Quantifying differences between flow fields is a key challenge in fluid mechanics, particularly when evaluating the effectiveness of flow control or other problem parameters. Traditional vector metrics, such as the Euclidean distance, provide straightforward pointwise comparisons but can fail to distinguish distributional changes in flow fields. To address this limitation, we employ optimal transport (OT) theory, which is a mathematical framework built on probability and measure theory. By aligning Euclidean distances between flow fields in a latent space learned by an autoencoder with the corresponding OT geodesics, we seek to learn low-dimensional representations of flow fields that are interpretable from the perspective of unbalanced OT. As a demonstration, we utilise this OT-based analysis on separated flows past a NACA 0012 airfoil with periodic heat flux actuation near the leading edge. The cases considered are at a chord-based Reynolds number of 23 000 and a free-stream Mach number of 0.3 for two angles of attack (AoA) of $6^\circ$ and $9^\circ$. For each angle of attack, we identify a two-dimensional embedding that succinctly captures the different effective regimes of flow responses and control performance, characterised by the degree of suppression of the separation bubble and secondary effects from laminarisation and trailing-edge separation. The interpretation of the latent representation was found to be consistent across the two AoA, suggesting that the OT-based latent encoding was capable of extracting physical relationships that are common across the different suites of cases. This study demonstrates the potential utility of optimal transport in the analysis and interpretation of complex flow fields.

Experimental deep reinforcement learning (DRL) control of a turbulent boundary layer is conducted for the first time at $Re_\tau$ = 1196, with the aim of friction-drag reduction. Two hot films, an impinging plasma jet actuator array and two wall hot wires act as the state detector, flow disturber and reward evaluator, respectively. The control law parametrised by a radial basis function network is executed in real time on a field programmable gate array and optimised using a classical value-based algorithm (deep Q-network). Results show that DRL control requires only 30 s to train a closed-loop control law with satisfactory drag-reduction performance. Compared with open-loop control where only fine-tuned periodical forcing can reduce the friction drag, the experimental efficiency is improved significantly. Proper setting of the hyper-parameters is crucial in DRL. Particularly, the reward time delay and control frequency need to match the convection time scale and the characteristic frequency of the turbulent boundary layer. The optimal DRL control setting achieves 6.7 % relative drag reduction, almost three times that of the best open-loop control (2.3 %). Physically, plasma actuation induces alternating low-speed and high-speed zones that confine the sidewise motion of turbulent streaks. The final control law optimised by DRL can be simplified as a threshold control, firing the plasma actuator after perceiving a streak burst event and a long-lasting high-speed zone. Control benefits are attributed to the increase in the occurrence probability of high-reward states and the elevation of mean reward at different clusters.

In this work, we propose a lattice Boltzmann model (LBM) to simulate diverse particle deposition patterns induced by isothermal droplet evaporation. The model is composed of two distributions, for the multiphase flow with phase change, and the particle transport with deposition, coupled with a contact angle hysteresis model for the contact line stick-slip dynamics. The model is validated by two benchmarks, and our simulations agree well with the theoretical solutions or experimental results. With the validated LBM, we first reproduced diverse deposition patterns, ranging from the coffee ring, uniform, to mountain-type patterns in single and multiple symmetrical/unsymmetrical forms. Then a parametric study is conducted to investigate how the solvent/particle/substrate properties affect the evaporation dynamics and resultant deposition patterns. Afterwards, we apply the average ratio ($r_{\phi ,a}$) of particles deposited at the droplet periphery and the centre to quantitatively classify the diverse emerging patterns. We show that $r_{\phi ,a}$ is controlled by the competition between the capillary transport and particle diffusion, leading to a linear dependence on the average Péclet number $\textit{Pe}_{a}$. Finally, we validated the scaling by lattice Boltzmann simulations with the proposed $\textit{Pe}_{a}$ spanning over three orders of magnitude, supplemented by discussions from the aspect of the particle transport equation.

