This article presents an overview of the dynamics of the human heart and the main goal is the discussion of its fluid mechanic features. We will see, however, that the flow in the heart can not be fully described without considering its electrophysiology and elastomechanics as well as the interaction with the systemic and pulmonary circulations with which it is strongly connected. Biologically, the human heart is similar to that of all warm-blooded mammals and it satisfies the same allometric laws. Since the Paleolithic Age, however, humans have improved their living conditions, have modified the environment to satisfy their needs and, more recently, have developed advanced medical knowledge which has allowed triple the number of heartbeats with respect to other mammals. In the last century, effective diagnostic tools, reliable surgical procedures and prosthetic devices have been developed and refined leading to substantial progress in cardiology and heart surgery with routine clinical practice which nowadays cures many disorders, once lethal. Pulse duplicators have been built to reproduce the pulsatile flow and ‘blood analogues’, have been realized. Heart phantoms, can attain deformations similar to the real heart although the active contraction and the tissue anisotropy still can not be replicated. Numerical models have also become a viable alternative for cardiovascular research: they do not suffer from limitations of material properties and device technologies, thus making possible the realization of truly digital twins. Unfortunately, a high-fidelity model for the whole heart consists of a system of coupled, nonlinear partial differential equations with a number of degrees of freedom of the order of a billion and computational costs become the bottleneck. An additional challenge comes from the inherent human variability and the uncertainty of the heart parameters whose statistical assessment requires a campaign of simulations rather than a single deterministic calculation; reduced and surrogate models can be employed to alleviate the huge computational burden and all possibilities are currently being pursued. In the era of big data and artificial intelligence, cardiovascular research is also advancing by exploiting the latest technologies: equation-based augmented reality, virtual surgery and computational prediction of disease progression are just a few examples among many that will become standard practice in the forthcoming years.

Wind tunnel experiments were performed to investigate turbulent flow over an array of heterogeneous roughness elements using stereoscopic particle image velocimetry. Nine streamwise planes, covering one periodic cell of a multi-scale roughness element that is arranged in a staggered pattern, are combined to quantify mean flow features and Reynolds stresses. Dispersive stresses, arising from spatial variations in the temporally averaged mean velocity, are also presented. The results highlight that the roughness elements create a large deficit pathway along the surfaces. Outer scaling of the time-averaged streamwise velocity presents features which are nearly independent of the roughness element type, with parameters of the flow revealing values near those observed in smooth wall boundary layers, such as the wake strength parameter, opposing the earlier work containing aligned patterns. The strength of the secondary motion is most accentuated at the ridges of the roughness, showing that the formation of structures is sensitive to the location investigated in the spanwise direction.

The propulsion of a pitching flexible plate in a uniform flow is investigated numerically. The effects of bending stiffness ($K$), pitching amplitude ($A_L$) and frequency ($St$) on the wake patterns, thrust generations and propulsive performances of the fluid–plate system are analysed. Four typical wake patterns, i.e. von Kármán, reversed von Kármán, deflected and chaotic wakes, emerge from various kinematics, and the $St-A_L$ wake maps are given for various $K$. The drag-to-thrust transitions (DTT) and the wake transitions (WT) between the von Kármán and reversed von Kármán wakes are examined. Results indicate that the WT and DTT boundaries can be scaled by the chord-averaged distance of travel, $\mathcal {L}$, which leads to $\mathcal {L}\times St \approx 1$ and $\mathcal {L}\times St \approx 1.2$, respectively. Further, the resonance mechanism for the performance enhancement is revealed and confirmed in a wide range of parameters. The dimensionless average speed of plate, $\mathcal {U^*}\left (=\mathcal {L}\times St\right )$, is adopted merely to characterize the propulsive performances. For the first time, the $\mathcal {U^*}$-based scaling laws for the thrust and power are revealed in pitching rigid and flexible plates for various $A_L$ and $St$. This study may deepen our understanding of biological swimming and flying, and provide a guide for bionic design.

