Plane Poiseuille flow of a Bingham fluid in a channel armed with a superhydrophobic (SH) lower wall is analysed via a semi-analytical model, accompanied by complementary direct numerical simulations (DNS). The SH surface represents a groovy structure with air trapped inside its cavities. Therefore, the fluid adjacent to the wall undergoes stick–slip conditions. The model is developed based on introducing infinitesimal wall-induced perturbations into the motion equations, followed by Fourier series expansions, and solving the resulting equations as a boundary value problem. The Navier slip law accounts for the slip at the liquid/air interface (assuming the Cassie state). The presented analysis is fairly comprehensive, covering the creeping and inertial regimes for thick channels (via the semi-analytical and DNS solutions). The main dimensionless numbers are the Reynolds ($Re$), Bingham ($Bi$) and slip ($b$) numbers, as well as the groove periodicity length ($\ell$) and the slip area fraction ($\varphi$). By increasing $Bi$, the perturbation and slip velocity fields grow. As $Re$ increases, the perturbation and slip velocity fields become asymmetric. For certain flow parameters, an unyielded plug zone may appear on the SH wall liquid/air interface, while its formation is accelerated by inertial effects. The results classify the regimes of creeping and inertial flows via predicting the onset of the unyielded plug zone formation at the SH wall.

The evaporation of multiple sessile droplets is both scientifically interesting and practically important, occurring in many natural and industrial applications. Although there are simple analytic expressions to predict evaporation rates of single droplets, there are no such frameworks for general configurations of droplets of arbitrary size, contact angle or spacing. However, a recent theoretical contribution by Masoud, Howell & Stone (J. Fluid Mech., vol. 927, 2021, R4) shows how considerable insight can be gained into the evaporation of arbitrary configurations of droplets without having either to obtain the solution for the concentration of vapour in the atmosphere or to perform direct numerical simulations of the full problem. The theoretical predictions show excellent agreement with simulations for all configurations, only deviating by 25 % for the most confined droplets.

It is generally believed that the temperature and the velocity fields are highly coupled in compressible wall-bounded turbulence. In the present study, we employ a linear model, i.e. the two-dimensional spectral linear stochastic estimation (SLSE), to study this coupling from the perspective of the multi-scale energy-containing eddies. Particular attention is paid to the relevant statistical characteristics of the temperature field. The connections of the two fields are found to be varied with the wall-normal position in the boundary layer. In a nutshell, their entanglement is strongest in the near-wall region, and only the extreme thermal events cannot be captured by SLSE. In the logarithmic region, only the scales that correspond to the attached eddies and the very large-scale motions (VLSMs) are firmly coupled. The near-wall footprints of the former are organized in an additive manner and fulfil the predictions of the celebrated attached-eddy model. In the outer region, the two fields are linearly coupled only at the scales corresponding to VLSMs. These findings are demonstrated to be insensitive to the Mach number effects and ascribed to the similarity between the momentum and the heat transfer in compressible wall turbulence. It is also shown that it is the Reynolds number rather than Mach number that acts as a key similarity parameter in constructing their coupling. The framework built in the present study may pave a way for investigating the multi-physics coupling in turbulence, and reinforcing our analysing and modelling capability to the interrelated problems.

Double diffusion occurs when the fluid density depends on two components that diffuse at different rates (e.g. heat and salt in the ocean). Double diffusion can lead to an up-gradient buoyancy flux and drive motion at the expense of potential energy. Here, we follow the work of Lorenz (Tellus, vol. 7 (no. 2), 1955, pp. 157–167) and Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115–128) for a single-component fluid and define the background potential energy (BPE) as the energy associated with an adiabatically sorted density field and derive its budget for a double-diffusive fluid. We find that double diffusion can convert BPE into available potential energy (APE), unlike in a single-component fluid, where the transfer of APE to BPE is irreversible. We also derive an evolution equation for the sorted buoyancy in a double-diffusive fluid, extending the work of Winters & D’Asaro (J. Fluid Mech., vol. 317, 1996, pp. 179–193) and Nakamura (J. Atmos. Sci., vol. 53 (no. 11), 1996, pp. 1524–1537). The criterion we develop for a release of BPE can be used to analyse the energetics of mixing and double diffusion in the ocean and other multiple-component fluids, and we illustrate its application using two-dimensional simulations of salt fingering.

When the average intermolecular distance is comparable to the size of gas molecules, the Boltzmann equation, based on the dilute gas assumption, becomes invalid. The Enskog equation was developed to account for this finite size effect that makes the collision non-local and increases the collision frequency. However, it is time-consuming to solve the Enskog equation due to its complicated structure of collision operator and high dimensionality. In this work, on the basis of the Shakhov model, a gas kinetic model is proposed to simplify the Enskog equation for non-ideal monatomic dense gases. The accuracy of the proposed Shakhov–Enskog model is assessed by comparing its solutions of the normal shock wave structures with the results of the Enskog equation obtained by the fast spectral method. It is shown that the Shakhov–Enskog model is able to describe non-equilibrium flow of dense gases, when the maximum local mean free path of gas molecules is still greater than the size of a molecular diameter. The accuracy and efficiency of the present model enable simulations of non-equilibrium flow of dense gases for practical applications.