This article presents experimental measurements of breaking wave impact loads on a vertical cylinder. The focus is on the influence of some of the breaking wave properties on the measured force. These properties are the distance to breaking, $\delta$, defined as the distance between the breaking location and the front face of the cylinder, and the breaking strength, characterised here by the $\varGamma$ parameter proposed by Derakhti et al. (J. Fluid Mech., 2018, vol. 848, p. R2). The wave characteristics are obtained through numerical simulations of the breaking waves using a fully nonlinear potential flow solver. Seven breaking waves with different breaking strengths have been considered. For each wave, the distance to breaking has been systematically varied and the resulting impact force time-history was measured. It is found that except for the two less intense breaking cases, corresponding to values of $\varGamma$ lower than one, there is a value of $\delta$ for which the magnitude of the impact force is maximum. Small variations of the distance to breaking $\delta$ strongly influence the impact force time-history and its maximum. A linear relationship is observed between the maximum force and the breaking strength $\varGamma$. For the wave cases with values of $\varGamma$ higher than one, the maximum impact force is observed when the distance to breaking $\delta$ is close to 5 % of the wavelength.

The acoustic fields of a contra-rotating propeller and isolated propellers producing the same overall thrust are compared at both design and off-design working conditions. The sound levels are reconstructed by using the Ffowcs Williams–Hawkings acoustic analogy, exploiting results of computations conducted on a cylindrical grid consisting of $4.6 \times {10}^9$ points and a large-eddy simulation technique. The analysis shows that, although the blades of the contra-rotating propeller are less loaded and produce less intense flow structures, the levels of radiated sound are reinforced, compared with the propellers working alone. This is due to the loading sound, originating from the pressure fluctuations on the surface of the blades of the propellers. The higher levels of linear sound are attributable to the interplay between the front and rear rotors of the contra-rotating system. This interaction is able to reinforce the unsteady component of the loads acting on the blades of the propellers and the resulting linear component of sound. While the shear occurring between the tip vortices shed by the front and rear rotors gives rise to a complex system of isolated vortex rings in the wake, increasing the quadrupole component of sound, these phenomena are balanced by the lower intensity of the vortices shed by the contra-rotating system.

We derive the far-field and near-field solutions for the Green’s function of a point force acting perpendicular to a no-slip wall in a Brinkman fluid, focusing on the regime where the distance between the force and the wall is much smaller than the screening length. The general solution is obtained in closed form up to a single integral, and can be systematically expanded in a Taylor series in both the far-field and near-field limits. The flow can then be expressed as a series of source-multipole singularities with an additional, analytically known, correction in the proximity of the wall. Comparisons with numerical integration demonstrate the accuracy and reliability of the asymptotic expansions. The results are also applicable to the unsteady Stokes flow driven by a localised assembly of forces, such as a beating cilium protruding from a flat surface.

This study experimentally investigates the bleeding flow characteristics downstream of isotropic porous square cylinders as a function of permeability and pore configuration across a broad range of Darcy numbers ($2.4 \times 10^{-5} \lt \textit{Da} \lt 2.9 \times 10^{-3}$). The porous cylinders, constructed with a simple cubic lattice design, were fabricated using a high-resolution three-dimensional printing technique. This novel design method, based on a periodic and scalable lattice structure, allows fine control over the number of lattice pores along the cylinder width, $D$, and the corresponding permeability, independently of porosity. Permeability was carefully determined by measuring the pressure drop and superficial velocity for each porous structure considered in this study. High-resolution particle image velocimetry measurements were conducted in an open-loop wind tunnel to characterize the downstream flow structures. The results reveal that bleeding flow characteristics near the cylinder trailing edge are strongly influenced by both permeability and pore configuration. These structural behaviours are further explored using an analogy to multiple plane turbulent jets. This approach identifies three distinct flow regions downstream of porous square cylinders, determined by the structural pattern of the bleeding flow. Additionally, an analytical framework is developed to model the longitudinal extent of the merging region by integrating the momentum equation, incorporating the Darcy–Brinkman–Forchheimer model, with a boundary layer assumption. The analytical model is validated against experimental data, demonstrating its capability to predict the key dynamics of bleeding flow evolution. Our results provide new insights into the fluid dynamics of porous bluff bodies, establishing pore configuration and permeability as dominant parameters governing downstream flow structures.

