In this paper, curved detonation equations with gradients for the pre-wave and post-wave are constructed followed by analysis, verification and applications. The study focuses on shock induced chemical reaction such as detonation, with the energy effect for the main attention. Equations consider both planar and transverse curvature to accommodate both planar and axisymmetric flow problems. Influence coefficients are derived and used to analyse the effect of energy and curvature on the post-wave gradient. Good agreement with the simulation results demonstrates that the equations presented in this paper can calculate various post-wave gradients accurately. After verification, the equations can be applied to applications, including not only solution and analysis but also in the inverse design. First, the method can be applied with polar analysis to provide a new perspective and higher order parameters for the study of detonation. Second, the equations can be used for the capture of detonation waves, where both planar and axisymmetric examples show better performance. Furthermore, the equations can be used in the inverse design of detonation waves in combination with the method of characteristics, which is one of the unique benefits of the present equations.

The dynamics of a shock-induced separation unit generated by a 20$^\circ$ sharp fin placed on a cylindrical surface in a Mach 2.5 flow was investigated. Specifically, the present work investigated the mechanisms that govern the mid-frequency range of separation shock unsteadiness in the fin shock wave–boundary layer interaction (SBLI) unit. Two-dimensional pressure fields were obtained over the cylinder surface spanning the entire fin SBLI unit using high-bandwidth pressure-sensitive paint at 40 kHz imaging rate that allowed probing the low- through mid-frequency ranges of the separation shock unsteadiness. The mean pressure field showed a progressive weakening of the separation shock with downstream distance, which is an artifact of the three-dimensional relief offered by the curved mounting surface. The root-mean-square (r.m.s.) pressure field exhibited a banded structure with elevated $p_{r.m.s.}$ levels beneath the intermittent region, separation vortex and adjacent to the fin root. The power spectral density (PSD) of the surface pressure fluctuations obtained beneath the intermittent region revealed that the separation shock oscillations exhibited the mid-frequency content over the majority of its length. Interestingly, neither the PSD nor the length of the intermittent region varied noticeably with downstream distance, revealing a constant separation shock foot velocity along the entire SBLI. The pressure fluctuation PSD beneath the separation vortex also exhibited the broadband peak at the mid-frequency range of the separation shock motions over the majority of its length within the measurement domain. By contrast, the region adjacent to the fin root exhibited pressure oscillations at a substantially lower frequency compared with the separation shock and the separation vortex. Two-point coherence and cross-correlation analysis provided unique insights into the critical sources and mechanisms that drive the separation shock unsteadiness. The separation vortex and separation shock dynamics were found to be driven by a combination of convecting perturbations that originated from the vicinity of the fin leading edge and the local interactions of the separated flow with the incoming boundary layer. The boundary layer locally strengthened or weakened the convecting pressure perturbations depending on the local momentum fluctuations within the boundary layer.

The present study offers a twofold contribution on counter-gradient transport (CGT) of turbulent scalar flux. First, by examining turbulent scalar mixing through synchronized particle image velocimetry and planar laser-induced fluorescence on an inclined jet in cross-flow, we clarify the previously unexplained phenomenon of CGT, revealing key flow structures, their spatial distribution and modelling implications. Statistical analysis identifies two distinct CGT regions: local cross-gradient transport in the windward shear layer and non-local effects near the wall after injection. These behaviours are driven by specific flow structures, namely Kelvin–Helmholtz vortices (local) and wake vortices (non-local), suggesting that scalar flux can be decomposed into a gradient-type term for gradient diffusion and a term for large-eddy stirring. Second, we propose a new approach for reconstruction of turbulent mean flow and scalar fields using continuous adjoint data assimilation (DA). By rectifying model-form errors through anisotropic correction under observational constraints, our DA model minimizes discrepancies between experimental measurements and numerical predictions. As expected, the introduced forcing term effectively identifies regions where traditional models fall short, particularly in the jet centreline and near-wall regions, thereby enhancing the accuracy of the mean scalar field. These enhancements occur not only within the observation region but also in unseen regions, underscoring present DA approach's reliability and practicality for reproducing mean flow behaviours from limited data. These findings lay a solid foundation for adjoint-based model-consistent data-driven methods, offering promising potential for accurately predicting complex flow scenarios like film cooling.

