Wall-resolved large-eddy simulations are used to investigate the effects of elevated surface temperature on the leading edge of a swept wing. Earlier stability analyses demonstrate a strong destabilising effect resulting from surface heating, but do not reveal the ultimate effects on transition and relaminarisation. The current results support the findings from stability analysis with elevated surface temperatures, but show a more complex dependency between instability and turbulence than is observed for adiabatic conditions. For attachment-line Reynolds numbers above the adiabatic relaminarisation threshold, instability leads to fully turbulent flow. At Reynolds numbers below the adiabatic relaminarisation threshold, instability leads to finite-amplitude equilibrium states and intermittent turbulent bursting, but not fully turbulent flow.

We study the onset of two-dimensional secondary shear instability (SSI) in the braid regions connecting primary Kelvin–Helmholtz billows in stratified shear flows. While strain induced by the billows stabilises the braids, it also compresses their tilted isopycnals, enhancing baroclinic shear that enables rapid perturbation growth. By modifying the classical analysis of Corcos & Sherman (1976 J. Fluid Mech. 73, 241–264) in braid-aligned coordinates and adding an instability criterion based on the ratio of strain rate to shear, we develop an inviscid, time-dependent model for the braid evolution and the onset of SSI. We show that the criterion for instability can be achieved significantly earlier than the saturation of the primary billow at sufficiently high initial Richardson number $ \textit{Ri}$, because increased stratification slows the rate of billow growth relative to the rate of baroclinic shear production. Two-dimensional direct numerical simulations up to Reynolds numbers $ \textit{Re}=10^7$ quantify the role of viscosity. At high $ \textit{Re}$, we find that SSI indeed develops early in the braid, as predicted by the inviscid model, while the primary billow is still growing and before viscosity slows braid thinning. These results provide a mechanistic explanation for field observations of braid-dominated mixing and suggest that, at geophysically relevant $ \textit{Ri}$ and $ \textit{Re}$, SSI can control the three-dimensional transition and ensuing mixing by preceding and pre-empting both vortex pairing instabilities and secondary convective instabilities in the billow core.

Minimising bacterial surface adhesion and subsequent biofilm formation in industrial and medical settings requires understanding how bacteria are transported and adhere to complex surface geometries in the presence of non-uniform flow. In this paper, we consider the transport of a dilute suspension of motile bacteria through a corrugated two-dimensional channel with perfectly adhesive walls. Considering flows much faster than the bacterial swimming speed, we asymptotically analyse the diffusive boundary layer that forms using a curvilinear coordinate system based on the fluid streamfunction. We present a similarity solution to the diffusivity-varying diffusion-type equation that arises, from which we derive an analytical expression for the bacterial adhesion rate as a function of surface arclength and the spatially varying wall shear rate. Our model predicts that bacterial adhesion becomes localised on curved surfaces, with bacteria showing preferential adhesion to wall ‘peaks’ at lower shear rates and preferential adhesion to wall ‘valleys’ at higher shear rates. More broadly, our results highlight how spatially varying flows generated by complex geometries can lead to localised bacterial adhesion, with potential implications for both enhancing and minimising biofilm formation.

A physics-based model is developed to predict the spatially varying sensitivity of turbulence to roughness geometry and the corresponding flow modifications. Resolvent analysis with a drag-normalized, Reynolds-decomposed volume penalization is formulated to efficiently link the mean flow profile to predictions of the drag force, wake field fluctuations, and dispersive stresses for broadband (sandgrain) surface roughness. An eddy viscosity approach models the modification of the stochastic stress arising due to the attenuation of convecting fluctuations within the roughness sublayer. Crucially, these components link the mean-flow profile to the force profiles and stresses which determine its shape, so that the model can be iteratively solved for a given bulk flow over a surface roughness, starting from an initial corresponding smooth wall flow condition. For engineering-relevant surface geometries in flows which respect outer similarity, the model output estimates turbulent-drag response, namely equivalent sandgrain roughness, Hama roughness-function and spatial variation of roughness sublayer (wake field) turbulent fluctuations. The approach is demonstrated on a channel flow with sandgrain roughness at an engineering-relevant Reynolds number.

