Synthetic-aperture radar images and mesoscale models show that wind-farm wakes differ from single-turbine wakes. For instance, wind-farm wakes often narrow and do not disperse over long distances, contrasting the broader and more dissipating wakes of individual turbines. In this work, we aim to better understand the mechanisms that govern wind-farm wake behaviour and recovery. Hence we study the wake properties of a $1.6$ GW wind farm operating in conventionally neutral boundary layers with capping-inversion heights $203$, $319$, $507$ and $1001$ m. In shallow boundary layers, we find strong flow decelerations that reduce the Coriolis force magnitude, leading to an anticlockwise wake deflection in the Northern Hemisphere. In deep boundary layers, the vertical turbulent entrainment of momentum adds clockwise-turning flow from aloft into the wake region, leading to a faster recovery rate and a clockwise wake deflection. To estimate the wake properties, we propose a simple function to fit the velocity magnitude profiles along the spanwise direction. In the vertical direction, the wake spreads up to the capping-inversion height, which significantly limits vertical wake development in shallow-boundary-layer cases. In the horizontal direction and for shallow boundary layers, the wake behaves as two distinct mixing layers located at the lateral wake edges, which expand and turn towards their low-velocity side, causing the wake to narrow along the streamwise direction. A detailed analysis of the momentum budget reveals that in deep boundary layers, the wake is predominantly replenished through turbulent vertical entrainment. Conversely, in shallow boundary layers, wakes are mostly replenished by mean flow advection in the spanwise direction.

Bubble–particle collisions in turbulence are key to the froth flotation process that is widely employed industrially to separate hydrophobic from hydrophilic materials. In our previous study (Chan et al., 2023 J. Fluid Mech. 959, A6), we elucidated the collision mechanisms and critically reviewed the collision models in the no-gravity limit. In reality, gravity may play a role since, ultimately, separation is achieved through buoyancy-induced rising of the bubbles. This effect has been included in several collision models, which have remained without a proper validation thus far due to a scarcity of available data. We therefore conduct direct numerical simulations of bubbles and particles in homogeneous isotropic turbulence with various Stokes, Froude and Reynolds numbers, and particle density ratios using the point-particle approximation. Generally, turbulence enhances the collision rate compared with the pure relative settling case by increasing the collision velocity. Surprisingly, however, for certain parameters the collision rate is lower with turbulence compared with without, independent of the history force. This is due to turbulence-induced bubble–particle spatial segregation, which is most prevalent at weak relative gravity and decreases as gravitational effects become more dominant, and reduced bubble slip velocity in turbulence. The existing bubble–particle collision models only qualitatively capture the trends in our numerical data. To improve on this, we extend the model by Dodin & Elperin (2002 Phys. Fluids 14, 2921–2924) to the bubble–particle case and found excellent quantitative agreement for small Stokes numbers when the history force is negligible and segregation is accounted for.

To elucidate the attenuation mechanism of wall-bounded turbulence due to heavy small particles, we conduct direct numerical simulations (DNS) of turbulent channel flow laden with finite-size solid particles. When particles cannot follow the swirling motions of wall-attached vortices, vortex rings are created around the particles. These particle-induced vortices lead to additional energy dissipation, reducing the turbulent energy production from the mean flow. This mechanism results in the attenuation of turbulent kinetic energy, which is more significant when the Stokes number of particles is larger or particle size is smaller under the condition that the volume fraction of particles is fixed. Moreover, we propose a method to quantitatively predict the degree of turbulence attenuation without using DNS data by estimating the additional energy dissipation rate in terms of particle properties.

Inspired by the need to theoretically understand the naturally occurring interactions between internal waves and mesoscale phenomena in the ocean, we derive a novel model equation from the primitive rotational Euler equations using the multi-scale asymptotic expansion method. By applying the classic balance $\epsilon =\mu ^2$ between nonlinearity (measured by $\epsilon$) and dispersion (measured by $\mu$), along with the assumption that variations in the transverse direction are of order $\mu$, which is smaller than those in the propagation direction, we arrive at terms from the classic Kadomtsev–Petviashvili equation. However, when incorporating background shear currents in two horizontal dimensions and accounting for Earth’s rotation, we introduce three additional terms that, to the best of the authors’ knowledge, have not been addressed in the previous literature. Theoretical analyses and numerical results indicate that these three terms contribute to a tendency for propagation in the transverse direction and an overall variation in wave amplitudes. The specific effects of these terms can be estimated qualitatively based on the signs of the coefficients for each term and the characteristics of the initial waves. Finally, the potential shortcomings of this proposed equation are illuminated.

