Variability of the rotor aspect ratio is one of the inherent characteristics of vertical-axis wind turbines (VAWTs) which differentiates them especially from the more conventional horizontal-axis wind turbines. In this study, we intend to investigate the effect of rotor aspect ratio on VAWT wakes. In particular, we aim to find out whether a common behaviour exists in the mean flow field of such wakes. In order to do so, we first design and perform a set of numerical experiments (using our already validated large-eddy simulation framework) to obtain the mean flow field of the wakes of three VAWTs of different aspect ratio (2, 1 and ${\textstyle \frac{1}{4}}$) and the same thrust coefficient ($C_{T}=0.8$). After observing the obvious differences in these three wakes, by using the classical momentum integral and the concept of momentum diameter, we come up with an appropriate normalization length scale $D_{eq}=\sqrt{(4/\unicode[STIX]{x03C0})DH}$, where $D$ is the rotor diameter and $H$ is the rotor height. By normalizing the lengths (both streamwise and lateral) involved in the mean velocity profiles by $D_{eq}$, we obtain a remarkable collapse of the wake profiles for the three aspect ratios. As a corollary, cross-sections of wakes of turbines with different aspect ratios eventually converge to a circular shape – not an elliptical one, for example, as one might presume intuitively. This result influences the modelling of VAWT far wakes and, in turn, has implications on the optimal configuration of VAWT farms.

We present an investigation on the onset of Darcy–Bénard instability in a two-dimensional porous medium saturated with a non-Newtonian fluid and heated from below in the presence of a uniform horizontal pressure gradient. The fluid is taken to be of power-law nature with constant rheological index $n$ and temperature-dependent consistency index $\unicode[STIX]{x1D707}^{\ast }$. A two-dimensional linear stability analysis in the vertical plane yields the critical wavenumber and the generalised critical Rayleigh number as functions of dimensionless problem parameters, with a non-monotonic dependence from $n$ and with maxima/minima at given values of $\unicode[STIX]{x1D6FE}$, a parameter representing the effects of consistency index variations due to temperature. A series of experiments are conducted in a Hele-Shaw cell of aspect ratio $H/b=13.3{-}20$ to provide a verification of the theory. Xanthan Gum mixtures (nominal concentration from 0.10 % to 0.20 %) are employed as working fluids with a parameter range $n=0.55{-}0.72$ and $\unicode[STIX]{x1D707}_{0}^{\ast }=0.02{-}0.10~\text{Pa}~\text{s}^{n}$. The experimental critical wavenumber corresponding to incipient instability of the convective cells is derived via image analysis for different values of the imposed horizontal velocity. Theoretical results for critical wavenumber favourably compare with experiments, systematically underestimating their experimental counterparts by 10 % at most. The discrepancy between experiments and theory is more relevant for the critical Rayleigh number, with theory overestimating the experiments by a maximum factor less than two. Discrepancies are attributable to a combination of factors: nonlinear phenomena, possible subcritical bifurcations, and unaccounted-for disturbing effects such as approximations in the rheological model, wall slip, ageing and degradation of the fluid properties.

We propose a one-way coupled model that tracks individual primary particles in a conceptually simple cellular flow set-up to predict flocculation in turbulence. This computationally efficient model accounts for Stokes drag, lubrication, cohesive and direct contact forces on the primary spherical particles, and allows for a systematic simulation campaign that yields the transient mean floc size as a function of the governing dimensionless parameters. The simulations reproduce the growth of the cohesive flocs with time, and the emergence of a log-normal equilibrium distribution governed by the balance of aggregation and breakage. Flocculation proceeds most rapidly when the Stokes number of the primary particles is $O(1)$. Results from this simple computational model are consistent with experimental observations, thus allowing us to propose a new analytical flocculation model that yields improved agreement with experimental data, especially during the transient stages.