The research on elasto-inertial turbulence (EIT), a new type of turbulent flow, has reached the stage of identifying the minimal flow unit (MFU). On this issue, direct numerical simulations of FENE-P fluid flow in two-dimensional channels with variable sizes are conducted in this study. We demonstrate with the increase of channel length that the simulated flow experiences several different flow patterns, and there exists an MFU for EIT to be self-sustained. At Weissenberg number ($Wi$) higher than the one required to excite EIT, when the channel length is relatively small, a steady arrowhead regime (SAR) flow structure and a laminar-like friction coefficient is achieved. However, as the channel length increases, the flow can fully develop into EIT characterized with high flow drag. Close to the size of the MFU, the simulated flow behaves intermittently between the SAR state with low drag and EIT state with high drag. The flow falling back to ‘laminar flow’ is caused by the insufficient channel size below the MFU. Furthermore, we give the relationship between the value of the MFU and the effective $Wi$, and explain its physical reasons. Moreover, the intermittent flow regime obtained based on the MFU gives us an opportunity to look into the origin and exciting process of EIT. Through capturing the onset process of EIT, we observed that EIT originates from the sheet-like extension structure located near the wall, which is maybe related to the wall mode rather than the centre mode. The fracture and regeneration of this sheet-like structure is the key mechanism for the self-sustaining of EIT.

Heavy particles suspended in turbulent flow possess inertia and are ejected from violent vortical structures by centrifugal forces. Once piled up along particle paths, this small-scale mechanism leads to an effective large-scale drift. This phenomenon, known as ‘turbophoresis’, causes particles to leave highly turbulent regions and migrate towards calmer regions, explaining why particles transported by non-homogeneous flows tend to concentrate near the minima of turbulent kinetic energy. It is demonstrated here that turbophoretic effects are just as crucial in statistically homogeneous flows. Although the average turbulent activity is uniform, instantaneous spatial fluctuations are responsible for inertial-range inhomogeneities in the particle distribution. Direct numerical simulations are used to probe particle accelerations, specifically how they correlate to local turbulent activity, yielding an effective coarse-grained dynamics that accounts for particle detachment from the fluid and ejection from excited regions through a space- and time-dependent non-Fickian diffusion. This leads to cast fluctuations in particle distributions in terms of a scale-dependent Péclet number ${\textit {Pe}}_\ell$, which measures the importance of turbulent advection compared with inertial turbophoresis at a given scale $\ell$. Multifractal statistics of energy dissipation indicate that $ {\textit {Pe}}_\ell \sim \ell ^\delta /\tau _{p}$ with $\delta \approx 0.84$. Numerical simulations support this behaviour and emphasise the relevance of the turbophoretic Péclet number in characterising how particle distributions, including their radial distribution function, depends on $\ell$. This approach also explains the presence of voids with inertial-range sizes, and the fact that their volumes have a non-trivial distribution with a power-law tail $p(\mathcal {V}) \propto \mathcal {V}^{-\alpha }$, with an exponent $\alpha$ that tends to 2 as ${\textit {Pe}}_\ell \to 0$.

This study investigates the influence of surface wave characteristics, specifically wave steepness and directional spreading, on intermittency in deep-water gravity wave turbulence through long-term numerical simulations of three-dimensional potential fully nonlinear periodic gravity waves. We conducted this investigation by estimating the scaling exponent of the surface elevation under different sea state conditions. With our numerical methods, we were able to evaluate the scaling exponents of the structure-function up to 12th order. The observed increased intermittency in directionally narrower sea states and in higher steepness conditions aligns with known effects of quasi-resonant wave–wave interactions and wave breaking. Comparative analyses reveal that both the conventional She–Leveque model and the multifractal models, also used to represent intermittency in wave turbulence of a different nature, exhibit a strong correlation in this study. This observation underscores the universality of intermittency phenomena within wave turbulence.

