We carry out direct numerical simulations (DNS) of flow in a turbulent square duct by focusing on heat transfer effects, considering the case of unit Prandtl number. Reynolds numbers up to ${{Re}}_{\tau } \approx 2000$ are considered that are much higher than in previous studies, and that yield clear scale separation between inner- and outer-layer dynamics. Close similarity between the behaviour of the temperature and the streamwise velocity fields is confirmed as in previous studies related to plane channels and pipes. Just like the mean velocity, the mean temperature is found to exhibit logarithmic layers as a function of the nearest wall, but with a different slope. The most important practical implication is the validity of the traditional hydraulic diameter as the correct reference length for reporting heat transfer data, as we show rigorously here. Temperature and velocity fluctuations also have similar behaviour, but apparently logarithmic growth of their inner-scaled peak variances is not observed here, unlike in canonical wall-bounded flows. Analysis of the split contributions to the heat transfer coefficient shows that mean cross-stream convection associated with secondary motions is responsible for about $5\,\%$ of the total. Finally, we use the DNS database to highlight shortcomings of traditional linear closures for the turbulent heat flux, and show that substantial modelling improvement in principle may be obtained by retaining at least the three terms in the vector polynomial integrity basis expansion.

Unstable periodic orbits are believed to underpin the dynamics of turbulence, but by their nature are hard to find computationally. We present a family of methods to converge such unstable periodic orbits for the incompressible Navier–Stokes equations, based on variations of an integral objective functional, and using traditional gradient-based optimisation strategies. Different approaches for handling the incompressibility condition are considered. The variational methods are applied to the specific case of periodic, two-dimensional Kolmogorov flow and compared against existing Newton iteration-based shooting methods. While computationally slow, our methods converge from very inaccurate initial guesses.

A droplet impinging on a superhydrophobic substrate in an electric field is an important process in droplet manipulation and electrostatic spraying. Here, the entire impinging dynamic of the droplet in a vertical electric field is studied by a visualization experiment and numerical simulation with OpenFOAM. We investigate the effect of an electrostatic force on droplet impact in depth, where four ejection modes and three rebound modes are found experimentally. In particular, the filamentous ejecting phenomenon occurs after a droplet impinging on a superhydrophobic substrate is first discovered. In the numerical simulation, the strong coupling between the dynamic distribution of the interface electric charge and the evolution of the droplet profile can lead to different ejection modes, and the different ejection behaviours are caused by the combined effects of electrostatic pressure, capillary pressure, dynamic pressure and static pressure on the droplet apex. A charge scaling law for the ejection droplets is proposed. Furthermore, a set of theoretical models is established, which can successfully predict the threshold electric capillary number for different droplet ejection modes. The results reveal some important characteristics for a droplet impinging on a superhydrophobic surface in an electric field, which could facilitate the design of electrically operated droplet equipment and guide the safe and stable operation of the device.

Potential flow has many applications, including the modelling of unsteady flows in aerodynamics. For these models to work efficiently, it is best to avoid Biot–Savart interactions. This work presents a grid-based treatment of potential flows in two dimensions and its use in a vortex model for simulating unsteady aerodynamic flows. For flows consisting of vortex elements, the treatment follows the vortex-in-cell approach and solves the streamfunction–vorticity Poisson equation on a Cartesian grid after transferring the circulation from the vortices onto the grid. For sources and sinks, an analogous approach can be followed using the scalar potential. The combined velocity field due to vortices, sinks and sources can then be obtained using the Helmholtz decomposition. In this work, we use several key tools that ensure the approach works on arbitrary geometries, with and without sharp edges. First, the immersed boundary projection method is used to account for bodies in the flow and the resulting body-forcing Lagrange multiplier is identified as the bound vortex sheet strength. Second, sharp edges are treated by decomposing the vortex sheet strength into a singular and non-singular part. To enforce the Kutta condition, the non-singular part can then be constrained to remove the singularity introduced by the sharp edge. These constraints and the Poisson equation are formulated as a saddle-point system and solved using the Schur complement method. The lattice Green's function is used to efficiently solve the discrete Poisson equation with unbounded boundary conditions. The method and its accuracy are demonstrated for several problems.

