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.

We generalise the classical Benjamin solution (Benjamin, J. Fluid Mech., vol. 31, 1968, pp. 209–248) modelling the flow in a horizontal duct of finite depth in situations where the flow contains a region spanning the depth of the duct, and a region in which the surface detaches from the ceiling of the duct as a free surface. It is shown that the Benjamin solution belongs to a one-parameter family of similar solutions, which are divided into two types: solutions that describe potential flows where the free surface of the fluid is deflected from the duct ceiling at a zero angle; and solutions that admit the formation of a vortex flow region in the vicinity of the point of fluid separation from the duct ceiling. It is shown that this one-parameter family of solutions is the limit of a two-parameter family of solutions in which part of the uniform flow energy is converted into energy of the small-scale fluid motion. Based on the local hydrostatic approximation, the applicability of the constructed solutions is discussed.

This work presents an experimental validation study of Isaacs’ incompressible unsteady-airfoil theory at Reynolds numbers above $10^{6}$, and explores the validity of the classical Kutta condition applied to surging flows. Harmonic variation of the free-stream velocity was produced by rotating choke vanes in an unsteady transonic wind tunnel, with time-resolved lift coefficients determined from surface pressure measurements on a NACA 0018 airfoil. Unsteady lift results demonstrate the same trends with reduced frequency and velocity amplitude ratio that are predicted by Isaacs’ theory. However, significant deviations of the lift magnitude and phase angle are observed. In order to understand the cause of these deviations, the background-oriented schlieren technique was used to visualize density gradients in the immediate vicinity of the airfoil trailing edge. The time-resolved background-oriented schlieren displacement field indicates oscillatory behaviour of the trailing-edge stagnation streakline, which violates the classical Kutta condition for this unsteady surging flow.

Liquid drop impact dynamics depends on the liquid–substrate interaction. In particular, when liquid–solid friction is decreased, the spreading of the impacting drop lasts longer. We characterise this effect by using two types of superhydrophobic surfaces, with similar wetting properties but different friction coefficients. It is found that, for large enough impact velocities, a reduced friction delays the buildup of a viscous boundary layer, and leads to an increase of the time required to reach the maximal radius of the impacting drop. An asymptotic analysis is carried out to quantify this effect, and agrees well with the experimental findings. Interestingly, this novel description complements the general picture of drop impact on solid surfaces, and more generally addresses the issue of drag reduction in the presence of slippage for non-stationary flows.

Classical and symmetrical horizontal convection is studied by means of direct numerical simulations for Rayleigh numbers $Ra$ up to 3 × 1012 and Prandtl numbers $Pr=0.1$, 1 and 10. For both set-ups, a very good agreement in global quantities with respect to heat and momentum transport is attained. Similar to Shishkina & Wagner (Phys. Rev. Lett., vol. 116, 2016, 024302), we find Nusselt number $Nu$ versus $Ra$ scaling transitions in a region $10^{8}\leqslant Ra\leqslant 10^{11}$. Above a critical $Ra$, the flow undergoes either a steady–oscillatory transition (small $Pr$) or a transition from steady state to a transient state with detaching plumes (large $Pr$). The onset of the oscillations takes place at $Ra\,Pr^{-1}\approx 5\times 10^{9}$ and the onset of detaching plumes at $Ra\,Pr^{5/4}\approx 9\times 10^{10}$. These onsets coincide with the onsets of scaling transitions.

We report a comparative experimental study of the reversal of the large-scale circulation in turbulent Rayleigh–Bénard convection in a quasi-two-dimensional corner-less cell where the corner vortices are absent and in a quasi-two-dimensional normal cell where the corner vortices are present. It is found that in the corner-less cell the reversal frequency exhibits a slow decrease followed by a fast decrease with increasing Rayleigh number $Ra$, separated by a transitional $Ra$ ($Ra_{t,r}$). The transition is similar to that in the normal cell, and $Ra_{t,r}$ is almost the same for both cells. Despite the similarities, the reversal frequency is greatly reduced in the corner-less cell. The reduction of the reversal frequency is more significant, in terms of both the amplitude and the scaling exponent, in the high-$Ra$ regime. In addition, we classified the reversals into main-vortex-led and corner-vortex-led, and found that both types exist in the normal cell while only the former exists in the corner-less cell. The frequency of main-vortex-led reversal in the normal cell is found to be in excellent agreement with the frequency of reversals in the corner-less cell. Our results reveal for the first time the quantitative role of the corner vortices in the occurrence of the reversals of the large-scale circulation.

