The finitely extensible nonlinear elastic with Peterlin closure (FENE-P) and simplified Phan-Thien–Tanner (sPTT) viscoelastic models are used widely for modelling of complex fluids. Although they are derived from distinct micro-structural theories, these models can become mathematically identical in steady and homogeneous flows with a particular choice of the values of the model parameters. However, even with this choice of parameter values, the model responses are known to differ from each other in transient flows. In this work, we investigate the responses of the FENE-P and sPTT constitutive models in large-amplitude oscillatory shear (LAOS). In steady shear, the shear stress scales with the non-dimensional group $Wi/(aL)$ ($Wi\,\sqrt {\epsilon }$) for the FENE-P (sPTT) model, where $Wi$ is the Weissenberg number, $L^2$ is the limit of extensibility in the FENE-P model ($a$ being $L^2/(L^2-3)$), and $\epsilon$ is the extensibility parameter in the sPTT model. Our numerical and analytical results show that in LAOS, the FENE-P model shows this universality only for large values of $L^2$, whereas the sPTT model shows it for all values of $\epsilon$. In the strongly nonlinear region, there is a drastic difference between the responses of the two models, with the FENE-P model exhibiting strong shear stress overshoots that manifest as self-intersecting secondary loops in the viscous Lissajous curves. We quantify the nonlinearity exhibited by each constitutive model using the sequence of physical processes framework. Despite the high degree of nonlinearity exhibited by the FENE-P model, we also show using fully nonlinear one-dimensional simulations that it does not shear band in LAOS within the range of conditions studied.

In nature and engineering applications, water jet plunging acts as a key process causing interface breaking and generating mixed-phase turbulence. In this paper, high-resolution numerical simulations of the plunging of a water jet into a quiescent pool were performed to investigate the statistical properties of mixed-phase turbulence, with a special focus on the closure problem of the Reynolds-averaged equation. We conducted phase-resolved simulations, with the air–water interface captured using a coupled level-set and volume-of-fluid method. Various cases were performed to analyse the effects of the Froude number and Reynolds number. The simulation results showed that the turbulence statistics are insensitive to the Reynolds number under investigation, while the Froude number influences the flow properties significantly. To investigate the closure problem of the mean momentum equation, the turbulent kinetic energy (TKE) and turbulent mass flux (TMF) and their transport equations were analysed further. It was discovered that the balance relationship of the TKE budget terms remained similar to many single-phase turbulent flows. The TMF is an additional unclosed term in mixed-phase turbulence over the single-phase turbulence. Our simulation results showed that the production term in its transport equation was highly correlated to TKE. Based on this finding, a closure model for the production term of TMF was further proposed.

The paper presents the results of an experimental and numerical study of turbulent thermal convection in a rectangular box containing an extended immersed free-floating plate. Varying the values of control parameters, such as Rayleigh number, aspect ratio and vertical position of the plate, provides a wide range of possible modes, from immobile and purely periodic to stochastic. We have shown that stable periodic motions occur when the plate floats close to one of the heat exchangers. An increase in the distance between the plate and the heat exchanger breaks the periodic motion and (at moderate Rayleigh numbers) leads to a pronounced asymmetry, when the plate stays close to one of the walls most of the time, makes rare excursions to the opposite wall and immediately returns. As the Rayleigh number increases, the plate motions from one edge of the box to the other reappear, but always have an irregular character. Regarding the dependence of the system behaviour on the geometry of the box, both lower and upper limits of periodic plate motions were found in the experiments. In the numerical simulations, the upper limit was not achieved – the plate moves quasi-periodically through the chain of vortices of different signs even at the largest aspect ratio being considered. The heat-insulating floating plate provides the spatial and temporal variation of the heat flux and reduces the integral heat flux, but the reduction in heat flux depends significantly on the vertical position of the plate.

