Using a physics-based approach, we infer the impact of the coherence of atmospheric turbulence on the power fluctuations of wind farms. Application of the random-sweeping hypothesis reveals correlations characterized by advection and turbulent diffusion of coherent motions. Those contribute to local peaks and troughs in the power spectrum of the combined units at frequencies corresponding to the advection time between turbines, which diminish in magnitude at high frequencies. Experimental inspection supports the results from the random-sweeping hypothesis in predicting spectral characteristics, although the magnitude of the coherence spectrum appears to be over-predicted. This deviation is attributed to the presence of turbine wakes, and appears to be a function of the turbulence approaching the first turbine in a pair.

A fluid stably stratified by a salinity gradient and enclosed between two vertical boundaries can become unstable when it is subjected to a temperature difference between the walls. The linear stability of such a fluid in a vertical slot is investigated. Errors in earlier results are found, confirming recent results of Young & Rosner (1998). Four different asymptotic regimes on the stability boundary are identified. One of these, the limit of a strong salinity gradient, has previously been analysed. The analyses of the separate asymptotic limits of weak salinity gradient, large temperature difference and small wavenumber are also given. These four cases make up much of the total boundary between stability and instability for double-diffusive instabilities in a vertical slot, and so most of this boundary can be mapped out for general Prandtl numbers and salt/heat diffusivity ratios using these results.

We study the shape and dynamics of cavities created by the explosion of firecrackers at the surface of a large pool of water. Without confinement, the explosion generates a hemispherical air cavity which grows, reaches a maximum size and collapses in a generic w-shape to form a final central jet. When a rigid open tube confines the firecracker, the explosion produces a cylindrical cavity that expands without ever escaping the free end of the tube. We discuss a potential flow model, which captures most of these features.

A semi-analytical theory for the scattering of plane sound waves by a compressible, non-homentropic, circular-cylindrical, single vortex is presented in this paper. As a special case, the scattering of sound by a cylindrical inhomogeneity (hot spot) is investigated. Contrary to the otherwise analogous quantum-mechanical scattering problem, there are singularities in the modified acoustic wave equation for radii xs ∈ (0, ∞) when the scattering by a vortex is considered. It will be shown how these singularities can be treated.

This sound-scattering theory is applied to the problem of the interaction of weak plane shock waves with a strong cylindrical vortex. The calculated scattered sound signal has a rather complicated structure in which a cylindrical wave with an essentially quadrupolar directivity pattern is discernible. In the case of shock–hot-spot interaction a scattered sound signal with dipole-like amplitude is obtained. Both results qualitatively agree with experimental findings.

We study experimentally the dynamics and statistics of capillary waves forced by random steep gravity waves mechanically generated in the laboratory. Capillary waves are produced here by gravity waves from nonlinear wave interactions. Using a spatio-temporal measurement of the free surface, we characterize statistically the random regimes of capillary waves in the spatial and temporal Fourier spaces. For a significant wave steepness (0.2–0.3), power-law spectra are observed both in space and time, defining a turbulent regime of capillary waves transferring energy from the large scale to the small scale. Analysis of temporal fluctuations of the spatial spectrum demonstrates that the capillary power-law spectra result from the temporal averaging over intermittent and strong nonlinear events transferring energy to the small scale in a fast time scale, when capillary wave trains are generated in a way similar to the parasitic capillary wave generation mechanism. The frequency and wavenumber power-law exponents of the wave spectra are found to be in agreement with those of the weakly nonlinear wave turbulence theory. However, the energy flux is not constant through the scales and the wave spectrum scaling with this flux is not in good agreement with wave turbulence theory. These results suggest that theoretical developments beyond the classic wave turbulence theory are necessary to describe the dynamics and statistics of capillary waves in a natural environment. In particular, in the presence of broad-scale viscous dissipation and strong nonlinearity, the role of non-local and non-resonant interactions should be reconsidered.

The flow past a flat plate at Reynolds numbers in the range 0·1 to 10·0 is investigated by an analogue method. The solution gives the stream function and the vorticity in the flow field surrounding the plate. From these are obtained the local coefficient of friction, the pressure distribution along the plate, and the total drag coefficient. The drag coefficient approaches the analytical values of Haaser (1950) and of Tomotika & Aoi (1953) as the Reynolds number decreases toward 0·1. The drag coefficient approaches the Blasius solution as the Reynolds number increases. At Reynolds number 10·0 the drag coefficient is still above the Blasius value, but is below the value obtained experimentally by Janour (1951). The difference from the experimental result is attributed for the most part to truncation error.