We examine the response of an inertialess viscoelastic channel flow to localized perturbations. We thus performed an experiment in which we perturbed the flow using a localized velocity pulse and probed the perturbed fluid packet downstream from the perturbation location. While for low Weissenberg numbers the perturbed fluid reaches the measurement location as a single velocity pulse, for sufficiently high Weissenberg numbers and perturbation strengths, a random number of pulses arrive at the measurement location. The average number of pulses observed increases with the Weissenberg number. This observation constitutes a transition to a novel elastic pulse-splitting regime. Our results suggest a possible new direction for studying the elastic instability of viscoelastic channel flows at high elasticity through the growth of localized perturbations.

We present a decomposition of the streamwise fluid force for in-line vortex-induced vibration (VIV) to provide insight into how the wake drag acts as a driving force in fluid–structure interaction. This force decomposition is an extension of that proposed in the recent work of Konstantinidis et al. (J. Fluid Mech., vol. 907, 2021, p. A34), and is applied to and validated by our experiments examining a circular cylinder freely vibrating in line with the free stream. It is revealed from the decomposition and linear analysis that two regimes of significant vibration are in phase synchronisation, while they are separated by a desynchronised regime marked by competition between non-stationary frequency responses of the cylinder vibration and the vortex shedding. Of interest, such a near-resonance desynchronisation regime is not seen in the transverse vibration case.

Bubble curtains are multiphase line plumes that are used to reduce buoyancy-driven flows between two water zones at different densities. They are similar to air curtains, plane turbulent jets, that are installed in doorways of buildings between two climatically different environments. In this study, we establish a formal analogy between bubble curtains and air curtains and unify the two frameworks for their description that had previously been used. By means of small-scale laboratory experiments conducted in a channel with freshwater and brine solutions, we study how effectively a bubble curtain acts as a separation barrier for a wide range of density differences as well as different air fluxes and water depths. Qualitatively, two regimes of operation of a bubble curtain are identified and we establish the optimum operating conditions on the basis of quantitative measurements and theoretical considerations. We develop a theoretical model to calculate the infiltration flux of dense water across the bubble curtain that is in very good agreement with experimental measurements and yields a theoretical upper limit on the effectiveness of the bubble curtain. We also study the zones of mixed fluid around the bubble curtain, provide a scaling law for their horizontal extent as well as theoretically predict the water density inside these mixed zones. We discuss how the theoretical models derived from our small-scale experiments apply to real-scale bubble curtains that are, for example, used in ship locks.

Taylor bubbles are a feature of the slug flow regime in gas–liquid flows in vertical pipes. Their dynamics exhibits a number of transitions such as symmetry breaking in the bubble shape and wake when rising in downward flowing and stagnant liquids, respectively, as well as breakup in sufficiently turbulent environments. Motivated by the need to examine the stability of a Taylor bubble in liquids, a systematic numerical study of a steadily moving Taylor bubble in stagnant and flowing liquids is carried out, characterised by the dimensionless inverse viscosity $( N_f )$, Eötvös $( Eo )$ and Froude numbers $( U_m )$, the latter being based on the centreline liquid velocity, using a Galerkin finite-element method. A boundary-fitted domain is used to examine the dependence of the steady bubble shape on a wide range of $N_f$, $Eo$ and $U_m$. Our analysis of the bubble nose and bottom curvatures shows that the intervals $Eo = [ 20,30 )$ and $N_f=[60,80 )$ are the limits below which surface tension and viscosity, respectively, have a strong influence on the bubble shape. In the interval $Eo = (60,100 ]$, all bubble features studied are weakly dependent on surface tension. A linear stability analysis of the axisymmetric base states shows that there exist regions of $(N_f,Eo,U_m)$ space within which the bubble is unstable and assumes an asymmetric shape. To elucidate the mechanisms underlying the instability, an energy budget analysis is carried out which reveals that perturbation growth is driven by the bubble pressure for $Eo \geq 100$, and by the tangential interfacial stress for $Eo < 100$. Examples of the asymmetric bubble shapes and their associated flow fields are also provided near the onset of instability for a wide range of $N_f$, $Eo$ and $U_m$.