The formation mechanism for the stopping vortex ring (SVR) and its effects on the development of starting jets have been systematically investigated. The radial inward flow near the nozzle exit, arising from the pressure difference caused by the deceleration of starting jets, is considered to be the main contributing factor to the formation of the SVR. The formation process can generally be divided into (i) the rapid accumulation stage ($t_d^*\leq 1$) and (ii) the development stage ($t_d^*>1$), where $t_d^*$ is the formation time defined by the duration of the deceleration stage. For starting jets with different $(L/D)_d$, the final circulation value and circulation growth rate of the SVR can be scaled by $[(L/D)_d]^{-0.5}$ and $[(L/D)_d]^{-1.5}$, respectively. Here $(L/D)_d$ represents the stroke ratio during the deceleration stage. Analysing the temporal evolution of fluid parcels in the vicinity of the nozzle exit reveals that SVR entrains fluid from both inside and outside of the nozzle. Additionally, the influence of the SVR on the leading vortex ring and the trailing jet has been examined, with particular attention to its effects on the propulsive performance of the starting jet. The SVR affects the profiles of axial velocity and gauge pressure at the nozzle exit, thereby enhancing the generation of total thrust during the deceleration stage. Analysis has shown that depending on the deceleration rate, SVR can enhance the average velocity thrust by at least $10\,\%$ and compensate for up to a $60\,\%$ reduction in pressure thrust due to deceleration.

This paper presents advances towards the data-based control of periodic oscillator flows, from their fully developed regime to their equilibrium stabilized in closed loop, with linear time-invariant (LTI) controllers. The proposed approach directly builds upon the iterative method of Leclercq et al. (J. Fluid Mech., vol. 868, 2019, pp. 26–65) and provides several improvements for an efficient online implementation, aimed at being applicable in experiments. First, we use input–output data to construct an LTI mean transfer functions of the flow. The model is subsequently used for the design of an LTI controller with linear quadratic Gaussian synthesis, which is practical to automate online. Then, using the controller in a feedback loop, the flow shifts in phase space and oscillations are damped. The procedure is repeated until equilibrium is reached, by stacking controllers and performing balanced truncation to deal with the increasing order of the compound controller. In this article, we illustrate the method for the classic flow past a cylinder at Reynolds number $Re=100$. Care has been taken such that the method may be fully automated and hopefully used as a valuable tool in a forthcoming experiment.

Resolvent analysis provides a framework to predict coherent spatio-temporal structures of the largest linear energy amplification, through a singular value decomposition (SVD) of the resolvent operator, obtained by linearising the Navier–Stokes equations about a known turbulent mean velocity profile. Resolvent analysis utilizes a Fourier decomposition in time, which has thus far limited its application to statistically stationary or time-periodic flows. This work develops a variant of resolvent analysis applicable to time-evolving flows, and proposes a variant that identifies spatio-temporally sparse structures, applicable to either stationary or time-varying mean velocity profiles. Spatio-temporal resolvent analysis is formulated through the incorporation of the temporal dimension to the numerical domain via a discrete time-differentiation operator. Sparsity (which manifests in localisation) is achieved through the addition of an $l_1$-norm penalisation term to the optimisation associated with the SVD. This modified optimisation problem can be formulated as a nonlinear eigenproblem and solved via an inverse power method. We first showcase the implementation of the sparse analysis on a statistically stationary turbulent channel flow, and demonstrate that the sparse variant can identify aspects of the physics not directly evident from standard resolvent analysis. This is followed by applying the sparse space–time formulation on systems that are time varying: a time-periodic turbulent Stokes boundary layer and then a turbulent channel flow with a sudden imposition of a lateral pressure gradient, with the original streamwise pressure gradient unchanged. We present results demonstrating how the sparsity-promoting variant can either change the quantitative structure of the leading space–time modes to increase their sparsity, or identify entirely different linear amplification mechanisms compared with non-sparse resolvent analysis.

