Modelling the noise emitted by turbulent jets is made difficult by their acoustic inefficiency: only a tiny fraction of the near-field turbulent kinetic energy is propagated to the far field as acoustic waves. As a result, jet-noise models must accurately capture this small, acoustically efficient component hidden among comparatively inefficient fluctuations. In this paper, we identify this acoustically efficient near-field source from large-eddy simulation data and use it to inform a predictive model. Our approach uses the resolvent framework, in which the source takes the form of nonlinear fluctuation terms that act as a forcing on the linearised Navier–Stokes equations. First, we identify the forcing that, when acted on by the resolvent operator, produces the leading spectral proper orthogonal decomposition modes in the acoustic field for a Mach 0.4 jet. Second, the radiating components of this forcing are isolated by retaining only portions with a supersonic phase speed. This component makes up less than 0.05 % of the total forcing energy but generates most of the acoustic response, especially at peak (downstream) radiation angles. Finally, we propose an empirical model for the identified acoustically efficient forcing components. The model is tested at other Mach numbers and flight-stream conditions and predicts noise within 2 dB accuracy for a range of frequencies, downstream angles and flight conditions.

We present a fully analytical solution for the natural oscillation of an inviscid sessile drop with small Bond number (surface tension dominates gravity) and a fixed contact line on a flat horizontal plate. The governing equations are expressed in terms of the toroidal coordinate system which yields solutions involving hypergeometric functions. Resonant frequencies are identified for zonal, sectoral and tesseral vibration modes. The predictions show excellent agreement with experimental data reported in the literature, particularly for flatter drops (lower $\theta _{c}$, but not so low as to incur significant viscous dissipation) and higher modes of vibration.

Non-isothermal reactive transport in complicated porous media is diverse in nature and industrial applications. There are challenges in the modelling of multiple physicochemical processes in multiscale pore structures with various length scales ranging from nanometres to micrometres. This study focuses on coke combustion during in situ crude oil combustion techniques. A micro-continuum model was developed to perform an image-based simulation of coke combustion through a multiscale porous medium. The simulation coupled weakly compressible gas flow, species transport, conjugate heat transfer, heterogeneous coke oxidation kinetics and structural evolution. The unresolved nanoporous coke region was treated as a continuum, for which the random pore model, permeability model and species diffusivity model were integrated as sub-grid models to account for the sub-resolution reactive surface area, Darcy flow and Knudsen diffusion, respectively. A Pe–Da diagram was provided to present five characteristic combustion regimes covering the ignition temperature and air flux in realistic field operations and laboratory measurements. The present model proved to achieve more accurate predictions of the feasible ignition temperature than previous models. Compared with the air flux of $\phi \sim O\textrm{(1) s}{\textrm{m}^\textrm{3}}(\textrm{air})\;{({\textrm{m}^\textrm{2}}\ \textrm{h})^{ - 1}}$ in the field, the increasing air flux in the laboratory transformed the combustion regime from diffusion-limited to convection-limited, which led to an overpredicted burning temperature. Reactive fingering combustion was analysed to understand the potential risks in some experimental measurements. The findings provide a better understanding of coke combustion and can help engineers design sustainable combustion methods. The developed image-based model allows other types of multiscale and nonlinear reactive transport to be simulated.

The linear stability of a liquid film falling down an inclined flexible plane under the influence of gravity is investigated using analytical and computational techniques. A general model for the flexible substrate is used leading to a modified Orr–Sommerfeld problem addressed numerically using a Chebyshev tau decomposition. Asymptotic limits of long waves and small Reynolds numbers are addressed analytically and linked to the computations. For long waves, the flexibility has a destabilising effect, where the critical Reynolds number decreases with decreasing stiffness, even destabilising Stokes flow for sufficiently small stiffness. To pursue this further, a Stokes flow approximation was considered, which confirmed the long-wave results, but also revealed a short wave instability not captured by the long-wave expansions. Increasing the surface tension has little effect on these instabilities and so they were characterised as wall modes. Wider exploration revealed mode switching in the dispersion relation, with the wall and surface mode swapping characteristics for higher wavenumbers. The zero-Reynolds-number results demonstrate that the long-wave limit is not sufficient to determine instabilities so the numerical solution for arbitrary wavenumbers was sought. A Chebyshev tau spectral method was implemented and verified against analytical solutions. Short wave wall instabilities persist at larger Reynolds numbers and destabilisation of all Reynolds numbers is achievable by increasing the wall flexibility, however increasing the stiffness reverts back to the rigid wall limit. An energy decomposition analysis is presented and used to identify the salient instability mechanisms and link them to their physical origin.