Various systems of mechanical devices and natural organisms use the repeated expansion and contraction of deformable structures to draw in and blow out fluid. Instead of such deformable continuous structures, a system that consists of multiple discrete bodies can also generate a directional flow through the cooperative movement of the individual bodies. In this study, we numerically investigate the collective effects of a multi-body system composed of eight circular cylinders, each of which oscillates separately in the radial direction to generate thrust. The cylinder array performs cooperative motion regulated by three motion parameters: phase difference, oscillation amplitude and frequency at a low Reynolds number ($Re = 10$). The phase difference between the cylinders is critical in determining the extent of the directional flow and the time-averaged thrust. The optimal phase difference that yields the maximum time-averaged thrust is consistent, regardless of the oscillation amplitude and frequency. However, the thrust generation performance becomes significantly weaker at a higher Reynolds number ($Re = 100$). This highlights that the hydrodynamic blockage in gaps between cylinders, which is induced by strong viscous diffusion at the low Reynolds number, is essential for the cooperative force generation of multiple closely spaced bodies. A new dimensionless geometric parameter based on the motion of the array is proposed to characterize the degree of bias in the generated flow and it successfully predicts the trend in the time-averaged thrust at the low Reynolds number with strong hydrodynamic blockage.

Direct numerical simulations are performed to study temporal variations of the wall shear stresses and flow dynamics in the turbulent pulsatile pipe flow. The mechanisms, responsible for the paradoxical phenomenon for which the amplitude of the oscillating wall shear stress in the turbulent flow is smaller than that in the laminar flow for the same pulsation conditions, are investigated. It is shown that the delayed response of turbulence in the buffer layer generates a large magnitude of the radial gradient of the Reynolds shear stress near the wall, which counteracts the effect of the oscillating pressure gradient on the change of the streamwise velocity and hence reduces the amplitude of the wall shear stress. Such a delayed response consists of two processes: the delayed development of near-wall streaks and the subsequent energy redistribution from the streamwise velocity fluctuation to the other two co-existing components. This is a dynamical manifestation of the viscoelasticity of turbulent eddies. As the frequency is reduced, the variation of the friction Reynolds number results in a phase-wise variation of the time scale and intensity of the turbulence response, causing the hysteresis of the wall shear stress. Such a phase asymmetry is amplified by the increase of the pulsation amplitude. An examination of the energy spectra reveals that the near-wall streaks are stretched in the streamwise direction during the acceleration phase, and then break up into small-scale structures in the deceleration phase, accompanied by the enhanced dissipation that transforms the turbulent kinetic energy into heat.

This article concerns the time-evolution, spectral structure and scaling of weak turbulence subject to rotation and stable stratification. The flow is expressed as a combination of particular solutions, referred to as modes, of the linearised governing equations without viscosity or diffusion. Modes are of two types: oscillatory ones which represent inertial-gravity waves and time-independent ones that express a non-propagating (NP) component of the flow. The presence of the NP component, which plays an active role in the dynamics apart from in the case of pure rotation, renders wave-turbulence analysis problematic because the NP mode is non-dispersive. Equations are derived for the time evolution of the modal amplitudes, evolution which is due to nonlinearity and visco-diffusion. Subsequent analysis assumes that one or other (or both) of the Rossby and Froude numbers is small (weak turbulence). Given this assumption, the NP component is found to evolve independently of the wave one and a numerical scheme, similar to, though significantly different from classical direct numerical simulation, is used to determine its time evolution. The treatment of the wave component assumes its amplitude is large compared with the NP one, otherwise there are seemingly intractable difficulties of closure in the analysis. Given this further assumption, the wave component decouples from the NP one. Evolution equations for the wave spectra are derived using wave-turbulence analysis and are integrated numerically. As might be expected, these equations indicate that nonlinear coupling of wave modes is dominated by resonances. Results are given for both the NP and wave components.

