The flow field of a bluff body, a circular disk, that moves horizontally in a stratified environment is studied using large-eddy simulations. Five levels of stratification (body Froude numbers of ${{Fr}} = 0.5, 1, 1.5, 2$ and $5$) are simulated at Reynolds number of ${{Re}} = 5000$ and Prandtl number of $Pr =1$. A higher ${{Re}} = 50\,000$ database at ${{Fr}} = 2, 10$ and $Pr =1$ is also examined for comparison. The wavelengths and amplitudes of steady lee waves are compared with a linear-theory analysis. Excellent agreement is found over the entire range of ${{Fr}}$ if an ‘equivalent body’ that includes the separation region is employed for the linear theory. For asymptotically large distances, the velocity amplitude varies theoretically as ${{Fr}}^{-1}$ but a correction owing to the dependence of the separation zone on ${{Fr}}$ is needed. The wake waves propagate in a narrow band of angles with the vertical, and have a wavelength that increases with increasing ${{Fr}}$. The envelope of wake waves, demarcated using buoyancy variance, exhibits self-similar behaviour. The higher ${{Re}}$ results are consistent with the buoyancy effects exhibited at the lower ${{Re}}$. The wake wave energy is larger at ${{Re}} = 50\,000$. Nevertheless, independent of ${{Fr}}$ and ${{Re}}$, the ratio of the wake wave potential energy to the wake turbulent energy increases to approximately 0.6–0.7 in the non-equilibrium stage showing their energetic importance besides suggesting universality in this statistic. There is a crossover of energetic dominance of lee waves at ${{Fr}} <2$ to wake-wave dominance at ${{Fr}} \approx 5$.

In the present study, we propose a Reynolds analogy model for compressible wall turbulence. This model is demonstrated to be able to alleviate the defects of the generalized Reynolds analogy model (GRA) (Zhang et al., J. Fluid Mech., vol. 739, 2014, pp. 392–420), and maintains its success in describing the mean velocity–temperature relation. Furthermore, the present model is superior to the GRA in depicting the relationship between their fluctuating fields and also bridges the gap between the phenomenological model and the mathematical representation of the Reynolds analogy. The key points of the present model are validated by analysing the data of compressible wall-bounded turbulence with different Mach numbers, Reynolds numbers and wall thermal conditions.

High-pressure fluid transport in nanoporous media such as shale formations requires further understanding because conventional continuum approaches become inadequate due to their ultralow permeability and complexity of transport mechanisms. We propose a species-based approach for modelling two partially miscible, multicomponent fluids in nanoporous media – one that does not rely on conventional bulk fluid transport frameworks but on species movement. We develop a numerical model for species transport of partially miscible, non-ideal fluid mixtures using the chemical potential gradient as the driving force. The model considers the binary friction concept to include the friction between fluid molecules as well as between fluid molecules and pore walls, and incorporates the key multicomponent transport mechanisms – Knudsen, viscous and molecular diffusion. Under single-phase conditions, the system under consideration is quantified by introducing multicomponent Sherwood number (Sh), Péclet number (Pe) and fluid–solid friction modulus (φ). Despite the complexity of fluid transport in nanopores, the steady-state single-phase transport results reveal the contribution of diffusion in nanopores, where all parameters collapse on a set of master curves for the multicomponent Sh with a dependence on multicomponent Pe and φ. Unsteady state, two-phase transport modelling of the codiffusion process shows that light and intermediate alkanes are produced much higher than heavy alkanes when the vapour phase appears. We demonstrate that the pressure gradient is also crucial in promoting CO2 and alkane mixing during counterdiffusion processes. These results stress the need for a paradigm shift from classical bulk flow modelling to species-based transport modelling in nanoporous media.