This paper investigates the existence and scaling of the so-called large-scale and very-large-scale motions (LSMs and VLSMs) in non-uniform turbulent open-channel flows developing over a smooth bed in a laboratory flume. A laser Doppler anemometry system was employed to measure vertical profiles of longitudinal and bed-normal velocity statistics over a wide range of hydraulic conditions. Pre-multiplied spectra of the longitudinal velocity fluctuations revealed the existence of two peaks occurring at wavelengths consistent with those associated with LSMs and VLSMs as detected in the past literature pertaining to wall turbulence. However, contrary to so-called canonical wall flows (i.e. flat-plate boundary layers, pipe and closed-channel flows), the LSM and VLSM peaks observed in the open-channel flows investigated herein are detectable over a much larger extent of the wall-normal coordinate. Furthermore, the VLSM peak appears at von Kármán numbers $Re_{\unicode[STIX]{x1D70F}}$ as low as 725, whereas in other wall flows much higher values are normally required. Finally, as conjectured by a recent study on uniform rough-bed open-channel flows, the present paper confirms that LSM wavelengths scale nicely with the flow depth, whereas the channel aspect ratio (i.e. the ratio between channel width and flow depth) is the non-dimensional parameter controlling the scaling of VLSM wavelengths. The intensity and wavelengths of the VLSM peaks were also observed to depend on the spanwise coordinate. This result suggests that VLSMs might be dynamically linked to secondary currents, as these are also known to vary in strength and size across the channel width.

Acoustic streaming in high-speed compressible channel flow and its impact on heat and momentum transfer is analysed numerically at two different Mach numbers, $M_{b}=0.75$ and 1.5, and moderate Reynolds numbers, $Re_{b}=3000$ and 6000. An external time-periodic forcing function is implemented to model the effect of acoustic drivers placed on the sidewalls. The excitation frequency is chosen according to the linear stability analysis of the background (unexcited) flow. High-fidelity numerical simulations performed at the optimal resonant condition reveal an initially exponential growth of perturbations followed by a nonlinear regime leading to the limit-cycle oscillations. In the last stage, we observe an acoustic (steady) streaming appearing as a result of nonlinear interactions between the periodic external wave and the background flow. This causes a steady enhancement in heat transfer at a rate higher than the skin-friction augmentation. We also show that perturbations of similar amplitude, but at suboptimal frequencies, may not lead to such limit-cycle oscillations and cannot make any noticeable modifications to the time-averaged flow quantities. The present research is the first study to demonstrate the acoustic streaming in compressible turbulent flows, and it introduces a novel technique towards enhancing the heat transfer with minimal skin-friction production.

We examine asymptotically the dynamics of two-dimensional, steady detonation wave propagation and failure for a strongly confined high explosive (HE), in which the width of the explosive is small relative to the reaction zone length. An energy balance equation is derived, which shows how the longitudinal acceleration of subsonic flow behind the detonation shock is influenced both by chemical reaction and by the effects of HE boundary streamline deflection, specifically via the induced rate of change of mass flux through the detonation wave. The latter serves to either counteract or reinforce the acceleration of longitudinal flow, depending on the sign of the gradient of the boundary streamline deflection at the detonation shock. The analysis is valid for general equations of state and chemical reaction rates in the HE. The asymptotically derived form of the energy equation represents an eigenvalue problem for the determination of the steady detonation propagation speed, solved via a shooting method. We explore specific results for ideal and stiffened equations of state, along with a pressure-dependent reaction rate for which changes in the pressure exponent and reaction order are also studied. We consider the influences of both straight and curved HE boundary streamline shapes. The asymptotic analysis reveals significant physical insights into how detonation propagation and failure are affected by strong confinement.

Quantifying accidental opaque discharges is a challenging task, since probing beyond their visible interfaces may be difficult or impossible. In this case, we show that the visible interface features near the jet exit can be used to gauge the flow. This work examines the interface in the near-field features of submerged homogeneous and immiscible turbulent jets. Experiments were carried out with water jets and immiscible silicone oil jets of two viscosities in a water tank. The jet Reynolds numbers are in the range of $Re\sim 4500{-}50\,000$ for homogeneous water jets and $Re\sim 3500{-}27\,000$ for silicone oil jets in water. The jet fluids are made visible by doping with fluorescent dye and excitation with directional illumination. The jet interfaces are continuous and convoluted for water jets, while convoluted and discontinuous with droplets and ligaments for oil jets. Direct flow visualization, schlieren photography, shadowgraph photography and particle image velocimetry are employed as appropriate. Interface length scales are characterized using various image processing techniques. Droplet sizes are quantified using Hough transformation. Interface length scales decrease with Reynolds number and increase gradually with distance from the exit plane for a given Reynolds number. These scales are isotropic for the homogeneous water jets and exhibit a streamwise-to-cross-stream ratio of approximately 1.3 for the oil jets. Interfacial tension, hence the Weber number, determines the average droplet size in the immiscible jets.