Physics-based eddy-viscosity and backscattering closures are widely used for large-eddy simulation (LES) of geophysical turbulence, but their key parameters are often chosen empirically. Here, we develop a semi-analytical framework for estimating these parameters in two-dimensional (2-D) geophysical turbulence by adapting a Lilly-type scaling argument. Specifically, we obtain closed-form estimates, up to an amplitude constant, for the coefficients of the Leith and Smagorinsky eddy-viscosity closures, a biharmonic eddy-viscosity closure and the Jansen–Held backscattering closure with a prescribed backscattering fraction. The amplitude constant appears in the turbulent kinetic energy direct-cascade spectrum and can be diagnosed from a few direct numerical simulation (DNS) or eddy-resolving snapshots. For isotropic cases ($\beta =0$), the diagnosed amplitude constant is consistent with previous theoretical estimates based on closure, renormalisation-group and mode-coupling methods. The resulting semi-analytical parameters closely match the online-learned values obtained using ensemble Kalman inversion across several 2-D geophysical turbulence set-ups (Guan et al. Phys. Rev. Res. vol. 8, 2026, p. 023215). LES using these parameters reproduces key DNS statistics, including the tails of the vorticity distribution, and robustly outperforms dynamic Leith and Smagorinsky baselines.

Horizontal convection (HC), the flow in a box heated and cooled from the bottom (or top), is a canonical model of buoyancy-driven transport with direct relevance to subglacial lakes, ocean circulation and climate in general. While HC is commonly characterised by the Rayleigh number $Ra$ and Prandtl number $Pr$, the role of the domain aspect ratio $\varGamma$ (defined as horizontal extension over height) has received less attention. Here, we perform three-dimensional direct numerical simulations over a broad range of Rayleigh numbers and aspect ratios at fixed Prandtl number in order to quantify how geometry affects flow organisation and heat transport. We find that $\varGamma$ strongly influences both the development of the overturning circulation and the scaling of heat transfer (i.e. in dimensionless form, the Nusselt number $Nu$). As the aspect ratio increases, the emergence of an organised overturning flow is shifted to higher Rayleigh numbers. Moreover, with increasing $\varGamma$ the Nusselt number shows a stronger dependence on the Rayleigh number. The results clarify the gradual cross-over from a finite intermediate range compatible with the Rossby-type scaling $\textit{Nu}\sim \textit{Ra}^{1/5}$ (Rossby 1965 Deep-Sea Res. Oceanogr. Abstr. vol. 12, pp. 9–16) towards the $\textit{Nu}\sim \textit{Ra}^{1/4}$ behaviour suggested by Shishkina, Grossmann & Lohse (2016 Geophys. Res. Lett. vol. 43, pp. 1219–1225). It reflects the progressively decreasing vertical confinement of the kinetic boundary layer with increasing aspect ratio. These results show that the aspect ratio is an important control parameter in laminar HC and should be incorporated explicitly into scaling arguments for geophysical flows.

We investigate the interaction between vortex rings and cylindrical obstacles using direct numerical simulations across a wide range of geometric and dynamical parameters. The flow is characterised in terms of the diameter ratio between ring and object $T_D = d/D$, the Reynolds number based on circulation $\textit{Re}_\varGamma$, and the slenderness ratio $\varLambda$. By systematically varying $T_D$ and $\textit{Re}_\varGamma$, we identify three distinct interaction regimes: the wire, cutting and wall regimes. In the wire regime ($T_D \lesssim 0.05$), the primary vortex ring survives the interaction with limited deformation, and carries a weak lobe of secondary vorticity generated by the object’s boundary layer. As $T_D$ increases, the interaction transitions to the cutting regime, where the ring is split into two secondary structures formed through the reconnection between boundary-layer and ring vorticity. For sufficiently large obstacles ($T_D \gtrsim 0.8$), the wall regime emerges, in which boundary-layer vorticity dominates, and the primary ring is deflected and stretched along the obstacle surface. The transition between regimes depends primarily on $T_D$, while increasing $\textit{Re}_\varGamma$ enhances vortical dynamics, producing additional small-scale and tertiary structures. Finally, by modifying the topology of the obstacle, we demonstrate that reconnection and recovery of the primary ring depend critically on the topology of the secondary vorticity. These results provide a unified framework for interpreting vortex–body interactions, bridging the gap between vortex ring, tube and wall collision dynamics.

