Turbulent mixing in a supercritical CO$_2$ shear layer is examined using both experimental and numerical methods. Boundary conditions are selected to focus on the rarely studied near-critical regime, where thermophysical properties vary nonlinearly with respect to temperature and pressure. Experimental results are obtained via Raman spectroscopy and shadowgraphy, while numerical results are obtained via direct numerical simulation. The shear layer growth rate is found to be 0.2. Additionally, density profiles indicate a relaxation of density gradients between the mixed fluid and heavy fluid as the flow evolves downstream, which runs counter to existing supercritical shear layer data in the literature. The computational results identify significant anisotropy in the turbulence in the shear layer, which is discussed in terms of the development of regions of high density gradient magnitude. The Reynolds-averaged enstrophy budget at various streamwise locations indicates no significant dilatational or baroclinic contribution within the shear layer.

Surface gravity waves induce a drift on objects floating on the water’s surface. This study presents laboratory experiments investigating the drift of large two-dimensional floating objects on deep-water, unidirectional, regular waves, with wave steepness ranging from 0.04 to 0.31 (0.04 $\lt k{a_w}\lt$ 0.31, where $k$ is the wavenumber and $a_w$ the wave amplitude). The objects were carefully designed to have a rectangular cross-section with a constant aspect ratio; their size varied from 2.6 $\%$ to 27 $\%$ of the incident wavelength. We observed Lagrangian behaviour for small objects. Small and large objects exhibited fundamentally different drift behaviour at high compared with low wave steepness, with a regime shift observed at a certain size and wave steepness. The scaling of object drift with steepness depends on the relative size of the object. For small objects, drift scales with steepness squared, whereas drift becomes a linear function of steepness as the object size increases. For objects that are relatively large but smaller than 13–16% of a wavelength (low to high steepness), we provide experimental evidence supporting the mechanisms of drift enhancement recently identified by Xiao et al. (J. Fluid Mech., vol. 980, 2024, p. A27) and termed the ‘diffraction-modified Stokes drift’. This enhanced drift behaviour, compared with the theoretical Stokes drift for infinitely small fluid parcels, is attributed to changes in the objects’ oscillatory motion and local wave amplitude distribution (standing wave pattern) due to the presence of the object. In the case of larger objects, similar to Harms (J. Waterw. Port Coast. Ocean Eng., vol. 113(6), 1987, pp. 606–622), we relate the critical size at which drift is maximised to their vertical bobbing motion. We determine the domain of validity for both Stokes drift and the diffraction-modified Stokes drift model of Xiao et al. (J. Fluid Mech., vol. 980, 2024, A27) in terms of relative size and wave steepness and propose an empirical parametrisation based on our experimental data.

Reflection of a rightward-moving shock over a steady oblique shock, equivalent to a shock-on-shock interaction, is typically studied with post-formed shock waves. Law, Felthun and Skews (Shock Waves, vol. 13, 2003, pp. 381–394) reported post-formed expansion fan (PFEF) reflection for second-family incident shock. Here, we show that PFEF reflection also exists for first-family incident shock. We derive the critical condition for PFEF reflection in the shock speed Mach number and incident shock angle plane. Our findings indicate PFEF emergence near type post-I region. Numerical simulations reveal that PFEF with rising incident angle can intersect the incident shock, triple point or Mach stem, echoing the Hillier (J. Fluid Mech., vol. 575, 2007, pp. 399–424) three-type classification of shock–expansion fan interactions. The complex shock reflection pattern is essentially composed of an upstream structure linked to the moving shock wave, and a downstream structure linked to the unperturbed oblique shock wave. Under the conditions investigated, the upstream structure is characterized by a Mach reflection of the incident shock over the wall, potentially featuring a triple point formed within the Mach stem. Below this triple point, there is a curved segment of the Mach stem that is close to the wall. As the inclined angle increases, the curved shock may expand and interact with the incident shock, leading to an asymmetric regular reflection, which is a phenomenon that has not been observed previously. The downstream structure is a double $\lambda$ shock structure, with the lower shock resulting from the generation of recompression shock waves.

