We present new unconstrained simulations and constrained experiments of a pair of pitching hydrofoils in a leader–follower in-line arrangement. Free-swimming simulations with matched pitching amplitudes show self-organisation into stable formations at a constant gap distance without any control. Over a wide range of phase synchronisation, amplitude and Lighthill number typical of biology, we discover that the stable gap distance scales with the actual wake wavelength of an isolated foil rather than the nominal wake wavelength. A scaling law for the actual wake wavelength is derived and shown to collapse data across a wide Reynolds number range of $200 \leqslant Re \leqslant 59\,000$. Additionally, vortex analysis uncovers that the leader’s wake wavelength-to-chord ratio, $\lambda /c$, is the key dimensionless variable to maximise the follower’s/collective efficiency. When $\lambda /c \approx 2$ it ensures that the follower’s leading edge suction force and the net force from a nearby vortex pair act in the direction with the foil’s motion thereby reducing the follower’s power. Moreover, in both simulations and experiments mismatched foil amplitudes are discovered to increase the efficiency of hydrofoil schools by 70 % while maintaining a stable formation without closed-loop control. This occurs by (i) increasing the stable gap distance between foils to push them into a high-efficiency zone and (ii) raising the level of efficiency in these zones. This study bridges the gap between constrained and unconstrained studies of in-line schooling by showing that constrained-foil measurements can map out the potential efficiency benefits of schooling. These findings can aid in the design of high-efficiency biorobot schools.

Aerodynamic loads play a central role in many fluid dynamics applications, and we present a method for identifying the structures (or modes) in a flow that make dominant contributions to the time-varying aerodynamic loads in a flow. The method results from the combination of the force partitioning method (Menon & Mittal, 2021, J. Fluid Mech., vol. 907, A37) and modal decomposition techniques such as Reynolds decomposition, triple decomposition and proper orthogonal decomposition, and is applied here to three distinct flows – two-dimensional flows past a circular cylinder and an aerofoil, and the three-dimensional flow over a revolving rectangular wing. We show that the force partitioning method applied to modal decomposition of velocity fields results in complex, and difficult to interpret inter-modal interactions. We therefore propose and apply modal decomposition directly to the $Q$-field associated with these flows. The variable $Q$ is a nonlinear observable that is typically used to identify vortices in a flow, and we find that the direct decomposition of $Q$ leads to results that are more amenable to interpretation. We also demonstrate that this modal force partitioning can be extended to provide insights into the far-field aeroacoustic loading noise of these flows.

In this study, we explore the evolution of instabilities in magneto-quasi-geostrophic (MQG) modons on the $f$-plane using a magnetohydrodynamic rotating shallow water model. The numerical experiments have been conducted using a recently proposed second-order flux-globalisation-based path-conservative central-upwind scheme. Our focus is on the evolution and interactions of three key configurations: singular, regular and hollow MQG modons, which represent cases where the magnetic field is confined within the separatrix, evenly distributed inside and outside the separatrix and localised outside the separatrix, respectively. The singular MQG modon emerges as the most stable configuration, demonstrating the greatest resilience to destabilising forces. A notable observation is its transition from a quadrupolar to a tripolar magnetic field structure before reverting to a quadrupole adjusted magnetic modon, accompanied by a clockwise rotation of the system. In terms of stability, singular modons are the most stable ones, while hollow modons are the least stable. As instabilities develop, southward or northward displacements become significantly more pronounced than eastward or westward movements, primarily due to the Coriolis force. Among the configurations, the hollow (singular) modons experience the biggest (smallest) displacements. Additionally, we investigate modon collisions and highlight three scenarios: interactions between cyclonic and anticyclonic components that form a composite modon with meridional bifurcation; collisions of cyclonic vortices that produce a tripolar structure with counterclockwise rotation; and collisions between anticyclonic components that result in a stable, quasi-stationary tripolar configuration. The resulting magnetic poles exhibit a checkered pattern, with their amplitude decreasing with increasing distance from the central vortex.

Viscous flow through high-permeability channels occurs in many environmental and industrial applications, including carbon sequestration, groundwater flow and enhanced oil recovery. In this work, we study the displacement of a less-viscous fluid by a more-viscous fluid in a layered porous medium in a rectilinear configuration, where two low-permeability layers sandwich a higher-permeability layer. We derive a theoretical model that is validated using corroborative laboratory experiments, when the influence of the density difference is negligible. We find that the location of the propagating front increases with time according to a power-law form $x_f \propto t^{1/2}$, while the fluid–fluid interface exhibits a self-similar shape, when the motion of the displaced fluid is negligible in an unconfined porous medium. In the experimental set-up, distinct permeability layers were constructed using various sizes of spherical glass beads. The working fluids comprised fresh water as the less-viscous ambient fluid, and a glycerine–water mixture as the more-viscous injecting fluid. Our experimental measurement show a better match with the theory for the experiments performed at low Reynolds numbers and with permeable boundaries in the far field.

