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.

‘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.

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.

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.

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.

The unsteady organization of a separated turbulent boundary layer was investigated upstream from the trailing edge of a NACA 4418 airfoil. The angle of attack was $9^{\circ }$ in the pre-stall regime. Two particle image velocimetry fields of view were of interest: a streamwise–wall-normal plane at midspan of the airfoil and a streamwise–spanwise plane parallel to and near the surface of the airfoil. In the near-surface streamwise–spanwise plane, the mean velocity field revealed a saddle point near midspan and a pair of counter-rotating foci at the sides. This pattern is reminiscent of a stall cell, which has been traditionally associated with flow separation on thick airfoils at and slightly beyond the angle of attack of maximum lift. Isolating the low frequencies showed that the instantaneous separation front consisted of several smaller structures that also resembled a stall cell pattern, but they were an order of magnitude smaller than the one found in the mean pattern. These instantaneous stall cells were of two types: forward and backward. The forward stall cells were formed by strong high-speed streaks from upstream, while backward stall cells formed as a result of strong backflow just downstream from the separation front, resulting in a foci pair and a saddle point on their upstream side. In both cases, the foci pairs acted to mobilize high-speed momentum of the associated streak into a rotational motion, causing these streaks to dissipate. Finally, proper orthogonal decomposition revealed that low-order modes were associated with the movement and distortion of the separation front.

Wave forecasting in ocean and coastal waters commonly relies on spectral models based on the spectral action balance equation. These models assume that different wave components are statistically independent and as a consequence cannot resolve wave interference due to statistical correlation between crossing waves, as may be found in, for instance, a focal zone. This study proposes a statistical model for the evolution of wave fields over non-uniform currents and bathymetry that retains the information on the correlation between different wave components. To this end, the quasi-coherent model (Smit & Janssen, J. Phys. Oceanogr., vol. 43, 2013, pp. 1741–1758) is extended to allow for wave–current interactions. The outcome is a generalized action balance model that predicts the evolution of the wave statistics over variable media, while preserving the effect of wave interferences. Two classical examples of wave–current interaction are considered to demonstrate the statistical contribution of wave interferences: (1) swell field propagation over a jet-like current and (2) the interaction of swell waves with a vortex ring. In both examples cross-correlation terms lead to development of prominent interference structures, which significantly change the wave statistics. Comparison with results of the SWAN model demonstrates that retention of cross-correlation terms is essential for accurate prediction of wave statistics in shear-current-induced focal zones.

A millimetric droplet of silicone oil may bounce and self-propel on the free surface of a vertically vibrating fluid bath due to the droplet’s interaction with its accompanying Faraday wave field. This hydrodynamic pilot-wave system exhibits many dynamics that were previously thought to be peculiar to the quantum realm. When the droplet is confined to a circular cavity, referred to as a ‘corral’, a range of dynamics may occur depending on the details of the geometry and the decay time of the subcritical Faraday waves. We herein present a theoretical investigation into the behaviour of subcritical Faraday waves in this geometry and explore the accompanying pilot-wave dynamics. By computing the Dirichlet-to-Neumann map for the velocity potential in the corral geometry, we can evolve the quasi-potential flow between successive droplet impacts, which, when coupled with a simplified model for the droplet’s vertical motion, allows us to derive and implement a highly efficient discrete-time iterative map for the pilot-wave system. We study the onset of the Faraday instability, the emergence and quantisation of circular orbits and simulate the exotic dynamics that arises in smaller corrals.

This paper considers the significant role of cross-sectional geometry on resistance in co-axial pipe flows. We consider an axially flowing viscous fluid in between two long and thin elliptical coaxial cylinders, one inside the other. The outer cylinder is stationary, while the inner cylinder (rod) is free to move. The rod poses a resistance to the axial flow, while the viscous fluid poses a resistance to any motion of the rod. We show that the equations for flow in the axial direction – driven by a prescribed flux – and for flow within the cross-section of the domain – driven by the motion of the rod – decouple in the asymptotic limit of small cylinder aspect ratio into axial Poiseuille flow and transverse Stokes flow, respectively. The objective of this paper is to calculate numerically the axial and cross-sectional resistances and to determine their dependence on cross-sectional geometry – i.e. rod position and the ellipticities of the rod and bounding cylinder. We characterise axial resistance, first for three reduced parameter spaces that have not been fully analysed in the literature: (i) a circle in an ellipse, (ii) an ellipse in a circle and (iii) an ellipse in an ellipse of equal eccentricity and orientation, before extending our geometric parameter space to determine the overall optimal geometry to minimise axial flow resistance for fixed cross-sectional area. Cross-sectional resistance is characterised via coefficients in a Stokes resistance matrix and we highlight the interdependent effects of cross-sectional ellipticity and boundary interactions.