Local shearing motions in turbulence form small-scale shear layers, which are unstable to perturbations at approximately 30 times the Kolmogorov scale. This study conducts direct numerical simulations of passive-scalar mixing layers in a shear-free turbulent front to investigate mixing enhancements induced by such perturbations. The initial turbulent Reynolds number based on the Taylor microscale is $ Re_\lambda = 72$ or 202. The turbulent front develops by entraining outer fluid. Weak sinusoidal velocity perturbations are introduced locally, either inside or outside the turbulent front, or globally throughout the flow. Perturbations at this critical wavelength promote small-scale shear instability, complicating the boundary geometry of the scalar mixing layer at small scales. This increases the fractal dimension and enhances scalar diffusion outward from the scalar mixing layer. Additionally, the promoted instability increases the scalar dissipation rate and turbulent scalar flux at small scales, facilitating faster scalar mixing. The effects manifest locally; external perturbations intensify mixing near the boundary, while internal perturbations affect the entire turbulent region. The impact of perturbations is consistent across different Reynolds numbers when the amplitudes normalised by the Kolmogorov velocity are the same, indicating that even weaker perturbations can enhance scalar mixing at higher Reynolds numbers. The mean scalar dissipation rate increases by up to 50 %, even when the perturbation energy is only 2.5 % of the turbulent kinetic energy. These results underscore the potential to leverage small-scale shear instability for efficient mixing enhancement in applications such as chemically reacting flows.

The greatest challenge in pressure reconstruction from the measured velocity fields is that the error of material acceleration is significantly contaminated due to error propagation. Particularly for flows with moving boundaries, accurate boundary velocities are difficult to obtain due to error propagation, and a complex boundary processing technique is needed to treat the moving boundaries. The present work proposes a machine-learning-based method to determine the pressure for incompressible flows with moving boundaries. The proposed network consists of two neural networks: one network, named the boundary network, is used to track the Lagrangian boundary points; the other physics-informed neural network, named the flow network, is adopted to approximate the flow fields. These two networks are coupled by imposing boundary conditions. We further propose a new dynamic weight strategy for the loss terms to guarantee convergence and stability. The performance of the proposed method is validated by two examples: the flow over an oscillating cylinder and the flow around a swimming fish. The proposed method can accurately determine the pressure fields and boundary motion from synthetic particle image velocimetry (PIV) flow fields. Moreover, this method can also predict the boundary and pressure at a given instant without supervised data. Finally, this method was applied to reconstruct the pressure from the two-dimensional and three-dimensional PIV velocities of the left ventricle. All of the results indicate that the proposed method can accurately reconstruct the pressure fields for flows with moving boundaries and is a novel method for surface pressure estimation.

Wave impact on solid structures is a well-studied phenomenon, but almost exclusively for the case that the impacting liquid (e.g. water) is surrounded by a non-condensable gas (such as air). In this study we turn to wave impact in a boiling liquid, a liquid that is in thermal equilibrium with its own vapour, which is of key relevance to the transport of cryogenic liquids, such as liquified natural gas and liquid hydrogen in the near future. More specifically, we use the Atmosphere facility at MARIN, NL, to prepare water/water vapour systems at different temperatures along the vapour curve. Here, we perform wave impact experiments by generating a soliton in a flume contained within the autoclave of the facility. A bathymetry profile interacts with the soliton, leading to a breaking wave that impacts onto a vertical wall, where we measure the pressures occurring during impact by means of $100$ embedded pressure sensors. In boiling liquids, we report wave impact pressures that are up to two orders of magnitude larger than those measured in comparable water–air experiments. We trace these pressures back to the collapse of the entrapped vapour pocket, which we semi-quantitatively describe using a simplified hemicylindrical vapour bubble model, which is in good agreement with the experimental findings. Finally, this allows us to predict the relevance of our findings for the transport of cryogenic liquids in huge overseas carriers where wave impact due to sloshing is the dominant cause of hydrodynamic load of containment systems in cargo tanks.