Although the process of hydrodynamic dispersion has been studied for many years, the description of solute spreading at early times has proved to be challenging. In particular, for some kinds of initial conditions, the solute evolution may exhibit a second moment that decreases (rather than increases, as is typically observed) in time. Most classical approaches would predict a negative effective hydrodynamic dispersion coefficient for such a situation. This creates some difficulties: not only does a negative dispersion coefficient lead to a violation of the second law of thermodynamics, but it also creates a mathematically ill-posed problem. We outline a set of four desirable qualities in a well-structured theory of unsteady dispersion as follows: (i) positivity of the dispersion coefficient, (ii) non-dependence upon initial conditions, (iii) superposability of solutions and (iv) convergence of solutions to classical asymptotic results. We use averaging to develop an upscaled result that adheres to these qualities. We find that the upscaled equation contains a source term that accounts for the relaxation of the initial configuration. This term decreases exponentially fast in time, leading to correct asymptotic behaviour while also accounting for the early-time solute dynamics. Analytical solutions are presented for both the effective dispersion coefficient and the source term, and we compare our upscaled results with averaged solutions obtained from numerical simulations; both averaged concentrations and spatial moments are compared. Error estimates are quantified, and we find good correspondence between the upscaled theory and the numerical results for all times.

Bounding walls or immersed surfaces are utilized in many industrial systems as the primary thermal source to heat a gas–solids mixture. Previous efforts to resolve the solids’ heat transfer near a boundary involve the extension of unbounded convection correlations into the near-wall region in conjunction with particle-scale theories for indirect conduction. Moreover, unbounded drag correlations are utilized in the near-wall region (without modification) to resolve the force exerted on a solid particle by the fluid. We rigorously test unbounded correlations and indirect conduction theory against outputs from direct numerical simulation of laminar flow past a hot plate and a static, cold particle. Here, local variables are utilized for consistency with unresolved computational fluid dynamics discrete element methods and lead to new unbounded correlations that are self-similar to those obtained with free-stream variables. The new drag correlation with local fluid velocity captures the drag force in both the unbounded system as well as the near-wall region while the classic, unbounded drag correlation with free-stream fluid velocity dramatically over-predicts the drag force in the near-wall region. Similarly, classic, unbounded convection correlations are found to under-predict the heat transfer occurring in the near-wall region. Inclusion of indirect conduction, in addition to unbounded convection, performs markedly better. To account for boundary effects, a new Nusselt correlation is developed for the heat transfer in excess of local, unbounded convection. The excess wall Nusselt number depends solely on the dimensionless particle–wall separation distance and asymptotically decays to zero for large particle–wall separation distances, seaming together the unbounded and near-wall regions.

Bottom-heated convection in rotating spherical shells provides a simple analogue for many astrophysical and geophysical fluid systems. We construct a database of 74 three-dimensional numerical convection models to investigate the scaling behaviour of seven diagnostics over a range of Ekman $(10^{-6}\leqslant E\leqslant 10^{-3})$ and Rayleigh $(15\leqslant \widetilde{Ra}\leqslant 18\,000)$ numbers while using a Prandtl number of unity. Our configuration is chosen to model Earth’s core as defined by the fixed flux thermal boundary conditions, radius ratio $r_{i}/r_{o}$ of $0.35$ and a gravity profile that varies linearly with radius. The quantities of interest are the viscous and thermal boundary layer thickness, mean temperature gradient, mean interior temperature, Nusselt number, horizontal flow length scale, and Reynolds number. We find four parameter regimes characterised by different scaling behaviour. For $E\leqslant 10^{-4}$ and low $Ra$ the weakly nonlinear regime is characterised by a balance between viscous, Archimedean and Coriolis forces and the heat transfer is described by weakly nonlinear theory. At low $E$ and moderate $Ra$, the rapidly rotating regime sees inertia take over from viscosity in the global force balance. In this regime the heat transfer scaling has increasing exponent with decreasing Ekman number and shows no saturation to the diffusion free $Ra^{3/2}E^{2}$ scaling. At high $Ra$ and all $E$ the importance of the Coriolis force gradually decreases and all diagnostics continually change in the transitional regime before approaching the scaling behaviour of non-rotating convection.