The performance benefits of deploying tidal turbines in close side-by-side proximity to exploit constructive interference effects are demonstrated experimentally using two 1.2 m diameter turbines. The turbines are arrayed side-by-side at 1/4 diameter tip-to-tip spacing, and their performance compared with that of a single rotor. Tests were completed in the 25 m diameter, 2 m deep wave and current FloWave Ocean Energy Research facility. A detailed assessment of inflow conditions at different control points is used to understand the impact that rotors, designed for high blockage conditions, have on the approach flow. After accounting for global blockage, a 10.8 % uplift in the twin-turbine-averaged power coefficient, relative to that for a single turbine, is found for the turbine design speed, at the expense of a 5.2 % increase in thrust coefficient and 3.1 % increase in tip-speed-ratio. Flowfield mapping demonstrated flow effects at array and device scale including array bypass flows and jetting between turbines. Azimuthal variation of blade root flapwise and edgewise bending moments show that the turbines interact in a beneficial manner, with additional and sustained loading peaks as the blades pass in close proximity to the neighbouring rotor. Peak performance for the twin turbines occurred at a higher tip-speed-ratio than for the single turbine, which is consistent with the twin turbines exerting a higher thrust on the flow to achieve maximum power. The twin turbine performance variation with tip-speed-ratio is found to be more gradual than for the single turbine. Using differential rotor speed control we observe that array performance is robust to small differences in neighbouring rotor operating point. Through these experiments we demonstrate that there is a substantial, achievable performance benefit from closely arraying turbines for side-by-side operation and designing them for constructive interference.

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.

Capillary flow of liquids plays a key role in many applications including lab-on-a-chip devices, heat pipes and printed electronics manufacturing. Open rectangular microchannels often appear in these applications, with the lack of a top resulting in a complex free-surface morphology and evaporation. In this work we develop a theoretical model based on lubrication theory and kinetically limited evaporation to examine capillary flow of evaporating liquid solutions in open rectangular microchannels connected to circular reservoirs. The model accounts for the complex free-surface morphology, solvent evaporation, Marangoni flows due to gradients in solute concentration and temperature and finite-size reservoir effects. Significant differences are predicted in flow behaviour between pure liquids and liquid solutions due to solvent evaporation and solute transport. Marangoni flows are found to promote more uniform solute deposition patterns after solvent evaporation. Model predictions of meniscus position evolution are in good agreement with prior capillary-flow experiments of aqueous poly(vinyl alcohol) solutions in the presence of evaporation. The model reveals that the principal mechanism through which evaporation influences the meniscus position in the experiments is the increase in viscosity with solute concentration.

We study the effects of interfacial kinetics on the electro-hydrodynamics of ion transport near an ion-selective surface using a combination of linear stability analysis and numerical simulation. The finite kinetics of the electrolyte–electrode interface affects the ion transfer and electroconvection in many ways. On a surface of fixed topography, such as a metal surface of slow and stable ion deposition or covered by a polymer membrane, the finite kinetics reduces the current in one-dimensional ion diffusion/migration, increases the critical voltage for the onset of the electroconvective instability, changes the dynamics of the electroconvection and the overlimiting current, and enhances the lateral ion diffusion within the interfacial layer. The first three effects are indirectly caused by the reaction kinetics and can be characterized by an effective voltage difference across the liquid electrolyte. In comparison, the last effect is controlled by a direct interplay between kinetics and nonlinear electroconvection. Scaling laws for ion transport and features of electroconvection are proposed. We also analyse the linear stability of a surface which evolves under ion deposition and find that the finite kinetics decreases the growth rate of both electroconvective and morphological instabilities and therefore modifies the wavenumber of the most unstable mode.

Wave shape (e.g. wave skewness and asymmetry) impacts sediment transport, remote sensing and ship safety. Previous work showed that wind affects wave shape in intermediate and deep water. Here, we investigate the effect of wind on wave shape in shallow water through a wind-induced surface pressure for different wind speeds and directions to provide the first theoretical description of wind-induced shape changes. A multiple-scale analysis of long waves propagating over a shallow, flat bottom and forced by a Jeffreys-type surface pressure yields a forward or backward Korteweg–de Vries (KdV)–Burgers equation for the wave profile, depending on the wind direction. The evolution of a symmetric, solitary-wave initial condition is calculated numerically. The resulting wave grows (decays) for onshore (offshore) wind and becomes asymmetric, with the rear face showing the largest shape changes. The wave profile's deviation from a reference solitary wave is primarily a bound wave and trailing, dispersive, decaying tail. The onshore wind increases the wave's energy and skewness with time while decreasing the wave's asymmetry, with the opposite holding for offshore wind. The corresponding wind speeds are shown to be physically realistic, and the shape changes are explained as slow growth followed by rapid evolution according to the unforced KdV equation.

