This paper investigates the aerodynamic and flow characteristics of a circular cylinder near the leading-edge separated flow of an elongated rectangular cylinder. The study varies the gap-to-diameter ratio (G/D) of 0 ≤ G/D ≤ 0.4 and distance-to-diameter ratio (L / D) of 0.6 ≤ L / D ≤ 5.8 in the subcritical Reynolds-number region. Here, D, G and L are the diameter of the circular cylinder, the gap between the two isomeric cylinders and the distance between the leading edge of the rectangular cylinder and the centre of the circular cylinder, respectively. Based on smoke-wire flow visualisations, particle image velocimetry test results, lift power spectral densities and pressure distributions, flow around the circular cylinder can be classified into three regimes, i.e. broadened body, body reattachment and co-shedding. In the broadened-body regime, gap flow is negligible, and the circular cylinder behaves as an extension of the rectangular cylinder. In the body-reattachment regime, the free shear layer separated from the rectangular cylinder’s leading edge reattaches to the circular cylinder forebody, significantly modifying its incoming flow. In the co-shedding regime, the free shear layer substantially alters the vortex shedding from the circular cylinder’s lower side, resulting in a distorted alternating vortex shedding from the circular cylinder. Both the drag and lift of the circular cylinder display distinct behaviours in the three flow regimes. Two primary flow modes are recognised through proper orthogonal decomposition analysis: an alternating vortex shedding mode and a one-sided shear flow mode, which result in two Strouhal numbers of 0.205 and 0.255, respectively.

Roll patterns on floating ice shelves have been suggested to arise from viscous buckling under compressive stresses. A model of this process is explored, allowing for a power-law fluid rheology for ice. Linear stability theory of uniformly compressing base flows confirms that buckling modes can be unstable over a range of intermediate wavelengths when gravity does not play a dominant role. The rate of compression of the base flow, however, ensures that linear perturbations have wavelengths that continually shorten with time. As a consequence, linear instability only ever arises over a certain window of time $t$, and its strength can be characterised by finding the net amplification factor a buckling mode acquires for $t\to \infty$, beginning from a given initial wavenumber. Bi-axial compression, in which sideways straining flow is introduced to prevent the thickening of the base flow, is found to be more unstable than purely two-dimensional (or uni-axial) compression. Shear-thinning enhances the degree of instability in both uni-axial and bi-axial flow. The implications of the theoretical results for the glaciological problem are discussed.

The compression waves/boundary layer interaction (CWsBLI) in high-speed inlets poses significant challenges for predicting flow separation, rendering traditional shock wave/boundary layer interaction (SWBLI) scaling laws inadequate due to unaccounted effects of the coverage range of compression waves. This study aims to establish a unified scaling framework for CWsBLIs and SWBLIs by proposing an equivalent interaction intensity. Experiments were conducted in a Mach 2.5 supersonic wind tunnel, employing schlieren imaging and pressure measurements to characterise flows induced by curved surfaces at two deflection angles ($10^{\circ }, 12^{\circ }$) and varying coverage ranges of compression waves ($d$). An equivalent transformation method was developed to convert the CWsBLI into an equivalent incident SWBLI (ISWBLI), with interaction intensity derived from pressure gradients considering the coverage range. Key results reveal a critical threshold based on the interaction length of ISWBLI ($L_{\textit{single}}$): when $d \leq L_{\textit{single}}$, the interaction scale remains comparable to ISWBLI; when $d \gt L_{\textit{single}}$, the weakened adverse pressure gradient leads to a reduction in the length scale. The proposed scaling framework unifies the CWsBLIs and SWBLIs, achieving better data collapse compared to the existing methods. This work advances our understanding of complex waves/boundary layer interactions, and provides a prediction method for the length scales of CWsBLIs.