We investigate pressure-driven turbulent flow in a square duct using wall-modelled large-eddy simulations (WMLES) of the Navier–Stokes equations. An equilibrium wall model based on the logarithmic law is employed, yielding results in close agreement with direct numerical simulations (DNS) over the range of overlapping Reynolds numbers. In particular, both the structure and intensity of the secondary motions are well captured, providing confidence in the ability of WMLES to extend to Reynolds numbers beyond the present reach of DNS, and to offer insight into the asymptotic state of turbulence. This expectation is reinforced by the good agreement with available experimental data at higher Reynolds numbers. The WMLES results confirm the validity of the classical Prandtl friction law, with the hydraulic diameter taken as the duct side, up to bulk Reynolds numbers $\textit{Re} \approx 10^8$. More interestingly, the simulations reveal an asymptotic regime in which the intensity of the secondary motions scales with the mean friction velocity. Consistent with the observations of Pullin et al. (2013, Phys. Fluids, vol. 25, 015116), this suggests that secondary motions are a finite-Reynolds-number effect, and that the asymptotic state of square-duct turbulence corresponds to a plug-like velocity profile with finite slip at the wall, when viewed in outer units.

Nonlinear interactions of internal ocean waves play an important role in ocean mixing and are of theoretical interest even in idealised settings. According to the resonant wave interaction approximation, which is commonly used to derive the kinetic equation for the energy spectrum, the dominant interactions are between wave triads whose wavevectors satisfy $\boldsymbol{k}=\boldsymbol{p}+\boldsymbol{q}$ and whose frequencies satisfy $\omega _{\boldsymbol{k}} = |\omega _{\boldsymbol{p}}\pm \omega _{\boldsymbol{q}}|$. To test the validity of the resonant wave interaction approximation, we examine several analytical derivations of the theory. The assumptions underlying each derivation are tested using idealised direct two-dimensional (2-D) non-rotating numerical simulations, representing near-observed energy levels of the oceanic internal wave field. We show that the slow-amplitude assumptions underlying the derivations are inconsistent with the simulated dynamics in this particular set of simulations. In addition, most of the triads satisfying the resonant conditions do not contribute significantly to nonlinear wave energy transfer in our idealised 2-D simulations, while some interactions that are dominant in nonlinear energy transfers do not satisfy the resonance conditions. Understanding the relevance of these results to the observed three-dimensional rotating ocean internal wave field requires further study.

Scalar turbulence, involving the chaotic transport of passive scalars such as temperature, pollution and chemical concentration, plays a critical role in scientific and engineering applications. Transport of scalars is largely affected by its inherent characteristic molecular diffusivity through the Schmidt number ($\textit{Sc}$, the ratio of fluid viscosity and diffusivity). This study explores the scalar convection speed and Taylor microscale in fully developed pipe flow with a localised scalar plume at Reynolds number $\textit{Re} = 10\,000$ for $10^{-3} \leqslant \textit{Sc} \leqslant 10$. We employ a hybrid approach combining direct numerical simulations for the velocity field with Lagrangian scalar tracking to capture the scalar field’s evolution under turbulent convection and molecular diffusion. The scalar concentration data are obtained in the Eulerian cylindrical coordinates. In a fixed range of streamwise positions, the scalar convection speed aligns with the convection speed of velocity fluctuations at $\textit{Sc} = 1$, but it is smaller and larger than the velocity convection speed for $\textit{Sc} \lt 1$ and $\textit{Sc} \gt 1$, respectively, and also for the radial positions close to the pipe wall. The Taylor microscale scales approximately as $\textit{Sc}^{-1/2}$ for $\textit{Sc} \geqslant 1$, while for $\textit{Sc} \lt 1$, it approximately follows $\textit{Sc}^{-0.03}$. Along the streamwise positions, the scalar convection speed is found to gradually approach the velocity convection speed, regardless of whether the Schmidt number is larger or smaller than unity, along with the decay of scalar fluctuations. Overall, both the scalar convection velocity and scalar Taylor microscale depend on $\textit{Sc}$ and the mixing state.

