We investigate the estimation of the turbulent flow around a canonical square cylinder at $Re=22\,000$ based on temporally resolved but spatially sparse velocity data and solving the unsteady Reynolds-averaged Navier–Stokes (URANS) equations. Flow reconstruction from sparse data is achieved through the application of a nudging data assimilation technique. It involves the introduction of a feedback control term in the momentum equations which allows us to drive URANS predictions towards reference data, which are here extracted from a direct numerical simulation. Such a data assimilation approach induces negligible supplementary computational cost compared with that of a standard URANS simulation. The influence of the spatial resolution of the reference data on the reconstruction performances is systematically investigated. Using a spacing of the order of one cylinder length between data points, we already observe synchronisation of the low-frequency vortex shedding between the full reference flow and the one that is estimated by URANS. The present data assimilation procedure allows us to compensate for deficiencies in standard URANS calculations and leads to a significant decrease in temporal and spectral errors as computed by spectral proper orthogonal decomposition. Furthermore, high accuracy in terms of mean-flow prediction by URANS is achieved. When considering spacings between measurements that are of the order of the wavelength of the Kelvin–Helmholtz vortices, such phenomena in the shear layers at the top and bottom of the cylinder are correctly estimated, while they are not self-sustained in standard URANS. The influence of the structure of the feedback control term in the data assimilation procedure is also investigated.

We study the shear-thinning mediation of elasto-inertial transitions in Taylor–Couette flow of viscoelastic polymer solutions. Two types of high molecular weight polymers are used at various concentrations and in water–glycerol solvents of various viscosities. This allows us to access a wide range of elastic numbers and effective shear-thinning indices. Conservative ramp-up (slow acceleration of the inner cylinder and subsequent increase in Reynolds number) and steady-state (constant rotation speed) experiments are performed, in which the flow is monitored continuously using flow visualisation. Depending on the shear-thinning and elastic properties of the working fluid, very different behaviours are observed. In almost constant-viscosity fluids (Boger fluids), or shear-thinning fluids with significant elasticity, the flow transitions from purely azimuthal Couette flow (CF) to a highly chaotic flow state referred to as elasto-inertial turbulence (EIT) via Taylor vortex flow (TVF) and elasto-inertial rotating spiral waves (RSW). When the degree of shear-thinning is increased and elasticity reduced, elastic waves or EIT may fade to a wavy Taylor vortex flow (WTVF) with increasing inertia. Significant shear-thinning leads to a delay in the onset of EIT. Remarkably, in some highly shear-thinning cases, even with a significant elasticity, elastic flow features (EIT, RSW) are completely suppressed, and the flow exhibits a ‘Newtonian-like’ transition sequence (CF–TVF–WTVF). Shear-thinning acts to modify, delay, or even completely suppress elasto-inertial behaviours (RSW, EIT), that would otherwise have existed in the absence of shear-thinning. It is, thus, possible to induce various hydrodynamic regimes by tuning the relative degrees of shear-thinning, elasticity and inertia.

Gradient ascent methods are developed to compute incompressible flows that maximize heat transport between two isothermal no-slip parallel walls. Parameterizing the magnitude of the velocity fields by a Péclet number $Pe$ proportional to their root-mean-square rate of strain, the schemes are applied to compute two-dimensional flows optimizing convective enhancement of diffusive heat transfer, i.e. the Nusselt number $Nu$ up to $Pe\approx 10^{5}$. The resulting transport exhibits a change of scaling from $Nu-1\sim Pe^{2}$ for $Pe<10$ in the linear regime to $Nu\sim Pe^{0.54}$ for $Pe>10^{3}$. Optimal fields are observed to be approximately separable, i.e. products of functions of the wall-parallel and wall-normal coordinates. Analysis employing a separable ansatz yields a conditional upper bound ${\lesssim}Pe^{6/11}=Pe^{0.\overline{54}}$ as $Pe\rightarrow \infty$ similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh–Bénard convection are discussed.