We present a two-dimensional numerical study of turbulent Rayleigh–Bénard convection with air as the working fluid over a multi-scale randomly roughened surface in a rectangular box of aspect ratio 2 over three decades of Rayleigh number ($10^8 \leq {\textit {Ra}} \leq 10^{11}$). With varied response of the roughness elements at different ${\textit {Ra}}$, enhanced heat transfer scaling ($\gamma =0.41$) is retained throughout the explored ${\textit {Ra}}$ range. The plume emission process is triggered from the taller roughness elements at a lower ${\textit {Ra}}$, while smaller elements contribute significantly at higher ${\textit {Ra}}$. Increased plume emission frequency compared to the smooth case is reflected from the enhanced volume fraction and thermal dissipation rate of plumes. The tip of the roughness elements exhibits the highest temperature and vertical velocity fluctuations, while washing out of the trapped fluid in the throat region holds the key to enhanced heat flux at higher ${\textit {Ra}}$. Increased localized pockets of fluid at higher ${\textit {Ra}}$ indicate better inter-scale energy transfer, which is reflected in higher energy content at all scales in the computed temperature spectra. The decomposition of the flow field into orthogonal modes reveals that heat transfer enhancement at higher ${\textit {Ra}}$ is associated with multiple small-scale structures. Owing to better energy transfer and intense localized fluctuations, modal distribution of energy is less severe for higher ${\textit {Ra}}$, and stable twin large-scale rolls do not favour an efficient heat transport process.

A hypersonic, spatially evolving turbulent boundary layer at Mach 12.48 with a cooled wall is analysed by means of direct numerical simulations. At the selected conditions, massive kinetic-to-internal energy conversion triggers thermal and chemical non-equilibrium phenomena. Air is assumed to behave as a five-species reacting mixture, and a two-temperature model is adopted to account for vibrational non-equilibrium. Wall cooling partly counteracts the effects of friction heating, and the temperature rise in the boundary layer excites vibrational energy modes while inducing mild chemical dissociation of oxygen. Vibrational non-equilibrium is mostly driven by molecular nitrogen, characterized by slower relaxation rates than the other molecules in the mixture. The results reveal that thermal non-equilibrium is sustained by turbulent mixing: sweep and ejection events efficiently redistribute the gas, contributing to the generation of a vibrationally under-excited state close to the wall, and an over-excited state in the outer region of the boundary layer. The tight coupling between turbulence and thermal effects is quantified by defining an interaction indicator. A modelling strategy for the vibrational energy turbulent flux is proposed, based on the definition of a vibrational turbulent Prandtl number. The validity of the strong Reynolds analogy under thermal non-equilibrium is also evaluated. Strong compressibility effects promote the translational–vibrational energy exchange, but no preferential correlation was detected between expansions/compressions and vibrational over-/under-excitation, as opposed to what has been observed for unconfined turbulent configurations.

A transition-predictive Reynolds-averaged Navier–Stokes (RANS) model is introduced as the Reynolds controlling condition to constrained large-eddy simulation (TrCLES) to improve the ability of this method to predict wall-bounded laminar–turbulent transition flows. In the TrCLES method, the constraint conditions for total Reynolds stress and heat flux are only imposed to the near-wall region in transitional and turbulent boundary layer. The newly proposed method recovers direct numerical simulation in the laminar boundary region, and retrieves the traditional large-eddy simulation method in the far-wall regions. The TrCLES method is validated in simulations of external flow around the Eppler 387 (E387) airfoil and internal flow past the ultrahigh-lift low-pressure turbine T106C cascade. The improved delayed detached-eddy simulation (IDDES) method, CLES method based on full-turbulence shear stress transport model and RANS method with a three-equation transition model are also evaluated in comparison with the available experimental and numerical data. As expected, the TrCLES method can predict laminar separation bubble and separation-induced transition process in both the E387 and T106C flows pretty well. In contrast, neither IDDES nor the original CLES can provide reasonable prediction for the laminar separation-induced transition phenomenon. The validity and fidelity of the TrCLES method are further verified by simulations of the T106C cascade flows in a wider range of exit Reynolds and Mach numbers. It is shown that the TrCLES method can not only predict the time-averaged aerodynamic quantities such as isentropic Mach number and exit kinetic energy loss very well, but also capture the laminar separation bubble and unsteady flow structures with satisfactory accuracy.