Systematic experimental data on local extreme velocities and unsteady features in high-Reynolds-number deep-cavity flows remain limited. In this study, two-component particle image velocimetry (PIV) measurements were performed to investigate the flow characteristics of an incoming-flow-driven deep cavity with $L/D$ = $1/3$ at Re = 2.55 × 105. Three streamwise $xy$ planes ($z/W$ = 0.10, 0.25, 0.50) and three spanwise yz planes ($x/L$ = 0.10, 0.25, 0.50) were measured systematically. The time-averaged velocity, root mean square velocity fluctuations, in-plane turbulent kinetic energy (TKE), instantaneous maximum velocity, and higher-order statistics including skewness and kurtosis were analysed. The results show that the vertical velocity fluctuation on the intermediate streamwise plane ($z/W$ = 0.25) reaches peak value approximately $v$rms/Ulid = 0.072 (where Ulid is the characteristic inflow velocity above the cavity), indicating the strongest local vertical unsteady response among the measured streamwise planes. On the near-wall plane ($z/W$ = 0.10), the instantaneous maximum velocity exhibits skewness 1.23 and kurtosis 7.47, reflecting pronounced local intermittent extreme events, whereas the extreme events on the middle plane ($z/W$ = 0.50) are weaker and closer to a Gaussian-like distribution. The spanwise planes also demonstrate clear non-Gaussian features, with high-fluctuation and high in-plane TKE regions primarily concentrated in areas associated with the interaction between the shear layer and recirculation structures, revealing significant spatial anisotropy in the flow. These results suggest that shear-layer development, primary recirculation, sidewall confinement and downstream flow turning jointly modulate local extreme events and intermittent velocity fluctuations. This study provides systematic experimental quantification of spatial non-uniformity and extreme-event characteristics in a high-Reynolds-number deep cavity using multi-plane PIV, offering useful experimental references for turbulence model validation and complex cavity-flow control.

Aerodynamic interactions between closely spaced lifting bodies are fundamental to many natural and engineered systems, including flapping insect wings, multi-rotor aerial vehicles and vertical-axis wind turbines. During operation, these systems undergo periodic counter-translating interactions (cross-over) in which adjacent wings or blades translate past each other in close proximity, producing strong unsteady coupling and transient load excursions. This study investigates the canonical form of this problem: two counter-translating lifting bodies in a hydrodynamic towing facility to isolate the fluid mechanics of a single cross-over event. Time-resolved load and flow-field measurements reveal that the unsteady forces arise from a combination of induced effects and transient pressure fields imposed by the mutual interaction between the wings. The resulting disturbance behaves as a self-induced gust, generated collectively by the coupled system rather than externally imposed. Comparison with linear unsteady aerodynamic models demonstrates that the measured response most closely follows Miles’s travelling-gust formulation. This connection establishes a continuum between externally imposed and self-induced gust interactions. A parametric analysis further identifies the primary non-dimensional groups governing the phenomenon, Reynolds number, aerofoil thickness, bound circulation and inter-wing spacing, and quantifies their influence on the transient lift response. Together, these results reveal a new regime of multi-body unsteady aerodynamics, where the interplay between circulatory and non-circulatory effects govern the short-time dynamics of interacting lifting surfaces.

Triboelectric charging in particle-laden flows is a complex interplay of fluid and particle dynamics, collision mechanics and electrostatics. In this study, we introduce triboFoam, an open-source solver built on the OpenFOAM framework, designed to simulate triboelectric charging in particle-laden turbulent flows. We validate triboFoam using direct numerical simulations (DNSs) of a fully developed turbulent channel flow at a friction Reynolds number of $\textit{Re}_\tau = 180$. The results demonstrate good agreement with DNS data for particle concentration profiles and charge distributions. Then, we investigate the influence of Reynolds number on particle distribution and charging behaviour using large-eddy simulations at varying friction Reynolds numbers up to $\textit{Re}_\tau = 550$. Our findings reveal that higher Reynolds numbers lead to increased near-wall particle concentrations and enhanced charging rates, attributed to intensified turbulent fluctuations and elevated impact velocities. Finally, an empirical correlation is proposed to predict the average particle charging rate as a function of Reynolds number and particle diameter. With this work, we provide a tool for simulating triboelectric charging in complex geometries and turbulent flows, advancing the understanding of electrostatic phenomena in particle-laden systems. The empirical correlation offers practical insights for predicting charging behaviour in industrial applications and thus can contribute to improved safety and efficiency in processes involving particulate matter.