We present direct numerical simulations of a three-layer Rayleigh–Taylor instability (RTI) problem with a configuration based on the experiments of Suchandra & Ranjan (J. Fluid Mech., vol. 974, 2023, A35) and Jacobs & Dalziel (J. Fluid Mech., vol. 542, 2005, pp. 251–279). The problem consists of a layer of light fluid between two layers of heavy fluid with an Atwood number of 0.3. These simulations are first validated through comparison with available experimental data. The validated simulations are then utilized to analyse statistics in this three-component flow. First, length scales are examined utilizing spectra and two-point spatial correlations of velocity and species concentration fluctuations. Next, joint probability density functions (p.d.f.s) of species concentration are compared against several model p.d.f.s representing generalizations of the bivariate beta distribution. Notably, the joint p.d.f.s do not appear to be accurately described by a Dirichlet distribution, indicating the marginal distributions do not conform to a beta distribution. Finally, similarity of the present configuration to three-component mixing found in inertial confinement fusion (ICF) applications is exploited to develop and validate an improved model for the impact of multicomponent mixing on thermonuclear (TN) reaction rates. A single time instant from the present simulations is chosen for a TN burn calculation under the hypothetical assumption of ICF materials and temperatures. Total TN output from this second calculation is then compared against the prediction of the improved model. The new model is found to accurately predict TN reaction rates in both premixed and non-premixed configurations.

We report the dynamics of a droplet levitated in a multi-emitter, single-axis acoustic levitator. The deformation and atomisation behaviour of the droplet in the acoustic field displays myriad complex phenomena, in a series of events. These include the primary breakup of the droplet, wherein it exhibits stable levitation, deformation, sheet formation and equatorial atomisation, followed by its secondary breakup, which could be of various types such as umbrella, bag, bubble or multi-stage breakup. A large number of tiny atomised droplets, formed as a result of the primary and secondary breakup, can remain levitated in the acoustic levitator and exhibit aggregation and coalescence. The visualisation of the interfacial instabilities on the surface of the liquid sheet using both side- and top-view imaging is presented. An approximate size distribution of the atomised droplets is also provided. The stable levitation of the droplet is due to a balance of acoustic and gravitational forces while the resulting ellipsoidal shape of the droplet is a consequence of the balance of the deforming acoustic force and the restoring capillary force. Stronger acoustic forces can no longer be balanced by capillary forces, resulting in a highly flattened droplet, with a thin liquid sheet at the edge (equatorial region). The thinning of the sheet is caused by the differential acceleration induced by the increasing pressure difference between the poles and the equator as the sheet deforms. When the sheet thickness reduces to of the order of a few microns, Faraday waves develop at the thinnest region (preceding the rim), which causes the generation of tiny-sized droplets that are ejected perpendicular to the sheet. The corresponding hole formation results in a perforated sheet that causes the detachment of the annular rim, which breaks due to Rayleigh–Plateau (RP) instability. The radial ligaments generated in the sheet, possibly due to Rayleigh–Taylor (RT) instability, break into droplets of different sizes. The secondary breakup exhibits Weber number dependency and includes umbrella, bag, bubble or multi-stage types, ultimately resulting in the complete atomisation of the droplets. Both the primary and the secondary breakup of the droplet involve interfacial instabilities such as Faraday, Kelvin–Helmholtz, RT and RP and are well supported by visual evidence.

