Turbulent open channel flows developing above submerged canopies made of slender cylinders mounted perpendicular to the channel bed are known to be largely governed by the solidity parameter $\lambda =dh/\Delta S^2$ ($d$ and $h$ being the diameter and height of the filament, and $\Delta S$ the average spacing between filaments). When the filaments are sufficiently slender, the ratio between the height of the stems and the spacing sets the hydrodynamic regime developing inside and outside the canopy. This ratio also establishes the conditions leading to the transition from a dense to a sparse canopy flow regime (Nepf, Annu. Rev. Fluid Mech., vol. 44, 2012, pp. 123–142). In a previous, companion numerical investigation, Monti et al. (J. Fluid Mech., vol. 891, 2020, A9) used large eddy simulation (LES) to study the influence of the canopy height on the onset of the different regimes without modifying the average spacing $\Delta S$ between the stems. In that LES study, we were looking at the complementary situation in which the height of the stems is constant while the filaments’ number density of the canopy is changed. It was found that for low values of $\lambda$ (i.e. sparse or moderately dense canopies: $\lambda \lessapprox 0.26$), the flows sharing the value of the solidity obtained by either varying $h$ or $\Delta S$ are very similar. Differently, for higher values of $\lambda$ (i.e. in denser canopies), the effects of $h$ and $\Delta S$ start to diverge although sharing the same nominal value of $\lambda$. In this paper, we analyse the different physical mechanisms that come into play for dense configurations obtained by varying either $\Delta S$ or $h$. In particular, we focus on the most relevant length scales and carry out a detailed analysis of the flows using a triple decomposition approach. We show that the inner region of dense canopy flows, characterised by tall stems, is dominated by wall-normal sweeps delivering high momentum in the wall vicinity. Here, the impenetrability condition of the bed redistributes the available momentum in the wall-parallel directions re-energising an otherwise stagnating flow. Differently, in densely packed canopies, the penetration of the outer jet and the momentum transfer from the external flow are limited by the decreasing value of the wall-parallel permeabilities leading to different behaviours, including a reduction of the total drag offered by the canopy.

Supersonic flow over a hollow cylinder/flare with a free-stream Mach number of 2.25 is numerically investigated in this study. Axisymmetric computational fluid dynamics simulations and global stability analysis (GSA) are performed for a wide range of cylinder radii and flare deflection angles. The onset of incipient and secondary separation is delayed as the cylinder radius is decreased due to the axisymmetric effects. The GSA reveals that a decrease in cylinder radius also postpones the emergence of global instability. The GSA results agree well with the results of direct numerical simulations for a supercritical case in the linear stage. The saturated flow exhibits pairs of unsteady streamwise streaks downstream of reattachment. The criterion of the global stability boundary established for supersonic flow over a compression corner (Hao et al., J. Fluid Mech, vol. 919, 2021, A4) is extended to its axisymmetric counterpart.

This work investigates the effect of surface roughness on cylinder flows in the postcritical regime and reexamines whether the roughness Reynolds number ($Re_{k_s}$) primarily governs the aerodynamic behaviour. It has been motivated by limitations of many previous investigations, containing occasionally contradictory findings. In particular, many past studies were conducted with relatively high blockage ratios and low cylinder aspect ratios. Both of these factors appear to have non-negligible effects on flow behaviour, and particularly fluctuating quantities such as the standard deviation of the lift coefficient. This study employs a 5 % blockage ratio and a span-to-diameter ratio of 10. Cylinders of different relative surface roughness ratios ($k_s/D$), ranging from $1.1\times 10^{-3}$ to $3\times 10^{-3}$, were investigated at Reynolds numbers up to $6.8 \times 10^5$ and $Re_{k_s}$ up to 2200. It is found that the base pressure coefficient, drag coefficient, Strouhal number, spanwise correlation length of lift and the standard deviation of the lift coefficient are well described by $Re_{k_s}$ in postcritical flows. However, roughness does have an effect on the minimum surface pressure coefficient (near separation) that does not collapse with $Re_{k_s}$. The universal Strouhal number proposed by Bearman (Annu. Rev. Fluid Mech., vol. 16, 1984, pp. 195–222) appears to be nearly constant over the range of $Re_{k_s}$ studied, spanning the subcritical through postcritical regimes. Frequencies in the separating shear layers are found to be an order of magnitude lower than the power law predictions for separating shear layers of smooth cylinders.

