We perform direct numerical simulations of surfactant-laden droplets in homogeneous isotropic turbulence with Taylor Reynolds number $Re_\lambda \approx 180$. The droplets are modelled using the volume-of-fluid method, and the soluble surfactant is transported using an advection–diffusion equation. Effects of surfactant on the droplet and local flow statistics are well approximated using a lower, averaged value of surface tension, thus allowing us to extend the framework developed by Hinze (AIChE J., vol. 1, no. 3, 1955, pp. 289–295) and Kolmogorov (Dokl. Akad. Navk. SSSR, vol. 66, 1949, pp. 825–828) for surfactant-free bubbles to surfactant-laden droplets. We find that surfactant-induced tangential stresses play a minor role in this set-up, thus allowing us to extend the Kolmogorov–Hinze framework to surfactant-laden droplets. The Kolmogorov–Hinze scale $d_H$ is indeed found to be a pivotal length scale in the droplets’ dynamics, separating the coalescence-dominated (droplets smaller than $d_H$) and the breakage-dominated (droplets larger than $d_H$) regimes in the droplet size distribution. We find that droplets smaller than $d_H$ have a rather compact, regular, spheroid-like shape, whereas droplets larger than $d_H$ have long, convoluted, filamentous shapes with a diameter equal to $d_H$. This results in very different scaling laws for the interfacial area of the droplet. The normalized area, $A/d_H^2$, of droplets smaller than $d_H$ is proportional to $(d/d_H)^2$, while the area of droplets larger than $d_H$ is proportional to $(d/d_H)^3$, where $d$ is the droplet characteristic size. We further characterize the large filamentous droplets by computing the number of handles (loops of the dispersed phase extending into the carrier phase) and voids (regions of the carrier fluid completely enclosed by the dispersed phase) for each droplet. The number of handles per unit length of filament scales inversely with surface tension. The number of voids is proportional to the droplet size and independent of surface tension. Handles are indeed an unstable feature of the interface and are destroyed by the restoring effect of surface tension, whereas voids can move freely in the interior of the droplets, unaffected by surface tension.

Two sets of measurements utilizing hot-wire anemometry and oil-film interferometry for flat-plate turbulent boundary layers, exposed to various controlled adverse and favourable pressure gradients, are used to evaluate history effects of the imposed and varying free-stream gradients. The results are from the NDF wind tunnel at Illinois Tech (IIT) and the MTL wind tunnel at KTH, over the range $800 < Re_\tau < 22\,000$ (where $Re_{\tau }$ is the friction Reynolds number). The streamwise pressure-gradient parameter $\beta \equiv (-\ell /\tau _{w}) \cdot (\partial P_{e}/\partial x)$ varied between $-2 < \beta < 7$, where $\ell$ is an outer length scale for boundary layers equivalent to the half-height of channel flow and the radius of pipe flow, and is estimated for each boundary-layer profile; note that $\tau_w$ is the wall-shear stress and $P_e$ is the free-stream static pressure. Extracting from each profile the three parameters of the overlap region, following the recent work of Monkewitz & Nagib (J. Fluid Mech., vol. 967, 2023, p. A15) that led to an overlap region of combined logarithmic and linear parts, we find minimum history effects in the overlap region. Thus, the overlap region in this range of pressure-gradient boundary layers appears to be in ‘quasiequilibrium’.

We use video footage of a water-tunnel experiment to construct a 2-D reduced-order model of the flapping dynamics of an inverted flag in uniform flow. The model is obtained as the reduced dynamics on a 2-D attracting spectral submanifold (SSM) that emanates from the two slowest modes of the unstable fixed point of the flag. Beyond an unstable fixed point and a limit cycle expected from observations, our SSM-reduced model also confirms the existence of two unstable fixed points for the flag, which were found by previous studies. Importantly, the model correctly reconstructs the dynamics from a small number of general trajectories and no further information on the system. In the chaotic flapping regime, we construct a 4-D SSM-reduced model that captures the system's chaotic attractor.