In this study, the effects of rotor–rotor interaction on wake characteristics were investigated experimentally for a twin-rotor configuration in axial descent. The wake velocities were measured at descent rates (descent speed/induced velocity at the rotor disk during hover) from 0.87 to 1.52, and the rotor–rotor interaction strength was controlled by adjusting the distance between the rotor tips. As the descent rate increased, the wake of the isolated rotor gradually entered the vortex ring state (VRS), where the flow established an extensive recirculation zone. Correlation analysis was performed to distinguish the rotor wake between tubular and VRS topologies. The flow states for the isolated rotor were classified into pre-VRS, incipient VRS, and fully developed VRS, depending on the probability of vortex ring formation. The results reveal that the effects of rotor–rotor interaction on the wake characteristics of twin rotors differ depending on the descent rate, distance between rotor tips, and wake region. In the outer region, the flow state of the rotor wake remains consistent with that of the isolated rotor, irrespective of the distance between rotor tips. Conversely, the strong rotor–rotor interaction changes the flow state in the inner region by disrupting the vortex ring structure, intensifying the wake asymmetry about the rotational axis. The thrust measurements show that under the VRS, as the two rotors get closer, the thrust coefficient increases until vortex ring disruption occurs, and then decreases after the vortex ring is disrupted.

Thermal convection in an inclined layer between two parallel walls kept at different fixed temperatures is studied for fixed Prandtl number $\mathit{Pr}=1.07$. Depending on the angle of inclination and the imposed temperature difference, the flow exhibits a large variety of self-organized spatio-temporal convection patterns. Close to onset, these patterns have been explained in terms of linear stability analysis of primary and secondary flow states. At a larger temperature difference, far beyond onset, experiments and simulations show complex, dynamically evolving patterns that are not described by stability analysis and remain to be explained. Here we employ a dynamical systems approach. We construct stable and unstable exact invariant states, including equilibria and periodic orbits of the fully nonlinear three-dimensional Oberbeck–Boussinesq equations. These invariant states underlie the observed convection patterns beyond their onset. We identify state space trajectories that, starting from the unstable laminar flow, follow a sequence of dynamical connections between unstable invariant states until the dynamics approaches a stable attractor. Together, the network of dynamically connected invariant states mediates temporal transitions between coexisting invariant states and thereby supports the observed complex time-dependent dynamics in inclined layer convection.

Modal global linear stability analysis of thermal convection is performed with the linearized lattice Boltzmann method (LLBM). The onset of Rayleigh–Bénard convection in rectangular cavities with conducting and adiabatic sidewalls and the instability of two-dimensional (2-D) and three-dimensional (3-D) natural convection in cavities are studied. The method of linearizing the local equilibrium probability distribution function that was first proposed by Pérez et al. (Theor. Comp. Fluid Dyn., vol. 31, 2017, pp. 643–664) is extended to solve the coupled linear Navier–Stokes equations together with the linear energy equation in this work. A multiscale analysis is also performed to recover the macroscopic linear Navier–Stokes equations from the discrete lattice Boltzmann equations for both the single and multiple relaxation time models. The present LLBM is implemented in the framework of the Palabos library. It is validated by calculating the linear critical value of 2-D natural convection that the LLBM with the multiple relaxation time model has an error less than 1 % compared with the spectral method. The instability mechanism of the flow is explained by kinetic energy transfer analysis. It is shown that the buoyancy mechanism and inertial mechanism tend to stabilize the Hopf bifurcation of the 2-D natural convection at Pr < 0.08 and Pr > 1, respectively. For 3-D natural convection, subcritical bifurcation of the Hopf type is found for low-Prandtl-number fluids (Pr < 0.1).

The flow over a forward-facing step (FFS) at $Ma_\infty =1.7$ and $Re_{\delta _0}=1.3718\times 10^{4}$ is investigated by well-resolved large-eddy simulation. To investigate effects of upstream flow structures and turbulence on the low-frequency dynamics of the shock wave/boundary layer interaction (SWBLI), two cases are considered: one with a laminar inflow and one with a turbulent inflow. The laminar inflow case shows signs of a rapid transition to turbulence upstream of the step, as inferred from the streamwise variation of $\langle C_f \rangle$ and the evolution of the coherent vortical structures. Nevertheless, the separation length is more than twice as large for the laminar inflow case, and the coalescence of compression waves into a separation shock is observed only for the fully turbulent inflow case. The dynamics at low and medium frequencies is characterized by a spectral analysis, where the lower frequency range is related to the unsteady separation region, and the intermediate one is associated with the shedding of shear layer vortices. For the turbulent inflow case, we furthermore use a three-dimensional dynamic mode decomposition to analyse the individual contributions of selected modes to the unsteadiness of the SWBLI. The separation shock and Görtler-like vortices, which are induced by the centrifugal forces in the separation region, are strongly correlated with the low-frequency unsteadiness in the current FFS case. Similarly as observed previously for the backward-facing steps, we observe a slightly higher non-dimensional frequency (based on the separation length) of the low-frequency mode than for SWBLI in flat plate and ramp configurations.