We use the recent frame-indifferent theory of diffusive momentum transport to identify internal barriers in wall-bounded turbulence. Formed by the invariant manifolds of the Laplacian of the velocity field, the barriers block the viscous part of the instantaneous momentum flux in the flow. We employ the level sets of single-trajectory Lagrangian diagnostic tools, the trajectory rotation average and trajectory stretching exponent, to approximate both vortical and internal wall-parallel momentum transport barrier (MTB) interfaces. These interfaces provide frame-indifferent alternatives to classic velocity-gradient-based vortices and high-shear boundaries between uniform momentum zones (UMZs). Indeed, we find that these elliptic manifold approximations and MTBs outperform standard vortices and UMZ interfaces in blocking diffusive momentum transport, which suggests our momentum barriers are physical features that may be the cause of coherence signatures in statistical and non-objective diagnostics. We also introduce normalized trajectory metrics that provide unprecedented visualizations of objective coherent structures by avoiding strong turbulence biases.

The decompositions of the skin-friction and heat-transfer coefficients based on the twofold repeated integration in hypersonic transitional and turbulent boundary layers are analysed to give some major reasons of the overshoot phenomena of the wall skin friction and heat transfer. It is shown that the overshoot of the skin-friction coefficient is mainly caused by the drastic change of the mean velocity profiles, especially the strong negative streamwise gradient of the mean streamwise velocity far from the wall; and the overshoot of the heat-transfer coefficient is primarily due to the viscous dissipation, especially the strong positive vertical gradient of the mean streamwise velocity near the wall. These observations are different from the previous observations that the Reynolds shear stress and Reynolds heat flux are the reasons, respectively. Further investigations show that the above observations are independent of the set-up of the wall blowing and suction parameters, which indicates the universality of the major reasons of the overshoot phenomena in our numerical simulations. In the hypersonic turbulent boundary layers, it is observed that the strongly cooled wall temperature and the high Mach number can slightly enhance the contribution of the Reynolds shear stress, and weaken the contribution of the mean convection, mainly due to the strong compressibility effect. Moreover, the magnitudes of the relative contributions of the mean convection, pressure dilatation, viscous dissipation and the Reynolds heat flux increase as the wall temperature increases.

We present a simple, novel kinematic criterion – that uses only the horizontal velocity fields and is free of arbitrary thresholds – to separate line plumes from local boundary layers in a plane close to the hot plate in turbulent convection. We first show that the horizontal divergence of the horizontal velocity field ($\boldsymbol {\nabla _H} \boldsymbol {\cdot } \boldsymbol {u}$) has negative and positive values in two-dimensional (2D), laminar similarity solutions of plumes and boundary layers, respectively. Following this observation, based on the understanding that fluid elements predominantly undergo horizontal shear in the boundary layers and vertical shear in the plumes, we propose that the dominant eigenvalue ($\lambda _D$) of the 2D strain rate tensor is negative inside the plumes and positive inside the boundary layers. Using velocity fields from our experiments, we then show that plumes can indeed be extracted as regions of negative $\lambda _D$, which are identical to the regions with negative $\boldsymbol {\nabla _H} \boldsymbol {\cdot } \boldsymbol {u}$. Exploring the connection of these plume structures to Lagrangian coherent structures (LCS) in the instantaneous limit, we show that the centrelines of such plume regions are captured by attracting LCS that do not have dominant repelling LCS in their vicinity. Classifying the flow near the hot plate based on the distribution of eigenvalues of the 2D strain rate tensor, we then show that the effect of shear due to the large-scale flow is felt more in regions close to where the local boundary layers turn into plumes. The lengths and areas of the plume regions, detected by the $\boldsymbol {\nabla _H}\boldsymbol {\cdot }\boldsymbol {u}$ criterion applied to our experimental and computational velocity fields, are then shown to agree with our theoretical estimates from scaling arguments. Using velocity fields from numerical simulations, we then show that the $\boldsymbol {\nabla _H}\boldsymbol {\cdot }\boldsymbol {u}$ criterion detects all the upwellings, while the available criteria based on temperature and flux thresholds miss some of these upwellings. The plumes detected by the $\boldsymbol {\nabla _H}\boldsymbol {\cdot }\boldsymbol {u}$ criterion are also shown to be thicker at Prandtl numbers ($Pr$) greater than one, expectedly so, due to the thicker velocity boundary layers of the plumes at $Pr>1$.