The paper is devoted to the study of the propulsive motion of a flapping wing of a circular cross-section performing translational–rotational oscillations in a viscous incompressible fluid. To describe the flow past the wing, the unsteady Navier–Stokes equation is solved. Using the method of asymptotic expansions for the case of small amplitudes of oscillations, an analytical solution of the problem is constructed in the first two terms. It is shown that the nonlinear interaction of time harmonics of translational and rotational oscillations causes secondary flows (steady streaming) that make the wing move in the direction perpendicular to the axis of translational oscillations. For the case of cruising motion, when the average hydrodynamic force acting on the wing is equal to zero, the dependence of the average speed on the dimensionless oscillation parameters is found. The results show that for relatively large angles of rotation the cruising speed of a flapping wing can be as high as the velocity amplitude of translational oscillations. The limits of applicability of the asymptotic theory are investigated using direct numerical simulations. Numerical data demonstrate that the theory well describes the flow past the wing in a wide range of dimensionless amplitudes, frequencies and angles of rotation. In conclusion, the efficiency of the propulsion system is evaluated. It is shown that in terms of relative energy consumption, a cylindrical flapping wing can be attributed to the most efficient propulsors in the range of Reynolds numbers $Re \sim 10^2\text {--}10^3$.

Experiments are carried out on a turbulent fountain to investigate the entrainment of ambient fluid and the dilution of scalar concentration in the fountain top (also referred to as the ‘cap’). The source Froude number ($Fr_o$), defined based on the density difference between the source and the ambient fluids, is varied between 10 and 30, while the Reynolds number ($Re_o$) is set to a minimum of 3000 to ensure a fully turbulent flow at the source. High-fidelity velocity and concentration measurements are obtained using particle image velocimetry and planar laser induced fluorescence techniques, respectively. The mean concentration field indicates that the cap is approximately hemispherical and its base is characterised by the local Froude number $Fr_z \approx 1.5$. It is observed that the ratio of the entrained ($Q_{top}$) volume flux in the fountain top and the volume flux supplied ($Q_{in}$) at the base of the cap varies between 1.5 and 3.5 at different $Fr_o$, exceeding the values of $Q_{top}/Q_{in} (= 0.5\text {--}0.8)$ for a fountain developing across a density interface (Lin & Linden, J. Fluid Mech., vol. 542, 2005, pp. 25–52). The present experimental results for $Q_{top}/Q_{in}$ are in excellent agreement with published numerical simulations of turbulent fountains at similar $Fr_o$. Lastly, a robust metric to estimate the dilution of scalar in the fountain top has been proposed. The results clearly indicate that dilution is not equal to the entrainment ratio; however, the self-similarity of non-dimensional dilution profiles at different $Fr_o$ suggests that the phenomenological model for entrainment in the fountain top put forth by Debugne & Hunt (J. Fluid Mech., vol. 796, 2016, pp. 195–210), can also be effectively used to model the dilution of scalar concentration in the cap. Further, it is found that the variation of the dilution ratio is closely linked to the local Reynolds number ($Re_z$) at the base of the cap. This is verified using an analytical model that describes the dependency of $Re_z$ on the local and source parameters – $Fr_z$, $Re_o$ and $Fr_o$.