A data-driven turbulence model for coarse-grained numerical simulations of two-dimensional Rayleigh–Bénard convection is proposed. The model starts from high-fidelity data and is based on adjusting the Fourier coefficients of the numerical solution, with the aim of accurately reproducing the kinetic energy spectra as seen in the high-fidelity reference findings. No assumptions about the underlying partial differential equation or numerical discretization are used in the formulation of the model. We also develop a constraint on the heat flux to guarantee accurate Nusselt number estimates on coarse computational grids and high Rayleigh numbers. Model performance is assessed in coarse numerical simulations at $Ra=10^{10}$. We focus on key features including kinetic energy spectra, wall-normal flow statistics and global flow statistics. The method of data-driven modelling of flow dynamics is found to reproduce the reference kinetic energy spectra well across all scales and yields good results for flow statistics and average heat transfer, leading to computationally cheap surrogate models. Large-scale forcing extracted from the high-fidelity simulation leads to accurate Nusselt number predictions across two decades of Rayleigh numbers, centred around the targeted reference at $Ra=10^{10}$.

The Richtmyer–Meshkov instability (Richtmyer, Commun. Pure Appl. Maths, vol. 13, issue 2, 1960, pp. 297–319; Meshkov, Fluid Dyn., vol. 4, issue 5, 1972, pp. 101–104) of a twice-shocked gas interface is studied using both high spatial resolution single-shot (SS) and lower spatial resolution, time-resolved, high-speed (HS) simultaneous planar laser-induced fluorescence and particle image velocimetry in the Wisconsin Shock Tube Laboratory's vertical shock tube. The initial condition (IC) is a shear layer with broadband diffuse perturbations at the interface between a helium–acetone mixture and argon. This IC is accelerated by a shock of nominal strength Mach number $M = 1.75$, and then accelerated again by the transmitted shock that reflects off the end wall of the tube. An ensemble of experiments is analysed after reshock while the interface mixing width grows linearly with time. The kinetic and scalar energy spectra and the terms of their evolution equation are calculated and compared between SS and HS experiments. The inertial range scaling of the scalar power spectrum is found to follow Gibson's relation (Gibson, Phys. Fluids, vol. 11, issue 11, 1968, pp. 2316–2327) as a function of Schmidt number when the effective turbulent Schmidt number is used in place of the material Schmidt number that controls equilibrium scaling. Further, the spatially integrated scalar flux follows similar behaviour observed for the kinetic energy in large eddy simulation studies by Zeng et al. (Phys. Fluids, vol. 30, issue 6, 2018, 064106) while the spatially varying scalar flux exhibits back scatter along the centre of the mixing layer and forward energy transfer in the spike and bubble regions.

Despite decades of research, a universal method for prediction of roughness-induced skin friction in a turbulent flow over an arbitrary rough surface is still elusive. The purpose of the present work is to examine two possibilities; first, predicting equivalent sand-grain roughness size $k_s$ based on the roughness height probability density function and power spectrum (PS) leveraging machine learning as a regression tool; and second, extracting information about relevance of different roughness scales to skin-friction drag by interpreting the output of the trained data-driven model. The model is an ensemble neural network (ENN) consisting of 50 deep neural networks. The data for the training of the model are obtained from direct numerical simulations (DNS) of turbulent flow in plane channels over 85 irregular multi-scale roughness samples at friction Reynolds number $Re_\tau =800$. The 85 roughness samples are selected from a repository of 4200 samples, covering a wide parameter space, through an active learning (AL) framework. The selection is made in several iterations, based on the informativeness of samples in the repository, quantified by the variance of ENN predictions. This AL framework aims to maximize the generalizability of the predictions with a certain amount of data. This is examined using three different testing data sets with different types of roughness, including 21 surfaces from the literature. The model yields overall mean error 5 %–10 % on different testing data sets. Subsequently, a data interpretation technique, known as layer-wise relevance propagation, is applied to measure the contributions of different roughness wavelengths to the predicted $k_s$. High-pass filtering is then applied to the roughness PS to exclude the wavenumbers identified as drag-irrelevant. The filtered rough surfaces are investigated using DNS, and it is demonstrated that despite significant impact of filtering on the roughness topographical appearance and statistics, the skin-friction coefficient of the original roughness is preserved successfully.