Direct numerical simulations of a turbulent spiral Poiseuille flow (SPF) in a narrow-gap geometry at low Taylor number have been performed to analyse the reverse transition dynamics. The presently investigated SPF results from a Taylor–Couette arrangement with a rotating inner cylinder and a stationary outer one, subject to a time-constant axial pressure gradient. Keeping fixed the Taylor number and reducing the axial Reynolds number, several flow regimes have been obtained until a complete laminarization occurred. In agreement with previous experimental evidence, it has been found that the laminar state is achieved at a Reynolds number significantly smaller than the corresponding non-rotating value. Moreover, the route to turbulence suppression has been shown to differ in the two cases, as confirmed by the increased Reynolds number friction coefficient envelope. The differences occurring in the reverse transition process between SPF and plain Poiseuille flow are attributed to a modification of the isotropy of the Reynolds stress tensor, caused by an alteration of the velocity pressure-strain redistribution mechanisms.

Linear three-dimensional instability is studied in the shock layer and the laminar separation bubble (LSB) induced by shock-wave/boundary-layer interactions in a Mach 7 flow of nitrogen over a double wedge with a $30^{\circ }\text {--}55^{\circ }$ cross-sectional profile. At a free-stream unit Reynolds number $Re=5.2\times 10^{4}\,{\rm m}^{-1}$ this flow exhibits rarefaction effects and has shock thicknesses comparable to the thickness of the boundary layer at separation. Flow features have been fully resolved using a high-fidelity massively parallel implementation of the direct simulation Monte Carlo method that captures the flow evolution from the inception of three-dimensionality, through linear growth of instabilities, to the early stages of nonlinear saturation. It is shown that the LSB sustains self-excited, small-amplitude perturbations that originate past the primary separation line and lead to spanwise-periodic wall striations inside the bubble and downstream of the primary reattachment line, as known from earlier experiments, simulations and instability analyses. A spanwise-periodic instability, synchronised with that in the separation zone, is identified herein for the first time, which exists in the internal structure of the separation and detached shock layers, and manifests itself as spanwise-periodic cats-eyes patterns in the global mode amplitude functions. The growth rate and the spanwise-periodicity length of linear disturbances in the shock layers and the LSB are found to be identical. Linear amplification of the most unstable three-dimensional flow perturbations leads to synchronised low-frequency unsteadiness of the triple point, with a Strouhal number of $St\approx 0.028$.

In this work, a transition process in a hypersonic flow over a cold-wall compression ramp is studied using direct numerical simulation (DNS) and global stability analysis (GSA). The free-stream Mach number and the Reynolds number based on the flat-plate length are 7.7 and $8.6 \times 10^5$, respectively. The shock-induced pressure rise causes the boundary layer to separate on the flat plate, forming a separation bubble around the corner. Without introducing any external disturbances, the DNS captures the transition to turbulence downstream of flow reattachment. The DNS results agree well with the experimental data as well as theoretical predictions. To uncover the intrinsic instability in the flow system, GSA is employed to investigate the three-dimensionality of the two-dimensional base flow. Several stationary and oscillatory unstable modes are revealed, which result in spanwise periodicity inside and downstream of the separation bubble. The GSA and DNS results indicate that the intrinsic instability of the flow system triggers the formation of streamwise counter-rotating vortices and boundary-layer streaks near reattachment. The downstream transition to turbulence starts from the breakdown of the streamwise vortices and streaks. Moreover, the second harmonic of the most unstable global mode and a broadband low-frequency unsteadiness occur in the saturated flow, which has a significant influence on the transition process. In summary, the present study demonstrates a transition process in a hypersonic compression-ramp flow as a result of the intrinsic instability of the flow system.

Liquid-infused surfaces can reduce friction drag in both laminar and turbulent flows. However, the heat transfer properties of such multi-phase surfaces have still not been investigated to a large extent. We use numerical simulations to study conjugate heat transfer of liquid-filled grooves. It is shown that heat transfer can increase for both laminar and turbulent liquid flows due to recirculation in the surface texture. Laminar flow simulations show that for the increase to be substantial, the thermal conductivity of the solid must be similar to the thermal conductivity of the fluids, and the recirculation in the grooves must be sufficiently strong (Péclet number larger than 1). The ratio of the surface cavity to the system height is an upper limit of the direct contribution from the recirculation. While this ratio can be significant for laminar flows in microchannels, it is limited for turbulent flows, where the system scale (e.g. channel height) usually is much larger than the texture height. However, heat transfer enhancement of the order of $10\,\%$ is observed (with a net drag reduction) in a turbulent channel flow at a friction Reynolds number ${{Re}}_\tau \approx 180$. It is shown that the turbulent convection in the bulk can be enhanced indirectly from the recirculation in the grooves.