We report on Lagrangian statistics of turbulent Rayleigh–Bénard convection under very different conditions. For this, we conducted particle tracking experiments in a $H=1.1$-m-high cylinder of aspect ratio $\varGamma =1$ filled with air (Pr = 0.7), as well as in two rectangular cells of heights $H=0.02$ m ($\varGamma =16$) and $H=0.04$ m ($\varGamma =8$) filled with water (Pr = 7.0), covering Rayleigh numbers in the range $10^6\le {\textit {Ra}}\le 1.6\times 10^9$. Using the Shake-The-Box algorithm, we have tracked up to 500 000 neutrally buoyant particles over several hundred free-fall times for each set of control parameters. We find the Reynolds number to scale at small Ra (large Pr) as $ {\textit{Re}} \propto {\textit{Ra}}^{0.6}$. Further, the averaged horizontal particle displacement is found to be universal and exhibits a ballistic regime at small times and a diffusive regime at larger times, for sufficiently large $\varGamma$. The diffusive regime occurs for time lags larger than $\tau _{co}$, which is the time scale related to the decay of the velocity autocorrelation. Compensated as $\tau _{co} {\textit {Pr}}^{-0.3}$, this time scale is universal and rather independent of $ {\textit {Ra}}$ and $\varGamma$. We have also investigated the Lagrangian velocity structure function $S^2_i(\tau )$, which is dominated by viscous effects for times smaller than the Kolmogorov time $\tau _\eta$ and hence $S^2_i\propto \tau ^2$. For larger times we find a novel scaling for the different components with exponents smaller than what is expected in the inertial range of homogeneous isotropic turbulence without buoyancy. Studying particle-pair dispersion, we find a Batchelor scaling (${\propto }\,t^2$) on small time scales, diffusive scaling (${\propto }\,t$) on large time scales and Richardson-like scaling (${\propto }\,t^3$) for intermediate time scales.

The gas dynamics of shock-induced gas filtration through densely packed granular columns with vastly varying shock intensity and the structural parameters are numerically investigated using a coupled Eulerian–Lagrangian approach. The results shed fundamental light on the thermal effects of the shock-induced gas filtration manifested by a distinctive self-heating hot gas layer traversing the medium. The characteristics of the thermal effects in terms of the thermal intensity and uniformity are found to vary with the shock Mach number, Ms, and the filtration coefficient of the granular media, Π. As the incident shock transitions from weak to strong, and (or) the filtration coefficient increases from O(10−5) to O(104), the heating mechanisms transition between three distinct heating modes. A phase diagram of heating modes is established on the parameter space (Ms, Π), which enables us to predict the characteristics of the thermal effect in different shock-induced gas filtrations. The thermal effects markedly accelerate the pressure diffusion due to the additional heat influx when the time scale of the former is smaller than or comparable to the latter. Based on the contour map displaying the coupling degree of the thermal effects and the pressure diffusion, we identify a decoupling criterion whereby the isothermal assumption holds if only the pressure diffusion is concerned. The thermal effects may well bring about considerable thermal shocks which pose a great threat to the integrity of the solid skeleton and further reduce the overall shock resistance performance of the porous media.

Elongated floaters drifting in propagating water waves slowly rotate towards a preferential orientation with respect to the direction of incidence. In this paper we study this phenomenon in the small floater limit $k L_x < 1$, with $k$ the wavenumber and $L_x$ the floater length. Experiments show that short and heavy floaters tend to align longitudinally, along the direction of wave propagation, whereas longer and lighter floaters align transversely, parallel to the wave crests and troughs. We show that this preferential orientation can be modelled using an inviscid Froude–Krylov model, ignoring diffraction effects. Asymptotic theory, in the double limit of a small wave slope and small floater, suggests that preferential orientation is essentially controlled by the non-dimensional number $F = k L_x^2 / \bar {h}$, with $\bar {h}$ the equilibrium submersion depth. Theory predicts the longitudinal-transverse transition for homogeneous parallelepipeds at the critical value $F_c = 60$, in fair agreement with the experiments that locate $F_c = 50 \pm 15$. Using a simplified model for a thin floater, we elucidate the physical mechanisms that control the preferential orientation. The longitudinal equilibrium for $F< F_c$ originates from a slight asymmetry between the buoyancy torque induced by the wave crests, that favours the longitudinal orientation, and that induced by the wave troughs, that favours the transverse orientation. The transverse equilibrium for $F>F_c$ arises from the variation of the submersion depth along the long axis of the floaters, which significantly increases the torque in the trough positions, when the tips are more submersed.