The nonlinear robustness of laminar plane Couette flow is considered under the action of in-phase spanwise wall oscillations by computing properties of the edge of chaos, i.e. the boundary of its basins of attraction. Three measures are used to quantify the chosen control strategy on laminar-to-turbulent transition: the kinetic energy of edge states (local attractors on the edge of chaos), the form of the minimal seed (least energetic perturbation on the edge of chaos), and the laminarization probability (the probability that a random perturbation from the laminar flow of given kinetic energy will laminarize). A novel Bayesian approach is introduced to enable the accurate computation of the laminarization probability at a fraction of the cost of previous methods. While the edge state and the minimal seed provide useful information about the dynamics of transition to turbulence, neither measure is particularly useful to judge the effectiveness of the control strategy since they are not representative of the global geometry of the edge. In contrast, the laminarization probability provides global information about the edge and can be used to evaluate the control effectiveness by computing a laminarization score (the expected laminarization probability) and the associated expected dissipation rate of the controlled flow. These two quantities allow for the determination of optimal control parameter values subject to desired constraints. The results discussed in the paper are expected to be applied to a wide range of transitional flows and control strategies aimed at suppressing or triggering transition to turbulence.

The rainbow reflection effect describes the broadband spatial separation of wave spectral components caused by a spatially graded array of resonators. Although mainly studied in optics and acoustics, this phenomenon has recently been demonstrated both theoretically and experimentally for water waves travelling through an array of vertical cylinders. The scattering of two-dimensional linear water waves by an array of vertical, surface-piercing barriers is considered here, in which both the submergence and spacing between the barriers are spatially graded. The rainbow reflection effect arises naturally as wave energy temporarily becomes amplified at different locations depending on frequency. Band diagram calculations are used to demonstrate that this is a consequence of the wave gradually slowing down throughout the array. The wave/barriers scattering problem is then augmented by positioning heave-restricted, rectangular floating bodies equipped with a linear damping mechanism between each adjacent pair of barriers. A solution to the resulting boundary value problem is obtained using an integral equation/Galerkin method. Using constrained optimisation, passive rainbow absorbers are designed that achieve near-perfect absorption (i) over a discrete set of frequencies and (ii) over an octave. This suggests potential applications of rainbow absorbers in the design of smart coastal technologies.

Many works have considered two-dimensional free-surface flow over the edge of a plate, forming a waterfall, and with uniform horizontal flow far upstream. The flow is assumed to be steady and irrotational, whilst the fluid is assumed to be inviscid and incompressible. Gravity is also taken into account. In particular, amongst these works, numerical solutions for supercritical flows have been computed, utilising conformal mappings as well as a series truncation and collocation method. We present an extension to this work where a more appropriate expression is taken for the assumed form of the complex velocity. The justification of this lies in the behaviour of the waterfall flow far downstream and the wish to better encapsulate the parabolic nature of such a free-falling jet. New numerical results will be presented, demonstrating the improved shape of the new free-surface profiles. These numerical solutions will also be validated through comparisons with asymptotic solutions, in particular for flows with larger Froude numbers. For flows with Froude numbers closer to 1, we demonstrate that the revised complex velocity ansatz should be employed in place of the asymptotic solution. We present further adjustments to the method that lead to enhanced coefficient decay. The aforementioned adjustments are also applied to supercritical weir flows and similar improvements to the jet shape can be observed.

The sound of a vortex ring passing near a semi-infinite porous edge is investigated analytically. A Green's function approach solves the associated vortex sound problem and determines the time-dependent pressure signal and its directivity in the acoustic far field as a function of a single dimensionless porosity parameter. At large values of this parameter, the radiated acoustic power scales on the vortex ring speed $U$ and the nearest distance between the edge and the vortex ring $L$ as $U^6 L^{-5}$, in contrast to the $U^5 L^{-4}$ scaling recovered in the impermeable edge limit. Results for the vortex ring configuration in a quiescent fluid furnish an analogue to scaling results from standard turbulence noise generation analyses, and permit a direct comparison to experiments described in Part 2 that circumvent contamination of the weak sound from porous edges by background noise sources that exist as a result of a mean flow.

The effect of hydrodynamic interactions on the collective locomotion of fish schools is still poorly understood. In this paper, the flow-mediated organization of two tandem flapping foils, which are free in both the longitudinal and lateral directions, is numerically studied. It is found that the tandem formation is unstable for two foils when they can self-propel in both the longitudinal and lateral directions. Three types of resultant regular formations are observed, i.e. semi-tandem formation, staggered formation and transitional formation. Which type of regular formation occurs depends on the flapping parameters and the initial longitudinal distance between the two foils. Moreover, there is a threshold value of the cycle-averaged longitudinal distance (which is approximately 0.55) below which both velocity enhancement and efficiency augmentation can be achieved by two foils in regular formations. The results obtained here may shed some light on understanding the emergence of regular formations of fish schools.