Analytical wind turbine wake models and wake superposition methods are prevailing tools widely adopted by the wind energy community to predict the power production of wind farms. However, none of the existing wake superposition methods conserve the streamwise momentum. In this study, a novel wake superposition method capable of conserving the total momentum deficit in the streamwise direction is derived theoretically, and its performance is validated with both particle imaging velocimetry measurements and large-eddy simulation results. Detailed inter-method comparisons show that the novel wake superposition method outperforms all the existing methods by delivering an accurate prediction of the power production and the centreline wake velocity deficit, with a typical error of less than 5 % (excluding the near-wake region). Additionally, the momentum-conserving wake superposition method is extended to combine the transverse velocities induced by yawed wind turbines, and the secondary wake steering effect crucial to the power optimization in active wake control is well reproduced.

Tumbling wings are one of Nature’s many tricks to enhance the dispersal efficiency of flying seedpods. However, the interplay between the seedpod morphology and its dispersal range is not well understood. Here, we investigate the question of how planform geometry affects two-dimensional tumbling flight by designing wings of various planform and length-to-width ratios. Through a combination of experiments and modelling, we compare the wings’ flight characteristics, specifically the rotation rate and descent angle, both of which are key parameters in the wing’s ability to drift away from its initial location. Starting from the quasi-steady flight model proposed by Wang et al. (J. Fluid Mech., vol. 733, 2013, pp. 650–679), we derive theoretical predictions of the performance of wings of arbitrary planform. Upon further simplifications, we arrive at a performance index based purely on wing geometry and we use it to obtain theoretically optimal wing shapes. These optimal predictions are then tested experimentally. We conclude by discussing the advantages and limitations of the theoretical approach and its utility in informing the design of aerodynamically efficient tumbling wings.

We present direct numerical simulations of non-Oberbeck–Boussinesq (NOB) Rayleigh–Bénard (RB) convection due to large temperature differences in two-dimensional (2-D) and three-dimensional (3-D) cells. Perfect air is chosen as the operating fluid and the Prandtl number ($Pr$) is fixed to 0.71 for the reference state $\hat{T}_{0}=300~\text{K}$. In the present system, we consider large temperature differences ranging from 60 K to 240 K, and relatively strong NOB effects are induced at moderate Rayleigh numbers ($Ra$) in the range $3\times 10^{6}\leqslant Ra\leqslant 5\times 10^{9}$. The large temperature difference also induces the turbulence system with large density variation. Due to top-down symmetry breaking under NOB conditions, an increase of the centre temperature $T_{c}$ is found compared to the arithmetic mean temperature $T_{m}$ of the top and bottom plates, and the shift of $T_{c}$ is strongly dependent on Rayleigh number $Ra$ and temperature differential $\unicode[STIX]{x1D716}$. The NOB effects on the Nusselt number ($Nu$) are quite small (${\lesssim}2\,\%$). The power-law scalings of $Nu$ versus $Ra$ are robust against NOB effects, even for the extremely large temperature difference 240 K, which has never been reached in previous experiments and simulations. The Reynolds numbers $Re$, as well as the scalings of $Re$ versus $Ra$, are also insensitive to NOB effects. It is noteworthy that the influence of NOB effects on $Nu$ and $Re$ in 3-D RB flow are weaker than its 2-D counterpart. Furthermore, the extended laminar boundary layer (BL) equations are developed based on the low-Mach-number Navier–Stokes equations, which qualitatively predicts the NOB effects on velocity profiles. Direct numerical simulation results indicate that the top and bottom thermal BLs can compensate each other much better than the velocity BLs under NOB conditions, which contribute to the robustness of $Nu$.

