We investigate a turbulent boundary layer (TBL) under uniform pressure force variations, focusing on understanding its response to local pressure force, local pressure force variation (local disequilibrating effect) and upstream history. The flow starts as a zero-pressure-gradient (ZPG) TBL, followed by a uniform increase in the ratio of pressure force to turbulent force in the outer region and concludes with a uniform decrease of the same magnitude. This last zone includes a subzone with a diminishing adverse pressure gradient (APG), followed by an increasing favourable pressure gradient (FPG). In both subzones, the impact remains the same: mean momentum gain and turbulence reduction. In the outer region, the mean flow responds to force balance changes with a considerable delay. The accumulated flow history leads to a FPG TBL at the domain's end with a momentum defect comparable to APG TBLs. Below $y^+=10$, the mean flow responds almost instantaneously to pressure force changes. In the overlap layer, velocity profiles deviate from the conventional logarithmic law of ZPG TBL. Outer-layer turbulence decays more slowly than it increases initially, the latter turbulence increase persisting even after the pressure force begins to decrease. As a result of the slow turbulence decay, the FPG TBL at the domain's end exhibits unusually high outer turbulence levels. Near the wall, turbulence responds with a delay to pressure force changes, partly due to the influence of large-scale turbulence. All these significant cumulative effects of continuous pressure force variation indicate that parameters based solely on local variables cannot fully describe the physics of non-equilibrium TBLs.

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.

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.

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.

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.

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 the present work, the tangential (swirl) velocity component is superimposed at the intake of a narrow fluidic cylindrical pipe to achieve the desired mixing of inelastic non-Newtonian fluids/solutes at the outlet. We discuss an analytical method for obtaining the swirl velocity profile, considering the nonlinear viscous effects for both shear-thinning and shear-thickening fluids, represented by the power-law model. We numerically solve the species transport equation, coupled with the analytically derived swirl velocity, using our in-house developed code for the concentration distribution in the flow field. The results show that the inlet swirl and an increase in the shear-thinning fluid property improve advection-dominated mixing. Additionally, higher Reynolds numbers significantly enhance advection's dominance, as more rotation leads to the generation of vortices, resulting in an engulfment flow (chaotic convection) based mixing. We demonstrate that considering the increase in the shear-thinning fluid property with swirl intake reduces the amount of mixing time required in the convective regime.

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 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.

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.