This paper presents bifurcation analyses characterising the nonlinear dynamics of fully developed laminar annular jets with respect to the centrebody diameter ${ {d}}$, Reynolds number ${ {Re}}$, and swirl ratio ${ {S}}$. Similar flows appear in numerous applications and feature a vibrant range of topological and dynamical characteristics associated with phenomena including shear layer separation and vortex breakdown. Our results begin by describing the non-monotonic evolution of the axisymmetric jet's steady topology under varying ${ {S}}$. In accord with earlier reports, the jet progresses through a sequence of wake, breakdown and wall jet regimes in a qualitatively similar manner across a wide span of ${ {d}}$ and ${ {Re}}$ values. In the wake regime, the non-swirling jet bifurcates to a plane-symmetric, but not axisymmetric, steady flow pattern beyond a ${ {d}}$-dependent critical ${ {Re}}$ value. With further increase in ${ {Re}}$, the steady non-swirling jet destabilises subsequently via multiple distinct Hopf bifurcations. Introducing ${ {S}}>0$ to the jet also induces unsteadiness by twisting the singly azimuthally periodic ($|m|=1$) asymmetric wake structure and causing it to precess periodically in time about the central axis. Intermediate swirl stabilises this unsteady dynamics and restores the jet's axisymmetry. This stabilising effect is then reversed in the breakdown regime at higher ${ {S}}$, where a variety of different $|m|=1$ and $|m|=2$ instabilities bifurcate from the steady flow as ${ {S}}$ is increased. Several instances of hysteresis and subcritical behaviour are reported and discussed, including one that manifests precessing vortex core oscillations.

The rebound of droplets impacting a deep fluid bath is studied both experimentally and theoretically. Millimetric drops are generated using a piezoelectric droplet-on-demand generator and normally impact a bath of the same fluid. Measurements of the droplet trajectory and other rebound metrics are compared directly with the predictions of a linear quasipotential model, as well as fully resolved direct numerical simulations of the unsteady Navier–Stokes equations. Both models resolve the time-dependent bath and droplet shapes in addition to the droplet trajectory. In the quasipotential model, the droplet and bath shape are decomposed using orthogonal function decompositions leading to two sets of coupled damped linear harmonic oscillator equations solved using an implicit numerical method. The underdamped dynamics of the drop are directly coupled to the response of the bath through a single-point kinematic match condition which we demonstrate to be an effective and efficient model in our parameter regime of interest. Starting from the inertio-capillary limit in which both gravitational and viscous effects are negligible, increases in gravity or viscosity lead to a decrease in the coefficient of restitution and an increase in the contact time. The inertio-capillary limit defines an upper bound on the possible coefficient of restitution for droplet–bath impact, depending only on the Weber number. The quasipotential model is able to rationalize historical experimental measurements for the coefficient of restitution, first presented by Jayaratne & Mason (Proc. R. Soc. Lond. A, vol. 280, issue 1383, 1964, pp. 545–565).

In this paper, we investigate the impingement of a two-dimensional (2-D) vortex pair translating downwards onto a horizontal wall with a wavy surface. A principal purpose is to compare the vortex dynamics with the complementary case of a wavy vortex pair (deformed by the long-wavelength Crow instability) impinging onto a flat surface. The simpler case of a 2-D vortex pair descending onto a flat horizontal ground plane leads to the well known ‘rebound’ effect, wherein the primary vortex pair approaches the wall but subsequently advects vertically upwards, due to the induced velocity of secondary vorticity. In contrast, a wavy vortex pair descending onto a flat plane leads to ‘rebounding’ vorticity in the form of vortex rings. A descending 2-D vortex pair, impinging on a wavy wall, also generates ‘rebounding’ vortex rings. In this case, we observe that the vortex pair interacts first with the ‘hills’ of the wavy wall before the ‘valleys’. The resulting secondary vorticity rolls up into a concentrated vortex tube, ultimately forming a vortex loop along each valley. Each vortex loop pinches off to form a vortex ring, which advects upwards. Surprisingly, these rebounding vortex rings evolve without the strong axial flows fundamental to the wavy vortex case. The present research is relevant to wing tip trailing vortices interacting with a non-uniform ground plane. A non-flat wall is shown to accelerate the decay of the primary vortex pair. Such a passive, ground-based method to diminish the wake vortex hazard close to the ground is consistent with Stephan et al. (J. Aircraft, vol. 50 (4), 2013a, pp. 1250–1260; CEAS Aeronaut. J., vol. 5 (2), 2013b, pp. 109–125).