Using linear stability analysis, we study the onset and formation mechanism of wall modes in confined magnetoconvection cells with the degree of confinement characterised by the cell aspect ratio $\varGamma$. We first outline the phase diagram of the dominating factors that determine the critical Rayleigh number $Ra_c$ for the onset of convection in the $\varGamma -Ha$ phase space, with $\textit{Ha}$ being the Hartmann number. Our study shows that $Ra_c$ is primarily determined by geometrical confinement, and bulk convection onset occurs with $Ra_c = 1090 \varGamma ^{-4.0}$ for $\varGamma \lt \varGamma _{c_1} = 1.21 \textit{Ha}^{-0.48}$. No wall modes form and $Ra_c$ depends on the strength of both the confinement and magnetic field for $\varGamma _{c_1} \leqslant \varGamma \lt \varGamma _{c_2} = 4.07 \textit{Ha}^{-0.53}$. For $\varGamma _{c_2}\leqslant \varGamma \lt \varGamma _{c_3}=0.99 \textit{Ha}^{-0.10}$, wall modes emerge and $Ra_c$ drops below the bulk onset Rayleigh number for magnetoconvection. When $\varGamma \geqslant \varGamma _{c_3}$, wall modes become fully developed with an onset Rayleigh number for wall modes $Ra_{c,w} \approx 65 \textit{Ha}^{1.5}$. In this fully developed regime, the radial velocity profile and $Ra_{c,w}$ become independent of $\varGamma$. Through analysing the length scales of wall modes and their interaction with spatial confinement, we show dynamically how wall modes emerge in confined cells: while the first layer with a characteristic length scale $\ell _1 = 1.04 \textit{Ha}^{-0.56}$ forms when $\varGamma \geqslant 5.39 \textit{Ha}^{-0.58}$, the second layer with a characteristic length scale $\ell _2 = 4.94 \textit{Ha}^{-0.56}$ emerges when $\varGamma \geqslant 9.07 \textit{Ha}^{-0.53}$. These scaling relations provide practical guidelines for experimental and numerical studies of the wall-mode dynamics.

This study experimentally investigates passive drag reduction on a sphere using azimuthally spaced surface protrusions under subcritical Reynolds numbers, focusing on the effects of the protrusion number at fixed surface coverage. The proposed surface modification strategy, termed partial protrusions, maintains a constant total protruded area while varying the number of protrusions $N$, thereby adjusting their azimuthal spacing. The objective is to determine whether such configurations can outperform the conventional full protrusion, in which protrusions continuously surround the azimuthal direction, and to elucidate the flow mechanisms behind any observed enhancement. Drag and flow field measurements reveal that increasing $N$ significantly improves aerodynamic performance. When $N$ exceeds a certain threshold, the partial-protrusion configuration achieves a greater drag reduction than the full-protrusion case, despite using only half the surface coverage. For low $N$, asymmetric pressure distributions across the protruded and smoothed sides induce unsteady separation delay, leading to shear-layer oscillations and elevated turbulent kinetic energy. As $N$ increases, the azimuthal spacing between protrusions decreases, promoting stable interaction between the two sides and leading to separation delay farther downstream than in the full-protrusion case, along with suppression of flow unsteadiness. These results demonstrate that a well-designed partial-protrusion configuration can outperform the full-protrusion configuration in drag reduction and unsteadiness control, offering new insights into effective passive flow control strategies for bluff body flows.

We investigate the unsteady lift response of compliant membrane wings in hovering kinematics by combining analytical inviscid theory with experimental results. An unsteady aerodynamic model is derived for a compliant thin aerofoil immersed in incompressible inviscid flow of variable free-stream velocity at high angles of attack. The model, representing a spanwise section of a hovering membrane wing, assumes small membrane deformation and attached flow. These assumptions are supported by experiments showing that passive membrane deformation suppresses flow separation when hovering at angles of attack up to $55^\circ$. An analytically derived expression is obtained for the unsteady lift response, incorporating the classical Wagner and Theodorsen functions and the membrane dynamic response. This theoretical expression is validated against experimental water-tank measurements that are performed on hovering membrane wings at angles of attack of $35^\circ$ and $55^\circ$. Data from membrane deformation measurements is applied to the theoretical lift expression, providing the theoretical lift response prediction for each of the available experimental scenarios. Results of the comparison show that the proposed theory accurately predicts unsteady lift contributions from membrane deformation at high angles of attack, provided the deformation remains small and the flow is attached. This agreement between inviscid theory and experimental measurements suggests that when flow separation is suppressed, the unsteady aerodynamic theory is valid well beyond the typical low-angle-of-attack regime.