The statistical properties of uniform momentum zones (UMZs) are extracted from laboratory and field measurements in rough wall turbulent boundary layers to formulate a set of stochastic models for the simulation of instantaneous velocity profiles. A spatiotemporally resolved velocity dataset, covering a field of view of $8 \times 9\,{\rm m}^2$, was obtained in the atmospheric surface layer using super-large-scale particle image velocimetry (SLPIV), as part of the Grand-scale Atmospheric Imaging Apparatus (GAIA). Wind tunnel data from a previous study are included for comparison (Heisel et al., J. Fluid Mech., vol. 887, 2020, R1). The probability density function of UMZ attributes such as their thickness, modal velocity and averaged vertical velocity are built at varying elevations and modelled using log-normal and Gaussian distributions. Inverse transform sampling of the distributions is used to generate synthetic step-like velocity profiles that are spatially and temporally uncorrelated. Results show that in the wide range of wall-normal distances and $Re_\tau$ up to $\sim O(10^6)$ investigated here, shear velocity scaling is manifested in the velocity jump across shear interfaces between adjacent UMZs, and attached eddy behaviour is observed in the linear proportionality between UMZ thickness and their wall normal location. These very same characteristics are recovered in the generated instantaneous profiles, using both fully stochastic and data-driven hybrid stochastic (DHS) models, which address, in different ways, the coupling between modal velocities and UMZ thickness. Our method provides a stochastic approach for generating an ensemble of instantaneous velocity profiles, consistent with the structural organisation of UMZs, where the ensemble reproduces the logarithmic mean velocity profile and recovers significant portions of the Reynolds stresses and, thus, of the streamwise and vertical velocity variability.

The presence of a mean magnetic field aligned with the direction of the acceleration greatly modifies the development of the Rayleigh–Taylor instability (RTI). High resolution direct numerical simulations of the Boussinesq–Navier–Stokes equations under the magnetohydrodynamics approximation reveal that, after an initial damping of the perturbations at the interface between the two miscible fluids, a rapid increase of the mixing layer is observed. Structures are significantly stretched in the vertical direction because magnetic tension prevents small-scale shear instabilities. When the vertical turbulent velocity exceeds the Alfvén velocity, the flow transitions to turbulence, structures break and an enhanced mixing occurs with strong dissipation. Afterwards, the mixing zone slows down and its growth rate is decreased compared to the hydrodynamic case. For larger magnitudes of the mean magnetic field, a strong anisotropy persists, and an increased fraction of potential energy injected into the system is lost into turbulent magnetic energy: as a consequence, the mixing zone growth rate is decreased even more. This phenomenology is embedded in a general buoyancy-drag equation, derived from simplified equations that reflect the large-scale dynamics, in which the drag coefficient is increased by the presence of turbulent magnetic energy.

Jetting of collapsing bubbles is a key aspect in cavitation-driven fluid–solid interactions as it shapes the bubble dynamics and additionally due to its direct interaction with the wall. We study experimentally and numerically the near-wall collapse and jetting of a single bubble seeded into the stagnation flow of a wall jet, i.e. a jet that impinges perpendicular onto a solid wall. High-speed imaging shows rich and rather distinct bubble dynamics for different wall jet flow velocities and bubble-to-wall stand-off distances. The simulations use a volume-of-fluid method that allows us to numerically determine the microscopic and transient pressures and shear stresses on the wall. It is shown that a wall jet at moderate flow velocities of a few metres per second already shapes the bubble ellipsoidally inducing a planar and convergent jet flow. The distinct bubble dynamics allow us to tailor the wall interaction. In particular, the shear stresses can be increased by orders of magnitude without increasing impact pressures the same way. Interestingly, at small seeding stand-offs, the bubble during the final collapse stage can lift off the wall and migrate against the flow direction of the wall jet such that the violent collapse occurs away from the wall.

The present study tests the efficacy of the well-known viscous drag reduction strategy of imposing spanwise wall oscillations to reduce pressure drag contributions in transitional and fully rough turbulent wall flow. This is achieved by conducting a series of direct numerical simulations of a turbulent flow over two-dimensional (spanwise-aligned) semi-cylindrical rods, placed periodically along the streamwise direction with varying streamwise spacing. Surface oscillations, imposed at fixed viscous-scaled actuation parameters optimum for smooth wall drag reduction, are found to yield substantial drag reduction ($\gtrsim$25 %) for all the rough wall cases, maintained at matched roughness Reynolds numbers. While the total drag reduction is due to a drop in both viscous and pressure drag in the case of transitionally rough flow (i.e. with large inter-rod spacing), it is associated solely with pressure drag reduction for the fully rough cases (i.e. with small inter-rod spacing), with the latter being reported for the first time. The study finds that pressure drag reduction in all cases is caused by the attenuation of the vortex shedding activity in the roughness wake, in response to wall oscillation frequencies that are of the same order as the vortex shedding frequencies. Contrary to speculations in the literature, this study confirms that the mechanism behind pressure drag reduction, achieved via imposition of spanwise oscillations, is independent of the viscous drag reduction. This mechanism is responsible for weakening of the Reynolds stresses and increase in base pressure in the roughness wake, explaining the pressure drag reduction observed by past studies, across varying roughness heights and geometries.