This study aims to examine the spatial evolution of the geometrical features of the turbulent/turbulent interface (TTI) in a cylinder wake. The wake is exposed to various turbulent backgrounds in which the turbulence intensity and the integral length scale are varied independently, and comparisons to a turbulent/non-turbulent interface (TNTI) are drawn. The turbulent wake was marked with a high Schmidt number ($Sc$) scalar, and a planar laser induced fluorescence experiment was carried out to capture the interface between the wake and the ambient flow from $x/d = 5$ to 40, where $x$ is the streamwise coordinate from the centre of the cylinder, and $d$ is the cylinder's diameter. It is found that the TTI generally spreads faster towards the ambient flow than the TNTI. A transition region of the interfaces’ spreading is found at $x/d \approx 15$, after which the interfaces propagate at a slower rate than previously (upstream), and the mean interface positions of both the TNTI and TTI scale with the local wake half-width. The locations of both the TNTI and TTI have non-Gaussian probability density functions (PDFs) in the near wake because of the influence of the large-scale coherent motions present within the flow. Further downstream, after the large-scale coherent motions have dissipated, the TNTI position PDF does become Gaussian. For the first time, we explore the spatial variation of the ‘roughness’ of the TTI, quantified via the fractal dimension, from near field to far field. The length scale in the background flow has a profound effect on the TTI fractal dimension in the near wake, whilst the turbulence intensity becomes important only for the fractal dimension farther downstream.

Piezoelectric macro-fibre composite (MFC) actuators are employed onto the flexible leeward surface of an airfoil for active control. Time-resolved aerodynamic forces, membrane deformations and flow fields are synchronously measured at low Reynolds number (Re = 6 × 104). Mean aerodynamics show that the actively controlled airfoil can achieve lift-enhancement and drag-reduction simultaneously in the angle of attack range of 10° ≤ α ≤ 14°, where the rigid airfoil encounters stall. The maximum increments of lift and lift-to-drag ratio are 27.1 % and 126 % at the reduced actuation frequency of ${f^ + } = 3.52$. The unsteady coupling features are further analysed at α = 12°, where the maximum lift-enhancement occurs. It is newly discovered that the membrane vibrations and flow fields are locked into half of the actuation frequency when ${f^ + } > 3$. The shift of the dominant vibration mode from bending to inclining is the reason for the novel ‘half-frequency lock-in’ phenomenon. To the fluid–structure interaction, there are three characteristic frequencies for the actively controlled airfoil: $S{t_1} = 0.5{f^ + }$, $S{t_2} = {f^ + }$, and $S{t_3} = 1.5{f^ + }$. Here, St1 and its harmonics (St2, St3) are coupled with the natural frequencies of the leading-edge shear layer, resulting in the generation of multi-scale flow structures and suppression of flow separation. The lift presents comparable dominant frequencies between St1 and St3, which means the instantaneous lift is determined by the flow structures of St1 and St3. The local membrane bulge and dent affect the instantaneous swirl strength of flow structures near the maximum vibration amplitude location, which is the main reason for the variation of instantaneous lift.

In a three-layer system, weakly nonlinear theory predicts that breathers exist under certain conditions which, under the Boussinesq approximation, include symmetric stratifications in which the density jump across each interface is the same and the upper and lower layer thicknesses are equal and less than 9/26 of the total water depth. The existence and characteristics of fully nonlinear breathers in this symmetric stratification are poorly understood. Therefore, this study investigates fully nonlinear breathers in a three-layer symmetric stratification in order to clarify their characteristics by making direct comparisons between numerical simulation results and theoretical solutions. A normalization of the breather profiles is introduced using theoretical solutions of a breather and a new energy scale is proposed to evaluate their potential and kinetic energy. We apply fully nonlinear and strongly dispersive internal wave equations in a three-layer system using a variational principle. The computational results show that the larger the amplitude, the shorter the length of the envelope of breathers, which agrees with the theoretical solution. However, breathers based on the theoretical solutions cannot progress without deformation and decay due to the emission of short small-amplitude internal waves. Furthermore we demonstrate that the shedding of larger amplitude waves occurs, and the amplitude of the envelope decays more strongly when the density interface crosses the critical depth where the ratio of the upper layer thickness and the total water depth is 9/26 suggesting a limiting amplitude for fully nonlinear breathers.

This paper presents the effects of two-dimensional bathymetry on the evolution of skewness and kurtosis of the surface elevation and velocity field in long-crested irregular waves propagating over a circular shoal and a submerged bar with a semicircular step. For a circular shoal, we show that the surface elevation has a local maximum of skewness and kurtosis on top of the shoal. We find that the skewness of the horizontal velocity field has similar behaviour to the surface elevation, but the kurtosis of the horizontal velocity field has two local maxima at different locations on the lee side of the shoal. For a semicircular step, which acts similar to a converging lens, we find that refraction contributes to the statistical properties of the surface elevation and horizontal velocity field to deviate from Gaussian statistics.