We demonstrate that gravity acting alone at large length scales (compared to the capillary length) can produce a jet from a sufficiently steep, axisymmetric surface deformation imposed on a quiescent, deep pool of liquid. Mechanistically, the jet owes it origin to the focusing of a concentric, surface wave towards the axis of symmetry, quite analogous to such focusing of capillary waves and resultant jet formation observed during bubble collapse at small scales. A weakly nonlinear theory based on the method of multiple scales in the potential flow limit is presented for a modal (single-mode) initial condition representing the solution to the primary Cauchy–Poisson problem. A pair of novel, coupled, amplitude equations are derived governing the modulation of the primary mode. For moderate values of the perturbation parameter $\epsilon$ (a measure of the initial perturbation steepness), our second-order theory captures the overshoot (incipient jet) at the axis of symmetry quite well, demonstrating good agreement with numerical simulation of the incompressible, Euler equation with gravity (Popinet 2014, Basilisk. http://basilisk.fr) and no surface tension. We demonstrate that the underlying wave focusing mechanism may be understood in terms of radially inward motion of nodal points of a linearised, axisymmetric, standing wave. This explanation rationalises the ubiquitous observation of such jets accompanying cavity collapse phenomena, spanning length scales from microns to several metres. Expectedly, our theory becomes inaccurate as $\epsilon$ approaches unity. In this strongly nonlinear regime, slender jets form with surface accelerations exceeding gravity by more than an order of magnitude. In this inertial regime, we compare the jets in our simulations with the inertial, self-similar, analytical solution by Longuet-Higgins (J. Fluid Mech., 1983, vol. 127, pp. 103–121) and find qualitative agreement with the same. This analysis demonstrates, from first principles, an example of a jet created purely under gravity from a smooth initial perturbation and provides support to the analytical model of Longuet-Higgins (J. Fluid Mech., 1983, vol. 127, pp. 103–121).

The present study focuses on two-dimensional direct numerical simulations of shallow-water breaking waves, specifically those generated by a wave plate at constant water depths. The primary objective is to quantitatively analyse the dynamics, kinematics and energy dissipation associated with wave breaking. The numerical results exhibit good agreement with experimental data in terms of free-surface profiles during wave breaking. A parametric study was conducted to examine the influence of various wave properties and initial conditions on breaking characteristics. According to research on the Bond number ($Bo$, the ratio of gravitational to surface tension forces), an increased surface tension leads to the formation of more prominent parasitic capillaries at the forwards face of the wave profile and a thicker plunging jet, which causes a delayed breaking time and is tightly correlated with the main cavity size. A close relationship between wave statistics and the initial conditions of the wave plate is discovered, allowing for the classification of breaker types based on the ratio of wave height to water depth, $H/d$. Moreover, an analysis based on inertial scaling arguments reveals that the energy dissipation rate due to breaking can be linked to the local geometry of the breaking crest $H_b/d$, and exhibits a threshold behaviour, where the energy dissipation approaches zero at a critical value of $H_b/d$. An empirical scaling of the breaking parameter is proposed as $b = a(H_b/d - \chi _0)^n$, where $\chi _0 = 0.65$ represents the breaking threshold and $n = 1.5$ is a power law determined through the best fit to the numerical results.

The efficiency of water electrolysis is significantly impacted by the generation of micro- and nanobubbles on the electrodes. Here molecular dynamics simulations are used to investigate the dynamics of single electrolytic nanobubbles on nanoelectrodes. The simulations reveal that, depending on the value of current, nucleated nanobubbles either grow to an equilibrium state or grow unlimitedly and then detach. To account for these findings, the stability theory for surface nanobubbles is generalized by incorporating the electrolytic gas influx at the nanobubble's contact line and adopting a real gas law, leading to accurate predictions for the numerically observed transient growth and stationary states of the nanobubbles. With this theory, the minimum current for bubble detachment can also be derived analytically. In the detachment regime, the radius of the nanobubble first increases with time (t) as $R\propto t^{1/2}$ and then as $R\propto t^{1/3}$, up to bubble detachment.

