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.

The evolution of turbulent spots in a flat plate boundary layer is examined using time-resolved tomographic particle image velocimetry (Tomo-PIV) experiments and direct numerical simulation (DNS). The characteristics of flow structures are examined using timelines and material surfaces. Both the numerical and experimental results reveal a notable behaviour in the developmental process of turbulent spots: the development of low-speed streaks at the spanwise edges of turbulent spots, followed by the subsequent formation of hairpin vortices. The behaviour of these low-speed streaks is further investigated using timelines and material surfaces generated for a series of regions and development times. The results indicate that these low-speed streaks exhibit characteristic wave behaviour. The low-speed streaks are observed to lift up as three-dimensional (3-D) waves, with high-shear layers forming at the interface of these waves. These induced high-shear layers become unstable and evolve into vortices, which contribute to the expansion of the turbulent spot. These findings show the significant role of 3-D waves in the development of turbulent spots, supporting the hypothesis that 3-D waves serve as initiators of vortices at the bounding surface of a turbulent spot.

Granular surface flows are frequently encountered in nature as well as during handling of powders in different industries. An experimental study of granular surface flow on a heap is carried out. The heap is formed by pouring nearly monodisperse spherical particles from the rectangular slit orifice of a hopper on a rough rectangular plate. A flow of particles is developed on the heap surface, which is planar in the central region, with particles flowing over the edge of the plate into a collection chamber. The geometry considered in this study is an example of a fully three-dimensional heap without side walls. The surface velocities of the particles are measured using high-speed videography and particle tracking velocimetry for different mass flow rates with steel balls and glass beads of two different sizes, for heaps of different aspect ratios. The flow is uni-directional and fully developed in a central zone on the heap surface. The flowing layer thickness is measured in this zone by immersing a soot-coated blade into the flow. The angle of inclination of the free surface of the heap is found to be nearly constant for a ten-fold increase of the mass flow rate. The scaled flowing layer thickness is found to vary linearly with the scaled flow rate and the data for all the particles collapse to a single line over a ten-fold increase in the scaled flow rate and an increase in the aspect ratio of the heap by a factor of 1.75. The predicted scaled surface velocity and scaled shear rate using this correlation match the measured values.

Thermal Marangoni effects play important roles in bubble dynamics such as bubbles generated by water electrolysis or boiling. As macroscopic bubbles often originate from nucleated nanobubbles, it is crucial to understand how thermocapillarity operates at the nanoscale. In this study, the motion of transient bulk gas nanobubbles in water driven by a vertical temperature gradient between two solid plates is investigated using molecular dynamics simulations and analytical theory. The simulation results show that due to the thermal Marangoni force, nanobubbles move towards the hot plate at a constant velocity, similar to the behaviour of macroscale gas bubbles. However, unlike macroscale gas bubbles whose thermal conductivity and viscosity are negligible compared to those of water, the thermal conductivity and viscosity of nanoscale gas bubbles are significantly increased due to their large gas density. The thermal resistance and the slip length are also found to matter at the liquid–gas interface, though they decrease with increasing gas densities. The previous thermocapillary theory for macroscale bubbles is extended by considering all these nanoscopic effects. Expressions of the Marangoni force and the drag force are derived. By balancing the Marangoni force and the drag force, the theoretical velocity of the nanobubble migration in a thermal gradient is obtained. When using the measured transport properties of liquid, gas, and their interfaces, the theoretically obtained velocity is consistent with the result of the molecular simulations. We find that the slip length is too small to have considerable effects on nanobubble motions in the current liquid–gas system.

In a quiescent medium, chemically active particles propel themselves by emitting or absorbing solutes, creating concentration gradients that induce a slip at the particle surface. This self-propulsion occurs when solute advection overcomes diffusion. However, an imposed flow field can alter these dynamics. This study explores the propulsion characteristics and the related rheological consequences of chemically active particles in an imposed uniaxial extensional flow analytically and numerically. An asymptotic solution is obtained for weak imposed flow relative to self-induced diffusiophoretic slip. Meanwhile, finite element simulations are carried out over a wide range of imposed flow strength and Péclet number. The results reveal that the interplay between solute advection, imposed flow and diffusiophoretic slip significantly affects particle propulsion and suspension rheology. While solute advection and diffusiophoretic slip tend to create asymmetric solute distributions, promoting self-propulsion, imposed extensional flow promotes symmetric distributions, hindering self-propulsion. This not only delays the start of self-propulsion but also results in an early transition from a propulsion state to a stationary state characterised by an abrupt halt at relatively lower Péclet number compared to a quiescent medium. Post the abrupt halt, a stirring effect induced by particle activity and imposed extensional flow results in an increased magnitude of stresslet, thus a sudden change in the effective viscosity of the active suspension. The effect of imposed extensional flow on active particle dynamics and suspension rheology can be described succinctly by categorising the overall dynamics into three separate regimes, determined by the Péclet number and the intensity of the extensional flow.