This paper develops scaling laws for wall-pressure root mean square and the streamwise turbulence intensity peak, accounting for both variable-property and intrinsic compressibility effects – those associated with changes in fluid volume due to pressure variations. To develop such scaling laws, we express the target quantities as an expansion series in powers of an appropriately defined Mach number. The leading-order term is represented using the scaling relations developed for incompressible flows, but with an effective Reynolds number. Higher-order terms capture intrinsic compressibility effects and are modelled as constant coefficients, calibrated using flow cases specifically designed to isolate these effects. The resulting scaling relations are shown to be accurate for a wide range of turbulent channel flows and boundary layers.

Numerical simulations and theoretical analysis are conducted to investigate the Atwood-number dependence of perturbation evolution in a shocked heavy fluid layer. For layers without reverberating waves, a higher Atwood number of one interface significantly enhances its coupling effect on the perturbation growth at the opposite interface. A theoretical model incorporating the startup, linear and nonlinear stages is developed to predict the interface mixing width. Dimensionless formulae are derived, identifying eight distinct modulation regimes of multi-interface instability. When reverberating waves are present, the individual effects of the upstream ($A_1$) and downstream ($A_2$) Atwood numbers are examined. The model is further modified to account for additional reverberating waves required at higher $A_2$ values for accurate amplitude prediction. Both theory and simulations demonstrate that perturbation growth at one interface can be actively controlled by adjusting the Atwood number of the opposite interface. These findings provide insights for mitigating instabilities in applications such as inertial confinement fusion through appropriate material selection.

We carry out an experimental study of granular flow in a quasi-two-dimensional wedge-shaped hopper, with glass front and back walls, using videography, along with image analysis and particle tracking. Results are presented for different orifice sizes and roughnesses of the sidewalls for nearly spherical glass and steel particles of different sizes. The data for the radial velocity in the hopper (wedge angle $2\theta _w$) are well described by $v_r(r,\theta )=v_{r0}(r)[1-F(r)(\theta /\theta _w)^2],$ in cylindrical coordinates $(r,\theta )$, with the origin at the apex of the wedge. The centreline velocity is given by $v_{r0}=(a_0/r+a_1)$, and the effective wall friction by $F=(b_0+b_1r)$, where $a_0$ and $a_1$ increase with orifice width, while $b_0$ increases with roughness. For the smooth wall system, we obtain $F\in (0,1)$, however, for the rough walls $F\gt 1$ for most cases, with the velocity at the wall being zero, and a few layers of slow-moving particles adjacent to the wall. The mass flow rate scaled by the particle density and the radial velocity profile are independent of the particle density, for a threefold increase in the density, implying insignificant inertial effects. Discrete element method simulations are carried out using glass particles for a system of the same size as the experimental hopper, with the simulation parameters calibrated to closely match the experimental results. The simulation results indicate that the variation in the direction normal to the plane of the flow is small and the radial velocity profiles without the front and back walls are similar to the experimental profiles.

The effects arising from interactions between two identical starting jets on their propulsive characteristics have been investigated numerically for different dimensionless distances $S/D$ (the distance between two jet axes normalised by nozzle diameter, from 1.1 to 4) and stroke ratios $L/D$ (the length-to-diameter ratio of jet column, from 2 to 5). The two jets are arranged in parallel and initiated simultaneously with identical conditions. Their leading vortical structures evolve from an axisymmetric to a plane-symmetric configuration, with deceleration in regions where the two jet wakes approach each other. The generation of axial thrust is affected, primarily dominated by variations in the pressure thrust component. This results from the combination of the mutually induced pressure (MIP) and the coupling effects of vortex rings (CEVR for $S/D\gt1.5$ and CEVR-R for $S/D\lt1.5$). The MIP governs the fluctuations introduced into thrust development, while CEVR (CEVR-R) is responsible for the reduction in average thrust. These effects become more pronounced as $S/D$ decreases, but remain almost unaffected by $L/D$. Adjusting the acceleration and deceleration rates of the velocity program shows limited effects on either the thrust fluctuations or the average thrust reduction. Furthermore, the interaction induces two lateral force components with equal magnitude but opposite directions on the outer walls of the two nozzles, with their magnitude exceeding $15\,\%$ of the axial thrust. The introduction of an additional vertical wall within the nozzle exit plane effectively eliminates the lateral force. However, it consequently enhances both the average thrust reduction and the thrust fluctuations induced by the interaction.