In this paper, we report that reversals of large-scale circulation in two-dimensional Rayleigh–Bénard convection could be suppressed or enhanced by imposing local constant-temperature control on sidewalls. When the control area is away from the centre of the sidewalls, the control can successfully eliminate the flow reversal if the size of the control region is large enough. With a proper location, the width can be as small as 1 % of the system size. When the control region is located around the centre, the control may enhance the flow reversal. It may also stimulate the occurrence of a double-roll mode when the control is located in the centre. Explanations are also discussed based on the twofold effects of the control region on the nearby plumes and the concept of symmetry. The present work provides a new way to control the flow reversals in Rayleigh–Bénard convection through modifying sidewall boundary conditions.

In this simulation-based study, we investigate the surface roughness signature induced by internal solitary waves in oceans. We present the first-ever effort to directly capture the surface roughness signature with a deterministic two-layer model to avoid the singularity encountered in the traditional wave–current interaction theory. By capturing over four million wave components, the simulation resolves the surface wave and internal wave dynamics simultaneously. The surface signature characterized by a rough region followed by a smooth region travelling with an internal wave is quantified by the local wave geometry variation and the wave energy change. The surface wave dynamics are analysed in the wavenumber–frequency slope spectrum calculated in the frame moving with the internal wave. The asymmetric behaviours of right-moving and left-moving surface waves are found to contribute to the surface signature formation. Our results show that the formation of the surface signature is essentially an energy-conservative process and justify the use of the wave-phase-resolved two-layer model.

‘Ground effect’ refers to the enhanced performance enjoyed by fliers or swimmers operating close to the ground. We derive a number of exact solutions for this phenomenon, thereby elucidating the underlying physical mechanisms involved in ground effect. Unlike previous analytic studies, our solutions are not restricted to particular parameter regimes, such as ‘weak’ or ‘extreme’ ground effect, and do not even require thin aerofoil theory. Moreover, the solutions are valid for a hitherto intractable range of flow phenomena, including point vortices, uniform and straining flows, unsteady motions of the wing, and the Kutta condition. We model the ground effect as the potential flow past a wing inclined above a flat wall. The solution of the model requires two steps: firstly, a coordinate transformation between the physical domain and a concentric annulus; and secondly, the solution of the potential flow problem inside the annulus. We show that both steps can be solved by introducing a new special function which is straightforward to compute. Moreover, the ensuing solutions are simple to express and offer new insight into the mathematical structure of ground effect. In order to identify the missing physics in our potential flow model, we compare our solutions against new experimental data. The experiments show that boundary layer separation on the wing and wall occurs at small angles of attack, and we suggest ways in which our model could be extended to account for these effects.

A weakly nonlinear time-dependent theory for the evolution of superharmonics generated by the nonlinear self-interaction of a mode-1 internal tide in non-uniform stratification is developed and compared to numerical simulations. The forcing by the internal tide is found to excite near-pure mode-1 superharmonics whose natural frequency is moderately different from twice the internal tide frequency. Consequently, the superharmonics undergo a slow periodic growth and decay that is comparable to an acoustic ‘beat’. At low latitudes the beat frequency is smaller and the superharmonics can grow to larger amplitude, allowing for the possibility of a superharmonic cascade.