New experiments in highly turbulent, steady, subcritical and uniform water open-channel flows have been carried out to measure the mean turbulent kinetic energy (TKE) budget of sediment-laden boundary layer flows with two sizes (dp = 3 mm and 1 mm) of Plexiglas particles $({\rm relative\ density}\ = 1.192)$. The experiments covered energetic sediment transport conditions (Shields number of $0.35 < \theta < 1.2$) ranging from non-capacity to full-capacity flows in bedload-to-suspension-dominated transport modes (suspension number of $0.5 < {w_s}/{u_\ast } < 1.3$ where ${w_s}$ is the settling velocity and $u_*$ is the friction velocity) and for weakly to highly inertial, finite size turbulence-particle conditions (Stokes number of 0.1 < St < 3.5 and dp/η > 10 where $\eta$ is the Kolmogorov length scale). It was shown that the effects of sediments on the TKE budget are very pronounced in all large particle experiments for which a bedload layer of several grain diameter thickness is developed above the channel bed. When compared with the corresponding reference clear-water flows, the TKE shear-production rate for the 3 mm particle flows is strongly reduced in the wall region corresponding to the bedload layer. This turbulence damping is seen to increase with sediment load until full capacity for flows with constant Shields value, as well as with Shields number value. Inside this damped TKE shear-production zone, a distinct peak of maximal turbulence production appears to coincide with the upper edge of the bedload layer delimited by a sharp gradient in mean sediment concentration. This vertically upshifted peak of TKE production is accompanied by an enhanced net downward oriented TKE flux when compared with the reference clear-water flows. The downward diffused TKE is found to act in the bedload layer as a local energy source in reasonable balance with the sediment transport term. The mechanism behind this downward TKE transport was further analysed on the basis of coherent flow structure dynamics controlled by ejection- and sweep-type events. The agreement between the height of downward directed mean TKE flux and the height below which sweep-type events dominate the Reynolds shear-stress contribution over ejections, revealed the leading role played by sweeps in mean TKE transport. This agreement holds for all reference clear-water flows supporting the well-known wall-roughness-induced dominance of the sweep contribution in turbulent, rough clear-water boundary layer flows. Furthermore, for all 3 mm particle flows, the two referred to transition levels were significantly and similarly upshifted to the upper edge of the bedload layer. Only for these sediment-laden flows, the bedload layer thickness is seen to exceed the wall-roughness sublayer of the reference clear-water flows. This supports a strong analogy between wall-roughness effects in clear-water flows and bedload layer effects in sediment-laden flows, on the mean TKE budget induced by a similarly modified coherent flow structure dynamics. The bedload layer-controlled wall roughness is finally confirmed by the good prediction of the wall-roughness parameter ks of the logarithmic velocity distribution. An empirical formulation fitting the presented measurements is presented, valid over the range of Shields number values covered herein.

A theoretical analysis is presented of peristaltic pumping down a narrow conduit with permeable walls, motivated by the flushing action of lugworms and other marine organisms in sandy burrows. Flow in the conduit is dealt with using lubrication theory; the leakage into the surrounding medium is taken into account by exploiting slender-body theory to solve the associated Darcy problem. By adopting a model for the local force balance on the pumping surface, we bridge between the limits in which the pump operates with either fixed load or displacement. In the latter limit we characterize peristaltic waves with either fixed form or ones that partially collapse the conduit. We construct pump characteristics (the relation between the mean flux and net pressure drop) when the burrow wall is impermeable and pressures are fixed at each end, and compare the results with existing laboratory experiments performed on lugworms. We then consider how the peristaltic dynamics is changed when the wall is made permeable. Last, we consider pumping along an impermeable burrow into a leaky head shaft. The results reveal that the permeability of the conduit wall or end can greatly impact the direction and strength of the recirculating flow.

Surface tension and wetting are dominating physical effects in microscale and nanoscale flows. We present an efficient and reliable model of surface tension and equilibrium contact angles in smoothed particle hydrodynamics for free-surface problems. We demonstrate its robustness and accuracy by simulating several three-dimensional free-surface flow problems driven by interfacial tension.