Bubble-type vortex breakdown in axial vortices is investigated numerically through a model problem of flow inside a cylinder with a top rotating lid, referred to as ‘Vogel–Escuider flow’. The parameters of the flow are Reynolds number ($Re$), based on the rotation speed of the top plate, and aspect ratio ($\unicode[STIX]{x1D6E4}$), which is the ratio of height to radius of the cylinder, depending on which the flow exhibits steady or unsteady breakdown bubble topologies. The flow is analysed for Reynolds number up to 5000 for $\unicode[STIX]{x1D6E4}=2.5$ using helicity density. In the absence of vortex breakdown, the helicity density does not change the sign in the bulk, while in the event of a breakdown, it changes the sign from positive to negative in the vicinity of the breakdown bubble. The three-dimensional flow is further represented as the sum of a two-dimensional velocity field in the $rz$ plane and an out-of-plane velocity vector based on the respective energies, referred to as two-dimensional three-component flow. Here $r$ is the radial coordinate, $z$ is the axial coordinate and $\unicode[STIX]{x1D703}$ is the azimuthal coordinate. Helicity density of the flow is then decomposed into planar helicity $(h_{r,z})$ and out-of-plane helicity $(h_{\unicode[STIX]{x1D703}})$. We show that a correlation exists between planar helicity and the vortex breakdown bubble. We also show that the topology of the breakdown bubbles is described by the planar helicity. Using only this planar helicity, the entire breakdown bubble is reconstructed for axisymmetric as well as non-axisymmetric flows.

In this paper, we propose artificial-neural-network-based (ANN-based) nonlinear algebraic models for the large-eddy simulation (LES) of compressible wall-bounded turbulence. An innovative modification is applied to the invariants and the tensor bases of the nonlinear algebraic models through using the local grid widths along each direction to normalise the corresponding gradients of the flow variables. Furthermore, the dimensionless model coefficients are determined by the ANN method. The modified ANN-based nonlinear algebraic model (MANA model) has much higher correlation coefficients and much lower relative errors than the dynamic Smagorinsky model (DSM), Vreman model and wall-adapting local eddy-viscosity model in the a priori test. The significantly more accurate estimations of the mean subgrid-scale (SGS) fluxes of the kinetic energy and temperature variance are also obtained by the MANA models in the a priori test. Furthermore, in the a posteriori test, the MANA model can give much more accurate predictions of the flow statistics and the mean SGS fluxes of the kinetic energy and the temperature variance than other traditional eddy-viscosity models in compressible turbulent channel flows with untrained Reynolds numbers, Mach numbers and grid resolutions. The MANA model has a better performance in predicting the flow statistics in supersonic turbulent boundary layer. The MANA model can well predict both direct and inverse transfer of the kinetic energy and temperature variance, which overcomes the inherent shortcoming that the traditional eddy-viscosity models cannot predict the inverse energy transfer. Moreover, the MANA model is computationally more efficient than the DSM.

Nucleate boiling, a ubiquitous heat transfer mode, involves multiple vapour bubble nucleations on the heater surface and offers high heat transfer coefficients. The bubble growth process on a heating substrate involves the formation of microlayer, a thin liquid film trapped between the growing bubble and the heating substrate, and contributes to the bubble growth phenomenon through evaporation. Microlayer dynamics for a single bubble have been widely investigated in the pool and flow boiling conditions. However, the literature on multiple bubbles interactions and their associated microlayer information is scarce. Notably, in the case of flow boiling, where the microlayer dynamics are not symmetric due to the bubble's movement, the bubbles’ interaction and its influence on associated microlayer dynamics have never been reported. Therefore, microlayer and bubble dynamics in multiple interactions have been investigated experimentally using simultaneous application of thin-film interferometry and high-speed videography techniques in flow boiling with water as the working fluid. Our experimental investigation revealed that the secondary nucleation could cause a reduction in lift-off time and may assist or hinder the movement of the first bubble. The experimental results also demonstrated that the secondary nucleation could deplete the microlayer of the first bubble hydrodynamically even when the bubbles are far apart. Furthermore, it has been found that the microlayer depletion rate depends on the growth rate and the location of the secondary nucleation. Hence, this experimental study emphasises the need to consider the interaction of bubbles while modelling boiling flows to avoid overestimating the contribution of microlayer evaporation.