We analyse the stability of viscoelastic Dean flow (flow of an elastic fluid through a curved two-dimensional channel, driven by an azimuthal pressure gradient) in the absence of fluid inertia. This configuration is well known to exhibit a hoop-stress-driven ‘purely elastic’ instability (referred to henceforth as the hoop-stress mode – ‘HSM’) on account of the base-flow streamline curvature. The objective of this study is to demonstrate the existence and importance of a distinct elastic instability in this flow configuration, which is not driven by hoop stresses, but instead is a continuation of a novel ‘centre-mode’ (CM) instability recently identified in rectilinear shear flows. On account of its origins, the CM instability in Dean flow is expected for two-dimensional (azimuthally varying) disturbances with no axial variation, but continues to exist for three-dimensional disturbances. In contrast, the HSM instability is expected primarily in the axisymmetric limit, and again continues to exist for three-dimensional disturbances. We use both the Oldroyd-B and FENE-P models to map out parameter regimes in the $\textit{Wi}$–$\epsilon$–$\beta$ space where the aforementioned instabilities are present. Here, $\textit{Wi}$ is a suitably defined Weissenberg number that characterises fluid elasticity, $\beta$ is the ratio of solvent to total solution viscosity and $\epsilon$ is the ratio of the gap (channel) width to the radius of curvature. While its origin in rectilinear shearing flows might lead one to expect the CM instability to only be present for small $\epsilon$ (the ‘narrow-gap’ limit), we show that it exists even for $O(1)$ values of $\epsilon$, and over a larger range of $\textit{Wi}$. Nevertheless, within the Oldroyd-B framework, the HSM determines the stability threshold for experimentally relevant gap-width ratios, corresponding to $0.1 \leqslant \epsilon \leqslant 1$, with the CM becoming the most unstable mode only for $\epsilon \lt 0.001$. For the more accurate FENE-P model, however, decreasing the finite extensibility parameter $L$ has opposing effects on the HSM and CM instabilities – stabilising the former, but destabilising the latter. In the dilute solution regime ($\beta \gt 0.95$) and for realistic values of $L \sim O(100)$, corresponding to polymer molecular weights of $O(10^{5-6})$ g mol−1, the CM remains the most unstable mode for $\epsilon \leqslant 0.25$, rendering it potentially relevant to the onset of elastic turbulence in the flow of such polymer solutions through curved channels.

We investigate the reconnection of two vortex bubble rings using direct numerical simulations of the incompressible two-phase Navier–Stokes equations with a volume-of-fluid method and a continuum surface force model. At fixed Reynolds number and varying Weber numbers ($\textit{We}$) spanning surface-tension-dominated to inertia-dominated regimes, the reconnection dynamics are divided into four stages, including approach, interfacial reconnection, helicity transport and pinch-off. Low $\textit{We}$ produces a larger reduction in interfacial area, leading to efficient reconnection and the formation of daughter bubbles. High $\textit{We}$ stretches the interface into slender filaments, producing residual fragments and secondary vortical structures. Simulations show that vorticity begins to rearrange towards the contact region before reconnection occurs, with surface tension modulating vorticity redistribution and interfacial contraction as $\textit{We}$ varies. These results relate $\textit{We}$ to interfacial morphology and the resulting reconnection topology, connecting vortex dynamics and interfacial physics in two-phase flows.

Secondary instabilities developing from the stationary primary global mode are investigated for an incident shock flow at Mach 2.15. Direct numerical simulations (DNS) and three-dimensional global stability analysis (3-D GSA) are combined to elucidate the evolution and characteristics of the associated secondary branches. Two spanwise domain widths are considered, $\lambda _m$ and $2\lambda _m$, where $\lambda _m$ denotes the spanwise wavelength of the primary global mode. In the $\lambda _m$ domain, a time-periodic unsteadiness emerges as a secondary instability after saturation of the primary global mode, which is identified as a supercritical Hopf bifurcation. In the $2\lambda _m$ domain, an additional secondary growth accompanied by wavelength doubling is observed following the onset of unsteadiness. A corresponding stationary secondary instability with wavelength $2\lambda _m$ is determined to be the mechanism underlying this secondary growth, which is a subcritical pitchfork bifurcation. Overall, DNS and 3-D GSA provide a consistent interpretation of the secondary-instability scenario, and highlight the sensitivity of the observed dynamics to the spanwise domain width.