Impedance spectroscopy is widely adopted for probing the charge and charge mobility of soft ion-conducting media, such as synthetic membranes and biological tissue. The spectra exhibit a variety of distinctive signatures, but the physical basis of these is not well understood, e.g. models have not previously accounted for viscoelasticity, hydrodynamics or microstructural heterogeneity. This study explores a physically grounded continuum model that captures how these factors shape conductivity spectra. Nonlinear thermodynamics and linearised dynamics of a viscous electrolyte and compressible, elastic polymer network are coupled under the forcing of an oscillatory electric field. The model is solved in a one-dimensional spatially periodic unit cell, reporting conductivity and dielectric permittivity spectra, including Nyquist representations. Whereas rigid microstructures exhibit ion-diffusion-controlled relaxation, which manifests as a low-frequency dielectric ‘constant’, hydrodynamic and elastic forces contribute to a strongly diverging dielectric permittivity at low frequencies for heterogeneous anionic microstructures. The model also captures distinctive characteristics of experimentally reported impedance spectra for films bearing alternating layers of cationic and anionic charge, again highlighting the role of coupled hydrodynamic, elastic and electrical forces. Sufficiently thin and highly charged bilayers exhibit a notably low high-frequency conductivity. This is explained by strong low-frequency electrostatic polarisation and counter-ion release. The one-dimensional solutions computed herein provide a foundation for much more challenging computations in two and three dimensions.

In the interaction of water waves with marine structures, the interplay between wave diffraction and drag-induced dissipation is seldom, if ever, considered. In particular, linear hydrodynamic models, and extensions thereof through the addition of a quadratic force term, do not represent the change in amplitude of the waves diffracted and radiated to the far field, which should result from local energy dissipation in the vicinity of the structure. In this work, a series of wave flume experiments is carried out, whereby waves of increasing amplitude impinge upon a vertical barrier, extending partway through the flume width. As the wave amplitude increases, the effect of drag – which is known to increase quadratically with the flow velocity – is enhanced, thus allowing the examination of the far-field effect of drag-induced dissipation, in terms of wave reflection and transmission. A potential flow model is proposed, with a simple quadratic pressure drop condition through a virtual porous surface, located on the sides of the barrier (where dissipation occurs). Experimental results confirm that drag-induced dissipation has a marked effect on the diffracted flow, i.e. on wave reflection and transmission, which is appropriately captured in the proposed model. Conversely, when diffraction becomes dominant as the barrier width becomes comparable to the incoming wavelength, the diffracted flow must be accounted for in predicting drag-induced forces and dissipation.

Droplet spreading is ubiquitous and plays a significant role in liquid-based energy systems, thermal management devices and microfluidics. While the spreading of non-volatile droplets is quantitatively understood, the spreading and flow transition in volatile droplets remains elusive due to the complexity added by interfacial phase change and non-equilibrium thermal transport. Here we show, using both mathematical modelling and experiments, that the wetting dynamics of volatile droplets can be scaled by the spatial–temporal interplay between capillary, evaporation and thermal Marangoni effects. We elucidate and quantify these complex interactions using phase diagrams based on systematic theoretical and experimental investigations. A spreading law of evaporative droplets is derived by extending Tanner's law (valid for non-volatile liquids) to a full range of liquids with saturation vapour pressure spanning from $10^1$ to $10^4$ Pa and on substrates with thermal conductivity from $10^{-1}$ to $10^3\ {\rm W}\ {\rm m}^{-1}\ {\rm K}^{-1}$. In addition to its importance in fluid-based industries, the conclusions also enable a unifying explanation to a series of individual works including the criterion of flow reversal and the state of dynamic wetting, making it possible to control liquid transport in diverse application scenarios.

We consider the process of convective dissolution in a homogeneous and isotropic porous medium. The flow is unstable due to the presence of a solute that induces a density difference responsible for driving the flow. The mixing dynamics is thus driven by a Rayleigh–Taylor instability at the pore scale. We investigate the flow at the scale of the pores using Hele-Shaw type experiment with bead packs, two-dimensional direct numerical simulations and physical models. Experiments and simulations have been specifically designed to mimic the same flow conditions, namely matching porosities, high Schmidt numbers and linear dependency of fluid density with solute concentration. In addition, the solid obstacles of the medium are impermeable to fluid and solute. We characterise the evolution of the flow via the mixing length, which quantifies the extension of the mixing region and grows linearly in time. The flow structure, analysed via the centreline mean wavelength, is observed to grow in agreement with theoretical predictions. Finally, we analyse the dissolution dynamics of the system, quantified through the mean scalar dissipation, and three mixing regimes are observed. Initially, the evolution is controlled by diffusion, which produces solute mixing across the initial horizontal interface. Then, when the interfacial diffusive layer is sufficiently thick, it becomes unstable, forming finger-like structures and driving the system into a convection-dominated phase. Finally, when the fingers have grown sufficiently to touch the horizontal boundaries of the domain, the mixing reduces dramatically due to the absence of fresh unmixed fluid. With the aid of simple physical models, we explain the physics of the results obtained numerically and experimentally. The solute evolution presents a self-similar behaviour, and it is controlled by different length scales in each stage of the mixing process, namely the length scale of diffusion, the pore size and the domain height.