The Enskog–Vlasov equation is a phenomenological kinetic equation that extends the Enskog equation for the dense (non-ideal) hard-sphere fluid by adding an attractive soft potential tail to the purely repulsive hard-sphere contribution. Simplifying assumptions about pair correlations lead to a Vlasov-like self-consistent force field that adds to the Enskog non-local hard-sphere collision integral. Within the limitations imposed by the underlying assumptions, the extension gives the Enskog–Vlasov equation the ability to give a unified description of ideal and non-ideal fluid flows as well as of those fluid states in which liquid and vapour regions coexist, being separated by a resolved interface. Furthermore, the Enskog–Vlasov fluid can be arbitrarily far from equilibrium. Thus the Enskog–Vlasov model equation provides an excellent, although approximate, tool for modelling processes with liquid–vapour interfaces and adjacent Knudsen layers, and allows us to look at slip, jump and evaporation coefficients from a different perspective. Here, a set of 26 moment equations is derived from the Enskog–Vlasov equation by means of the Grad moment method. The equations provide a meaningful approximation to the underlying kinetic equation, and include the description of Knudsen layers. This work focuses on the – rather involved – derivation of the moment equations, with only a few applications shown.

The unstable fluid–fluid displacement patterns in porous media with rough invasion fronts and trapping of the defending phase are often observed in drainage, i.e. when the solid is non-wetting to the invading phase. On the other hand, during imbibition, compact and faceted growth is expected in regular porous media with geometrically homogeneous pore structure. Here, we report the irregular growth of invading fluid during the imbibition process in two-dimensional regular porous media. The ramified invasion morphology associated with thin fingers is reminiscent of capillary fingering. Through examining the capillary pressure signals and the type of pore-scale invasion mechanisms, the fundamental differences between faceted growth and irregular invasion are revealed. By analysing the pore-scale invasion mechanisms, a phase diagram describing the dominance of different invasion events is proposed. Through conducting systematic quasi-static radial injection simulations across a wide range of porosity and wettability, excellent agreement is observed on the transition boundary from faceted and compact displacement patterns to irregular and dendritic invasion morphologies. This is reflected by the overlap of the transition boundaries from analytical prediction, type of pore-scale invasion events, and macroscopic morphology quantified by the fractal dimension. This work provides new insights on the role of geometrical features of solid structures during multiphase flow with emphasis on the porosity and wettability. The findings could assist in guiding the design of microfluidic devices to control deterministically the multiphase flow, transport and reaction processes.

Depth-integrated mathematical models for simulating waves and currents from deep to shallow water are presented. These models are derived from Euler’s equations in the $\unicode[STIX]{x1D70E}$-coordinate system, mapping the total water depth in Cartesian coordinates onto a specified range of $\unicode[STIX]{x1D70E}$-coordinates. The horizontal velocity is approximated as a truncated infinite series of products of prescribed shape functions of $\unicode[STIX]{x1D70E}$ and unknown functions of horizontal coordinates and time. Adopting the method of weighted residuals, the new models are obtained by minimizing the residuals of the horizontal momentum equations with either the Galerkin method or the subdomain method. These models’ linear and nonlinear water wave properties are investigated. The new models are implemented numerically. A hierarchy of numerical models with different degree of polynomial approximation is developed and checked against several benchmarked experiments and a new set of experiments of self-focusing wave groups. For both the Galerkin and subdomain models, excellent agreements are observed for both the free surface elevations and the velocity profiles. The new models are superior to the existing Boussinesq-type models for their applicability to a wide range of physical scenarios, including the interactions between a wave package of multiple frequency components and a linearly sheared current. The new Galerkin models have similar characteristics and accuracy as the Green–Naghdi models, but the new models are more efficient computationally. Finally, for the same degree of polynomial approximation the subdomain models perform better than the Galerkin models and require less computational time.

Fish schools are ubiquitous in marine life. Although flow interactions are thought to be beneficial for schooling, their exact effects on the speed, energetics and stability of the group remain elusive. Recent numerical simulations and experimental models suggest that flow interactions stabilize in-tandem formations of flapping foils. Here, we employ a minimal vortex sheet model that captures salient features of the flow interactions among flapping swimmers, and we study the free swimming of a pair of in-line swimmers driven with identical heaving or pitching motions. We find that, independent of the flapping mode, heaving or pitching, the follower passively stabilizes at discrete locations in the wake of the leader, consistent with the heaving foil experiments, but pitching swimmers exhibit tighter and more cohesive formations. Further, in comparison to swimming alone, pitching motions increase the energetic efficiency of the group while heaving motions result in a slight increase in the swimming speed. A deeper analysis of the wake of a single swimmer sheds light on the hydrodynamic mechanisms underlying pairwise formations. These results recapitulate that flow interactions provide a passive mechanism that promotes school cohesion, and afford novel insights into the role of the flapping mode in controlling the emergent properties of the school.