Droplet impingement on a heated substrate is the fundamental process underlying various technologies, ranging from spray cooling to inkjet printing. Understanding the coupled effects of fluid dynamics and heat transfer patterns during droplet jumping, boiling and evaporation, which determine the outcomes of the impingement process, is essential. Here, we developed two-colour planar laser-induced fluorescence and micro-particle image velocimetry technologies to measure quantitatively the velocity and temperature distributions inside the droplet during an impingement process with high temporal and spatial resolution. With our novel measuring system, the hot spots at the solid–liquid interface are discovered for the first time. The influence of contact boiling on the droplet internal mixing, which impedes droplet recoiling and reduces the rebounding velocity, is discussed. A significant enhancement in heat absorption for partially rebounding droplets is discovered, where the impingement heat transfer rate is doubled compared to other vapour-layer-covered droplets. The scaling correlations of viscous dissipation rate and contact time of rebounding droplets, as well as the time variation of droplet temperature rise, are proposed. More detailed patterns inside droplets can be captured by these experimental methods, which will help to reveal more intrinsic mechanisms lying in thermally induced flow, complex fluids and droplet-impacting-based technologies.

Bounds on heat transfer have been the subject of previous studies concerning convection in the Boussinesq approximation: in the Rayleigh–Bénard configuration, the first result obtained by Howard (J. Fluid Mech., vol. 17, issue 3, 1963, pp. 405–432) states that the dimensionless heat flux $\textit {Nu}$ carried out by convection is such that $\textit {Nu} < (3/64 \ Ra)^{1/2}$ for large values of the Rayleigh number $Ra$, independently of the Prandtl number $Pr$. This is still the best-known upper bound, only with the prefactor improved to $\textit {Nu} -1 < 0.02634 \ Ra^{1/2}$ by Plasting & Kerswell (J. Fluid Mech., vol. 477, 2003, pp. 363–379). In the present paper, this result is extended to compressible convection. An upper bound is obtained for the anelastic liquid approximation, which is similar to an anelastic model used in astrophysics based on a turbulent diffusivity for entropy. The anelastic bound is still scaling as $Ra^{1/2}$, independently of $Pr$, but depends on the dissipation number $\mathcal {D}$ and on the equation of state. For monatomic gases and large Rayleigh numbers, the bound is $\textit {Nu} < 25.8\, Ra^{{1}/{2}} / (1-\mathcal {D}/2 )^{{5}/{2}}$.

Flows over a disc are studied in a wind tunnel over incidence angles between $0^\circ \text { and }36^\circ$, a Reynolds number of $2.7 \times 10^4$ and rotational speed ratios of $0\unicode{x2013}10$. Smoke-wire visualization, particle image velocimetry and hot-film anemometry are employed. Two vortex shedding modes originating from the upstream surface of the disc are observed. The first is dominant at incidence angles up to ${\sim }21^\circ$. Beyond $21^\circ$, the second mode dominates. It appears as a soliton on the vortices and has a shedding frequency nearly twice that of the first at the highest incidence angle. The Strouhal number monotonically increases with incidence angle from approximately 0.2 to 0.4. Spectral analysis of the hot-film measurements confirms the findings from flow visualization experiments. Flows over the spinning disc generally mimic the stationary disc flows; however, centrifugal forces lead to cross-stream instability features that may be attributed to spiral wave instabilities intrinsic to the boundary layers in rotating flows. Velocity measurements are used to construct streamline patterns to compare with the smoke streaklines. The unsteadiness of the flows results in large variances. Mean strain rates are extracted from velocity data, where the fixed disc case at normal incidence compares well with theoretical predictions. The unsteady boundary layer thickness over the fixed disc, however, is approximately twice that predicted by theory for steady flow, stemming from the dominance of large unsteady vortices. Limited comparisons are made of the Strouhal numbers from experiments and numerical calculations in the literature.