The thermal protection system in atmospheric-entry and hypersonic-cruise vehicles are oftentimes designed to ablate during their operation, thus injecting a mixture of various gases with distinct properties into the boundary layer. Such outgassing affects the propagation of instabilities within the boundary layer that ultimately originates the transition to turbulence. This work uses linear stability theory, in combination with the e$^{N}$ method, to establish the underlying reason for the experimentally observed advancement/delay of transition in sharp slender hypersonic cones, when injecting lighter/heavier gases. Contrary to the current understanding and experimental correlations, this numerical analysis suggests that such a behaviour is not linked to the isolated effect of the injected gas’ molar weight, but to its combination with the blowing discontinuity, porosity and the appearance of a shocklet, consequence of the injected gas composition. The shocklet constitutes a density gradient that acts on second-mode instabilities like a thermoacoustic impedance.

We use a multiple-scale expansion to average the wave action balance equation over an ensemble of sea-surface velocity fields characteristic of the ocean mesoscale and submesoscale. Assuming that the statistical properties of the flow are stationary and homogeneous, we derive an expression for a diffusivity tensor of surface-wave action density. The small parameter in this expansion is the ratio of surface current speed to gravity wave group speed. For isotropic currents, the action diffusivity is expressed in terms of the kinetic energy spectrum of the flow. A Helmholtz decomposition of the sea-surface currents into solenoidal (vortical) and potential (divergent) components shows that, to leading order, the potential component of the surface velocity field has no effect on the diffusivity of wave action: only the vortical component of the sea-surface velocity results in diffusion of surface-wave action. We validate our analytic results for the action diffusivity by Monte Carlo ray-tracing simulations through an ensemble of stochastic velocity fields.

The permanent precession of a baroclinic geophysical vortex is reproduced, under the quasi-geostrophic approximation, using three potential vorticity anomaly modes in spherical geometry. The potential vorticity modes involve the spherical Bessel functions of the first kind $\text{j}_{l}(\unicode[STIX]{x1D70C})$ and the spherical harmonics $\text{Y}_{l}^{m}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$, where $l$ is the degree, $m$ is the order, and $(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ are the spherical coordinates. The vortex precession is interpreted as the horizontal and circular advection by a large-amplitude background flow associated with the spherical mode $c_{0}\text{j}_{0}(\unicode[STIX]{x1D70C})$ of the small-amplitude zonal mode $c_{2,0}\text{j}_{2}(\unicode[STIX]{x1D70C})\text{Y}_{2}^{0}(\unicode[STIX]{x1D703})$ tilted by a small-amplitude mode $c_{2,1}\text{j}_{2}(\unicode[STIX]{x1D70C})\text{Y}_{2}^{1}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$, where $\{c_{0},c_{2,0},c_{2,1}\}$ are constant potential vorticity modal amplitudes. An approximate time-dependent, closed-form solution for the potential vorticity anomaly is given. In this solution the motion of the potential vorticity field is periodic but not rigid. The vortex precession frequency $\unicode[STIX]{x1D714}_{0}$ depends linearly on the amplitudes $c_{0}$ and $c_{2,0}$ of the modal components of order 0, while the slope of the precessing axis depends on the ratio between the modal amplitude $c_{2,1}$ and $\unicode[STIX]{x1D714}_{0}$.

In this paper, we present the two-dimensional (2-D) energy cross-spectrum of the streamwise velocity ($u$) component and use it to test the notion of self-similarity in turbulent boundary layers. The primary focus is on the cross-spectrum ($\unicode[STIX]{x1D6F7}_{cross}^{w}$) measured across the logarithmic ($z_{o}$) and near-wall ($z_{r}$) wall-normal locations, providing the energy distribution across the range of streamwise ($\unicode[STIX]{x1D706}_{x}$) and spanwise ($\unicode[STIX]{x1D706}_{y}$) wavelengths (or length scales) that are coherent across the wall-normal distance. $\unicode[STIX]{x1D6F7}_{cross}^{w}$ may thus be interpreted as a wall-filtered subset of the full 2-D $u$-spectrum ($\unicode[STIX]{x1D6F7}$), the latter providing information on all coexisting eddies at $z_{o}$. To this end, datasets comprising synchronized two-point $u$-signals at $z_{o}$ and $z_{r}$, across the friction Reynolds number range $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{3}){-}O(10^{4})$, are analysed. The published direct numerical simulation (DNS) dataset of Sillero et al. (Phys. Fluids, vol. 26 (10), 2014, 105109) is considered for low-$Re_{\unicode[STIX]{x1D70F}}$ analysis, while the high-$Re_{\unicode[STIX]{x1D70F}}$ dataset is obtained by conducting synchronous multipoint hot-wire measurements. High-$Re_{\unicode[STIX]{x1D70F}}$ cross-spectra reveal that the wall-attached large scales follow a $\unicode[STIX]{x1D706}_{y}/z_{o}\sim \unicode[STIX]{x1D706}_{x}/z_{o}$ relationship more closely than seen for $\unicode[STIX]{x1D6F7}$, where this self-similar trend is obscured by coexisting scales. The present analysis reaffirms that a self-similar structure, conforming to Townsend’s attached eddy hypothesis, is ingrained in the flow.