Generalised two-dimensional (2-D) fluid dynamics is characterised by a relationship between a scalar field $q$, called generalised vorticity, and the stream function $\psi$,namely $q = (-\nabla ^2)^{\frac {\alpha }{2}} \psi$. We study the transition of cascades in generalised 2-D turbulence by systematically varying the parameter $\alpha$ and investigating its influential role in determining the directionality (inverse, forward or bidirectional) of these cascades. We derive upper bounds for the dimensionless dissipation rates of generalised energy $E_G$ and enstrophy $\Omega _G$ as the Reynolds number tends to infinity. These findings corroborate numerical simulations, illustrating the inverse cascade of $E_G$ and forward cascade of $\Omega _G$ for $\alpha \gt 0$, contrasting with the reverse behaviour for $\alpha \lt 0$. The dependence of dissipation rates on system parameters reinforces these observed transitions, substantiated by spectral fluxes and energy spectra, which hint at Kolmogorov-like scalings at large scales but discrepancies at smaller scales between numerical and theoretical estimates. These discrepancies are possibly due to non-local transfers, which dominate the dynamics as we go from positive to negative values of $\alpha$. Intriguingly, the forward cascade of $E_G$ for $\alpha \lt 0$ reveals similarities to three-dimensional turbulence, notably the emergence of vortex filaments within a 2-D framework, marking a unique feature of this generalised model.

An analytical theory is developed that describes acoustic microstreaming produced by the interaction of an oscillating gas bubble with a viscoelastic particle. The bubble is assumed to undergo axisymmetric oscillation modes, which can include radial oscillation, translation and shape modes. The oscillations of the particle are excited by the oscillations of the bubble. No restrictions are imposed on the ratio of the bubble and the particle radii to the viscous penetration depth and the separation distance, as well as on the ratio of the viscous penetration depth to the separation distance. Capabilities of the developed theory are illustrated by computational examples. The shear stress produced by the acoustic microstreaming on the particle’s surface is calculated. It is shown that this stress is much higher than the stress predicted by Nyborg’s formula (1958 J. Acoust. Soc. Am. 30, 329–339), which is commonly used to evaluate the time-averaged shear stress produced by a bubble on a rigid wall.

The description of the generation mechanism of impulse surface waves remains an important challenge in environmental fluid mechanics, owing to the need for a better understanding of large-scale phenomena such as landslide-generated tsunamis. In the present study, we investigated the generation phase of laboratory-scale water waves induced by the impulsive motion of a rigid piston, whose maximum velocity $U$ and total stroke $L$ are independently varied, as well as the initial liquid depth $h$. By doing so, the influence of two dimensionless numbers is studied: the Froude number $\mathrm {Fr}_p$ = $U/(gh)^{1/2}$, with $g$ the gravitational acceleration, and the relative stroke $\Lambda _p =L/h$ of the piston. During the constant acceleration phase of the vertical wall, a transient water bump forms and remains localised in the vicinity of the piston, for all investigated parameters. Experiments with a small relative acceleration $\gamma /g$, where $\gamma =U^2/L$, are well captured by a first-order potential flow theory established by Joo et al. (1990), which provides a fair estimate of the overall free surface elevation and the maximum wave amplitude reached at the contact with the piston. For large Froude numbers, however, wave breaking hinders the use of such an approach. In this case, an unsteady hydraulic jump theory is proposed, which accurately predicts the time evolution of the wave amplitude at the contact with the piston throughout the generation phase. At the end of the formation process, the dimensionless volume of the bump evolves linearly with $\Lambda _p$ and the wave aspect ratio is found to be governed, at first-order, by the relative acceleration $\gamma /g$. As the piston begins its constant deceleration, the water bump evolves into a propagating wave and several regimes such as dispersive, solitary-like and bore waves, as well as water jets are then reported and mapped in a phase diagram in the ($\mathrm {Fr}_p$, $\Lambda _p$) plane. While the transition from waves to water jets is observed if the typical acceleration of the piston is close enough to the gravitational acceleration $g$, the wave regimes are found to be mainly selected by the relative piston stroke $\Lambda _p$. On the other hand, the Froude number determines whether the generated wave breaks or not.

