We present new constrained and free-swimming experiments and simulations in the inertial regime, with Reynolds number $\mbox{Re} = O(10^4)$, of a pair of two-dimensional and three-dimensional pitching hydrofoils interacting in a minimal school. The hydrofoils have an out-of-phase synchronisation, and they are varied through in-line, staggered and side-by-side formations within the two-dimensional interaction plane. It is discovered that there is a two-dimensionally stable equilibrium point for a side-by-side formation. This formation is super-stable, meaning that hydrodynamic forces will passively maintain this formation even under external perturbations, and the school as a whole has no net forces acting on it that cause it to drift to one side or the other. Previously discovered one-dimensionally stable equilibria driven by wake vortex interactions are shown to be, in fact, two-dimensionally unstable, at least for an out-of-phase synchronisation. Additionally, it is discovered that a trailing-edge vortex mechanism provides the restorative force to stabilise a side-by-side formation. The stable equilibrium is further verified by experiments and simulations for freely swimming foils where dynamic recoil motions are present. When constrained, swimmers in compact side-by-side formations experience collective efficiency and thrust increases up to 40 % and 100 %, respectively, whereas slightly staggered formations output an even higher efficiency improvement of 84 %, with an 87 % increase in thrust. Freely swimming foils in a stable side-by-side formation show efficiency and speed enhancements of up to 9 % and 15 %, respectively. These newfound schooling performance and stability characteristics suggest that fluid-mediated equilibria may play a role in the control strategies of schooling fish and fish-inspired robots.

The fingers known as bubbles (spikes) resulting from the penetration of light (heavy) fluids into heavy (light) fluids are significant large-scale features of Richtmyer–Meshkov instability (RMI). Through shock-tube experiments, we study finger collisions in light fluid layers under reshock conditions. Four unperturbed fluid layers with varying thicknesses are created to analyse the motion of waves and interfaces during finger collisions. The wave dynamics, sensitive to initial layer thicknesses, are characterized by a one-dimensional theory. Eight perturbed fluid layers, with four thicknesses and two interface phase combinations, are generated to explore the finger collision mechanism. It is shown that after reshock, the initial in-phase and anti-phase cases undergo spike–bubble rear-end collisions (SBCs) and spike–spike head-on collisions (SSCs), respectively. Compared with SBCs, SSCs significantly suppress spike growth, leading to the attenuation of perturbation growth, especially for larger thicknesses. As the initial thickness decreases, an SSC impedes the downstream interface from reversing its phase, resulting in abnormal RMI, thereby reducing the SSC's effectiveness in attenuating growth. The effects of rarefaction waves enhance both interfaces’ amplitudes and the whole layer's thickness, diminishing the intensity of finger collisions, while the second reshock exerts an opposing influence. Linear and nonlinear models, incorporating the influence of reshocks and rarefaction waves, are developed to predict the interface perturbation growth before and after finger collisions.

This study introduces vector autoregression (VAR) as a linear procedure that can be used for synthesizing turbulence time series over an entire plane, allowing them to be imposed as an efficient turbulent inflow condition in simulations requiring stationary and cross-correlated turbulence time series. VAR is a statistical tool for modelling and prediction of multivariate time series through capturing linear correlations between multiple time series. A Fourier-based proper orthogonal decomposition (POD) is performed on the two-dimensional (2-D) velocity slices from a precursor simulation of a turbulent boundary layer at a momentum thickness-based Reynolds number, $Re_{\theta }=790$. A subset of the most energetic structures in space are then extracted, followed by applying a VAR model to their complex time coefficients. It is observed that VAR models constructed using time coefficients of 5 and 30 most energetic POD modes per wavenumber (corresponding to $66\,\%$ and $97\,\%$ of turbulent kinetic energy, respectively) are able to make accurate predictions of the evolution of the velocity field at $Re_{\theta }=790$ for infinite time. Moreover, the 2-D velocity fields from the POD–VAR when used as a turbulent inflow condition, gave a short development distance when compared with other common inflow methods. Since the VAR model can produce an infinite number of velocity planes in time, this enables reaching statistical stationarity without having to run an extremely long precursor simulation or applying ad hoc methods such as periodic time series.

