce a length scale considering both the tip vortex strength and the core radius. This length scale is then employed to non-dimensionalize the spatial resolution in the tip vortex region, the results of which serve as a basis for estimation of the required mesh resolution in large eddy simulations of TVC. We next perform simulations to analyse the dynamical modes of tip vortex cavity oscillation at different cavitation numbers, and compare them with the semi-analytical solution. The breathing mode of cavity surface oscillation is extracted from the spatial-temporal evolution of the cavity's effective radius. The temporally averaged effective radius demonstrates that the columnar cavity experiences a growth region followed by decay as it progresses away from the tip. Further examination of the characteristics of local breathing mode oscillations in the growth and decay regions indicates the alteration of the cavity's oscillatory behaviour as it travels from the growth region to the decay region, with the oscillations within the growth region being characterized by lower frequencies. For representative cavitation numbers $\sigma \in [1.2,2.6]$, we find that pressure fluctuations exhibit a shift of the spectrum towards lower frequencies as the cavitation number decreases, similar to its influence on breathing mode oscillations. The results indicate the existence of correlations between the breathing mode oscillations and the pressure fluctuations. While the low-frequency pressure fluctuations are found to be correlated with the growth region, the breathing mode oscillations within the decay region are related to higher-frequency pressure fluctuations. The proposed mechanism can play an important role in developing mitigation strategies for TVC, which can reduce the underwater radiated noise by marine propellers.

In a horizontally heated melting system, where a solid substance is subject to melting by a warmer liquid beneath, the presence of solute in the liquid introduces a complex interplay between temperature and concentration dynamics. Employing a recently developed sharp interface method (Xue et al., J. Comput. Phys., vol. 491, 2023), we conduct direct numerical simulations to investigate the transient behaviour of the system across a broad range of Rayleigh numbers and solute concentrations. Our observations reveal distinct flow regimes: at low concentrations, the system resembles a temperature-driven melting problem, characterized by vortex rolls beneath the melting interface. As the solute concentration increases, a stably stratified layer emerges beneath the interface, leading to the transition from thermal convection to penetrative convection, which resembles those flow characteristics observed in the double-diffusive convection. This shift results from the competition between the stabilizing effect induced by solute concentration gradient and the destabilizing effect caused by temperature gradient. Otherwise in the diffusion regime, characterized by very high solute concentrations, the flow becomes static due to the complete suppression of convection by the stably stratified layer. This regime further exhibits two distinct patterns: ‘melting’ and ‘dissolution’. Beyond characterizing diverse flow patterns, our study conducts a quantitative analysis, examining heat/mass transfer, melting rates, and the evolution of temperature and concentration at the interface. These insights contribute to a better understanding of the intricate interplay between temperature and solute concentration during phase change, with implications for accurately estimating melting rates in binary fluid systems.

Structural damage in turbomachinery is a primary origin of aeronautic accidents, which is receiving increased attention. This study is thus focused on the aeroelastic analysis of damaged blades, including the onset of flutter and underlying mechanisms. First, a high-fidelity fluid–solid coupling system is established with computational fluid dynamics (CFD) and computational structural dynamics (CSD) technologies, via which the dynamic aeroelastic analysis is conducted based on static aeroelastic deformation. Second, a damaged rotor blade is parametrically modelled with variable damage levels, extents, and positions. Finally, the modal identification method of spectral proper orthogonal decomposition (SPOD) is applied to observe flow details and provide physical insight into the flutter mechanism for damaged blades. Numerical analysis finds that there is a critical damage level below which the aeroelastic stability is positively improved with increasing damage level; otherwise, a significant loss of stability is induced. The damage location and extent further affect this critical damage level and the change rate crossing the threshold. The simulation with CFD/CSD finds that the high pressure near the trailing edge induced from boundary layer separation suppresses vibrations in stable conditions, but motivates vibrations during flutter, which is because of the high-pressure spread to nearing blades. SPOD modes reveal that high-frequency disturbances with large scale are primary factors inducing flutter, which is further stimulated by the high-order disturbances with small scale. This study provides a crucial foundation for the fatigue prediction for rotor blades in service and the optimisation design for high-performance turbomachinery in the near future.

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a circular pipe is investigated using asymptotic and numerical methods. Attention is focused on the low-frequency disturbances that induce streamwise elongated structures. A pair of vortical modes with opposite azimuthal wavenumbers is used to model the free-stream disturbances. Their amplitude is assumed to be intense enough for nonlinear interactions to occur inside the pipe. The formation and evolution of the perturbation flow are described by the nonlinear unsteady boundary-region equations in the cylindrical coordinate system, derived and solved herein for the first time. Matched asymptotic expansions are employed to construct appropriate initial conditions and the initial–boundary value problem is solved numerically by a marching procedure in the streamwise direction. Numerical results show the stabilising effect of nonlinearity on the intense algebraic growth of the disturbances and an increase of the wall-shear stress due to the nonlinear interactions. A parametric study is carried out to evince the effect of the Reynolds number, the streamwise and azimuthal wavelengths, and the radial length scale of the inlet disturbance on the nonlinear flow evolution. Elongated pipe-entrance nonlinear structures (EPENS) occupying the whole pipe cross-section are discovered. EPENS with $h$-fold rotational symmetry comprise $h$ high-speed streaks positioned near the wall, and $h$ low-speed streaks centred around the pipe core. These distinct structures display a striking resemblance to nonlinear travelling waves found numerically and observed experimentally in fully developed pipe flow. Good agreement of our mean-flow and root mean square data with experimental measurements is obtained.

We present a new derivation of the kinetic equation for weak, non-hydrostatic internal gravity wave turbulence. The equation is equivalent to the one obtained by Caillol & Zeitlin (Dyn. Atmos. Oceans, vol. 32, issue 2, 2000, pp. 81–112), but it takes a canonical form. We show that it conserves the energy without involving the resonance condition in frequency, and look for the isotropic part of the steady, scale-invariant solutions. We provide a parametrization of the resonant manifold of non-hydrostatic internal gravity wave triadic interactions. This allows us to simplify the collision integral, and to evaluate the transfer coefficients of all triadic interactions. In the hydrostatic limit, our equation is equivalent to the Hamiltonian description of Lvov & Tabak (Phys. Rev. Lett., vol. 87, issue 16, 2001, 168501).