We present direct numerical simulations of the splashing process between two cylindrical liquid rims. This belongs to a class of impact and collision problems with a wide range of applications in science and engineering, and motivated here by splashing of breaking ocean waves. Interfacial perturbations with a truncated white noise frequency profile are introduced to the rims before their collision, whose subsequent morphological development is simulated by solving the two-phase incompressible Navier–Stokes equation with the adaptive mesh refinement technique, within the Basilisk software environment. We first derive analytical solutions predicting the unsteady interfacial and velocity profiles of the expanding sheet forming between the two rims, and develop scaling laws for the evolution of the lamella rim under capillary deceleration. We then analyse the formation and growth of transverse ligaments ejected from the lamella rims, which we find to originate from the initial corrugated geometry of the perturbed rim surface. Novel scaling models are proposed for predicting the decay of the ligament number density due to the ongoing ligament merging phenomenon, and found to agree well with the numerical results presented here. The role of the mechanism in breaking waves is discussed further and necessary next steps in the problem are identified.

This paper reports on the efficacy of the Görtler number in scaling the laminar-turbulent boundary-layer transition on rotating cones facing axial inflow. Depending on the half-cone angle $\psi$ and axial flow strength, the competing centrifugal and cross-flow instabilities dominate the transition. Traditionally, the flow is evaluated by using two parameters: the local meridional Reynolds number $Re_l$ comparing the inertial versus viscous effects and the local rotational speed ratio $S$ accounting for the boundary-layer skew. We focus on the centrifugal effects, and evaluate the flow fields and reported transition points using Görtler number based on the azimuthal momentum thickness of the similarity solution and local cone radius. The results show that Görtler number alone dominates the late stages of transition (maximum amplification and turbulence onset phases) for a wide range of investigated $S$ and half-cone angle ($15^{\circ } \leq \psi \leq 50^{\circ }$), although the early stage (critical phase) seems to be not determined by the Görtler number alone on the broader cones ($\psi =30^{\circ }$ and $50^{\circ }$) where the primary cross-flow instability dominates the flow. Overall, this indicates that the centrifugal effects play an important role in the boundary-layer transition on rotating cones in axial inflow.

Polymer chains in turbulent flows are generally modelled as dumbbells, i.e. two beads joined by a nonlinear spring. The dumbbell only maps a single spatial configuration, described by the polymer end-to-end vector, thus a multi-bead FENE (finitely extensible nonlinear elastic) chain seems a natural improvement for a more accurate characterisation of the polymer spatial conformation. At a large Weissenberg number, a comparison with the more accurate Kuhn chain reveals that the multi-bead FENE chain drastically overestimates the probability of folded configurations. Surprisingly, the dumbbell turns out to be the only meaningful bead-spring model to coarse-grain a polymer macromolecule in turbulent pipe flows.

We develop a mean-field model to examine the stability of a ‘quasi-2-D suspension’ of elongated particles embedded within a viscous membrane. This geometry represents several biological and synthetic settings, and we reveal mechanisms by which the anisotropic mobility of particles interacts with long-ranged viscous membrane hydrodynamics. We first show that a system of slender rod-like particles driven by a constant force is unstable to perturbations in concentration – much like sedimentation in analogous 3-D suspensions – so long as membrane viscous stresses dominate. However, increasing the contribution of viscous stresses from the surrounding 3-D fluid(s) suppresses such an instability. We then tie this result to the hydrodynamic disturbances generated by each particle in the plane of the membrane and show that enhancing subphase viscous contributions generates extensional fields that orient neighbouring particles in a manner that draws them apart. The balance of flux of particles aggregating versus separating then leads to a wave number selection in the mean-field model.