The study presents observations on the interaction of double-blade propeller tip vortices with a smooth-wall turbulent boundary layer (TBL). The wall-bounded helicoidal vortices from the propeller modify the velocity profiles and turbulence statistics. The effects of two different tip clearances, $\epsilon = 0.1\delta _0$ and $0.5\delta _0$, at a matched thrust, are explored with particle image velocimetry to understand the dynamics of tip-vortex formation within the logarithmic and wake regions of the boundary layer. The measurements are performed with $\lambda =U_{tip}/U_{\infty }$ in the range 5.3–5.9, and a blade passing frequency ($\,f_{prop}$) of the same order of the boundary-layer time scale ($\,f_{TBL}$). Observations indicate a reduction in the extent of the log region and an enhancement of the wake parameter $\varPi$, mirroring the behaviour seen in TBLs under adverse pressure gradient conditions. Notably, the slipstream most contracted region exhibits a significant reduction in the skin friction coefficient $C_f$ and an amplification of the velocity fluctuation statistics across the entire boundary layer. At a clearance of $\epsilon = 0.1\delta _0$, there is evidence of the formation of paired coherent wall-bounded structures. The presence of the wall decreases the amplitude of both periodic and stochastic fluctuations obtained with a phase-locked triple decomposition. An exception is observed behind the propeller for the stochastic fluctuations of the wall-normal component of the flow, which become amplified as the blades move away from the wall. This leads to the creation of a more intense phase-locked two-point spatial coherence than that observed in fluctuations aligned with the streamwise direction. Furthermore, results reveal that reduced tip clearances lead to higher viscous dissipation and more active energy exchange between the mean flow and organized motions.

Direct numerical simulation is performed for flow separation over a bump in a turbulent channel. Comparisons are made between a smooth bump and one where the lee side is covered with replicas of shark denticles – dermal scales that consist of a slender base (the neck) and a wide top (the crown). As flow over the bump is under an adverse pressure gradient (APG), a reverse pore flow is formed in the porous cavity region underneath the crowns of the denticle array. Remarkable thrust is generated by the reverse pore flow as denticle necks accelerate the fluid passing between them in the upstream direction. Several geometrical features of shark denticles, including some that had not previously been considered hydrodynamically functional, are identified to form the two-layer denticle structure that enables and sustains the reverse pore flow and thrust generation. The reverse pore flow is activated by the APG before massive flow detachment. The results indicate a proactive, on-demand drag reduction mechanism that leverages and transforms the APG into a favourable outcome.

We explore the application of the reference map technique, originally developed for Eulerian simulation of solid mechanics, in Lagrangian kinematics of turbulent flows. Unlike traditional methods based on explicit particle tracking, the reference map facilitates the calculation of flow maps and gradients without the need for particles. This is achieved through an Eulerian update of the reference map, which records the take-off positions of fluid particles. This approach is found to be mathematically equivalent to the work of Leung (J. Comput. Phys., vol. 230, issue 9, 2011, pp. 3500–3524), who computed the flow map of simple two-dimensional flows using an Eulerian approach. We discuss important modifications necessary for its first application to complex three-dimensional turbulent flows, including the conservative, low-dissipation update of the flow map and the treatment of periodic boundary conditions. We first demonstrate the accuracy of finite-time Lyapunov exponent (FTLE) calculations based on the reference map against the standard particle-based approach in a two-dimensional Taylor–Green vortex. Then we apply it to turbulent channel flow at $Re_\tau =180$, where Lagrangian coherent structures identified as ridges of the backward-time FTLE are found to bound vortical regions of flow, consistent with Eulerian coherent structures from the $Q$-criterion. The reference map also proves suitable for material surface tracking despite not explicitly tracking particles. This capability can provide valuable insights into the Lagrangian landscape of turbulent momentum transport, complementing Eulerian velocity field analysis. The evolution of initially wall-normal material surfaces in the viscous sublayer, buffer layer and log layer sheds light on the Reynolds stress-generating events from a Lagrangian perspective.

We investigate the effect of high wind speeds on the breakup mechanisms that govern the formation of a spray from nozzles that form liquid sheets, which subsequently break up. The fragmentation mechanism of liquid sheets from spray nozzles has recently been described in detail under quiescent conditions. With high wind speeds, measurements of the droplet size distribution reveal two rather than one characteristic drop sizes, suggesting the existence of two distinct breakup mechanisms. High-speed images of the spray are used to identify these two mechanisms. We show that the smaller droplets result from the breakup of ‘bags’ formed in the spray sheet by the wind, while the larger droplets result from the breakup of the remaining perforated sheet. Based on the two mechanisms, a probability density function is constructed and fitted to the measured droplet size distributions. We show that the spray sheet destabilises due to the Rayleigh–Taylor instability induced by the airflow, and that the experimentally observable breakup length and size of the holes blown in the sheet are predicted by the fastest growing wavenumber. From this, a theoretical prediction for the droplet size from bag breakup and remaining sheet breakup is derived.