The dynamics of turbulent flows is chaotic and difficult to predict. This makes the design of accurate reduced-order models challenging. The overarching objective of this paper is to propose a nonlinear decomposition of the turbulent state to predict the flow based on a reduced-order representation of the dynamics. We divide the turbulent flow into a spatial problem and a temporal problem. First, we compute the latent space, which is the manifold onto which the turbulent dynamics live. The latent space is found by a series of nonlinear filtering operations, which are performed by a convolutional autoencoder (CAE). The CAE provides the decomposition in space. Second, we predict the time evolution of the turbulent state in the latent space, which is performed by an echo state network (ESN). The ESN provides the evolution in time. Third, by combining the CAE and the ESN, we obtain an autonomous dynamical system: the CAE-ESN. This is the reduced-order model of the turbulent flow. We test the CAE-ESN on the two-dimensional Kolmogorov flow and the three-dimensional minimal flow unit. We show that the CAE-ESN: (i) finds a latent-space representation of the turbulent flow that has ${\lesssim }1\,\%$ of the degrees of freedom than the physical space; (ii) time-accurately and statistically predicts the flow at different Reynolds numbers; and (iii) takes ${\lesssim }1\,\%$ computational time to predict the flow with respect to solving the governing equations. This work opens possibilities for nonlinear decomposition and reduced-order modelling of turbulent flows from data.

In this paper, transient granular flows are examined both numerically and experimentally. Simulations are performed using the continuous three-dimensional (3-D) granular model introduced in Daviet & Bertails-Descoubes (ACM Trans. Graph., vol. 35, no. 4, 2016b, p. 102), which represents the granular medium as an inelastic and dilatable continuum subject to the Drucker–Prager yield criterion in the dense regime. One notable feature of this numerical model is to resolve such a non-smooth rheology without any regularisation. We show that this non-smooth model, which relies on a constant friction coefficient, is able to reproduce with high fidelity various experimental granular collapses over inclined erodible beds, provided the friction coefficient is set to the avalanche angle – and not to the stop angle, as generally done. In order to better characterise the range of validity of the fully plastic rheology in the context of transient frictional flows, we further revisit scaling laws relating the shape of the final collapse deposit to the initial column aspect ratio, and accurately recover established power-law dependences up to aspect ratios of the order of 10. The influence of sidewall friction is then examined through experimental and simulated collapses with varying channel widths. The analysis offers a comprehensive framework for estimating the effective flow thickness in relation to the channel width, thereby challenging previously held assumptions regarding its estimation in the literature. Finally, we discuss the possibility to extend the constant coefficient model with a hysteretic model in order to refine the predictions of the early-stage dynamics of the collapse. This illustrates the potential effects of such phenomenology on transient flows, paving the way to more elaborate analysis.

We study a fifty-year-old problem of fast acoustic streaming, that is, the generation of moderate or large hydrodynamic Reynolds number ($\textit {Re}$) acoustic streaming (or steady flow) by the convection of momentum in an acoustic wave (or another periodic flow), while the latter is simultaneously altered by the former. The intrinsic disparity of length and time scales makes a brute-force solution of the full Navier–Stokes and continuity equations a formidable problem. Circumventing this difficulty, we split the problem into a time-averaged system of equations for the steady flow component and a dynamic system of equations for its quasi-periodic flow counterpart. The latter system of equations is obtained by subtracting the time-averaged Navier–Stokes equation from its original dynamic form, and is rendered a nonlinear wave equation using the continuity equation and an adiabatic connection between density and pressure. The resulting equations are compatible with the theory by Eckart for small $\textit {Re}$ flow, and capture large-$\textit {Re}$ effects. Scaling analysis and a case study show that acoustic streaming is weak and does not contribute to the acoustic wave close to the wave source, relevant to many microfluidic systems. At small $\textit {Re}$, the streaming magnitude is proportional to an inverse Strouhal number, a small quantity in experiments. Moderate and large $\textit {Re}$ render the streaming magnitude comparable to the pre-attenuating periodic flow (or particle velocity of the wave) at approximately a wave attenuation length away from the wave source or further; the wave is altered by the streaming that it generates, and the streaming dominates the flow far from the wave source.