We study the homogeneous isotropic turbulence of a shear-thinning fluid modelled by the Carreau model, and show how the variable viscosity affects the multiscale behaviour of the turbulent flow. We show that Kolmogorov theory can be extended to such non-Newtonian fluids, provided that the correct choice of average is taken when defining the mean Kolmogorov scale and dissipation rate, to properly capture the effect of the variable viscosity. Thus the classical phenomenology à la Kolmogorov can be observed in the inertial range of scale, with the energy spectra decaying as $k^{-5/3}$, with $k$ being the wavenumber, and the third-order structure function obeying the $4/5$ law. The changing viscosity instead strongly alters the small scale of turbulence, leading to an enhanced intermittent behaviour of the velocity field.

A nonlinear Schrödinger equation for pure capillary waves propagating at the free surface of a vertically sheared current has been used to study the stability and bifurcation of capillary Stokes waves on arbitrary depth. A linear stability analysis of weakly nonlinear capillary Stokes waves on arbitrary depth has shown that (i) the growth rate of modulational instability increases as the vorticity decreases whatever the dispersive parameter $kh$, where $k$ is the carrier wavenumber and $h$ the depth; (ii) the growth rate is significantly amplified for shallow water depths; and (iii) the instability bandwidth widens as the vorticity decreases. Particular attention has been paid to damping due to viscosity and forcing effects on modulational instability. In addition, a linear stability analysis to transverse perturbations in deep water has been carried out, demonstrating that the dominant modulational instability is two-dimensional whatever the vorticity. Near the minimum of linear phase velocity in deep water, we have shown that generalised capillary solitary waves bifurcate from linear capillary Stokes waves when the vorticity is positive. Moreover, we have shown that the envelope of pure capillary waves in deep water is unstable to transverse perturbations. Consequently, deep-water generalised capillary solitary waves are expected to be unstable to transverse perturbations.

Görtler vortices induced by concave curvature in supersonic turbulent flows are investigated using resolvent analysis and large-eddy simulations at Mach 2.95 and Reynolds number $ Re_{\delta }=63\,500$ based on the boundary-layer thickness $ \delta$. Resolvent analysis reveals that the most amplified coherent structures manifest as streamwise counter-rotating vortices with optimal spanwise wavelength $ 2.4\delta$ at cut-off frequency $f\delta /{u}_{\infty } =0.036$, where $ {u}_{\infty }$ is the freestream velocity. The leading spectral proper orthogonal decomposition modes with spanwise wavelength approximately $ 2\delta$ align well with the predicted coherent structures from resolvent analysis at $f\delta /{u}_{\infty } =0.036$. These predicted and extracted coherent structures are identified as Görtler vortices, driven by the Görtler instability. The preferential spanwise scale of the Görtler vortices is further examined under varying geometric and freestream parameters. The optimal spanwise wavelength is insensitive to the total turning angle beyond a critical value, but sensitive to the concave curvature $ K$ at the same turning angle. A limit spanwise wavelength $ 1.96\delta$, corresponding to an infinite concave curvature as $ K\rightarrow \infty$, is identified and validated. Increasing the freestream Mach number or decreasing the ratio of wall temperature to freestream temperature reduces the optimal wavelength normalised by $ \delta$, while variations in freestream Reynolds number have negligible impact. Additionally, a modified definition of the turbulent Görtler number $ G_{T}$ based on the peak eddy viscosity in boundary layers is proposed and employed to assess the occurrence of Görtler instability.

The supersonic wake of a circular cylinder in Mach 3 flow was studied through spectral proper orthogonal decomposition (SPOD) of high-speed focussing schlieren datasets. A wavenumber decomposition of the SPOD eigenvectors was found to be an effective tool for isolating imaging artefacts from the flow features, resulting in a clearer interpretation of the SPOD modes. The cylinder wake consists of both symmetric and antisymmetric instabilities, with the former being the dominant type. The free shear layers that form after the flow separates from the cylinder surface radiate strong Mach waves that interact with the recompression shocks to release significant disturbances into the wake. The wake shows a bimodal vortex shedding behaviour with a purely hydrodynamic instability mode around a Strouhal number of 0.2 and an aeroacoustic instability mode around Strouhal number of 0.42. The hydrodynamic mode, which is presumably the same as the incompressible case, is weaker and decays rapidly as the wake accelerates due to increasing compressibility. The aeroacoustic mode is the dominant shedding mode and persists farther into the wake because of an indirect energy input received through free-stream acoustic waves. A simple aeroacoustic feedback model based on an interaction between downstream propagating shear-layer instabilities and upstream propagating acoustic waves within the recirculation region is shown to accurately predict the shedding frequency. Based on this model, the vortex shedding in supersonic flows over a circular cylinder occurs at a universal Strouhal number (based on approach free-stream velocity and feedback path length) of approximately 0.3.