Direct numerical simulations are performed to explore the impact of surface roughness on inter-scale energy transfer and interaction in a turbulent open-channel flow over differently arranged rough walls. With friction Reynolds number approximately 540, six distinct configurations of roughness arrangements are examined. The results show that the clustered roughness arrangements yield notable changes in large-scale secondary-flow structures, which manifest in the profiles of dispersive stresses, predominantly near the roughness elements. They are marked by the presence of spanwise alternating high-momentum pathways and low-momentum pathways. From the outer peak in the spanwise energy spectra, the size and intensity of turbulent secondary flows are shown to be related to the spanwise spacing of the roughness heterogeneity. The emergence of turbulent secondary flows serves to suppress the original large-scale structures in the outer region of smooth-wall turbulence, paving the way for the development of new turbulent structures at the second harmonic scale. Furthermore, the spanwise triadic interaction analysis reveals the mutual energy exchange between the secondary harmonic scale and the secondary-flow scale. These findings elucidate the underlying mechanisms behind the attenuation of large-scale structures in the outer region influenced by roughness, offering new insights into the dynamic interplay of scale interactions in rough-wall turbulence.

Several hypotheses are employed to describe the fluctuating motions within nozzles and to analytically predict combustion noise generation mechanisms. One of these assumptions is that of a calorifically perfect gas mixture, where $c_p$ is constant. Nonetheless, a realistic flow rather encompasses heat capacities $c_p$ that vary with temperature, i.e. $c_p = c_p(T)$, such that the mixture is called thermally perfect. The influence of the mixture assumptions on noise generation mechanisms is re-examined in this paper. To do so, the quasi one-dimensional Euler equations for multi-species, isentropic and non-reactive flow are considered within the nozzle. Their linearisation yields a new prediction model in addition to showing a new entropy-to-entropy coupling mechanism. Relying on either the assumption of low frequencies or the Magnus’ expansion methodology, two analytical solutions are derived and studied. Validation of these two prediction models is then provided relying on unsteady simulations of axisymmetric nozzles with superimposed incident waves. To generalise previous results, parametric studies are performed considering various nozzle flow geometries. Variations of up to $10\,\,\%$ are exhibited in a choked flow nozzle between the two mixtures, especially for the indirect entropy noise and the entropy-to-entropy transmission moduli.

The physical fidelity of turbulence models can benefit from a partial resolution of fluctuations, but doing so often comes with an increase in computational cost. To explore this trade-off in the context of wall-bounded flows, this paper introduces a framework for turbulence-resolving integral simulations (TRIS) with the goal of efficiently resolving the largest motions using a two-dimensional, three-component representation of the flow defined by instantaneous wall-normal integrals of velocity and pressure. Self-sustaining turbulence with qualitatively realistic large-scale structures is demonstrated for TRIS on an open-channel (half-channel) flow configuration using moment-of-momentum integral equations derived from Navier–Stokes with relatively simple closure approximations. Evidence from direct numerical simulations (DNS) suggests that TRIS can theoretically resolve $35\,\%{-}40\,\%$ of the turbulent skin friction enhancement for friction Reynolds numbers between $180$ and $5200$, without a noticeable decrease or increase as a function of Reynolds number. The current implementation of TRIS can match this resolution while simulating one flow through time in ${\sim}1$ minute on a single processor, even for very large Reynolds numbers. The framework facilitates a detailed apples-to-apples comparison of predicted statistics against data from DNS. Comparisons at friction Reynolds numbers of $395$ and $590$ show that TRIS generates a relatively accurate representation of the flow, while highlighting discrepancies that demonstrate a need for improving the closure models. The present results for open-channel flow represent a proof of concept for TRIS as a new approach for wall-bounded turbulence modelling, motivating extension to more general flow configurations such as boundary layers on immersed objects.

The aerodynamic sound generated by the oblique collision of two vortex rings is featured by the asymmetric emission associated with the octupole mode, which differs from the symmetric emission associated with the quadrupole mode observed in the coaxial collision of two vortex rings. This distinctive feature of aerodynamic sound is closely related to the tilting and reconnecting of the vortex rings. While previous studies have explored the effects of reconnecting on aerodynamic sound, this study specifically addresses the impact of vortex ring tilting. We propose a novel vortex sound formula to quantitatively assess the role of tilting in aerodynamic sound generation. The proposed formula relates the far-field sound pressure to equivalent circulations and vorticity centroids by referring to Truesdell’s consistency conditions for vorticity moments. The variations of the equivalent circulations and vorticity centroids in the oblique collision of two vortex rings under different configurations are analysed based on the numerical solution of the Navier–Stokes equations in the source region. It is found that the tilting of vortex rings results in a rapid change of the equivalent circulation associated with the vorticity in the collision direction. However, the change caused by titling is almost out of phase with that caused by reconnecting and deforming. The vortex tilting significantly reduces the aerodynamic sound associated with the longitudinal quadrupole and octupole modes, which is opposite to the role of vortex reconnecting that was reported in the oblique collision of vortex rings.