We present the results of a combined experimental and theoretical investigation of different fluid sheet structures formed during the impingement of a laminar liquid jet on a vial, with a slightly larger diameter than the jet and filled with the same liquid. The present set-up produces all diverse fluid sheet structures, unlike previous experiments that required a deflector disk resulting in no-slip and no-penetration boundary conditions. The water sheet structures are classified into four regimes; regime I: pre-sheet; regime II: puffing, characterized by the periodic formation and destruction of the upward-rising water sheet, an interesting observation not reported earlier; regime III, steady upward, inverted umbrella-like sheet structures; and regime IV, the formation of downward, umbrella-like sheet structures, which can be either open or closed, classically referred to as ‘water bells’. The water sheet structures observed are governed by the non-dimensional parameters: the ratio of vial diameter to the jet diameter at impact ($X$), the capillary number ($Ca$), the Weber number ($We$) and the Froude number ($Fr$). The parametric spaces $X-Ca$, $X-We$ and $X-Fr$ exhibit the demarcation of the four regimes. A semi-empirical expression for the ejection angle of the liquid sheet, primarily responsible for different shapes, is derived in a control volume that provides a theoretical basis for the identified regime diagrams. The puffing water bells in regime II are found to be quasi-steady as the experimental trajectories are in good agreement with the steady-state theory. The rise time of puffing water bells that determines the puffing frequency has been modelled.

Even without breaking or wind influence, ocean surface waves are observed to produce turbulence in the water, possibly influencing ocean surface dynamics and air–sea interactions. Based on the water-side free-surface simulations, recent studies suggest that such turbulence is produced through the interaction between the waves and the near-surface Eulerian current associated with the viscous attenuation of waves. To clarify the dynamical role of the air–water interface in the turbulence production, the attenuating interfacial gravity waves were simulated directly using a newly developed two-phase wave-resolving numerical model. The air–water coupling enhanced the wave energy dissipation through the formation of a strong shear at the air-side viscous boundary layer. This led to an enhancement of the wave-to-current momentum transfer and the formation of the down-wave Eulerian mean sheared current, which is favourable for the CL2 instability responsible for the production of Langmuir circulations. As a result, the water-side turbulence grew stronger compared with the corresponding free surface (water-only) wave-resolving simulation. The evolution of the wave-averaged field was well reproduced with the Craik–Leibovich equation with the upper boundary condition provided with the virtual wave stress based on linear theory. The wave energy dissipation by air–water coupling plays a significant role in the quantitative understanding of the wave-induced turbulence at the laboratory and field scales.

We investigate the coupling effects of the two-phase interface, viscosity ratio and density ratio of the dispersed phase to the continuous phase on the flow statistics in two-phase Taylor–Couette turbulence at a system Reynolds number of $6\times 10^3$ and a system Weber number of 10 using interface-resolved three-dimensional direct numerical simulations with the volume-of-fluid method. Our study focuses on four different scenarios: neutral droplets, low-viscosity droplets, light droplets and low-viscosity light droplets. We find that neutral droplets and low-viscosity droplets primarily contribute to drag enhancement through the two-phase interface, whereas light droplets reduce the system's drag by explicitly reducing Reynolds stress due to the density dependence of Reynolds stress. In addition, low-viscosity light droplets contribute to greater drag reduction by further reducing momentum transport near the inner cylinder and implicitly reducing Reynolds stress. While interfacial tension enhances turbulent kinetic energy (TKE) transport, drag enhancement is not strongly correlated with TKE transport for both neutral droplets and low-viscosity droplets. Light droplets primarily reduce the production term by diminishing Reynolds stress, whereas the density contrast between the phases boosts TKE transport near the inner wall. Therefore, the reduction in the dissipation rate is predominantly attributed to decreased turbulence production, causing drag reduction. For low-viscosity light droplets, the production term diminishes further, primarily due to their greater reduction in Reynolds stress, while reduced viscosity weakens the density difference's contribution to TKE transport near the inner cylinder, resulting in a more pronounced reduction in the dissipation rate and consequently stronger drag reduction. Our findings provide new insights into the physics of turbulence modulation by the dispersed phase in two-phase turbulence systems.