Many of the cell membrane's vital functions are achieved by the self-organization of the proteins and biopolymers embedded in it. The protein dynamics is in part determined by its drag. A large number of these proteins can polymerize to form filaments. In vitro studies of protein–membrane interactions often involve using rigid beads coated with lipid bilayers, as a model for the cell membrane. Motivated by this, we use slender-body theory to compute the translational and rotational resistance of a single filamentous protein embedded in the outer layer of a supported bilayer membrane and surrounded on the exterior by a Newtonian fluid. We first consider the regime where the two layers are strongly coupled through their inter-leaflet friction. We find that the drag along the parallel direction grows linearly with the filament's length and quadratically with the length for the perpendicular and rotational drag coefficients. These findings are explained using scaling arguments and by analysing the velocity fields around the moving filament. We then present and discuss the qualitative differences between the drag of a filament moving in a freely suspended bilayer and a supported membrane as a function of the membrane's inter-leaflet friction. Finally, we briefly discuss how these findings can be used in experiments to determine membrane rheology. In summary, we present a formulation that allows computation of the effects of membrane properties (its curvature, viscosity and inter-leaflet friction), and the exterior and interior three-dimensional fluids’ depth and viscosity on the drag of a rod-like/filamentous protein, all in a unified theoretical framework.

Viscous streaming is an efficient rectification mechanism to exploit flow inertia at small scales for fluid and particle manipulation. It typically entails a fluid vibrating around an immersed solid feature that, by concentrating stresses, modulates the emergence of steady flows of useful topology. Motivated by its relevance in biological and artificial settings characterized by soft materials, recent studies have theoretically elucidated, in two dimensions, the impact of body elasticity on streaming flows. Here, we generalize those findings to three dimensions, via the minimal case of an immersed soft sphere. We first improve existing solutions for the rigid-sphere limit, by considering previously unaccounted terms. We then enable body compliance, exposing a three-dimensional, elastic streaming process available even in Stokes flows. Such effect, consistent with two-dimensional analyses but analytically distinct, is validated against direct numerical simulations and shown to translate to bodies of complex geometry and topology, paving the way for advanced forms of flow control.

Flow-induced vibration (FIV) of a flexible cylinder with a triangular cross-section, allowed to oscillate in the cross-flow, inline and torsional direction, is studied experimentally through water tunnel tests. The dynamic response of the cylinder was studied for three different angles of attack ($0^\circ, 30^\circ, 60^\circ$), at reduced velocities of 0.9–16.27, corresponding to Reynolds numbers of 364–3600. At the angle of attack of $0^\circ$, vortex-induced vibration at low reduced velocity was observed, which transitioned to galloping at higher reduced velocities. At the angles of attack of $30^\circ$ and $60^\circ$, galloping-type response was observed over the range of the reduced velocities tested. Our results show that the cylinder's torsional oscillation breaks the system's symmetry and affects the structural response at higher reduced velocities regardless of the angle of attack. The FIV response of the flexible triangular cylinder is distinct from that of a rigid flexibly mounted triangular cylinder due to torsional oscillation, spanwise flexibility and the two fixed boundary conditions limiting the amplitude of oscillation. Flow field analysis in the wake of the cylinder was done qualitatively and quantitatively using hydrogen bubble flow visualisation and time-resolved volumetric particle tracking velocimetry techniques, respectively. Our results show the existence of highly three-dimensional vortex structures in the wake of the cylinder. We studied the coupling between the vortex shedding modes in the wake of the cylinder and the structural vibration modes through the spatiotemporal mode analysis using the proper orthogonal decomposition technique to distinguish between different types of the FIV response observed.