In this article, we consider the modelling of stationary incompressible and isothermal one-dimensional fluid flow through a long pipeline. The approximation of the average pressure in the developed model by the arithmetic mean of inlet and outlet pressures leads to the known empirical Darcy–Weisbach equation. Most importantly, we also present another improved approach that is more accurate because the average pressure is estimated by integrating the pressure along the pipeline. Through appropriate transformation, we show the difference between the Darcy–Weisbach equation and the improved model that should be treated as a Darcy–Weisbach model error, in multiplicative and additive form. This error increases when the overall pressure drop increases. This symptomatic phenomenon is discussed in detail. In addition, we also consider four methods of estimating the coefficient of friction, assess the impact of pressure difference on the estimated average flow velocity and, based on experimental data, we show the usefulness of new proposals in various applications.

The interactions between finite-size spheroidal particles and upward turbulent flows in a vertical channel are numerically simulated with a direct-forcing fictitious domain method at two particle settling coefficients $u_{s}$ (the ratio of the particle Stokes free-fall velocity to the bulk velocity) of 0.1 and 0.3, a bulk Reynolds number of 2873, a ratio of the particle equivalent diameter to the channel width of 0.05, a particle volume fraction of 2.36 % and particle aspect ratios of $1/3$, 1 and 2. Our results show that the flow friction is largest for the case of a sphere, and smallest for the oblate case when the particle sedimentation effect is weak ($u_{s}=0.1$), whereas the flow friction is smallest for the case of a sphere, and largest for the oblate case when the particle sedimentation effect is moderately strong ($u_{s}=0.3$). The reason for the lower flow friction of the spherical particles is that the large-scale vortices are more strongly attenuated by the spherical particles than by the non-spherical particles in the case of $u_{s}=0.3$. The settling particles tend to migrate towards the channel centre due to the Saffman effect, and the migration is strongest for the spherical particles. The non-spherical particles tend to align their long axes with the streamwise direction in the near-wall region, and perpendicular to the streamwise direction in the bulk region due to the significant settling effect.

The dynamics of air bubbles in turbulent Rayleigh–Bénard (RB) convection is described for the first time using laboratory experiments and complementary numerical simulations. We performed experiments at $Ra=5.5\times 10^{9}$ and $1.1\times 10^{10}$, where streams of 1 mm bubbles were released at various locations from the bottom of the tank along the path of the roll structure. Using three-dimensional particle tracking velocimetry, we simultaneously tracked a large number of bubbles to inspect the pair dispersion, $R^{2}(t)$, for a range of initial separations, $r$, spanning one order of magnitude, namely $25\unicode[STIX]{x1D702}\leqslant r\leqslant 225\unicode[STIX]{x1D702}$; here $\unicode[STIX]{x1D702}$ is the local Kolmogorov length scale. Pair dispersion, $R^{2}(t)$, of the bubbles within a quiescent medium was also determined to assess the effect of inhomogeneity and anisotropy induced by the RB convection. Results show that $R^{2}(t)$ underwent a transition phase similar to the ballistic-to-diffusive ($t^{2}$-to-$t^{1}$) regime in the vicinity of the cell centre; it approached a bulk behavior $t^{3/2}$ in the diffusive regime as the distance away from the cell centre increased. At small $r$, $R^{2}(t)\propto t^{1}$ is shown in the diffusive regime with a lower magnitude compared to the quiescent case, indicating that the convective turbulence reduced the amplitude of the bubble’s fluctuations. This phenomenon associated to the bubble path instability was further explored by the autocorrelation of the bubble’s horizontal velocity. At large initial separations, $R^{2}(t)\propto t^{2}$ was observed, showing the effect of the roll structure.

We study the shape and the geometrical properties of sessile drops with translational invariance (namely ‘liquid cylinders’) deposited upon a flat superhydrophobic substrate. We account for the flattening effects of gravity on the shape of the drop using a pendulum rotation motion analogy. In the framework of the inviscid Saint-Venant equations, we show that liquid cylinders are always unstable because of the Plateau–Rayleigh instability. However, a cylindrical drop deposited upon a superhydrophobic non-flat channel (here, wedge-shaped channels) is stabilised beyond a critical cross-sectional area. The critical threshold of the Plateau–Rayleigh instability is analytically computed for various profiles of the channel. The stability analysis is performed in terms of an effective propagation speed of varicose waves. Experiments are performed in order to test these analytical results. We measure the critical drop size at which breakup occurs, together with the decreasing effective propagation speed of varicose waves as the threshold is approached. Our theoretical predictions are in excellent agreement with the experimental measurements.