In this work we focus on expected flow in porous formations with highly conductive isolated fractures, which are of non-negligible length compared with the scales of interest. Accordingly, the definition of a representative elementary volume (REV) for flow and transport predictions may not be possible. Recently, a non-local kernel-based theory for flow in such formations has been proposed. There, fracture properties like their expected pressure are represented as field quantities. Unlike existing models, where fractures are assumed to be small compared with the scale of interest, a non-local kernel function is used to quantify the expected flow transfer between a point in the fracture domain and a potentially distant point in the matrix continuum. The transfer coefficient implied by the kernel is a function of the fracture characteristics that are in turn captured statistically. So far the model has successfully been applied for statistically homogeneous cases. In the present work we demonstrate the applicability for heterogeneous cases with spatially varying fracture statistics. Moreover, a scaling law is presented that relates the transfer coefficient to the fracture characteristics. Test cases involving discontinuously and continuously varying fracture statistics are presented, and the validity of the scaling law is demonstrated.

Fluids at supercritical pressures exhibit large variations in density near the pseudo-critical line, such that buoyancy plays a crucial role in their fluid dynamics. Here, we experimentally investigate heat transfer and turbulence in horizontal hydrodynamically developed channel flows of carbon dioxide at $88.5$ bar and $32.6\,^{\circ }\rm C$, heated at either the top or bottom surface to induce a strong vertical density gradient. In order to visualise the flow and evaluate its heat transfer, shadowgraphy is used concurrently with surface temperature measurements. With moderate heating, the flow is found to strongly stratify for both heating configurations, with bulk Richardson numbers $Ri$ reaching up to 100. When the carbon dioxide is heated from the bottom upwards, the resulting unstably stratified flow is found to be dominated by the increasingly prevalent secondary motion of thermal plumes, enhancing vertical mixing and progressively improving heat transfer compared with a neutrally buoyant setting. Conversely, stable stratification, induced by heating from the top, suppresses the vertical motion, leading to deteriorated heat transfer that becomes invariant to the Reynolds number. The optical results provide novel insights into the complex dynamics of the directionally dependent heat transfer in the near-pseudo-critical region. These insights contribute to the reliable design of heat exchangers with highly property-variant fluids, which are critical for the decarbonisation of power and industrial heat. However, the results also highlight the need for further progress in the development of experimental techniques to generate reliable reference data for a broader range of non-ideal supercritical conditions.

The energy-harvesting performance of two oscillating hydrofoil turbines in tandem configuration is experimentally studied at a $Re$ of $20\,000$ to determine the array’s optimal kinematics. By characterising interactions between the leading foil’s wake and the trailing foil, the kinematic configuration required to maximise array power extraction is identified. This is done by prescribing leading-foil kinematics that produce specific wake regimes, identified by the maximum effective angle of attack, $\alpha _{T/4}$, parameter. The kinematics of the trailing foil are varied significantly from those of the leading foil, with heave and pitch amplitudes of $0.6c\lt h_{0,{\textit{tr}}}\lt 1.8c$ and $65^{\circ} \lt \theta _{0,{\textit{tr}}}\lt 75^{\circ}$, and inter-foil phase of $-110^{\circ} \lt \psi _{1-2}\lt 180^{\circ}$. Configurations with reduced frequencies of $0.11$ and $0.12$, and foil separations of $4c$ and $6c$ are tested within each wake regime. The power extracted by each foil over an oscillation cycle is measured through force and torque measurements. Wake–foil interactions that improve trailing foil performance are analysed with time-resolved particle image velocimetry. Constructive and destructive wake–foil interactions are compared, showing that trailing-foil performance improves by either avoiding wake vortices or interacting directly with them. By interacting with the primary wake vortex, the latter configuration sees no power loss during the cycle. System power from the two foils is found to be maximised when the leading foil operates at an intermediate $\alpha _{T/4}$ range, and when the trailing foil avoids wake vortices. This optimal array configuration sees both foils operating with different kinematics compared with the optimal kinematics of a single oscillating foil.