Tom et al. (J. Fluid Mech., vol. 947, 2022, p. A7) investigated the impact of two-way coupling (2WC) on particle settling velocities in turbulence. For the limited parameter choices explored, it was found that (i) 2WC substantially enhances particle settling compared with the one-way coupled case, even at low mass loading $\varPhi _m$ and (ii) preferential sweeping remains the mechanism responsible for the particles settling faster than the Stokes settling velocity in 2WC flows. However, significant alterations to the flow structure that can occur at higher mass loadings mean that the conclusions from Tom et al. (J. Fluid Mech., vol. 947, 2022, p. A7) may not generalise. Indeed, even under very low mass loadings, the influence of 2WC on particle settling might persist, challenging the conventional assumption. We therefore explore a much broader portion of the parameter space, with simulations covering cases where the impact of 2WC on the global fluid statistics ranges from negligible to strong. We find that, even for $\varPhi _m=7.5\times 10^{-3}$, 2WC can noticeably increase the settling for some choices of the Stokes and Froude numbers. When $\varPhi _m$ is large enough for the global fluid statistics to be strongly affected, we show that preferential sweeping continues to be the mechanism that enhances particle settling rates. Finally, we compare our results with previous numerical and experimental studies. While in some cases there is reasonable agreement, discrepancies exist even between different numerical studies and between different experiments. Future studies must seek to understand this before the discrepancies between numerical and experimental results can be adequately addressed.

An additional distant wall is known to highly alter the jetting scenarios of wall-proximal bubbles. Here, we combine high-speed photography and axisymmetric volume of fluid (VoF) simulations to quantitatively describe its role in enhancing the micro-jet dynamics within the directed jet regime (Zeng et al., J. Fluid Mech., vol. 896, 2020, A28). Upon a favourable agreement on the bubble and micro-jet dynamics, both experimental and simulation results indicate that the micro-jet velocity increases dramatically as $\eta$ decreases, where $\eta =H/R_{max}$ is the distance between two walls $H$ normalized by the maximum bubble radius $R_{max}$. The mechanism is related to the collapsing flow, which is constrained by the distant wall into a reverse stagnation-point flow that builds up pressure near the bubble's top surface and accelerates it into micro-jets. We further derive an equation expressing the micro-jet velocity $U_{jet}=87.94\gamma ^{0.5}(1+(1/3)(\eta -\lambda ^{1.2})^{-2})$, where ${\gamma =d/R_{max}}$ is the stand-off distance to the proximal wall with $d$ the distance between the initial bubble centre and the wall, $\lambda =R_{y,m}/R_{max}$ with $R_{y,m}$ the distance between the top surface and the proximal wall at the bubble's maximum expansion. Viscosity has a minimal impact on the jet velocity for small $\gamma$, where the pressure buildup is predominantly influenced by geometry.

The growth and characteristics of linear, oblique instabilities on a highly swept fin on a straight cone in Mach 6 flow are examined. Large streamwise pressure gradients cause doubly inflected cross-flow profiles and reversed flow near the wall, which necessitates using the harmonic linearized Navier–Stokes equations. The cross-flow instability is responsible for the most-amplified disturbances, however, not all disturbances show typical cross-flow characteristics. Distinct differences in perturbation structure are shown between small ($\sim$3–5 mm) and large ($\sim$10 mm) wavelength disturbances at the unit Reynolds number $Re' = 11 \times 10^6$ m$^{-1}$. As a result, amplification measurements based solely on wall quantities bias a most-amplified disturbance assessment towards larger wavelengths and lower frequencies than would otherwise be determined by an off-wall total-energy approach. A spatial-amplification energy-budget analysis demonstrates (i) that wall-normal Reynolds-flux terms dictate the local growth rate, despite other terms having a locally larger magnitude and (ii) that the Reynolds-stress terms are responsible for large-wavelength disturbances propagating closer to the wall compared with small-wavelength disturbances. Additionally, the effect of free-stream unit Reynolds number and small yaw angles on the perturbation amplification and energy budget is considered. At a higher Reynolds number ($Re' = 22 \times 10^6$ m$^{-1}$), the most-amplified wavelength shrinks. Perturbations do not behave self-similarly in the thinner boundary layer, and the shift in most-amplified wavelength is due to decreased dissipation relative to the lower-Reynolds-number case. Small yaw angles produce a streamwise shift in the boundary layer and disturbance amplification. The yaw results quantify a potential uncertainty source in experiments and flight.

