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.

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.

In this paper, we propose a numerical model to simulate gas–liquid–solid interaction problems, coupling the lattice Boltzmann method and discrete element method (LBM–DEM). A cascaded LBM is used to simulate the liquid–gas flow field using a pseudopotential interaction model for describing the liquid–gas multiphase behaviour. A classical DEM resorting to fictitious overlaps between the particles is used to simulate the multiple-solid-particle system. A multiphase fluid–solid two-way coupling algorithm between LBM and DEM is constructed. The model is validated by four benchmarks: (i) single disc sedimentation, (ii) single floating particle on a liquid–gas interface, (iii) sinking of a horizontal cylinder and (iv) self-assembly of three particles on a liquid–gas interface. Our simulations agree well with the numerical results reported in the literature. Our proposed model is further applied to simulate droplet impact on deformable granular porous media at pore scale. The dynamic droplet spreading process, the deformation of the porous media (composed of up to 1277 solid particles) as well as the invasion of the liquid into the pores are well captured, within a wide range of impact Weber number. The droplet spreading dynamics on particles is analysed based on the energy budget, which reveals mechanisms at play, showing the evolution of particle energy, surface energy and viscous dissipation energy. A scaling relation based on the impact Weber number is proposed to describe the maximum spreading ratio.

A typical feature of thermal convection is the formation of large-scale flow (LSF) structures of the order of system size. How this structure affects global heat transport is an important issue in the study of thermal convection. We present an experimental study of the coupling between the flow structure and heat transport in liquid metal convection with different degrees of spatial confinement, characterized by the aspect ratio $\varGamma$ of the convection cell. Combining measurements in two convection cells with $\varGamma =1.0$ and 0.5, the study shows that a large-scale circulation (LSC) transports ${\sim }35\,\%$ more heat than a twisted LSC. It is further found that when the LSF is in the form of the LSC state, the system is in a fully developed turbulence state with a $Nu\sim Ra^{0.29}$ scaling for the heat transport. However, the twisted LSC state with a heat transport scaling of $Nu\sim Ra^{0.37}$ appears when the system is not in the fully developed turbulence state. Bistability is observed when the system evolves from the twisted-LSC-dominated to the LSC-dominated state.

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).

In this study, the effects of antagonistic muscle actuation on the propulsion of a bilaminar-structure fish fin ray were investigated using a two-dimensional computational flow–structure interaction (FSI) model. The structure and material properties of the model were based on the realistic biological data of the sunfish fin. The effect of muscle actuation was modelled using root displacement offset between the two hemitrichs. Parametric FSI simulations were conducted by assuming a sinusoidal function of the offset over a cycle and varying the amplitude and phase difference between the actuations and pitching/plunging motions. The results show that the phase of muscle actuation is a critical factor affecting its effects. Three performance regions can be identified with different phase ranges, including a thrust-favour region, an efficiency-favour region and a thrust-efficiency-unfavour region. In each region, the relationships among the root actuations, fin-ray kinematics, vortex dynamics and resulting performance are studied and discussed. Furthermore, a strong positive correlation between the trailing–leading amplitude ratio and thrust coefficient as well as a negative relationship between the efficiency and angle of attack at the centre of mass of the fin ray are observed.

Insights gained from modal analysis are invoked for predictive large-eddy simulation (LES) wall modelling. Specifically, we augment the law of the wall (LoW) by an additional mode based on a one-dimensional proper orthogonal decomposition (POD) applied to a two-dimensional turbulent channel. The constructed wall model contains two modes, i.e. the LoW-based mode and the POD-based mode, and the model matches with the LES at two, instead of one, off-wall locations. To show that the proposed model captures non-equilibrium effects, we perform a priori and a posteriori tests in the context of both equilibrium and non-equilibrium flows. The a priori tests show that the proposed wall model captures extreme wall-shear stress events better than the equilibrium wall model. The model also captures non-equilibrium effects due to adverse pressure gradients. The a posteriori tests show that the wall model captures the rapid decrease and the initial decrease of the streamwise wall-shear stress in channels subjected to suddenly imposed adverse and transverse pressure gradients, respectively, both of which are missed by currently available wall models. These results show promise in applying modal analysis for turbulence wall modelling. In particular, the results show that employing multiple modes helps in the modelling of non-equilibrium flows.

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.