Non-stationary, rotational, linear surface waves are considered where the underlying sheared current has constant vorticity. A time-dependent study is presented on the formation and persistence of a Kelvin cat-eye structure in the presence of bottom topography. The flow domain is two-dimensional, which allows for the use of a conformal mapping and working in a computational flat-bottom domain. In some cases an initial disturbance is prescribed, while in others the waves are generated from rest. Submarine particle dynamics numerically captures the horizontal critical layer, defined by closed orbits separating the fluid domain into two disjoint regions. In the wave’s moving frame, these recirculation regions are structured in the form of Kelvin cat-eyes. Owing to the interaction with topography, the usual travelling-wave formulation is abandoned and the critical layer is identified through a non-stationary set of equations. The respective time-dependent Kelvin cat-eye structure dynamically adjusts itself at the onset of wave–topography interaction, without losing its integrity. The formation of a Kelvin cat-eye structure is also studied in the case where the surface is initially undisturbed. Surface waves are generated from either the current–topography interaction or by a pressure distribution suddenly imposed along the free surface. Under the pressure forcing, an isolated cat-eye forms with a single recirculation region beneath the wave.

The stability of the jet in cross-flow is investigated using a complete set-up including the flow inside the pipe. First, direct simulations were performed to find the critical velocity ratio as a function of the Reynolds number, keeping the boundary-layer displacement thickness fixed. At all Reynolds numbers investigated, there exists a steady regime at low velocity ratios. As the velocity ratio is increased, a bifurcation to a limit cycle composed of hairpin vortices is observed. The critical bulk velocity ratio is found at approximately $R=0.37$ for the Reynolds number $Re_{D}=495$, above which a global mode of the system becomes unstable. An impulse response analysis was performed and characteristics of the generated wave packets were analysed, which confirmed results of our global mode analysis. In order to study the sensitivity of this flow, we performed transient growth computations and also computed the optimal periodic forcing and its response. Even well below this stability limit, at $R=0.3$, large transient growth ($10^{9}$ in energy amplification) is possible and the resolvent norm of the linearized Navier–Stokes operator peaks above $2\times 10^{6}$. This is accompanied with an extreme sensitivity of the spectrum to numerical details, making the computation of a few tens of eigenvalues close to the limit of what can be achieved with double precision arithmetic. We demonstrate that including the meshing of the jet pipe in the simulations does not change qualitatively the dynamics of the flow when compared to the simple Dirichlet boundary condition representing the jet velocity profile. This is in agreement with the recent experimental results of Klotz et al. (J. Fluid Mech., vol. 863, 2019, pp. 386–406) and in contrast to previous studies of Cambonie & Aider (Phys. Fluids, vol. 26, 2014, 084101). Our simulations also show that a small amount of noise at subcritical velocity ratios may trigger the shedding of hairpin vortices.

The linear amplification mechanisms leading to streamwise-constant large-scale structures in laminar and turbulent channel flows are considered. A key feature of the analysis is that the Orr–Sommerfeld and Squire operators are each considered separately. Physically, this corresponds to considering two separate processes: (i) the response of wall-normal velocity fluctuations to external forcing; and (ii) the response of streamwise velocity fluctuations to wall-normal velocity fluctuations. The analysis is performed for both plane Couette flow and plane Poiseuille flow; and for each we consider linear amplification mechanisms about both the laminar and turbulent mean velocity profiles. The analysis reveals two things. First, that the most amplified structures (with a spanwise spacing of approximately $4h$, where $h$ is the channel half-height) are to an important degree encoded in the Orr–Sommerfeld operator alone, thus helping to explain their prevalence. Second – and consistent with numerical and experimental observations – that Couette flow is significantly more efficient than Poiseuille flow in leveraging the mean shear to produce channel-wide streamwise streaks.