We perform a direct numerical simulation (DNS) of interacting Kelvin–Helmholtz instabilities (KHI) that arise at a stratified shear layer where KH billow cores are misaligned or exhibit varying phases along their axes. Significant evidence of these dynamics in early laboratory shear-flow studies by Thorpe (Geophys. Astrophys. Fluid Dyn., vol. 34, 1985, pp. 175–199) and Thorpe (J. Geophys. Res., vol. 92, 1987, pp. 5231–5248), in observations of KH billow misalignments in tropospheric clouds (Thorpe, Q. J. R. Meteorol. Soc., vol. 128, 2002, pp. 1529–1542) and in recent direct observations of such events in airglow and polar mesospheric cloud imaging in the upper mesosphere reveals that these dynamics are common. More importantly, the laboratory and mesospheric observations suggest that these dynamics lead to more rapid and more intense instabilities and turbulence than secondary convective instabilities in billow cores and secondary KHI in stratified braids between and around adjacent billows. To date, however, no simulations exploring the dynamics and energetics of interacting KH billows (apart from pairing) have been performed. Our DNS performed for Richardson number $Ri=0.10$ and Reynolds number $Re=5000$ demonstrates that KHI tubes and knots (i) comprise strong and complex vortex interactions accompanying misaligned KH billows, (ii) accelerate the transition to turbulence relative to secondary instabilities of individual KH billows, (iii) yield significantly stronger turbulence than secondary KHI in billow braids and secondary convective instabilities in KHI billow cores and (iv) expand the suite of secondary instabilities previously recognized to contribute to KHI dynamics and breakdown to turbulence in realistic geophysical environments.

Fritts et al. (J. Fluid Mech., vol. xx, 2022, xx) describe a direct numerical simulation of interacting Kelvin–Helmholtz instability (KHI) billows arising due to initial billow cores that exhibit variable phases along their axes. Such KHI exhibit strong ‘tube and knot’ dynamics identified in early laboratory studies by Thorpe (Geophys. Astrophys. Fluid Dyn., vol. 34, 1985, pp. 175–199). Thorpe (Q.J.R. Meteorol. Soc., vol. 128, 2002, pp. 1529–1542) noted that these dynamics may be prevalent in the atmosphere, and they were recently identified in atmospheric observations at high altitudes. Tube and knot dynamics were found by Fritts et al. (J. Fluid. Mech., 2022) to drive stronger and faster turbulence transitions than secondary instabilities of individual KH billows. Results presented here reveal that KHI tube and knot dynamics also yield energy dissipation rates $\sim$2–4 times larger as turbulence arises and that remain $\sim$2–3 times larger to later stages of the flow evolution, compared with those of secondary convective instabilities (CI) and secondary KHI accompanying KH billows without tube and knot influences. Elevated energy dissipation rates occur due to turbulence transitions by tube and knot dynamics arising on much larger scales than secondary CI and KHI where initial KH billows are misaligned. Tube and knot dynamics also excite large-scale Kelvin ‘twist waves’ that cause vortex tube and billow core fragmentation, more energetic cascades of similar interactions to smaller scales and account for the strongest energy dissipation events accompanying such KH billow evolutions.

Boundary-layer transition is accompanied by a significant increase in skin friction whose origin is rigorously explained using the stochastic Lagrangian formulation of the Navier–Stokes equations. This formulation permits the exact analysis of vorticity dynamics in individual realizations of a viscous incompressible fluid flow. The Lagrangian reconstruction formula for vorticity is here extended for the first time to Neumann boundary conditions (Lighthill source). We can thus express the wall vorticity, and, therefore, the wall stress, as the expectation of a stochastic Cauchy invariant in backward time, with contributions from (a) wall vorticity flux (Lighthill source) and (b) interior vorticity that has been evolved by nonlinear advection, viscous diffusion, vortex stretching and tilting. We consider the origin of stress maxima in the transitional region, examining a sufficient number of events to represent the increased skin friction. The stochastic Cauchy analysis is applied to each event to trace the origin of the wall vorticity. We find that the Lighthill source, vortex tilting, diffusion and advection of the outer vorticity make minor contributions. They are less important than spanwise stretching of near-wall spanwise vorticity, which is the dominant source of skin-friction increase during laminar-to-turbulent transition. Our analysis should assist more generally in understanding drag generation and reduction strategies and flow separation in terms of near-wall vorticity dynamics.