Smooth surface features were recently found to stabilise stationary cross-flow instability (CFI) of swept-wing boundary layers, thus holding potential for passive laminar flow control. Notably, the effect of surface features on the transition location exhibited a significant dependence on the CFI amplitude. In this work, numerical solutions of the harmonic Navier–Stokes (HNS) equations are used to explore the impact of a smooth surface hump on the linear and nonlinear development of stationary CFI under various perturbation amplitudes. Linear simulations identify regions of successive inhibited and enhanced perturbation growth. Despite the recovery of the base flow and perturbation kinetic energy to the reference (i.e. no-hump) state, significantly reduced perturbation growth is observed. The distorted perturbation profile due to the interaction with the hump is postulated to be responsible for this. Increasing the perturbation amplitude results in a response of the flow that is qualitatively similar to the linear case, albeit with increasing local destabilisation of new fundamental (i.e. primary wavelength) structures and higher-order harmonics near the wall. An energy budget analysis reveals that the growth of the fundamental incoming CFI is inhibited through the reduced effectiveness of the lift-up mechanism downstream of the hump. This is preceded by a spatial perturbation shape deformation, governed by (spanwise) transport terms. The results suggest that stabilisation of incoming stationary CFI via smooth surface humps is most effective at low incoming perturbation amplitudes. At higher perturbation amplitudes, newly formed near-wall structures, pre-conditioned by the incoming CFI, overtake the incoming CFI and could anticipate the transition process.

Fluids at supercritical pressure (SCP) exhibit significant real-fluid effects across the pseudo-critical point, which challenges the validity of the existing wall-scaling laws developed under atmospheric pressure condition. This study revisits prior efforts on the temperature-based transformation for the collapse of mean scalar profiles, emphasising the difficulties in accurately describing universal characteristics of thermal boundary layers at SCP. To address this, a novel thermal scaling law using enthalpy transformation is proposed by incorporating the chain rule and heat flux balance. This transformation effectively accounts for variations in the near-wall thermophysical properties associated with the scalar profile while excluding the gradient of isobaric specific heat capacity-related terms. The proposed scaling law demonstrates substantially improved alignment of transformed mean scalar profiles in SCP channel flows at different wall-temperature differences and Reynolds numbers. Additionally, the enthalpy transformation shows superior performance compared with the existing enthalpy–velocity relations, particularly near the heated-wall region where the fluid thermodynamic states undergo the pseudo-boiling process. The present work could facilitate the development of universal wall model in supercritical flows, enabling rapid and reliable heat transfer predictions in practical applications.

Direct numerical simulation is performed to study the effects of spanwise curvature on transitioning and turbulent boundary layers. Turbulent transition is induced with an array of resolved cuboids. Spanwise curvature is prescribed using a novel approach with a body force that is applied orthogonally to the bulk flow to curve the mean free-stream streamlines at a set radius. The flows are analysed in a streamline-aligned coordinate system. Although the radius of curvature is large compared with the size of the boundary layer, its effects on the development of the boundary layer are appreciable. The results indicate that spanwise curvature induces a non-uniform mean secondary flow and alters the structure of turbulence within the boundary layer. Analytical expressions for the crossflow are derived in the viscous sublayer and log layer. These alterations are visible as changes in the distribution of the turbulent stresses and alignment of the vortical structures with the mean flow. These modifications are responsible for a misalignment between the Reynolds stress tensor and the velocity gradient tensor, which has important consequences for the validity of the widely used Boussinesq turbulent viscosity hypothesis in Reynolds-averaged Navier–Stokes models. Spanwise curvature was observed to decrease turbulent kinetic energy. These results have important implications on the development of turbulence in general applications, such as the flow over a prolate spheroid.