We have performed fully resolved simulations of turbulent flows over various submerged rigid canopies covering the wall of an open channel. All the numerical predictions have been obtained considering the same nominal bulk Reynolds number (i.e. $Re_{b}=U_{b}H/\unicode[STIX]{x1D708}=6000$, $H$ being the channel depth and $U_{b}$ the bulk velocity). The computations directly tackle the region occupied by the canopy by imposing the zero-velocity condition on every single stem, while the outer flow is dealt with a highly resolved large-eddy simulation. Four canopy configurations have been considered. All of them share the same in-plane solid fraction while the canopy to channel height ratios have been selected to be $h/H=(0.05,0.1,0.25,0.4)$. The lowest and the highest values lead to flow conditions approaching the two asymptotic states that in the literature are usually termed the sparse and dense regimes (see Nepf (Annu. Rev. Fluid Mech., vol. 44, 2012, pp. 123–142)). The other two $h/H$ selected ratios are representative of transitional regimes, a generic category that incorporates all the non-asymptotic states. While the interaction of a turbulent flow with a filamentous canopy in the two asymptotic conditions is relatively well understood, not much is known on the transitional flows and on the physical mechanisms that are responsible for the variations of flow regimes when the canopy solidity is changed. The effects of the latter on the flow developing in the intra-canopy region, on the outer flow and on their mutual interactions have been numerically explored and are reported in this work. By systematically varying the canopy height, we have unravelled the main character of the different regimes that are generated by the interplay between the outer flow structures, the emerging instabilities driven by the canopy drag and the interstitial flow between the canopy stems. The key role played by the relative positions of the inflection points of the mean velocity profile and the location of the virtual wall origin (as seen from the outer flow) is put forward and used to define a new condition to infer the canopy flow regime when the solidity is changed. Finally, the presence and the effects of an instability occurring close to the bed, nearby the interior inflectional point of the mean velocity profile is highlighted together with its consequences on the flow structure within the canopy region. To the best of our knowledge, this is the first time that the emergence of close-to-the-bed coherent structures induced by the inner inflection point is reported in the literature.

This paper investigates the properties of the mean momentum balance (MMB) equation in the azimuthal $\unicode[STIX]{x1D719}$ direction of a turbulent Taylor–Couette flow (TCF). The MMB-$\unicode[STIX]{x1D719}$ equation is integrated to determine the properties of the Reynolds shear stress. An approximation is developed for the Reynolds shear stress in the core region of a turbulent TCF, and the dependence of the peak Reynolds shear stress location on the Reynolds number and the gap geometry is revealed. The properties of the global integral of the turbulent Coriolis force are also revealed. For a turbulent TCF with a small gap or at high Reynolds numbers, the global integral of the turbulent Coriolis force is found to be only weakly influenced by the rotation ratio of the cylinders. Two controlling non-dimensional numbers are derived directly from the scaling analysis of the MMB-$\unicode[STIX]{x1D719}$ equation. The first is a geometry Atwood number $A_{t}\stackrel{\text{def}}{=}\unicode[STIX]{x1D6FF}/r_{ctr}$ to characterize the gap geometry, where $\unicode[STIX]{x1D6FF}$ is the gap half-width, and $r_{ctr}$ is the mid-gap radial location. The second is the friction Reynolds number $Re_{\unicode[STIX]{x1D70F},i}$ defined as $Re_{\unicode[STIX]{x1D70F},i}\stackrel{\text{def}}{=}\unicode[STIX]{x1D6FF}u_{\unicode[STIX]{x1D70F},i}/\unicode[STIX]{x1D708}$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $u_{\unicode[STIX]{x1D70F},i}$ is the friction velocity at the inner cylinder. A new three-layer structure is proposed for the inner half of a turbulent TCF at sufficiently high Reynolds number, based on the force balance in the MMB-$\unicode[STIX]{x1D719}$ equation. Layer I is an inner layer, where the force balance is between the viscous force and the Reynolds shear force: $F_{visc}\approx -F_{turb}$. Layer III occupies the core of the gap, where the force balance is between the turbulent Coriolis force and the Reynolds shear force: $F_{cori}\approx -F_{turb}$. In Layer II, all three forces contribute to the balance. An inner scaling is developed for Layer I, and an outer scaling is developed for Layer III. The inner and outer scalings are verified against direct numerical simulation data. Similarities and differences between the turbulent TCF and a pressure-driven turbulent channel flow are elucidated.