The impact of a chemical reaction, $A+B \rightarrow C$, on the stability of a miscible radial displacement in a porous medium is established. Our study involves a comprehensive analysis employing both linear stability analysis and nonlinear simulations. Through linear stability analysis, the onset of instability for monotonic as well as non-monotonic viscosity profiles corresponding to the same end-point viscosity are discussed and compared. We establish a $(R_b,R_c)$ phase plane for a wide range of Damköhler number ($Da$) and Péclet number ($Pe$) into stable and unstable regions. Here, $R_b=\ln (\mu _B/ \mu _A)$ and $R_c=\ln (\mu _C/ \mu _A)$ and $\mu _{i}$ is the viscosity of fluid $i$ $\in \{A$, $B$, $C$}. The stable zone in the $(R_b, R_c)$ phase plane contracts with increased $Da$ and $Pe$ but never vanishes. It exists even for $Da \rightarrow \infty$. Interestingly, we obtain a $Da$ independent stable region in the neighbourhood of $R_c=R_b$ where no transition occurs in stability despite changes in reaction rate. The study allows us to acquire knowledge about the transition of the stability for varying $Da, Pe$ and different reactions classified using $R_b, R_c$.

The breaking and energy distribution of mode-1 depression internal solitary wave interactions with Gaussian ridges are examined through laboratory experiments. A series of processes, such as shoaling, breaking, transmission and reflection, are captured completely by measuring the velocity field in a large region. It is found that the maximum interface descent ($a_{max}$) during wave shoaling is an important parameter for diagnosing the type of wave–ridge interaction and energy distribution. The wave breaking on the ridge depends on the modified blockage parameter $\zeta _m$, the ratio of the sum of the upper layer depth and $a_{max}$ to the water depth at the top of the ridge. As $\zeta _m$ increases, the interaction type transitions from no breaking to plunging and mixed plunging–collapsing breaking. Within the scope of this experiment, the energy distribution can be characterized solely by $\zeta _m$. The transmission energy decreases monotonically with increasing $\zeta _m$, and there is a linear relationship between $\zeta _m^2$ and the reflection coefficient. The value of $a_{max}$ can be determined from the basic initial parameters of the experiment. Based on the incident wave parameters, the depth of the upper and lower layers, and the topographic parameters, two new simple methods for predicting $a_{max}$ on the ridge are proposed.

We theoretically and experimentally study gravity currents of a Newtonian fluid advancing in a two-dimensional, infinite and saturated porous domain over a horizontal impermeable bed. The driving force is due to the density difference between the denser flowing fluid and the lighter, immobile ambient fluid. The current is taken to be in the Darcy–Forchheimer regime, where a term quadratic in the seepage velocity accounts for inertial contributions to the resistance. The volume of fluid of the current varies as a function of time as $\sim T^{\gamma }$, where the exponent parameterizes the case of constant volume subject to dam break ($\gamma =0$), of constant ($\gamma =1$), waning ($\gamma <1$) and waxing inflow rate ($\gamma >1$). The nonlinear governing equations, developed within the lubrication theory, admit self-similar solutions for some combinations of the parameters involved and for two limiting conditions of low and high local Forchheimer number, a dimensionless quantity involving the local slope of the current profile. Another parameter $N$ expresses the relative importance of the nonlinear term in Darcy–Forchheimer's law; values of $N$ in practical applications may vary in a large interval around unity, e.g. $N\in [10^{-5},10^{2}]$; in our experiments, $N\in [2.8,64]$. Sixteen experiments with three different grain sizes of the porous medium and different inflow rates corroborate the theory: the experimental nose speed and current profiles are in good agreement with the theory. Moreover, the asymptotic behaviour of the self-similar solutions is in excellent agreement with the numerical results of the direct integration of the full problem, confirming the validity of a relatively simple one-dimensional model.

Compound flows consist of two or more parallel compressible streams in a duct and their theoretical treatment has gained attention for the analysis and modelling of ejectors. Recent works have shown that these flows can experience choking upstream of the geometric throat. While it is well known that friction can push the sonic section downstream of the throat, no mechanism has been identified yet to explain its displacement in the opposite direction. This study extends the existing compound flow theory and proposes a one-dimensional (1-D) model, including friction between the streams and the duct walls. The model captures the upstream and downstream displacements of the sonic section. Through an analytical investigation of the singularity at the sonic section, it is demonstrated that friction between the streams is the primary driver of upstream displacement. The 1-D formulation is validated against axisymmetric Reynolds averaged Navier–Stokes simulations of a compound nozzle for various inlet pressures and geometries. The effect of friction is investigated using an inviscid simulation for the isentropic case and viscous simulations with both slip and no-slip conditions at the wall. The proposed extension accurately captures the displacement of the sonic section, offering a new tool for in-depth analysis and modelling of internal compound flows.