This study focuses on numerically investigating the impact of fluid viscoelasticity on the flow dynamics around a transversely forced oscillating cylinder operating in the laminar vortex shedding regime at a fixed Reynolds number of $Re = 100$. Specifically, we explore how fluid viscoelasticity affects the boundary between the lock-in and no lock-in regions and the corresponding wake characteristics compared with a simple Newtonian fluid. Our findings reveal that fluid viscoelasticity enables the synchronization of the vortex street with the cylinder motion at lower oscillation frequencies than those required for a Newtonian fluid. Consequently, the lock-in region boundary for a viscoelastic fluid differs from that of a Newtonian fluid and expands in the non-dimensional cylinder oscillation amplitude and frequency parameter space. In the primary synchronization region, the wake of a Newtonian fluid exhibits ‘2S’ (two single vortices) and ‘P+S’ (a pair of vortices and a single vortex) shedding modes. In contrast, a ‘2P’ (two pairs of vortices) vortex mode is observed for a viscoelastic fluid within the same region. To gain a deeper understanding of the differences in the coherent flow structures and their associated frequencies between the two fluids, we employ the data-driven reduced-order modelling technique, known as the dynamic mode decomposition (DMD) technique. Utilizing this technique, we successfully extract and visualize the two competing fundamental frequencies (cylinder oscillation and natural vortex shedding frequencies) and their associated flow structures in the case of the no lock-in state, whereas only the dominant cylinder oscillation frequency and associated flow structure in the case of the lock-in state. Furthermore, we propose that the presence of excess strain resulting from the stretching of polymer molecules in viscoelastic fluids leads to a distinct difference in the wake structure compared with Newtonian fluids. This observation aligns with the findings obtained from the $Q$-criterion and vorticity transport analysis of the wake.

Using a combination of proper variable transformation and integral methods, we rigorously derive an analytical formulation for the mean wall-normal velocity in turbulent boundary layers (TBLs) subjected to arbitrary pressure gradients. The accuracy and robustness of this novel formulation are validated extensively through comparisons with two independent sets of numerical simulation data, demonstrating excellent agreement in both near-equilibrium and non-equilibrium TBLs. In addition, the robustness of the analytical formulations to various choices of boundary-layer edge definition is further confirmed in non-equilibrium TBLs. Our formulation includes a streamwise derivative term, which has minimal significance in near-equilibrium TBLs but plays a crucial role in determining the mean wall-normal velocity in non-equilibrium TBLs. Moreover, we investigate the physical significance of the pre-factors associated with the mean wall-normal velocity components, and unveil a close connection between a previously defined pressure gradient parameter and the ratio of these pre-factors in the analytical equation governing the mean wall-normal velocity. The insights gained from the examination of the pre-factors and their connection to the pressure gradient parameter offer valuable knowledge for interpreting and predicting the behaviour of turbulent boundary layers in various practical applications.

We investigate the effects of fluid elasticity on the flow forces and the wake structure when a rigid cylinder is placed in a viscoelastic flow and is forced to oscillate sinusoidally in the transverse direction. We consider a two-dimensional, uniform, incompressible flow of viscoelastic fluid at $Re=100$, and use the FENE-P model to represent the viscoelastic fluid. We study how the flow forces and the wake patterns change as the amplitude of oscillations, $A^*$, the frequency of oscillations (inversely proportional to a reduced velocity, $U^*$), the Weissenberg number, $Wi$, the square of maximum polymer extensibility, $L^2$, and the viscosity ratio, $\beta$, change individually. We calculate the lift coefficient in phase with cylinder velocity to determine the range of different system parameters where self-excited oscillations might occur if the cylinder is allowed to oscillate freely. We also study the effect of fluid elasticity on the added mass coefficient as these parameters change. The maximum elastic stress of the fluid occurs in between the vortices that are observed in the wake. We observe a new mode of shedding in the wake of the cylinder: in addition to the primary vortices that are also observed in the Newtonian flows, secondary vortices that are caused entirely by the viscoelasticity of the fluid are observed in between the primary vortices. We also show that, for a constant $Wi$, the strength of the polymeric stresses increases with increasing reduced velocity or with decreasing amplitude of oscillations.

Richtmyer–Meshkov instability (RMI) at a light–heavy single-mode interface over a wide range of post-shock Atwood numbers $A_1$ is studied systematically through elaborate experiments. The interface generation and $A_1$ variation are achieved by the soap-film technology and gas-layer scheme, respectively. Qualitatively, the nonlinear interface evolution features, including spike, bubble and roll-up structures, are more significant in RMI with higher $A_1$. Quantitatively, both the impulsive model and an analytical linear model perform well in predicting the linear growth rate under a wide range of $A_1$ conditions. For the weakly nonlinear stage, the significant spike acceleration occurring when $A_1$ is high, which is observed experimentally for the first time, results in the evolution law of RMI with high $A_1$ being different from the counterpart with low or intermediate $A_1$. None of the considered nonlinear models is found to be applicable for RMI under all $A_1$ conditions, and the predictive capabilities of these models are analysed and summarized. Based on the present experimental results, an empirical nonlinear model is proposed for RMI over a wide range of $A_1$. Further, modal analysis shows that in RMI with high (low or intermediate) $A_1$, high-order harmonics evolve rapidly (slowly) and cannot (can) be ignored. Accordingly, for RMI with high (low or intermediate) $A_1$, the modal model proposed by Zhang & Sohn (Phys. Fluids, vol. 9, 1997, pp. 1106–1124) is less (more) accurate than the one proposed by Vandenboomgaerde et al. (Phys. Fluids, vol. 14, 2002, pp. 1111–1122), since the former ignores perturbation solutions higher than fourth order (the latter retains only terms with the highest power in time).