Viscoplastic fluids flow only when stress exceeds the yield stress, confining swimmer-induced flows within the yield surface and limiting far-field hydrodynamic interactions. We investigate how yield stress tunes collective dynamics of microswimmers in monolayer suspensions. Higher yield stress shifts the phase transition in puller suspensions to higher densities and weakens flocking of spherical neutral swimmers, governed by the ratio of yield stress to swimmer density. In ellipsoidal swimmers, collisions dominate, making yield stress less influential. These systems constitute a distinct category in which hydrodynamic interactions are cut off, providing new insight into the long-standing debate about the impact of far- and near-field hydrodynamics on collective dynamics of microswimmers.

In high-Reynolds-number wall-bounded flows, the inner-scaled wall-pressure variance is often represented as a logarithmic increase with frictional Reynolds number. We consider the two sources of the incompressible pressure–Poisson equation: a linear (rapid) term linked to mean shear and a nonlinear (slow) term composed of quadratic velocity fluctuations. This paper establishes a link between the sources and the coefficients in a logarithmic inner-scaled variance representation. To leading order we posit that the linear source provides a Reynolds-number-independent offset, while the nonlinear source contributes the logarithmic coefficient. The illustrative dataset is direct numerical simulation (DNS) at frictional Reynolds number $\delta ^+\approx 550$, although the principal contribution is the establishment of a mechanistic link to well-known high-$\delta ^+$ scalings of wall-bounded turbulence. Through consideration of the sources and the integral solution method of the Poisson equation, we find that the linear source contribution sits predominantly in the buffer layer and maps to the near-wall cycle. To leading order, this contribution becomes $\delta ^+$ invariant under inner scaling, thus contributing an offset in the logarithmic representation. The interfacial regions between uniform momentum zones characteristic of the inertial layer (vortical fissures) spatially localise strain and vorticity contributions and contain an increasingly large proportion of the strain and vorticity. We show that fissures act as a compact carrier for the source terms, with the nonlinear term especially prominent in these regions. By considering the inertial layer statistics, we link the changing nonlinear contribution to the $\ln \delta ^+$ growth, in agreement with previous empirical observations.

The two-layer quasi-geostrophic (2LQG) model and the Eady model are two idealised systems illustrating the baroclinic instability of atmospheric jets and ocean currents. The two set-ups share many ingredients – background vertically sheared zonal flow of density-stratified fluid in a rapidly rotating frame – while differing in complexity and dimensionality. The Eady model has a continuous vertical direction, with baroclinic turbulence induced by boundary potential vorticity (PV) gradients at top and bottom. By contrast, the 2LQG sytem typically models baroclinic instability induced by interior PV gradients. This distinction challenges our ability to clearly identify a couple of ‘modes’ through which the Eady dynamics could be inferred from a simpler 2LQG system. In the present study, we show that this difficulty can be circumvented in the turbulent regime arising for weak bottom drag. Namely, guided by the common organisation of both systems into a gas of coherent vortices, we identify a quantitative mapping between the Eady and 2LQG models. The mapping allows for parameter-free predictions of the eddy diffusivity of the Eady model based on knowledge of the 2LQG diffusivity. We illustrate these results using numerical simulations of the Eady and 2LQG models with linear or quadratic bottom drag.

The interactions between two identical starting jets (twin-starting jets) are numerically investigated, focusing on the influence of initiation with temporal differences (the dimensionless time difference $\Delta \widetilde {t}\gt 0$) and spatial differences (the dimensionless axial distance $\Delta \widetilde {X}\gt 0$). Temporal differences mainly weaken the mutual interaction between the two leading vortex rings, while spatial differences weaken both the vortex-ring interactions and the influence of mutually induced pressure. In both cases, two leading vortex rings become non-coplanar, increasing their spatial distance along the jet direction and thereby weakening their mutual interaction. For propulsive characteristics, both temporal and spatial differences mitigate the reduction in average thrust caused by jet interactions, whereas only spatial differences significantly suppress the overall thrust fluctuations. In addition, both types of difference induce a net lateral force on the nozzle outer walls. The combined effects of $\Delta \widetilde {X}$ and the dimensionless nozzle spacing $S/D$, where $S$ is the distance between the axes of the two nozzles, and is the nozzle diameter. are incorporated into the mutually induced pressure model. Finally, the influence distance $S_3/D$ is introduced to quantify the weakening of vortex-ring interactions by temporal and spatial differences. This parameter also unifies the effects of $\Delta \widetilde {t}$ and $\Delta \widetilde {X}$, which contribute comparably to mitigating the reduction in the average thrust.