Natural flyers and swimmers employ flexible wings or fins to propel. While the complex interaction between the foil with deformation and the surrounding non-steady fluid environment defines the propulsion performance of the propellers, elucidating the interaction mechanism through theoretical models earns much challenge. Based on elastokinetics and linear potential flow theory, this study proposes a simplified analytical model to clarify the kinematics and the propulsion performance of a flexible thin foil pitching in flow. The dynamical forces, including the inertial force of the foil and the non-steady fluid pressure, are used to determine the averaged deformation angle of the foil. Combining the averaged deformation angle and the prescribed driving pitching motion, the kinematics of the foil is resolved analytically. Based on the analytical expressions for the corresponding pitching motion, analytical relations among the physical parameters of the stiffness and the mass of the foil and the driving frequency are given to these critical conditions, including resonance of the flow–structure system, equal pitching amplitude between the flexible foil and the rigid counterpart, phase angle transition from ${\rm \pi}/2$ to $- {\rm \pi}/2$. Subsequently, the performance of the foil, including the thrust, the power and the propulsive efficiency, as a function of the flexibility of the foil are derived, together with the introduction of a bluff body type offset drag to the thrust. The formulated analytical theory, which matches nicely with previous reports, will help to interpret the effect of the flexibility and regulate the propulsive performance of the flexible foil when pitching in fluid.

In this paper, we study non-isochoric models for mixtures of solid particles, at high volume concentration, and a gas. One of the motivations for this work concerns geophysics and more particularly the pyroclastic density currents, whose dense basal parts are precisely mixtures of pyroclasts and lithic fragments and air. They are extremely destructive phenomena, capable of devastating urbanised areas, and are known to propagate over long distances, even over almost flat topography. Fluidisation of these dense granular flows by pore gas pressure is one response that could explain this behaviour and must therefore be taken into account in the models. Starting from a solid–gas mixing model and invoking the compressibility of the gas through a law of state, we rewrite the conservation of mass equation of the gas phase into an equation of the pore gas pressure whose net effect is to reduce the friction between the particles. The momentum equation of the solid phase is completed by generic constitutive laws, specified as in Schaeffer et al. (J. Fluid Mech., vol. 874, 2019, pp. 926–951) by a yield function and a dilatancy function. Therefore, the divergence of the velocity field, which reflects the ability of the granular flow to expand or compress, depends on the volume fraction, pressure, strain rate and inertial number. In addition, we require the dilatancy function to describe the rate of volume change of the granular material near an isochoric equilibrium state, i.e. at constant volume. This property ensures that the volume fraction, which is the solution to the conservation of mass equation, is positive and finite at all times. We also require that the non-isochoric fluidised model be linearly stable and dissipate energy (over time). In this theoretical framework, we derive the dilatancy models corresponding to classical rheologies such as Drucker–Prager and $\mu (I)$ (with or without expansion effects). The main result of this work is to show that it is possible to obtain non-isochoric and fluidised granular models satisfying all the properties necessary to correctly account for the physics of granular flows and which are well posed, at least linearly stable.

An efficient numerical method for calculation of the sea surface skewness and kurtosis for arbitrary wave spectra, taking up to third-order nonlinear effects into account, is developed. The skewness and kurtosis are calculated for a large number of sea states, covering a wide range of sea-state parameters in terms of wave steepness, water depth, directional spreading and frequency bandwidth. The results are used to propose new accurate expressions for skewness and kurtosis, valid over a wide range of sea states. Existing expressions for skewness and kurtosis reported in the literature are reviewed, and their accuracy is evaluated. Comparison to model-test results and phase-resolved numerical simulation are presented. It is suggested that the new improved parametrizations for skewness and kurtosis, which include dependence on wave steepness, water depth, spectral bandwidth and directional spreading, represent a convenient way to include these covariates into higher-order distributions for crest heights, wave heights and surface elevation.