When a saturated brine layer is cooled from above, both a convective temperature front as well as a front of sedimenting salt crystals can form. We employ direct numerical simulations to investigate the evolution and interaction of these two density fronts. Depending on the ratio of the temperature front velocity and the crystal settling velocity, which is governed by a dimensionless parameter in the form of a Rayleigh number, we find that either two separate fronts exist for all times, two initially separate fronts combine into a single front after some time or a single front exists at all times. We furthermore propose approximate scaling laws for the propagation of the thermal and crystal fronts in each regime and compare them with the simulation data, with generally good agreement.

Trip-resolved large-eddy simulations of the DARPA SUBOFF are performed to investigate the development of turbulent boundary layers (TBLs) in model-scale studies. The primary consideration of the study is the extent to which the details of tripping affect statistics in large-eddy simulations of complex geometries, which are presently limited to moderate Reynolds number TBLs. Two trip wire configurations are considered, along with a simple numerical trip (wall-normal blowing), which serves as an exemplar of artificial computational tripping methods often used in practice. When the trip wire height exceeds the laminar boundary layer thickness, shedding from the trip wire initiates transition, and the near field is characterized by an elevation of the wall-normal Reynolds stress and a modification of the turbulence anisotropy and mean momentum balance. This trip wire also induces a large jump in the boundary layer thickness, which affects the way in which the TBL responds to the pressure gradients and streamwise curvature of the hull. The trip-induced turbulence decays along the edge of the TBL as a wake component that sits on top of the underlying TBL structure, which dictates the evolution of the momentum and displacement thicknesses. In contrast, for a trip wire height shorter than the laminar boundary layer thickness, transition is initiated at the reattachment point of the trip-induced recirculation bubble, and the artificial trip reasonably replicates the resolved trip wire behaviour relatively shortly downstream of the trip location. For each case, the inner layer collapses rapidly in terms of the mean profile, Reynolds stresses and mean momentum balance, which is followed by the collapse of the Reynolds stresses in coordinates normalized by the local momentum thickness, and finally against the 99 % thickness. By this point, the lasting impact of the trip is the offset in boundary layer thickness due to the trip itself, which becomes a diminishing fraction of the total boundary layer thickness as the TBL grows. The importance of tripping the model appendages is also highlighted due to their lower Reynolds numbers and susceptibility to laminar separations.

The sinusoidal roughness effect is investigated using a direct numerical simulation (DNS) of a spatially developing turbulent boundary layer (TBL) over three-dimensional sinusoidal roughness. The validity of Townsend's outer-layer similarity hypothesis is assessed based on comparisons of mean and second-order flow statistics, with a DNS of smooth-wall TBL data set at a similar Reynolds number. The total, Reynolds and dispersive stress tensors are calculated using the double-averaging procedure. The mean and second-order statistical similarities in the outer layer between rough-wall and smooth-wall TBLs are generally observed. The transport between total, turbulent and dispersive kinetic energy is investigated utilising triple-decomposed kinetic energy transports equations. The transport behaviour of turbulent kinetic energy (TKE) is significantly affected by the local mean shear induced by the surface roughness. However, the TKE transport shows good collapse with the smooth-wall case in the outer region of the flow. On the other hand, the transport of dispersive kinetic energy, including local production, redistribution and dissipation, are confined within the roughness sublayer. The intercomponent transfer between TKE and dispersive kinetic energy is quantified from the triple-decomposed kinetic energy transport equations. The intercomponent energy transfer is associated with the local spatial gradients of the turbulent momentum fluxes generated near the roughness canopy.

We develop a theoretical model to study (dense) two-dimensional gravity current flow in a laterally extensive porous medium experiencing leakage through a discrete fissure situated along this boundary at some finite distance from the injection point. Our model, which derives from the depth-averaged mass and buoyancy equations in conjunction with Darcy's law, considers dispersive mixing between the gravity current and the surrounding ambient by allowing two different gravity current phases. Thus do we define a bulk phase consisting of fluid whose density is close to that of the source fluid and a dispersed phase consisting of fluid whose density is close to that of the ambient. We characterize the degree of dispersion by estimating, as a function of time, the buoyancy of the dispersed phase and the separation distance between the bulk nose and the dispersed nose. On this basis, it can be shown that the amount of dispersion depends on the flow conditions upstream of the fissure, the fissure permeability and the vertical and horizontal extents of the fissure. We also show that dispersion is larger when the gravity current propagates along an inclined barrier rather than along a horizontal barrier. Model predictions are fitted against numerical simulations. The simulations in question are performed using COMSOL and consider different inclination angles and fissure and upstream flow conditions. Our study is motivated by processes related to underground $\mathrm {H}_2$ storage e.g. an irrecoverable loss of $\mathrm {H}_2$ when it is injected into the cushion gas saturating an otherwise depleted natural gas reservoir.