Flow control of a low-aspect-ratio flat-plate heaving wing at an average angle of attack of $10^{\circ }$ by a steady-blowing jet is numerically studied by using a feedback immersed boundary–lattice Boltzmann method. Blowing jets at the leading edge, mid-chord and trailing edge are considered. The wing enjoys the highest lift production with the trailing-edge downstream blowing jet, which improves the average lift by 50.0 % at $Re = 1000$ and 22.9 % at $Re = 5000$ through the enhancement of the tip vortex circulation caused by the increase in the mass flux of the shear layer at the wing tips. This increase in mass flux decreases as $Re$ increases from 1000 to 5000 due to its self-limiting mechanism. A mid-chord vertical blowing jet induces a middle vortex which enhances the lift production but the enhancement is smaller than that of trailing-edge downstream blowing jet. Other jet arrangements do not significantly increase the lift coefficient, but the mid-chord upstream blowing jet experiences a significant reduction in the drag coefficient, leading to an increase of 50.6 % in the average lift-to-drag ratio. The effectiveness of the flow control is not significantly affected by the aspect ratio.

This study comprehensively investigates the response of a combusting droplet during its interaction with a high-speed transient flow imposed by a coaxially propagating blast wave. The blast wave is generated using a specially designed miniature shock generator that produces blast waves using the wire-explosion technique, facilitating a wide range of Mach numbers (1.03 < Ms < 1.8). The experiments are performed in two configurations: open field and focused blast wave. The charging voltage and the configuration determine the Mach number (Ms) and flow characteristics. The flame is found to exhibit two major response patterns: partial extinction followed by reignition and full extinction. Increasing the Mach number (Ms > 1.1) makes the droplet flame more vulnerable to extinction. Additionally, the flame exhibits stretching and shedding, followed by reignition at lower Mach numbers (Ms < 1.06). In all cases, the flame base lifts off in response to the imposed flow, and the advection of the flame base interacting with the flame tip results in flame extinction. The entire interaction occurs in two stages: (i) interaction with the blast wave and the decaying velocity profile associated with it, and (ii) interaction with the induced flow behind the blast wave as a result of the entrainment (delayed response). Alongside the flame's response, the droplet also interacts with the flow imposed by the blast wave, exhibiting different response modes including pure deformation, Rayleigh–Taylor piercing bag breakup and shear-induced stripping.

Pressure gradient over topography will significantly affect wind-farm flow. However, knowledge gaps still exist on how to superpose wind-turbine wakes in the wind-farm flow analytical model to account for this effect, leading to systematic errors in evaluating wind-farm wake effects. To this end, we derive an implicit momentum-conserving wake superposition method under pressure gradient (PG-IMCM) based on the total momentum deficit equation, which is linearised by the convection velocity introduced by Zong & Porté-Agel (J. Fluid Mech., vol. 889, 2020, A8). The PG-IMCM method consists of the linear-weighted sum of individual velocity deficits, the sum of the individual pressure correction terms and the total pressure correction term. Based on a sensitivity analysis, we demonstrate that the last two terms nearly cancel out and, thus, can be neglected, resulting in a simplified form, which has the same form as its counterpart under zero pressure gradient but with the single-wake quantities redefined based on the wake model under pressure gradient. This motivates us to further examine the performance of the combination of five empirical superposition methods and the stand-alone wake model under pressure gradient. Validation results based on large-eddy simulation show that PG-IMCM has an overall satisfactory performance in both the magnitude and shape of the velocity-deficit profiles, provided that the stand-alone turbine wake can be modelled accurately, which is virtually identical with its simplified form. Further comparison with empirical superposition methods shows that local linear and wind product superposition methods based on the updated base flow also have comparable performance, with only discernable differences with the PG-IMCM method in the near-wake region of downstream turbines. Therefore, they are two attractive methods for engineering applications.