Estimation of near-wall turbulence in channel flow from outer observations is investigated using adjoint-variational data assimilation. We first consider fully resolved velocity data, starting at a distance from the wall. By enforcing the estimated flow to exactly satisfy the Navier–Stokes equations, we seek a statistically stationary turbulent state that reproduces the instantaneous outer measurements. Such an estimated state provides full access to the unknown near-wall turbulence, including the wall shear stresses and pressure. When the first observation is within 50 viscous units from the wall, the correlation coefficient between the true and estimated state exceeds 95 %. As the observations are further separated from the wall, at 90 viscous units, the accuracy of the assimilated wall stresses decreases to 40 % at the wall. This trend is nearly independent of the Reynolds number. The Fourier spectrum of the estimation error is qualitatively consistent with the coherence spectrum between the outer and the inner state variables: observed long wavelength structures in the outer flow have deeper coherence into the unobserved near-wall region and, therefore, the error is lowest at large scales. Nevertheless, the adjoint-variational approach provides a more rigorous quantification of the capacity to accurately predict the instantaneous near-wall turbulence from outer measurements. Lastly, we demonstrate the robustness of the estimation accuracy using filtered and sub-sampled outer observations.

An important parameter characterising the synchronisation of turbulent flows is the threshold coupling wavenumber. This study investigates the relationship between the threshold coupling wavenumber and the leading Lyapunov vector using large eddy simulations and the SABRA model. Various subgrid-scale stress models, Reynolds numbers and different coupling methods are examined. A new scaling relation is identified for the leading Lyapunov exponents in large eddy simulations, showing that they approximate those of filtered direct numerical simulations. This interpretation provides a physical basis for results related to the Lyapunov exponents of large eddy simulations, including those related to synchronisation. Synchronisation experiments show that the peak wavenumber of the energy spectrum of the leading Lyapunov vector coincides with the threshold coupling wavenumber, in large eddy simulations of box turbulence with standard Smagorinsky or dynamic mixed models as well as in the SABRA model, replicating results from direct numerical simulations of box turbulence. Although the dynamic Smagorinsky model exhibits different behaviour, the totality of the results suggests that the relationship is an intrinsic property of a certain class of chaotic systems. We also confirm that conditional Lyapunov exponents characterise the synchronisation process in indirectly coupled systems as they do in directly coupled ones, with their values insensitive to the nature of the master flow. These findings advance the understanding of the role of the Lyapunov vector in the synchronisation of turbulence.

The reflection of multiple incident shock waves that converge to a single point on the reflecting surface is studied in this paper. The number of the incident shocks, denoted $K$, is arbitrary. The interaction between the reflected shock of one incident shock and the other incident shocks may produce various possible configurations, such as type-I, type-II and type-IV shock interferences. The number of possible reflection configurations is shown to be an exponential function of ($K-1$) with base 2. The possibility of pre-, middle- and post-Mach reflections, which means Mach reflection occurs for the first, middle and last incident shock, is revealed through numerical simulation for $K=3$. For the particular case where the incident shocks are produced by equal variation of wedge surface deflection, the conventional von Neumann condition and detachment condition for the $k\mathrm{th}$ incident shock to have Mach reflection are derived. It is shown that the von Neumann condition for regular reflection is lowered and the detachment condition for Mach reflection is elevated as $k$ increases. The shock reflection patterns for $ K=1,2,\ldots ,10$ are obtained by numerical simulations. We observe a shock interaction train structure, where we have pre-Mach reflection followed by ($K-1$) type-I or type-II shock interferences. We also observe that the Mach stem height decreases with $K$ well above the von Neumann condition and becomes non-monotonic near the von Neumann condition.

An important feature of the dynamics of double-diffusive fluids is the spontaneous formation of thermohaline staircases, where wide regions of well-mixed fluid are separated by sharp density interfaces. Recent developments have produced a number of one-dimensional reduced models to describe the evolution of such staircases in the salt fingering regime relevant to mid-latitude oceans; however, there has been significantly less work done on layer formation in the diffusive convection regime. We aim to fill this gap by presenting a new model for staircases in diffusive convection based on a regularisation of the $\gamma$-instability (Radko 2003 J. Fluid Mech. vol. 805, 147–170), with a range of parameter values relevant to both polar oceans and astrophysical contexts. We use the results of numerical simulations to inform turbulence-closure parametrisations as a function of the horizontally averaged kinetic energy $e$, and ratio of the haline to thermal gradients $R_0^*$. These parametrisations result in a one-dimensional model that reproduces the critical value of $R_0^*$ for the layering instability, and the spatial scale of layers, for a wide range of parameter values, although there is a mismatch between the range of $R_0^*$ for layer formation in the model and observational values from polar oceans. Staircases form in the one-dimensional model, evolving gradually through layer merger events that closely resemble simulations.