Previous studies show that at the small scales of stably stratified turbulence, the scale-dependent buoyancy flux reverses sign, corresponding to a conversion of turbulent potential energy (TPE) back into turbulent kinetic energy (TKE). Moreover, the magnitude of the reverse flux becomes stronger with increasing Prandtl number $\textit{Pr}$. Using a filtering analysis we demonstrate analytically how this flux reversal is connected to the mechanism identified in Bragg & de Bruyn Kops (2024 J. Fluid Mech. vol. 991, A10) that is responsible for the surprising observation that the TKE dissipation rate increases while the TPE dissipation rate decreases with increasing $\textit{Pr}$ in stratified turbulence. The mechanism identified by Bragg & de Bruyn Kops, which is connected to the formation of ramp–cliff structures in the density field, is shown to give the scale-local contribution to the buoyancy flux. At the smallest scales this local contribution dominates and explains the flux reversal, while at larger scales a non-local contribution is important. Direct numerical simulations of three-dimensional statistically stationary, strongly stably stratified turbulence confirm the theoretical analysis, and indicate that, while on average the local contribution only dominates the buoyancy flux at the smallest scales, it remains strongly correlated with the buoyancy flux at all scales. The results show that ramp cliffs are not only connected to the reversal of the local buoyancy flux but also the non-local part. At the small scales (approximately below the Ozmidov scale), ramp structures contribute exclusively to reverse buoyancy flux events, whereas cliff structures contribute to both forward and reverse buoyancy flux events.

This work uses large-eddy simulations to study the transition of a tulip flame stabilised by bubble vortex breakdown (BVB) mode towards a V flame stabilised by a conical vortex breakdown (CVB). The transition is triggered when the equivalence ratio is increased, resulting in a rise in temperature within the central recirculation zone (CRZ). Simultaneously, the pressure inside the CRZ bubble increases, while the average pressure inside the combustion chamber remains constant. This increase in pressure causes the CRZ bubble to open radially and expand, changing the vortex breakdown mode from BVB to CVB, and the flame shape from a tulip shape to a V shape. A criterion for a limit pressure inside the CRZ was then devised based on the radial momentum equation and the balance between centrifugal force and radial pressure gradient, found to control the radial motion of the CRZ. This criterion helped us to understand the main events of the transition, showing that, once the pressure inside the CRZ exceeds the limit given by the criterion, the flow topology changes from a BVB mode to a CVB mode. This transition highlights the differences between a BVB mode and a CVB mode, showing for the first time that there are characteristic pressure and velocity profiles for each mode in their swirling jets and CRZ. Finally, a significant achievement of this work is the identification of a novel mechanism for the controlled transition of vortex breakdown mode, a combustion-driven transition of vortex breakdown mode.

The modal and non-modal stability of laminar flow in a rectangular microchannel is investigated by incorporating the effects of Coriolis forces due to rotation, cross-sectional aspect ratio and superhydrophobic wall slip. The full Navier–Stokes equations are linearised into modified Orr–Sommerfeld and Squire equations, which are then formulated as an eigenvalue problem using small disturbances of the Tollmien–Schlichting type. These equations are subsequently solved by the spectral collocation method. The transition to instability in rotating microchannel flows, influenced by aspect ratio and slip conditions, is analysed through eigenvalue spectra and neutral stability curves. For non-modal analysis, we express the solution in matrix exponential form and then, using the singular value decomposition method, calculate the maximum energy growth. The study reveals that the flow becomes unstable in the presence of rotation at a critical Reynolds number of $ Re_c \approx 40$ for a low aspect ratio and $ Re_c \approx 50.4$ for a high aspect ratio. We find that instability is more pronounced in spanwise-rotating flows at higher aspect ratios compared with those at lower aspect ratios. Rotation induces disturbances from both walls along the spanwise direction, forming secondary flow structures near the centreline. Furthermore, we examine the influence of anisotropic slip by separately considering streamwise and spanwise slip as limiting cases. The numerical results demonstrate that while streamwise slip has a stabilising effect on rotating flows at small scales, a sufficiently large spanwise slip length can trigger instability at Reynolds numbers lower than those observed in the no-slip case. Rotation has the potential to enhance non-modal transient energy growth, while streamwise slip can effectively suppress this instability. These findings suggest that the onset of instability and transient energy growth can be effectively regulated by adjusting the aspect ratio and spanwise slip of the channel walls.