We present a self-propelled axisymmetric low-Reynolds-number swimmer (force- and torque-free) that, while unable to swim in a Newtonian fluid, propels itself in a non-Newtonian fluid as a result of fluid elasticity. The propulsion gait is pure ‘swirling’ and is demonstrated using a robotic swimmer, comprised of a ‘head’ sphere and a ‘tail’ sphere, whose swimming speed is shown to demonstrate good agreement with a microhydrodynamic asymptotic theory and numerical simulations. Schlieren imaging demonstrates that propulsion of the swimmer is driven by a strong viscoelastic jet at the tail, which develops due to the fore–aft asymmetry of the swimmer. Optimized cylindrical and conic tail geometries are shown to double the propulsive signal, relative to the optimal spherical tail. We show that we can use observations of this robot to infer rheological properties of the surrounding fluid. We measure the primary normal stress coefficient $\varPsi _1$ at shear rates ${<}1\,{\rm Hz}$, and show reasonable agreement with extrapolated bench-top measurements ($0.8\unicode{x2013}1.2\,{\rm Pa}\,{\rm s}^2$ difference). We also derive how the head rotation may be used to measure the second normal stress coefficient, $\varPsi _2$. The study demonstrates the exciting potential for a ‘swimming rheometer’, examining the intersection between swimming and sensing in complex fluids.

An artificial neural-network-based subgrid-scale (SGS) model, which is capable of predicting turbulent flows at untrained Reynolds numbers and on untrained grid resolution is developed. Providing the grid-scale strain-rate tensor alone as an input leads the model to predict a SGS stress tensor that aligns with the strain-rate tensor, and the model performs similarly to the dynamic Smagorinsky model. On the other hand, providing the resolved stress tensor as an input in addition to the strain-rate tensor is found to significantly improve the prediction of the SGS stress and dissipation, and thereby the accuracy and stability of the solution. In an attempt to apply the neural-network-based model trained for turbulent flows with a limited range of the Reynolds number and grid resolution to turbulent flows at untrained conditions on untrained grid resolution, special attention is given to the normalisation of the input and output tensors. It is found that the successful generalization of the model to turbulence for various untrained conditions and resolution is possible if distributions of the normalised inputs and outputs of the neural network remain unchanged as the Reynolds number and grid resolution vary. In a posteriori tests of the forced and the decaying homogeneous isotropic turbulence and turbulent channel flows, the developed neural-network model is found to predict turbulence statistics more accurately, maintain the numerical stability without ad hoc stabilisation such as clipping of the excessive backscatter, and to be computationally more efficient than the algebraic dynamic SGS models.

Vortex-induced vibration (VIV) of a two degree-of-freedom (DOF) circular cylinder, placed in the test section of a recirculating water tunnel and free to oscillate in its first two vibrational modes in the crossflow direction, is studied experimentally. The dynamic response of the cylinder is studied for a reduced velocity range of $U^*=4\unicode{x2013}30$ for eigenfrequency ratios in the range of 1.3–3.0. For the two DOF system, while the onset of the VIV response followed a similar lock-in region as those observed for a classical VIV response of a single DOF system, by increasing the reduced velocity a secondary lock-in region was observed over which the oscillations of the cylinder were locked into the system's second mode. In addition, there existed an intermediate range of reduced velocity over which the VIV response consisted of oscillations at a combination of the first two natural modes of the system. As the eigenfrequency ratio between the first two modes increased, the secondary lock-in range was extended to higher reduced velocities and the reduced velocity range over which multi-modal oscillations were observed was decreased. A full map of vortex dynamics in the wake of the cylinder was developed qualitatively and quantitatively using hydrogen bubble flow visualization and time-resolved volumetric particle tracking velocimetry techniques, respectively. A Q-criterion analysis revealed the existence of highly three-dimensional vortex structures in the wake of the cylinder. The spatiotemporal mode analysis using the proper orthogonal decomposition technique revealed strong coupling between the vortex shedding modes in the wake of the cylinder and the structural vibration modes.

A millimetre-size superhydrophobic sphere impacting on the free surface of a quiescent bath can be propelled back into the air by capillary effects and dynamic fluid forces, whilst transferring part of its energy to the fluid. We report the findings of a thorough investigation of this phenomenon, involving different approaches. Over the range from minimum impact velocities required to produce rebounds to impact velocities that cause the sinking of the solid sphere, we focus on the dependence of the coefficient of restitution, contact time and maximum surface deflection on the different physical parameters of the problem. Experiments, simulations and asymptotic analysis reveal trends in the rebound metrics, uncover new phenomena at both ends of the Weber number spectrum, and collapse the data. Direct numerical simulations using a pseudo-solid sphere successfully reproduce experimental data whilst also providing insight into flow quantities that are challenging to determine from experiments. A model based on matching the motion of a perfectly hydrophobic impactor to a linearised fluid free surface is validated against direct numerical simulations and used in the low-Weber-number regime. The hierarchical and cross-validated models in this study allow us to explore the entirety of our target parameter space within a challenging multi-scale system.