When turbulent convection interacts with a turbulent shear flow, the cores of convective cells become aligned with the mean current, and these cells (which span the height of the domain) may interact with motions closer to the solid boundary. In this work, we use coupled Eulerian–Lagrangian direct numerical simulations of a turbulent channel flow to demonstrate that, under conditions of turbulent mixed convection, interactions between motions associated with ejections and low-speed streaks near the solid boundary and coherent superstructures in the interior of the flow interact and lead to significant vertical transport of strongly settling Lagrangian particles. We show that the primary suspension mechanism is associated with strong ejection events (canonical low-speed streaks and hairpin vortices characterised by $u'\lt 0$ and $w'\gt 0$, where $u'$ and $w'$ are the streamwise and vertical turbulent velocity fluctuations), whereas secondary suspension is strongly associated with large-scale plume structures aligned with the mean shear (characterised by $w'\gt 0$ and $\theta '\gt 0$, where $\theta$ represents temperature fluctuations). This coupling, which is absent in the limiting cases (pure channel flow or free convection) is shown to lead to a sudden increase in the interior concentration profiles as ${Ri}_\tau$, the friction Richardson number, increases, resulting in concentrations that are larger by roughly an order of magnitude at the channel midplane.

The linear stability of miscible displacement for radial source flow at infinite Péclet number in a Hele-Shaw cell is calculated theoretically. The axisymmetric self-similar flow is shown to be unstable to viscous fingering if the viscosity ratio $m$ between ambient and injected fluids exceeds $3/2$, and to be stable if $m\lt {3/2}$. If $1\lt m\lt {3/2}$, then small disturbances decay at rates between $t^{-3/4}$ and $t^{-1}$ (the exact range depending on $m$) relative to the $t^{1/2}$ radius of the axisymmetric base-state similarity solution; if $m\lt 1$, then they decay faster than $t^{-1}$. Asymptotic analysis confirms these results and gives physical insight into various features of the numerically determined relationship between the growth rate and the azimuthal wavenumber and viscosity ratio.

The acoustofluidic method holds great promise for manipulating micro-organisms. When exposed to the steady vortex structures of acoustic streaming flow, these micro-organisms exhibit intriguing dynamic behaviours, such as hydrodynamic trapping and aggregation. To uncover the mechanisms behind these behaviours, we investigate the swimming dynamics of both passive and active particles within a two-dimensional acoustic streaming flow. By employing a theoretically calculated streaming flow field, we demonstrate the existence of stable bounded orbits for particles. Additionally, we introduce rotational diffusion and examine the distribution of particles under varying flow strengths. Our findings reveal that active particles can laterally migrate across streamlines and become trapped in stable bounded orbits closer to the vortex centre, whereas passive particles are confined to movement along the streamlines. We emphasise the influence of the flow field on the distribution and trapping of active particles, identifying a flow configuration that maximises their aggregation. These insights contribute to the manipulation of microswimmers and the development of innovative biological microfluidic chips.

The laminar flow past rectangular prisms is studied in the space of length-to-height ratio ($1 \leqslant L/H \leqslant 5$), width-to-height ratio ($1.2 \leqslant W/H \leqslant 5$) and Reynolds number ($Re \lessapprox 700$); $L$ and $W$ are the streamwise and cross-flow dimensions of the prisms. The primary bifurcation is investigated with linear stability analysis. For large $W/L$, an oscillating mode breaks the top/bottom planar symmetry. For smaller $W/L$, the flow becomes unstable to stationary perturbations and the wake experiences a static deflection, vertical for intermediate $W/L$ and horizontal for small $W/L$. Weakly nonlinear analysis and nonlinear direct numerical simulations are used for $L/H = 5$ and larger $Re$. For $W/H = 1.2$ and 2.25, the flow recovers the top/bottom planar symmetry but loses the left/right one, via supercritical and subcritical pitchfork bifurcations, respectively. For even larger $Re$, the flow becomes unsteady and oscillates around either the deflected (small $W/H$) or the non-deflected (intermediate $W/H$) wake. For intermediate $W/H$ and $Re$, a fully symmetric periodic regime is detected, with hairpin vortices shed from the top and bottom leading-edge (LE) shear layers; its triggering mechanism is discussed. At large $Re$ and for all $W/H$, the flow approaches a chaotic state characterised by the superposition of different modes: shedding of hairpin vortices from the LE shear layers, and wake oscillations in the horizontal and vertical directions. In some portions of the parameter space the different modes synchronise, giving rise to periodic regimes also at relatively large $Re$.