Ostwald ripening is a pore-scale phenomenon that coarsens a dispersed phase until thermodynamic equilibrium. Based on our previous finding that multi-bubble equilibrium is possible and likely in complex porous media, we develop a new continuum-scale model for Ostwald ripening in heterogeneous porous media. In this model, porous media with two different capillary pressure curves are put into contact, allowing only diffusive flow through the aqueous phase to redistribute a trapped gas phase. Results show that Ostwald ripening can increase the gas saturation in one medium while decreasing the gas saturation in the other, even when the gas phase is trapped in pore spaces by capillary forces. We develop an analogous retardation factor to show that the characteristic time for Ostwald ripening is about $10^{5}$ times slower than a single-phase diffusion problem due to the fact that separate-phase gas requires a much larger amount of mass transfer before equilibrium is established. An approximate solution has been developed to predict the saturation redistribution between the two media. The model has been validated by numerical simulation over a wide range of physical parameters. Millimetre to centimetre-scale systems come to equilibrium in years, ranging up to 10 000 years and longer for metre-scale systems. These findings are particularly relevant for geological $\text{CO}_{2}$ storage, where residual trapping is an important mechanism for immobilizing $\text{CO}_{2}$. Our work demonstrates that Ostwald ripening due to heterogeneity in porous media is slow and on a similar time scale compared to other processes that redistribute trapped $\text{CO}_{2}$ such as convective mixing.

We study the formation of liquid plugs in a vertical heated tube in contact with a reservoir filled with a binary liquid mixture. Various morphologies, such as liquid films, rings and plugs, are observed. A key phenomenon is the transition between a liquid ring and a plug, which is described using the concept of a quasi-static minimal energy surface that becomes unstable when the liquid volume exceeds a specific value. The critical diameter of the liquid ring and the volume and the position of the formed plug are obtained from an analytical model. The inner diameter of the liquid ring obeys a $d_{l}\sim (t_{0}-t)^{0.57\pm 0.02}$ scaling law shortly before forming a plug at time $t_{0}$. The height of the liquid column created develops according to $X\sim (t-t_{0})^{0.5\pm 0.01}$ in the first moments. The subsequent time evolution is described by a damped harmonic oscillator based on a scaling analysis. The discoveries presented in this work could be of great importance for our understanding of thermally induced interfacial phenomena in confined space.

The stability of a thin falling film with both surface elasticity and surface viscosities induced by insoluble surfactants on its free surface is studied. Based on the full Navier–Stokes equations and surfactant concentration equation with corresponding boundary conditions, a weighted residual model (WRM) is derived to investigate the long-wave instability of the thin film incorporating the influence of surfactants. The Chebyshev spectral collocation method is employed to solve the linear stability of the film. The results show good agreement between the WRM and full equations. It is found that surface elasticity decreases the temporal growth rate and increases the critical Reynolds number, showing a stabilizing impact on the film. And the surface viscosity effect slightly reduces the growth rate and cutoff wavenumber while it does not alter the critical Reynolds number. Nonlinear travelling wave solutions are obtained using the WRM equations. As the surface elasticity is enhanced, the speed of travelling waves gradually approaches the corresponding linear neutral value, implying that the dispersion effect is damped; and the amplitudes of both fast waves and slow waves are suppressed by surface elasticity. Moreover, the bifurcation diagram of travelling waves is influenced by the surface viscosity, which basically promotes the speed of travelling waves with relatively large wavelengths. As the surface viscosity effect becomes stronger, for fast waves the amplitude of the humps slightly increases while that of the troughs becomes smaller for slow waves.

Sprays produced by pressure atomization of various liquids are investigated experimentally, showing the self-similar flow fields of both the liquid and the gas phases. Phase-Doppler measurements are conducted in the sprays at varying radial and axial distances from the atomizer orifice. The theoretical description of the gas flow field based on boundary-layer theory reveals a self-similar velocity field driven by momentum transfer from the liquid phase ejected into the gaseous environment. The momentum loss of the liquid droplet phase is also found to be self-similar, which was to be expected, but not shown in the literature before. The analytical self-similar description of the two-phase flow field is in excellent agreement with the experimental data.