Studies in the literature on two-dimensional, fully developed, turbulent wall jets on flat surfaces, have invariably reckoned on either the nozzle initial conditions or the asymptotic conditions far downstream, as scaling parameters for the streamwise variations of length and velocity scales. These choices, however, do not square with the notion of self-similarity, which is essentially a ‘local’ concept. We first demonstrate that the streamwise variations of velocity and length scales in wall jets show remarkable scaling with local parameters, i.e. there appear to be no imposed length and velocity scales. Next, it is shown that the mean velocity profile data suggest the existence of two distinct layers – the wall (inner) layer and the full-free jet (outer) layer. Each of these layers scales on the appropriate length and velocity scales and this scaling is observed to be universal, i.e. independent of the local friction Reynolds number. Analysis shows that the overlap of these universal scalings leads to a Reynolds-number-dependent power-law velocity variation in the overlap layer. It is observed that the mean-velocity overlap layer corresponds well to the momentum-balance mesolayer and there appears to be no evidence for an inertial overlap; only the meso-overlap is observed. Introduction of an intermediate variable absorbs the Reynolds-number dependence of the length scale in the overlap layer and this leads to a universal power-law overlap profile for mean velocity in terms of the intermediate variable.

Pressure and shear-driven flows of a confined film of fluid overlying a periodic one-dimensional topography of arbitrary shape are considered for prediction of the effective hydraulic permeability in the Stokes flow regime. The other surface confining the fluid may be a planar no-slip wall, an identically patterned wall, a free surface or a surface with prescribed shear. Analytical predictions are obtained using spectral analysis and the domain perturbation method under the assumption of small pattern size to pitch ratio. Using a novel decomposition of the channel height effects into exponentially and algebraically decaying components, a simple surface-metrology-dependent relationship which connects the eigenvalues of the effective permeability tensor is obtained. Two representative topographies are assessed numerically: the infinitely differentiable topography of a phase-modulated sinusoid which has multiple local extrema and zero crossings and the non-differentiable triangular-wave topography. Non-differentiability in the form of corners of triangular patterns and the cusps of scalloped patterns are not found to be an impediment to meaningful and numerically accurate asymptotic predictions of effective permeability and effective slip, contradicting an earlier suggestion from the literature. Several distinct applications of the theory to the friction-reduction and shear-stability performance of the recently developed lubricant impregnated patterned surfaces as well as to scalloped and trapezoidal drag-reduction riblets are discussed, with comparison to experimental data from the literature for the last application. Analytical approximations which have an extended domain of numerical accuracy are also proposed.

In this paper, a vortex moment map (VMM) method is proposed to predict the pitching moment on a body from the vorticity field. VMM is designed to identify the moment contribution of each given vortex in the flow field. Implementing this VMM approach in starting flows of a NACA0012 airfoil, it is found that, due to the rolling up of leading-edge vortices (LEVs) and trailing-edge vortices (TEVs), the unsteady nose-down moment about the quarter chord is higher than the steady-state value. The time variation of the unsteady moment is closely related to the LEVs and TEVs near the body and the VMM gives an intuitive understanding of how each part of the vorticity field contributes to the pitching moment on the body.

Flow past three identical circular cylinders is numerically investigated using the immersed boundary method. The cylinders are arranged in an equilateral-triangle configuration with one cylinder placed upstream and the other two side-by-side downstream. The focus is on the effect of the spacing ratio $L/D(=1.0{-}6.0)$, Reynolds number $Re(=50{-}300)$ and three-dimensionality on the flow structures, hydrodynamic forces and Strouhal numbers, where $L$ is the cylinder centre-to-centre spacing and $D$ is the cylinder diameter. The fluid dynamics involved is highly sensitive to both $Re$ and $L/D$, leading to nine distinct flow structures, namely single bluff-body flow, deflected flow, flip-flopping flow, steady symmetric flow, steady asymmetric flow, hybrid flow, anti-phase flow, in-phase flow and fully developed in-phase co-shedding flow. The time-mean drag and lift of each cylinder are more sensitive to $L/D$ than $Re$ while fluctuating forces are less sensitive to $L/D$ than $Re$. The three-dimensionality of the flow affects the development of the wake patterns, changing the $L/D$ ranges of different flow structures. A diagram of flow regimes, together with the contours of hydrodynamic forces, in the $Re-L/D$ space, is given, providing physical insights into the complex interactions of the three cylinders.

We present results from direct numerical simulations of flows in spherical and oblate spheroidal shells, driven both by precession and thermal convection, with Ekman number $Ek=10^{-4}$, non-diffusive Rayleigh numbers from $Ra=0.1$ to $Ra=10$ and unity Prandtl number. The applied precessional forcing spans seven orders of magnitude. Our experiments show a clear transition between a convective state and a precessing flow that can be approximated by a reduced dynamical model. The change in the flow is apparent in visualizations and a decomposition of the velocity into symmetric and antisymmetric components. For the flow dominated by precession, some parameter combinations show two stable solutions that can be realized by a hysteresis or a strong thermal forcing. An increase of the Rayleigh number at a constant precession rate exhibits established scaling properties for the heat transfer, with exponents $2/7$ and $6/5$.