This work is devoted to the study of the conditions under which a drop directed normally towards a superheated or isothermal smooth substrate prevents the initial contact with the solid by skating over a micrometre-sized vapour or air layer. The results have been obtained analysing the gas flow at the spatio-temporal region where the maximum liquid pressure is attained, which is also where and when the minimum values of the film thickness are reached. For the common case in which $We St^{-1/6}\gtrsim 1$, where $We=\rho _l U^{2} R/\gamma$ and $St=\rho _l U R/\eta _a$ denote, respectively, the Weber and Stokes numbers, we find that capillary effects are negligible and the ratio between the minimum film thickness and the local drop radius of curvature is $h_m/R\propto St^{-7/6}$, with $\rho _l$, $\gamma$, $\eta _a$, $U$ and $R$ indicating the liquid density, interfacial tension coefficient, gas viscosity, impact velocity and drop radius, respectively. In contrast, when $We St^{-1/6}\lesssim 1$, capillary effects can no longer be neglected and $h_m/R\propto We^{-1/3} St^{-10/9}$. The predicted values of the minimum film thickness are compared with published experimental data, finding good agreement between predictions and measurements for the cases of both isothermal and superheated substrates. In addition, using mass conservation, we have also deduced an equation providing the minimum value of the substrate temperature for which a cylindrical central vapour bubble of constant height $h_d/R\propto St^{-2/3}$, with $h_d\gg h_m$, grows radially at the wetting velocity deduced in Riboux & Gordillo (Phys. Rev. Lett., vol. 113, 2014, 024507). The predicted values are in good agreement with the dynamic Leidenfrost temperatures reported by Shirota et al. (Phys. Rev. Lett., vol. 116, 2016, 064501).

The recent field measurements of katabatic winds on steep alpine slopes provide a unique database for theoretical analysis of the mean flow development and the determination of mixing properties. The theory is based on the depth-integrated momentum and heat equations, and demonstrates an increase in mean velocity $U$ with downstream distance $x$ according to $x^{n}$ ($n\leq 1/2$). An equation for the mean wind velocity is established, expressing its dependency on the buoyancy flux, related to the heat flux to the ground, entrainment and bottom friction. No ambient stratification, and ambient wind and constant ground surface temperature, lead to $U{\sim} x^{1/2}$, while constant heat flux to the ground leads to $U{\sim} x^{1/3}$ and requires that the reduced gravity decreases as $x^{-1/3}$. Stable ambient stratification $N$ causes, in addition to small-amplitude mean flow oscillations, a decrease in reduced gravity with $x$, in which case the assumption of constant surface heat flux along $x$ is only an approximation. The turbulent fluxes are a function of gradient Richardson number $Ri$ with the ratio of turbulent diffusivity to viscosity $K_h/K_m$ changing from nearly $1.4$ to approximately $0.5$ at $Ri\approx 0.5$. A new mixing efficiency is introduced that includes turbulence kinetic energy production or consumption by along-slope turbulent buoyancy flux. It increases with $Ri$ up to $0.25$ at $Ri\approx 0.5$ and then remains nearly constant. The measurements allowed us to determine the bottom drag coefficients and interfacial entrainment, with the ground surface heat flux being determined from the mean buoyancy flux.