Direct numerical simulations are performed to explore the effects of the rotating direction of the vertically asymmetric rough wall on the transport properties of Taylor–Couette (TC) flow, up to a Taylor number of ${Ta} = 2.39\times 10^{7}$. It is shown that, compared with the smooth wall, the rough wall with vertical asymmetric strips can enhance the dimensionless torque ${Nu}_{\omega }$. More importantly, at high Ta, clockwise rotation of the inner rough wall (where the fluid is sheared by the steeper slope side of the strips) results in a significantly greater torque enhancement compared to counter-clockwise rotation (where the fluid is sheared by the smaller slope side of the strips), due to the larger convective contribution to the angular velocity flux. However, the rotating direction has a negligible effect on the torque at low Ta. The larger torque enhancement caused by the clockwise rotation of the vertically asymmetric rough wall at high Ta is then explained by the stronger coupling between the rough wall and the bulk, attributed to the larger biased azimuthal velocity towards the rough wall at the mid-gap of the TC system, the increased turbulence intensity manifested by larger Reynolds stress and a thinner boundary layer, and the more significant contribution of the pressure force on the surface of the rough wall to the torque.

Particle motion near non-plane surfaces can exhibit intricate hydrodynamics, making it an attractive tool for manipulating particles in microfluidic devices. To understand the underlying physics, this work investigates the Stokesian dynamics of a sphere near a sinusoidal surface, using a combination of perturbation analysis and boundary element simulation. The Lorentz reciprocal theorem is employed to solve the particle mobility near a small-amplitude surface. Compared with a plane wall, the curved topography induces additional translation and rotation velocity components, with the direction depending on the location of the sphere and the wavelength of the surface. At a fixed distance from the surface, the longitudinal and vertical mobilities of the sphere are strongly affected by the wavelength and amplitude of the surface, whereas its transverse mobility is only mildly influenced. When a sphere settles perpendicular to a sinusoidal surface, the far-field hydrodynamic effect drives the particle towards the local hill, while the near-field effect attracts the particle to the valley. These results provide valuable insights into the particle motion near surfaces with complex geometry.

A dual scaling of the turbulent longitudinal velocity structure function $\overline {{(\delta u)}^n}$, i.e. a scaling based on the Kolmogorov scales ($u_K$, $\eta$) and another based on ($u'$, $L$) representative of the large scale motion, is examined in the context of both the Kármán–Howarth equation and experimental grid turbulence data over a significant range of the Taylor microscale Reynolds number $Re_\lambda$. As $Re_\lambda$ increases, the scaling based on ($u'$, $L$) extends to increasingly smaller values of $r/L$ while the scaling based on ($u_K$, $\eta$) extends to increasingly larger values of $r/\eta$. The implication is that both scalings should eventually overlap in the so-called inertial range as $Re_\lambda$ continues to increase, thus leading to a power-law relation $\overline {{(\delta u)}^n} \sim r^{n/3}$ when the inertial range is rigorously established. The latter is likely to occur only when $Re_\lambda \to \infty$. The use of an empirical model for $\overline {{(\delta u)}^n}$, which complies with $\overline {{(\delta u)}^n} \sim r^{n/3}$ as $Re_\lambda \to \infty$, shows that the finite Reynolds number effect may differ between even- and odd-orders of $\overline {{(\delta u)}^n}$. This suggests that different values of $Re_\lambda$ may be required between even and odd values of $n$ for compliance with $\overline {{(\delta u)}^n} \sim r^{n/3}$. The model describes adequately the dependence on $Re_\lambda$ of the available experimental data for $\overline {{(\delta u)}^n}$ and supports indirectly the extrapolation of these data to infinitely large $Re_\lambda$.

Preferential flow in a porous medium is commonly encountered in many practical applications. Our previous studies have discovered the preferential flow-induced non-monotonic wettability effect on displacement (J. Fluid Mech, vol. 942, 2022a, R5), but whether this non-monotonic rule is consistent for different disordered media and the impact of the interplay between the disorder and wettability under preferential flow conditions is still not well understood. Here, we combine microfluidic experiments, pore-scale simulations and theoretical analysis to study the impact of the disorder on the invading process where wettability varies from strongly water wet to strongly oil wet. Even though the strongly disordered matrix varies to a uniform state, the generality of the preferential flow-induced non-monotonic wettability effect is still proved. However, the previous pore-scale dynamics based on the strongly disordered matrix cannot explain the invading behaviour in the uniform matrix under preferential flow conditions. New mechanisms for the uniform matrix are further investigated by pore-scale modelling, which indicates that the balance of microscopic imbibition ability and the macroscopic interfacial stability dominate the invading process. We derive a theoretical model to describe the wettability effect and predict the optimal contact angle, which fits well with experimental and simulation results. Our work extends the understanding of the impact of preferential flow conditions on the wettability effect and is also of practical significance for engineering applications, such as geological CO2 sequestration, enhanced hydrocarbon recovery, soil wetting, liquid-infused material fabrication and microfluidic device design.

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.

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 $Ra=10^{10}$.

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.

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.