This work studies the three-dimensional flow dynamics around a rotating circular cylinder of finite length, whose axis is positioned perpendicular to the streamwise direction. Direct numerical simulations and global stability analyses are performed within a parameter range of Reynolds number $Re=DU_\infty /\nu <500$ (based on cylinder diameter $D$, uniform incoming flow velocity $U_\infty$), length-to-diameter ratio ${\small \text{AR}}=L/D\leq 2$ and dimensionless rotation rate $\alpha =D\varOmega /2U_\infty \leq 2$ (where $\varOmega$ is rotation rate). By solving Navier–Stokes equations, we investigated the wake patterns and explored the phase diagrams of the lift and drag coefficients. For a cylinder with ${\small \text{AR}}=1$, we found that when the rotation effect is weak ($0\leq \alpha \lesssim 0.3$), the wake pattern is similar to the unsteady wake past the non-rotating finite-length cylinder, but with a new linear unstable mode competing to dominate the saturation state of the wake. The flow becomes stable for $0.3\lesssim \alpha \lesssim 0.9$ when $Re<360$. When the rotation effect is strong ($\alpha \gtrsim 0.9$), new low-frequency wake patterns with stronger oscillations emerge. Generally, the rotation effect first slightly decreases and then sharply increases the $Re$ threshold of the flow instability when $\alpha$ is relatively small, but significantly decreases the threshold at high $\alpha$ ($0.9<\alpha \leq 2$). Furthermore, the stability analyses based on the time-averaged flows and on the steady solutions demonstrate the existence of multiple unstable modes undergoing Hopf bifurcation, greatly influenced by the rotation effect. The shapes of these global eigenmodes are presented and compared, as well as their structural sensitivity, visualising the flow region important for the disturbance development with rotation. This research contributes to our understanding of the complex bluff-body wake dynamics past this critical configuration.

An experimental study of long interfacial gravity waves was conducted in a closed wave tank containing two layers of viscous immiscible fluids. The study focuses on the development in time of the mean particle drift that occurs close to the interface where the two fluids meet. From a theoretical analysis by Weber & Christensen (Eur. J. Mech.-B/Fluids, vol. 77, 2019, pp. 162–170) it is predicted that the growing drift in this region is associated with the action of the virtual wave stress. This effect has not been explored experimentally before. Interfacial waves of different amplitudes and frequencies were produced by a D-shaped paddle. Particle tracking velocimetry was used to track the time development of the Lagrangian mean drift. The finite geometry of the wave tank causes a mean return flow that is resolved by mass transport considerations. The measurements clearly demonstrate the importance of the virtual wave stress mechanism in generating wave drift currents near the interface.

To draw concrete pictures of the hierarchy of multiscale coherent vortices in turbulence behind a cylinder and to reveal their sustaining mechanism, we conduct direct numerical simulations of the turbulence at the Reynolds number, which is defined by the uniform inflow velocity and the cylinder diameter, $5000$. The turbulence consists of three kinds of hierarchies of coherent vortices in three distinct regions: namely, the downstream region, the recirculation region just behind the cylinder and the separate shear layers. By tracking the temporal evolution of multiscale vortices in each of these regions, we demonstrate that, in all the three regions, smaller coherent vortices tend to align in the direction perpendicular to larger ones. This implies that smaller vortices are stretched and amplified in the strain-rate fields around larger ones. We also show the relevance of this observation to the energy cascade. Smaller-scale vortices receive the kinetic energy in the regions where they are stretched by larger-scale vortices; and, at the same time, they tend to compress larger ones, thus reducing larger-scale kinetic energy.