Fine fibre immersed in different flows is ubiquitous. For a fibre in shear flows, most motion modes appear in the flow-gradient plane. Here the two-dimensional behaviours of an individual flexible flap in channel flows are studied. The nonlinear coupling of the fluid inertia ($\textit {Re}$), flexibility of the flap ($K$) and channel width ($W$) is discovered. Inside a wide channel (e.g. $W=4$), as $K$ decreases, the flap adopts rigid motion, springy motion, snake turn and complex mode in sequence. It is found that the fluid inertia tends to straighten the flap. Moreover, $\textit {Re}$ significantly affects the lateral equilibrium location $y_{eq}$, therefore affecting the local shear rate and the tumbling period $T$. For a rigid flap in a wide channel, when $\textit {Re}$ exceeds a threshold, the flap stays inclined instead of tumbling. As $\textit {Re}$ further increases, the flap adopts swinging mode. In addition, there is a scaling law between $T$ and $\textit {Re}$. For the effect of $K$, through the analysis of the torque generated by surrounding fluid, we found that a smaller $K$ slows down the tumbling of the flap even if $y_{eq}$ is comparable. As $W$ decreases, the wall confinement effect makes the flap easier to deform and closer to the centreline. The tumbling period would increase and the swinging mode would be more common. When $W$ further decreases, the flaps are constrained to stay inclined, parabolic-like or one-end bending configurations moving along with the flow. Our study may shed some light on the behaviours of a free fibre in flows.

The problem of hydroelastic wave diffraction by a surface-piercing vertical circular cylinder mounted on the bottom of an ice-covered channel is considered. The ice sheet is modelled as an elastic thin plate with homogeneous properties, while the linearized velocity potential theory is adopted to describe the motion of the fluid. The solution starts from the Green function satisfying all other boundary conditions apart from that on the body surface. This is obtained through applying a Fourier transform in the longitudinal direction of the channel and adopting an eigenfunction expansion in the vertical direction. The boundary conditions on the side walls and ice edges are imposed through an orthogonal product. Through the Green function, the velocity potential due to a surface-piercing structure with arbitrary shape can be expressed through a source distribution formula derived in this work, in which only integrals over the body surface and its interaction line with the ice sheet need to be retained. For a vertical circular cylinder, the unknown source distribution can be expanded further into a Fourier series in the circumferential direction, and then the analytical solution of the velocity potential can be obtained further. Extensive results and discussions are provided for the hydrodynamic forces and vertical shear forces on the cylinder, as well as the deflection and strain of the ice sheet. In particular, the behaviour of the solution near one of the natural frequencies of the channel is investigated in detail.

The Kármán–Howarth–Monin–Hill equation is employed to study the production and interscale energy transfer in a boundary layer undergoing bypass transition due to free-stream turbulence. The energy flux between different length scales is calculated at several streamwise locations covering the laminar, transitional and turbulent regimes. Maps of scale energy production and flux vectors are visualised on two-dimensional planes and three-dimensional hyper-planes that comprise both physical and separation spaces. In the transitional region, the maps show strong inverse cascade in the streamwise direction near the wall. The energy flux vectors emanate from a region of strong production and transfer energy to larger streamwise scales. To provide deeper insight into the origin of the inverse cascade process, we decompose the energy flux vector into components arising from nonlinear interactions between velocity fluctuations, mean flow inhomogeneity, pressure and viscous effects. The inverse cascade is mainly due to the nonlinear interaction component, and in the earliest stages of transition this component competes with that due to mean flow inhomogeneity. By superposing the instantaneous velocity fields and the energy flux vectors, we relate the inverse cascade process to the growth of turbulent spots. Once the transition process is complete, the maps become very similar to those observed in other fully developed turbulent flows, such as channel flow. Finally we characterise the nonlinear interaction term using probability density functions (PDFs) evaluated at different wall-normal heights. The PDFs are asymmetric and wide-skirted as in homogeneous isotropic turbulence, but are skewed towards positive values reflecting the inverse cascade.

A class of exact solutions of the magnetohydrodynamic quasi-geostrophic equations (MQG), which result from rotating shallow water magnetohydrodynamics in the limit of small Rossby and magnetic Rossby numbers is constructed analytically. These solutions are magnetic modons, steady-moving dipolar vortices, which are generalizations of the well-known quasi-geostrophic modons. It is shown that various configurations of magnetic modons are possible: with or without external magnetic field, and with or without internal magnetic field trapped inside the dipole. By using the modon solutions as initial conditions for direct numerical simulations of the MQG equations, it is shown that they remain coherent for a long time, running over about a hundred deformation radii without change of form, provided the external and internal magnetic fields are not too strong, and even if a small-amplitude noise is added to initial conditions.