The dual-tone transition phenomenon and its formation mechanism in the flow around a heptagonal cylinder (side number $N= 7$) are experimentally investigated in depth over a range of free-stream velocities corresponding to Reynolds numbers of the order of $10^4$–$10^5$. Dual tone in this context refers to the emergence of two dominant peaks in the far-field acoustic spectrum when a flow in transition regime passes over a polygonal cylinder in principal orientations. The dual-tone phenomenon is also observed in an $N = 9$ cylinder in the face orientation and an $N = 11$ in the corner orientation, which contrasts with all the other polygonal cylinders of $N\in {\mathbb{Z}}[3,12]$ systematically investigated in the present study, where only a single tonal peak dominates the spectrum, similar to the Aeolian tone observed in the circular cylinder in the subcritical regime. The emergence of the dual tone is found to be responsible for the reduction of far-field noise. Continuous wavelet transform analysis reveals that the occurrence of the two competing tones in the time domain can be empirically modelled by Gaussian distributions. Additional proper orthogonal decomposition based phase averaging using time-resolved planar particle image velocimetry enables coherent vortex structure identification for the two quasi-stable shedding modes, which are responsible for the formation of the two tones. Near-wall flow and pressure fluctuation analysis further confirms that the two tones originate from stochastic shear-layer separation–reattachment switching, thereby generating two patterns of dipole sound sources through distinct vortex formation pathways.

This study presents an active flow control framework for fluid–structure interaction (FSI) systems involving a flexible plate in the wake of a cylinder, by integrating Koopman-based reduced-order models (ROMs) with model predictive control (MPC). Specifically, a novel switched-system control strategy is developed, wherein kernel dynamic mode decomposition (DMD) and residual DMD are jointly employed to construct accurate ROMs capable of capturing strongly nonlinear FSI dynamics. This approach ensures accurate low-order representations across multiple control inputs, while suppressing spurious modes. The resulting Koopman ROMs provide fast state predictions over a receding horizon, enabling an MPC optimiser to determine real-time actuation. To improve control performance, resolvent analysis is utilised to optimise the actuator–probe placement. Remarkably, only three strategically placed structural probes are sufficient to capture dominant Koopman modes and enable effective closed-loop control. The proposed framework is then applied to regulate synthetic jets on the cylinder to suppress the plate’s flapping. It successfully stabilises large-amplitude (LAF), small-deflection (SDF) and small-amplitude (SAF) flapping regimes within a unified control strategy. By combining Koopman modal decomposition with an analysis of system energy evolution, we elucidate distinct control mechanisms across these regimes. For LAF and SAF cases, control is achieved primarily through local modulation of existing saturated modes, which requires relatively low actuation energy. In contrast, stabilisation of the SDF case involves the emergence of entirely new Koopman modes that disrupt the original symmetry-breaking dynamics, resulting in increased control input. The framework matches the control performance of reinforcement learning while markedly reducing computational cost.

Aeroacoustic noise generated by wings under the wing-in-ground (WIG) effect is prevalent in various industrial applications, such as WIG vehicles, tower–blade interactions, and slat device noise issues. At chord-based Reynolds number 50 000 and freestream Mach number 0.3, the introduction of an engine jet transforms the separated stall noise of an NACA 4412 aerofoil under the influence of the WIG effect into laminar boundary layer vortex shedding (LBL-VS) noise. This study investigates the underlying mechanisms of this LBL-VS noise. Instead of relying on acoustic analogy, the unapproximated acoustic field is captured with high fidelity using direct numerical simulations. We identify the vorticity transfer process around the trailing edge as a key mechanism in the generation of LBL-VS noise. The results show that as the ground clearance decreases, the overall noise intensity is reduced. When the clearance becomes sufficiently small (10 % chord length), the well-organised vortex structures above the aerofoil break down under high adverse pressure, transitioning into a turbulent state that disrupts the vorticity transfer process. At this clearance, the dominant noise frequency drops from the vortex shedding frequency to an intermittent bursting frequency. This intermittent behaviour arises because only when certain vortices are amplified by acoustic feedback can they shed from the trailing edge, triggering the vorticity transfer process and generating pressure fluctuations. These findings provide new insights into the LBL-VS noise mechanisms under WIG conditions, and can inform strategies for noise reduction in relevant applications.