Turbulent flows over rough surfaces can be encountered in a wide range of engineering applications. Despite the progress made after several decades of studies, the prediction of drag and roughness function from the surface geometrical parameters remains an open question. Several methods have shown encouraging results. However, they lack generality and present some scatter in the data. In this paper we propose a new parameter, the effective distribution ($ED$), which lays foundation on the effective slope with some changes to take into account the sheltering effect of large roughness elements and the drag induced by pinnacles higher than the average roughness elements. To develop this new correlation between geometrical features of the wall and the drag, we performed a set of simulations of the turbulent flow over a rough surface made of triangular elements varying their height and spatial distribution. The $ED$ correlates quite well both with the drag and the roughness function for a wide range of cases having different mean roughness height, skewness and kurtosis. To further validate the $ED$, and assessing how it can be generalized to real rough wall, an irregular wall made from the superposition of random sinusoidal function was considered. Results were consistent with the correlation here presented.

Turbulent convection in the interiors of the Sun and the Earth occurs at high Rayleigh numbers $Ra$, low Prandtl numbers $Pr$, and different levels of rotation rates. To understand the combined effects better, we study rotating turbulent convection for $Pr = 0.021$ (for which some laboratory data corresponding to liquid metals are available), and varying Rossby numbers $Ro$, using direct numerical simulations in a slender cylinder of aspect ratio 0.1; this confinement allows us to attain high enough Rayleigh numbers. We are motivated by the earlier finding in the absence of rotation that heat transport at high enough $Ra$ is similar between confined and extended domains. We make comparisons with higher aspect ratio data where possible. We study the effects of rotation on the global transport of heat and momentum as well as flow structures (a) for increasing rotation at a few fixed values of $Ra$, and (b) for increasing $Ra$ (up to $10^{10}$) at the fixed, low Ekman number $1.45 \times 10^{-6}$. We compare the results with those from unity $Pr$ simulations for the same range of $Ra$ and $Ro$, and with the non-rotating case over the same range of $Ra$ and low $Pr$. We find that the effects of rotation diminish with increasing $Ra$. These results and comparison studies suggest that for high enough $Ra$, rotation alters convective flows in a similar manner for small and large aspect ratios, so useful insights on the effects of high thermal forcing on convection can be obtained by considering slender domains.

The analysis of Scott (J. Fluid Mech., vol. 741, 2014, pp. 316–349) is implemented numerically. Decaying turbulence is confined to a channel between two infinite, parallel, rotating walls. The Rossby and Ekman numbers are supposed small, the former condition making nonlinearity small, while the latter allows the turbulence to persist for the many rotational periods needed for the small nonlinearity to be effective. The flow is expressed as a combination of inertial waveguide modes, indexed by a two-dimensional wave vector $\boldsymbol{k}$ and an integer n. The $n = 0$ modes form a two-dimensional component of the flow, whereas the remainder is the wave component, on which attention is focused in this article. Assuming statistical axisymmetry and homogeneity in directions parallel to the walls, the second-order moments of the mode amplitudes yield a spectral matrix ${A_{nm}}(k,t)$ (where $k = |\boldsymbol{k} |$), of which the diagonal elements describe the distribution of energy over different modes. Wave-turbulence analysis provides an equation governing the time evolution of ${A_{nn}}$, $n \ne 0$, the wave spectra, which forms the basis for the present work. The initial distribution of energy is Gaussian and depends on a parameter $\varXi$, the initial spectral width. The problem has two other parameters, ${\beta _w}$ and ${\beta _v}$, which correspond to two distinct viscous dissipative mechanisms: wall damping due to boundary layers and volumetric damping by viscous effects throughout the flow. Results obtained by numerical solution include the time evolution of the total wave energy, E, and the detailed description of its distribution over k and n provided by ${A_{nn}}(k)$.