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.

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.

Over the past two decades several attempts have been made to formulate constitutive models for slow granular flow to remedy the deficiencies of classical plasticity. All the proposed models assume the medium to be incompressible, though it is well known that density change accompanies deformation in granular materials. A particularly important aspect of density change that is distinctive of granular materials is dilatancy, or volume deformation caused by shear deformation. No constitutive model for sustained flow has thus far captured dilatancy. Here we present a non-local constitutive model wherein the deformation rate and density at a point depend on the state of stress in a mesoscopic region around it. Apart from incorporating dilatancy, our model has a physical origin that is distinct from that of the previously proposed non-local models. We test our model on simple shear flow in the absence and presence of gravity, and find its predictions to be in good agreement with particle dynamics simulations.

A new methodology to construct three-dimensional, temporally stationary but spatially inhomogeneous, incompressible turbulence is presented. The method combines use of the data-driven spectral proper orthogonal decomposition (SPOD) to identify and isolate large-scale coherent motions of the flow, together with a physics-based enrichment algorithm using spatiotemporally localized Gabor modes that capture the inertial subrange turbulence. This fusion of data-driven and physics-based methods enables a statistically correct reconstruction of broadband turbulent flows using fewer modes than would be required using SPOD alone. To demonstrate the approach, we consider the problem of reconstructing wake turbulence on a plane downstream of a dragging actuator disk impinged by homogeneous isotropic turbulence. The reconstructed flow has single- and two-point correlations that are consistent with the reference high-resolution simulation data and could be used to generate statistically consistent inflow boundary conditions for subsequent simulations.

Recognizing the fact that the finite-time singularity of the Navier–Stokes equations is widely accepted as a key issue in fundamental fluid mechanics, and motivated by the recent model of Moffatt & Kimura (J. Fluid Mech., vol. 861, 2019a, pp. 930–967; J. Fluid Mech., vol. 870, 2019b, R1) on this issue, we have performed direct numerical simulation (DNS) for two colliding slender vortex rings of radius $R$. The separation between the two tipping points $2s_{0}$ and the scale of the core cross-section $\unicode[STIX]{x1D6FF}_{0}$ are chosen as $\unicode[STIX]{x1D6FF}_{0}=0.1s_{0}=0.01R$; the vortex Reynolds number ($Re=\text{circulation/viscosity}$) ranges from 1000 to 4000. In contrast to the claim that the core remains compact and circular, there is notable core flattening and stripping, which further increases with $Re$ – akin to our previous finding in the standard anti-parallel vortex reconnection. Furthermore, the induced motion of bridges arrests the curvature growth and vortex stretching at the tipping points; consequently, the maximum vorticity grows with $Re$ substantially slower than the exponential scaling predicted by the model – implying that, for this configuration, even physical singularity is unlikely. Our simulations not only shed light on the longstanding question of finite-time singularities, but also further delineate the detailed mechanisms of reconnection. In particular, we show for the first time that the separation distance $s(\unicode[STIX]{x1D70F})$ before reconnection follows 1/2 scaling exactly – a significant DNS result.