We integrate a discrete vortex method (DVM) with complex network analysis to strategise dynamic stall mitigation over aerofoils with active flow control. The objective is to inform the actuator placement and the timing to introduce control inputs during the highly transient process of dynamic stall. To this end, we treat a massively separated flow as a network of discrete vortical elements and quantify the interactions among the vortical nodes by tracking the spread of displacement perturbations between each pair of vortical elements using a DVM. This allows us to perform network broadcast mode analysis to identify an optimal set of discrete vortices, the critical timing and the direction to seed perturbations as control inputs. Motivated by the objective of dynamic stall mitigation, the optimality is defined as maximising the reduction of total circulation of the free vortices generated from the leading edge over a prescribed time horizon. We demonstrate the use of the analysis on a two-dimensional flow over a flat plate aerofoil and a three-dimensional turbulent flow over an SD$7003$ aerofoil. The results from the network analysis reveal that the optimal timing for introducing disturbances occurs slightly after the onset of flow separation, before the shear layer rolls up and forms the core of the dynamic stall vortex. The broadcast modes also show that the vortical nodes along the shear layer are optimal for introducing disturbances, hence providing guidance to actuator placement. Leveraging these insights, we perform nonlinear simulations of controlled flows by introducing flow actuation that targets the shear layer slightly after the separation onset. We observe that the network-guided control results in a $21 \,\%$ and $14\,\%$ reduction in peak lift for flows over the flat plate and SD$7003$ aerofoil, respectively. A corresponding decrease in vorticity injection from the aerofoil surface under the influence of control is observed from simulations, which aligns with the objective of the network broadcast analysis. The study highlights the potential of integrating the DVMs with the network analysis to design an effective active flow control strategy for unsteady aerodynamics.

This direct numerical simulation study analyses the transition-to-turbulence in plane Couette flow (PCF) with three-dimensional (3-D) roughness. Square ribs of height $k=0.2h$ (where $h$ is the half-channel height) and a streamwise pitch separation of $\lambda =10k$, classified as k-type roughness, are mounted on the stationary wall. This configuration features alternating rough and smooth zones in the spanwise direction and is referred to as 3-D k-type roughness. We compare the behaviour of 3-D k-type roughness in the transitional regime with that of two-dimensional (2-D) k-type roughness (Gokul & Narasimhamurthy 2024 J. Fluid Mech. vol. 1000, p. A40). The route-to-transition in 3-D k-type roughness confirms the formation of laminar–turbulent patterns, which exist in the transitional Reynolds number range $\textit{Re} \in [325, 350]$. This range interestingly overlaps with those for smooth PCF ($\textit{Re} \in [325, 400]$) and 2-D k-type roughness ($\textit{Re} \in [300, 325]$), thereby indicating that 3-D k-type roughness amalgamates the characteristics of both the rough and smooth zones. In striking contrast to the 2-D k-type roughness, the laminar–turbulent bands in the 3-D k-type configuration are of non-uniform bandwidth. The 3-D k-type roughness surprisingly introduces continuous competition among bands of opposite orientations, a characteristic unique in this case. Due to this competition, the large-scale flow associated with the oblique bands is never fully aligned with the diagonal direction. The smooth zones in the 3-D k-type roughness exhibit complex interactions, evident from oscillatory velocity signals and multiple high-energy peaks in the frequency spectra, which likely contribute to the competing patterns with opposite orientations.