The motion of a sphere freely rising or falling in a 5d (d is the diameter of the sphere) square tube was numerically studied for the sphere-to-fluid density ratio ranging from 0.1 to 2.3 (0.1 ≤ ρs/ρ ≤ 2.3, ρs is the density of spheres and ρ the fluid density) and Galileo number from 140 to 230 (140 ≤ Ga ≤ 230). We report that Hopf bifurcation occurs at Gacrit ≈ 157, where both the heavy and light spheres lose stability. The helical motion is widely seen for all spheres at Ga > 160 resulting from a double-threaded vortex interacting with the tube walls, which becomes irregular at Ga ≥ 190 where heavy spheres act differently from their counterparts; that is, heavy spheres change their helical directions alternately while light spheres exhibit helical trajectories with jaggedness in connection with the shedding of the double-threaded vortices. This is because of the difference in inertia between the heavy and light spheres. We also checked the oscillation periods for the helical motion of the spheres. They show opposite variations with ρs/ρ for the two types of spheres. Light spheres (ρs/ρ ≤ 0.7) reach a zigzagging regime at Ga ≥ 200 where a vortex loop (hairpin-like vortical structure) is formed which may develop into a vortex ring downstream at small ρs/ρ. This might be the first time a transition from the helical motion to the zigzagging motion for heavy spheres (ρs/ρ ≥ 1.8) has been reported. Finally, we examined the dependence of both the terminal Reynolds number and the drag coefficient of the spheres on the Galileo number.

The flow-induced oscillation of an S-shaped buckled flexible filament was explored using the penalty immersed boundary method. As the length and bending rigidity of the filament were varied, three distinct modes emerged: the equilibrium mode, streamwise oscillation (SO) mode and transverse oscillation (TO) mode. A transition region between the SO and TO modes was identified. Notably, the filament exhibited a 3P wake pattern under SO and a 2S wake pattern under TO. The former was induced by fluid–elastic instability, while the latter was attributed to vortex-induced oscillation. The interaction between the filament's motion and vortex shedding was examined for both modes. To elucidate the disparity between the TO of the S-shaped buckled filament and snap-through oscillation (STO), a ball-on-a-hill analogy was introduced. The performance of energy harvesting was evaluated using metrics including the elastic energy and power coefficient. The TO mode was found to show significantly higher energy harvesting performance than the SO and STO modes. The majority of the strain energy was concentrated at the upper and lower midpoints of the filament.

We study numerically the flow around a spherical droplet set fixed in a linear shear flow with moderate shear rates ($Sr\leq 0.5$, $Sr$ being the ratio between the velocity difference across the drop and the relative velocity) over a wide range of external Reynolds numbers ($0.1<{{Re}}\leq 250$, ${{Re}}$ based on the slip velocity and the viscosity of the external fluid) and drop-to-fluid viscosity ratios ($0.01\leq \mu ^\ast \leq 100$). The flow structure, the vorticity field and their intrinsic connection with the lift force are analysed. Specifically, the results on lift force are compared with the low-${{Re}}$ solution derived for droplets of arbitrary $\mu ^\ast$, as well as prior data at finite ${{Re}}$ available in both the clean-bubble limit ($\mu ^\ast \to 0$) and the solid-sphere limit ($\mu ^\ast \to \infty$). Notably, at ${{Re}}=O(100)$, the lift force exhibits a non-monotonic transition from $\mu ^\ast \to 0$ to $\mu ^\ast \to \infty$, peaking at $\mu ^\ast \approx 1$. This behaviour is related to an internal three-dimensional flow bifurcation also occurring under uniform-flow conditions, which makes the flow to evolve from axisymmetric to biplanar symmetric. This flow bifurcation occurs at low-but-finite $\mu ^\ast$ when the internal Reynolds number (${{Re}}^i$, based on the viscosity of the internal fluid) exceeds approximately 300. In the presence of shear, the corresponding imperfect bifurcation enhances the extensional rate of the flow in the wake. Consequently, the streamwise vortices generated behind the droplet can be more intense compared with those behind a clean bubble. Given the close relation between the lift and these vortices, a droplet with ${{Re}}=O(100)$ and $\mu ^\ast \approx 1$ typically experiences a greater lift force than that in the inviscid limit.