Turbulent flames in practical devices are subject to a superposition of broadband turbulence and narrowband harmonic flow oscillations. In such cases, flames have a superposition of space–time correlated wrinkles, superposed with broadband turbulent disturbances that interact nonlinearly. This paper extends our prior experimental work to characterise and quantify these flame dynamics. We extract ensemble-averaged flame edge and velocity by ensemble-averaging the instantaneous data at the same phase with respect to the forcing cycle. This paper shows that the ensemble-averaged spatio-temporal dynamics of the flame changes significantly with turbulence intensity. From a spatial viewpoint, the ensemble-averaged flame at weak turbulence intensities exhibits clear cusps and a large ratio between curvature in concave and convex regions. In contrast, at high turbulence intensities, the concave and convex parts of the ensemble-averaged flame are nearly symmetric. From a temporal viewpoint, increasing turbulence intensity monotonically suppresses higher harmonics of the forcing frequency that are manifestations of flame nonlinearities. Taken together, these both point to the interesting observation that the ensemble-averaged flame exhibits increasingly linear dynamics with increasing turbulence intensities, in contrast to its very strong nonlinear behaviours at weak turbulence intensities and juxtaposed with the increasingly nonlinear nature of its instantaneous dynamics with increasing turbulence intensity. In addition, prior studies have shown clear coherent modulation of turbulent flame speed correlated with coherent curvature modulation and that this relationship could be quantified via a ‘turbulent Markstein number’, $M_{T}$. We develop correlations for $M_{T}$ showing how it scales with turbulent and narrowband disturbance quantities, such as turbulent flame brush thickness and convective length scale.

Steady-state Bloch wave systems at resonance with fixed frequencies and amplitudes are investigated using the homotopy analysis method. Nonlinear waves propagate over a stationary undulating bottom topography of infinite extent, modelled as a superposition of two waveforms. The wave systems are classified as type 1 if the primary transmitted and resonant wave components have equal energies, and type 2 if the energy distribution is unequal. Two subtypes of type 2 are identified, distinguished by their responses to frequency detuning and bottom topography: the wave steepness in subtype 1 shows monotonic variations with detuning, while in subtype 2 it exhibits a peak at a particular detuning value, indicating downward resonance that intensifies with greater wave steepness. A pair of peaks in wave steepness arises in each subtype at certain values of the angle $\theta$ between the waveforms of the bottom topography. In both subtypes, the peaks are slightly affected by the ratio $k_{{b}1}/k_{{b}2}$ of the two bottom wave vectors, and significantly affected by the propagation angle $\alpha$ of the primary transmitted wave, but remain stable under changes to other topographic parameters. As the topography amplitude and $\theta$ vary, significant additional contributions to the total energy of the wave system appear from components other than resonant and primary transmitted waves. The most pronounced effects occur with changes in $\theta$, with the additional components accounting for up to 12 % of the total energy. This study provides an enriched understanding of resonant Bloch wave systems and a basis for improving the effectiveness of wave energy converters.

This paper presents detailed analyses of the Reynolds stresses and their budgets in temporally evolving stratified wakes using direct numerical simulation. Ensemble averaging is employed to mitigate statistical errors in the data, and the results are presented as functions of both the transverse and vertical coordinates – at time instants across the near-wake, non-equilibrium, and quasi-two-dimensional regimes for wakes in weakly and strongly stratified environments. Key findings include the identification of dominant terms in the Reynolds stress transport equations and their spatial structures, the generation and destruction processes of the Reynolds stresses, and the energy transfer between the Reynolds stress and the mean flow. The study also clarifies the effects of the Reynolds number and the Froude number. Additionally, we assess the validity of the eddy-viscosity type models and some existing closures for the Reynolds stress model, highlighting the limitations of isotropy and return-to-isotropy hypotheses in stratified flows.