Pulsed gravity currents are generated by the sequential release of dense material into a lighter ambient. We investigate the dynamics of pulsed gravity currents using physical scale experiments, two-dimensional depth-averaged shallow water equation (SWE) based models and three-dimensional lattice Boltzmann method (LBM) simulations. Integrating these results we show for the first time that short duration pulsed releases generate intrusive layers, which accelerate front propagation relative to an instantaneously released current of the same total volume. Conversely, a long delay time between pulses produces a current that propagates slower than an equivalent instantaneous release. This finding is supported by physical experiments and depth-resolving LBM simulations. The depth-resolving simulations show that intrusions in pulsed flows experience less drag resistance than those generated by instantaneous releases. The depth-averaged model considered in the present study does not accurately capture the intrusive flow dynamics of pulsed currents. However, the limitations of the finite-depth SWE model may be mitigated by extensions to incorporate entrainment and density stratification. The results also motivate further research into the impact of buoyancy Reynolds number and channel slope on the propagation of pulsed currents.

We investigate the role of slippery boundaries, quantified by the Navier boundary friction coefficient $\beta$, in regulating heat transport and flow structures in rotating Rayleigh–Bénard convection. Owing to the Ekman pumping effect arising from viscous boundary layers that is intensified with increasing boundary friction, it is found that the properties of global heat transport exhibit two distinct parameter regimes separated by a transitional Rayleigh number ($ \textit{Ra}_t$). In the rotation-dominated regime ($ \textit{Ra} \lt \textit{Ra}_t$), enhanced viscous friction increases the efficiency of Ekman pumping, significantly elevating the Nusselt number and lowering the convection onset threshold. Conversely, in the buoyancy-dominated regime ($ \textit{Ra} \gt \textit{Ra}_t$), boundary-induced viscous dissipation suppresses convective motions, thereby reducing heat transport. Large-scale vortices (LSVs), prevalent under free-slip conditions, progressively dissipate as $\beta$ increases, revealing that viscous friction disrupts the inverse energy cascade from baroclinic to barotropic modes. Through kinetic energy partitioning analysis, the transition between quasi-two-dimensional and three-dimensional turbulent states is identified, with the parameter $\beta _{\textit{cr}}$ following a generic scaling relation on the Prandtl (Pr) and Ekman (Ek) numbers $\beta _{\textit{cr}}\sim \textit{Pr}^{-0.67}\textit{Ek}^{-1.18}$. This relation enables us to predict LSV emergence across different parameter spaces. Furthermore, it is reported that the heat-transport scaling exponent, the convection onset and the partitioning of kinetic energy between barotropic and baroclinic components undergo a smooth flow transition at $\beta _{\textit{cr}}$. These results also indicate a direct correlation between Ekman pumping efficacy and the friction coefficient $\beta$, demonstrating that controlling boundary friction can modulate global transport properties and reshape flow structures.

To advance understanding of the influence hill-slope and hill-shape have on neutrally stratified turbulent air flow over isolated forested hills, we interrogate four turbulence-resolving simulations. A spectrally friendly fringe technique enables the use of periodic boundary conditions to simulate flow over isolated two-dimensional (2D) and three-dimensional (3D) hills of cosine shape. The simulations target recently conducted wind tunnel (WT) experiments that are configured to fall outside the regimes for which current theory applies. Simulation skill for flow over isolated 3D hills is demonstrated through matching the canopy and hill configuration with the recently conducted WT experiments and comparing results. The response of the mean and turbulent flow components to 2D versus 3D hills along the hill-centreline are discussed. The phase and amplitude of spatially varying flow perturbations over forested hills are evaluated for flows outside the regime valid for current theory. Flow over isolated 2D forested hills produces larger amplitude vertical motions on a hill’s windward and leeward faces and the speed-up of the mean wind compared with that over isolated 3D forested hills at the hill-centreline. The 3D hills generate surface pressure minima over hill-crests that are only half the magnitude of those over 2D hills. The spatial region over which hill-induced negative pressure drag acts increases with increasing hill steepness. Assumptions in partitioning the flow into an upper layer with an inviscid response to the hill’s pressure field are robust and lead to solid predictions of hill-induced perturbations to the mean flow; however, applying those assumptions to predict the evolution of the turbulent moments only provides approximate explanations at best.