The two-dimensional, steady, homogeneous flow field proposed by Astarita (J. Rheol., vol. 35, 1991, pp. 687–689) is studied for a range of viscoelastic constitutive equations of differential form including the models due to Oldroyd (the upper and lower convected Maxwell; UCM/LCM), Phan-Thien and Tanner (simplified, linear form; sPTT) and Giesekus. As the flow is steady and homogeneous, the sPTT model results also give the FENE-P model solutions via a simple transformation of parameters. The flow field has the interesting feature that a scalar parameter may be used to vary the flow ‘type’ continuously from solid-body rotation to simple shearing to planar extension whilst the rate of deformation tensor $\boldsymbol{\mathsf{D}}$ remains constant (i.e. independent of flow type). The response of the models is probed in order to determine how a scalar ‘viscosity’ function may be rigorously constructed which includes flow-type dependence. We show that for most of these models – the Giesekus being the exception – the first and second invariants of the resulting extra stress tensor are linearly related, and for models based on the upper convected derivative, this link is simply via a material property, i.e. half the modulus. By defining a frame-invariant coordinate system with respect to the eigenvectors of $\boldsymbol{\mathsf{D}}$, we associate a ‘viscosity’ for each of the flows to a deviatoric stress component and show how this quantity varies with the flow-type parameter. For elliptical motions, rate thinning is always observed and all models give essentially the UCM response. For strong flows, i.e. flow types containing at least some extension, thickening occurs and only a small element of extension is required to remove any shear thinning inherent in the model (e.g. as occurs in steady simple shearing for the sPTT/Giesekus models). Finally, a functional form of a viscosity equation which could incorporate flow type, but be otherwise inelastic, the so-called GNFFTy (generalised Newtonian fluid model incorporating flow type, pronounced ‘nifty’), is proposed. In the frame-invariant coordinate system proposed, this model is also capable of capturing normal-stress differences.

There exists much research examining the role of surface-attached air bubbles in drag reduction. Most of this literature considers isothermal flows and so ignores temperature differences, e.g. with the solid boundary. Here, we relax this assumption and ask whether surface-attached air bubbles may prove useful as thermal insulators, e.g. when the solid temperature differs from that of the cargo liquid (water). Theoretical and numerical solutions, e.g. for the variation of the Nusselt number with bubble thickness, are presented for cases characterized by a uniform surface heat flux (USF). We examine channel and pipe flow geometries, and consider instances where the net mass flow rate within the (continuous) air bubble is zero or non-zero. When the thermal boundary condition is changed to uniform surface temperature (UST), our analysis limits attention to numerical solutions. We identify and discuss a remarkable connection between the drag reduction problem and the USF thermal insulation problem: the proportional change of water temperature with bubble thickness is identical to the proportional change of drag. Also, and although our analysis is conducted in the ‘perfect plastron limit’, we can, e.g. by evaluating hydrodynamic and thermal slip lengths, contrast our work against related studies where heat transfer occurs through the ridges or pillars that affix the air layer in place. This comparison indicates that the oft-applied adiabatic interface assumption may prove restrictive in some regions of the parameter space. We conclude by examining the implications of our work in the context of UST micro-channels, which are relevant to various lab-on-a-chip technologies.

The twin-jet configuration allows two different scenarios to close the screech feedback. For each jet, there is one loop involving disturbances which originate in that jet and arrive at its own receptivity point in phase (self-excitation). The other loop is associated with free-stream acoustic waves that radiate from the other jet, reinforcing the self-excited screech (cross-excitation). In this work, the role of the free-stream acoustic mode and the guided-jet mode as a closure mechanism for twin rectangular jet screech is explored by identifying eligible points of return for each path, where upstream waves propagating from such a point arrive at the receptivity location with an appropriate phase relation. Screech tones generated by these jets are found to be intermittent with an out-of-phase coupling as a dominant coupling mode. The instantaneous phase difference between the twin jets computed by the Hilbert transform suggests that a competition between out-of-phase and in-phase coupling is responsible for the intermittency. To model wave components of the screech feedback while ensuring perfect phase locking, an ensemble average of leading spectral proper orthogonal decomposition modes is obtained from several segments of large-eddy simulation data that correspond to periods of invariant phase difference between the two jets. Each mode is then extracted by retaining relevant wavenumber components produced via a streamwise Fourier transform. Spatial cross-correlation analysis of the resulting modes shows that most of the identified points of return for the cross-excitation are synchronised with the guided jet mode self-excitation, supporting that it is preferred in closing rectangular twin-jet screech coupling.