For a fluid system with an interface between two-layer liquids and a free surface, there exist two series of natural frequencies. Under vertical excitations, the interface and the free surface can be excited separately or simultaneously. Laboratory experiments were conducted to excite the first- and second-mode Faraday waves on the free surface, which forced the in-phase motion of the interface. The experimental data were used to validate the extended numerical model NEWTANK. The secondary surface instability in the form of period tripling breaking occurred on the free surface when second-mode surface resonant response was large. Further numerical investigations were conducted to study the resonance of interface and in addition to the primary Faraday instability, the secondary interface instability occurred due to Rayleigh–Taylor (RT) instability. The analysis revealed that this was due to the contribution from the combined effects of vertical fluid acceleration and external excitation, which made the upper layer liquid essentially ‘heavier’ than the lower layer at certain instances, even when the amplitude of acceleration of the external excitation was lower than that of gravity at all times. A dimensionless parameter ‘normalised effective gravitational acceleration’ (NEGA) was proposed to quantify the occurrence criterion of RT instability in a dynamic fluid system. Furthermore, simultaneous resonances at both the interface and the free surface were studied and reverse flows could be formed across the interface as the upper Faraday waves and lower Faraday waves moved in opposite directions, which generated the secondary interface instability of Kelvin–Helmholtz (KH) type. When KH instability took place, more energy was transferred to higher frequency modes as time evolved.

The subtle but crucial effects of large-scale forcing on the small-scale velocity-gradient (VG) dynamics is examined using direct numerical simulation (DNS) data of incompressible turbulence. The interplay among large-scale forcing, inertia, pressure and viscous effects is characterised as a function of local streamline geometry and VG magnitude (Frobenius norm). When conditioned on local topology, forcing: (i) counteracts inertial and viscous action in the strain-dominated nodal topologies; and (ii) balances pressure action in the rotation-dominated unstable focal topologies. Unexpectedly, forcing acts to reduce VG magnitudes in regions of strong dissipation. In these regions, forcing balances the non-local pressure effects whereas viscous action offsets the nonlinear inertial effects. In regions of very low dissipation, forcing combines with inertia and pressure effects to offset viscous action. With regard to the probability distribution of the normalised VG invariants, the primary role of forcing is to nullify certain features (dilatational probability currents) of inertia, pressure and viscous action. This results in the emergence of universal statistical features (solenoidal probability currents) that are determined largely by inertia–pressure–viscous interactions. These findings serve to enhance our understanding of small-scale processes in turbulence and guide VG model development.

The seabed in coastal regions is often covered by a dense submerged canopy formed by benthic organisms. The in-canopy flow is of great interest to coastal researchers. In this study, a semi-analytical model is proposed for the in-canopy current profile under the influence of coexisting waves. The current is driven by a mean shear stress at the canopy top and a mean streamwise pressure gradient. Wave–current interaction arises from the mean canopy drag, which is described by the conventional quadratic law. An algebraic eddy viscosity model, which accounts for the turbulence generated by a canopy-scale vortex and stem-scale wake, is proposed. The model is calibrated against unidirectional flume experiments, and is subsequently validated against a wave–current flume experiment. It is found that the mean canopy drag experienced by the current can be significantly amplified by a co-existing wave-driven oscillatory flow. Under the same current driving force, the current in wave–current flow is reduced from that in a pure-current flow. The reduction increases with canopy density and wave strength. For a wave and current at an angle, veering of the current velocity vector is suggested by the model, which makes the mean canopy drag align with the driving force of the current.