A permeable disk serves as a simplified model for the conversion of wind energy by a horizontal axis wind turbine. In this study, we investigate how inflow turbulence intensity (TI), $I_\infty$, and inflow turbulence integral length scale, $L_\infty$, influence the flow recovery in the wake, the capability of a permeable disk in extracting turbulence kinetic energy (TKE) of the incoming flow, and the statistics of wake-added turbulence using large-eddy simulation. The simulated inflows include various TIs (i.e. $I_\infty =2.5\,\%$–$25\,\%$) and integral length scales (i.e. $L_\infty / D =0.5$–$2.0$) for two thrust coefficients. Simulation results show that both inflow TI and integral length scale influence flow recovery via enhanced ejections and sweeps across the wake boundary, with the former strongly affecting the position where the wake starts to recover and the latter mainly on the recovery rate. Moreover, it is shown that increasing $I_\infty$ and $L_\infty$ increases the TKE extraction by the disk, occurring mainly at scales ($s$) greater than $0.5D$ and frequencies depending on the inflow integral length scale. As for the wake-added TKE, the inflow TI mainly affects its intensity, while the inflow integral length scale affects both its intensity and the sensitive frequencies, with the spectral distributions in scale space ($s$) being similar and the peak located around $s/D=1.0$ for the considered inflows.

The effects of reshock conditions, including the interface evolution state before reshock and the second shock intensity, on interface instability induced by two successive shocks propagating in the same direction are investigated via shock-tube experiments. It is observed that the reshock promotes the interface instability, and the post-reshock perturbation evolution relates to both the pre-reshock interface evolution state and second shock intensity. For the linear evolution of the twice-shocked interface, existing models perform poorly when either the pre-reshock interface shape effect or the secondary compression effect is pronounced, as current reduction factors fail to accurately describe these effects. Besides, the reshock-induced linear amplitude growth rate shows a non-monotonic dependence on the scaled pre-reshock amplitude, primarily due to the shape effect of the pre-reshock interface. For the post-reshock nonlinear evolution, the model proposed by Zhang & Guo (J. Fluid Mech., vol. 786, 2016, pp. 47–61) offers reasonable predictions when the second shock is weak. However, when the second shock is moderately strong, the model overestimates the bubble growth and underestimates the spike evolution under the influence of the significant secondary compression effect. Furthermore, empirical linear and nonlinear models capable of describing the dependence of the post-reshock evolution on reshock conditions are proposed based on the present experimental results and existing models.

Oscillatory boundary layers over flat and rippled seabeds are well described in the literature. However, the presence of protruding vegetation stems has received no theoretical or experimental attention. The present work establishes an analytical constant viscosity model akin to the Stokes oscillatory boundary layer solution and a nonlinear varying-viscosity numerical model with a turbulence closure. The two models are used to describe the importance of vegetation and free stream velocity characteristics on spatially averaged oscillatory boundary layers: their friction factors, thickness and phase leads over the free-stream velocity. The models are periodic in time and resolve boundary and shear layers over the vertical, contrary to past efforts applying two-layer models. The models are extended to investigate the importance of finite wavelengths with steady streaming stresses and their associated mean velocity profile. Steady streaming is quantified both for the near-bottom streaming within the canopy and for the streaming in the shear layer above the canopy. Finally, akin to theoretical and experimental works on mean flows over unvegetated and flat seabeds due to oscillatory and nonlinear free-stream velocities, the numerical model investigates varying degree of nonlinearity for velocity- and acceleration-skewed velocity signals, and it is identified that the presence of vegetation stems gives rise to an additional contribution to the horizontal momentum balance which is not present for unvegetated conditions. Finally, it is discussed how the presence of a free surface, contrary to purely oscillatory conditions, alters the horizontal momentum balance within and above the canopy.

The reflection of a shock pulse at a liquid–gas interface occurs in many applications, from lithotripsy to underwater explosions and additive manufacturing. In linear theory, reflection and transmission at an interface depend only on the impedance difference, but this does not hold for a nonlinear pulse. This work develops an analytical framework for computing the reflection and transmission coefficients for an impulsive shock wave at a liquid–gas interface. The problem is treated analytically by considering idealised pulses and solving a series of consecutive Riemann problems. These correspond to the initial interaction with the interface and important subsequent wave interactions that enable a complete description of the process to be obtained. Comparisons with numerical and existing analytical approaches are made for the case of a water–air interface. In the acoustic limit, the method produces results identical to those of linear acoustic theory. As the pulse strength increases, the proposed method agrees well with numerical simulation results, whereas existing analytical methods that consider only the interface fail. We detail how a reflecting pulse can put water into tension without any incident negative pressure. It is further shown that the magnitude of the reflection coefficient decreases with increasing incident shock pressure, and the reflected pulse widens. Reflections of pulses with positive and negative pressures temporarily create negative pressure regions with greater magnitude than the incident pulse. Finally, we consider non-idealised waves. Comparisons with simulations show that the reflection characteristics can be explained qualitatively using the analytical method, and the reflection coefficients are predicted accurately.

Unlike in solids, heat transfer in fluids can be greatly enhanced due to the presence of convection. Under gravity, an unevenly distributed temperature field results in differences in buoyancy, driving fluid motion that is seen in Rayleigh–Bénard convection (RBC). In RBC, the overall heat flux is found to have a power-law dependence on the imposed temperature difference, with enhanced heat transfer much beyond thermal conduction. In a bounded domain of fluid such as a cube, how RBC responds to thermal perturbations from the vertical sidewall is not clear. Will sidewall heating or cooling modify flow circulation and heat transfer? We address these questions experimentally by adding heat to one side of the RBC. Through careful flow, temperature and heat flux measurements, the effects of adding side heating to RBC are examined and analysed, where a further enhancement of flow circulation and heat transfer is observed. Our results also point to a direct and simple control of the classical RBC system, allowing further manipulation and control of thermal convection through sidewall conditions.

To characterize fluid flow in the slip regime, the use of Navier–Stokes–Fourier (NSF) equations with slip boundary conditions is prevalent. This trend underscores the necessity of developing reliable and accurate slip boundary conditions. According to kinetic theory, slip behaviours are intrinsically linked to the gas scattering processes at the surface. The widely used Maxwell scattering model, which employs a single accommodation coefficient to describe gas scattering processes, reveals its limitations when the difference between accommodation coefficients in the tangential and normal directions becomes significant. In this work, we provide a derivation of velocity slip and temperature jump boundary conditions based on the Cercignani–Lampis–Lord scattering model, which applies two independent accommodation coefficients to describe the gas scattering process. A Knudsen layer correction term is introduced to account for the impact of the surface on the velocity distribution function, which is associated with the scattering model. The governing equation of the correction term is established based on the linearized Boltzmann equation. Additionally, two moments are derived to capture the collision effect in the Knudsen layer: a conserving moment of collision invariants, and an approximate higher-order conserving moment. These moments are then employed to determine the coefficients in the correction term. We demonstrate that the derived slip coefficients align closely with numerical results obtained by solving the Boltzmann equation in the Knudsen layer. Besides, we apply the derived slip boundary conditions within the framework of the NSF equations, yielding numerical results that exhibit excellent consistency with those obtained through molecular-level simulations.

Several transition scenarios are present in a hypersonic compression-ramp flow. In our previous work (Cao et al., J. Fluid Mech., vol. 941, 2022, p. A8), a complete transition process induced by the global instability of a compression-ramp flow was revealed. In a globally stable flow, however, the transition to turbulence can be promoted by convective instabilities, which is the focus of this work. The same flow conditions as in our previous work (Mach number 7.7, Reynolds number $8.6\times 10^5$ based on the flat-plate length) are considered here. Owing to a smaller ramp angle, a weakly separated flow forms on the compression ramp, which supports no global instability. Resolvent analysis identifies low-frequency streamwise streaks as the optimal response of base flow to upstream forcing. Local stability analysis reveals Mack's second mode in the boundary layer downstream of reattachment. By introducing random disturbances upstream of separation in direct numerical simulations, we observe breakdown to turbulence downstream of reattachment. Two transition scenarios are revealed, and they are highly dependent on the amplitude of upstream disturbances. For a large amplitude, strong streamwise streaks develop near the reattachment region, which break down to turbulence quickly. However, when the disturbance amplitude is reduced, the second-mode instability dominates the transition to turbulence.