This paper considers the problem of the propagation of a shock wave down a non-uniform tube. Linearized solutions to the problem (Chester 1954) do not hold when the velocity behind the shock is near or at the sonic speed. By retaining appropriate non-linear terms of the flow equations, a solution is obtained which holds for all conditions behind the shock, and reduces to the linearized solution for conditions away from sonic.

The behaviour of supersonic or subsonic flow entering regions of expanding or contracting area changes is discussed. It is found that subsidiary shocks may be formed; these can be located and described using the present solution. Explicit solutions are given for the cases of supersonic or subsonic flow entering a region of linearly expanding or contracting area. The point of shock formation as well as the path of the subsidiary shock is obtained for the case in which the area contracts.

Globally unstable wakes with co-flow at intermediate Reynolds numbers are studied, to quantify important spatial regions for the development and control of the global instability. One region of high structural sensitivity is found close to the inlet for all wakes, in agreement with previous findings for cylinder wakes. A second, elongated region of high structural sensitivity is seen downstream of the first one for unconfined wakes at . When base-flow modifications are considered, a spatially oscillating sensitivity pattern is found inside the downstream high-structural-sensitivity region. This implies that the same change in the base flow can either destabilize or stabilize the flow, depending on the exact position where it is applied. It is shown that the sensitivity pattern remains unchanged for different choices of streamwise boundary conditions and numerical resolution. The actual base-flow is modified in selected configurations, and the linear global modes recomputed. It is confirmed that the linear global eigenvalues move according to the predicted sensitivity pattern for small-amplitude base-flow modifications, for which the theory applies. We also look at the implications of a small control cylinder for the flow. Only the upstream high-sensitivity region proves to be robust in terms of control, but one should be careful not to disturb the flow in the downstream high-sensitivity region, in order to achieve control. The findings can have direct implications for the numerical resolution requirements for wakes at higher Reynolds numbers. Furthermore, they provide one more possible explanation for why confined wakes have a more narrow frequency spectrum than unconfined wakes.

The extrusion of a yield stress fluid from the space between two parallel plates is investigated experimentally. Oscillating the magnitude of the squeezing force about a mean value (F=f[1+αcos(ωt)]) was observed to significantly enhance the flow rate of yield stress fluids, while having no effect on the flow rate of Newtonian fluids. This is a novel result. The enhancement depends on the magnitude of the force, the oscillatory frequency and amplitude, the fluid being squeezed, and the thickness of the fluid layer. Non-dimensional results for the various flow quantities have been presented by using the flow predicted for the constant-force squeezing of a Herschel–Bulkley yield stress fluid as the reference. In the limit of constant-force squeezing, the present experimental results compare very well with those of our earlier theoretical model for this situation (Zwick, Ayyaswamy & Cohen 1996). The results presented in this paper have significance, among many applications, for injection moulding, in the adhesive bonding of microelectronic chips, and in surgical procedures employed in health care.

The trailing edge region of a finite flat plate in laminar, incompressible flow is examined for the limit of high Reynolds numbers.

It is shown that the trailing edge region is an elliptic region of O(R−¾) and therefore a correct mathematical description must be based upon the full Navier–Stokes equations.

The ‘method of series truncation’ is used to reduce the full Navier–Stokes equations, written in parabolic co-ordinates, to an infinite set of non-linear, coupled, ordinary differential equations. Two sets of asymptotic boundary conditions, called simplified and exact boundary conditions, are determined by matching the Navier–Stokes region downstream with Goldstein's near wake solution.

By numerical integration the solutions for the first and second truncations are obtained for both sets of asymptotic boundary conditions. The results confirm that the size of the trailing edge region is of O(R−¾).

This experimental and numerical study considers the two-dimensional stability of a circular cylinder wake, whose axis is tilted with respect to a stable density gradient. When the Reynolds number increases, the wake transitions from a steady flow to a periodic von Kármán vortex street as in a homogeneous fluid. However, the presence of a moderate stratification delays the appearance of the von Kármán vortex street, in agreement with the stabilization of shear flows by a density gradient. This stabilization, which does not occur for a vertical cylinder, increases with the tilt angle of the cylinder and is maximum for a horizontal cylinder. The critical Reynolds number increases when the stratification increases and diverges at a Froude number of order one for a horizontal cylinder. This critical Reynolds number can be predicted using the Richardson number based on the projection of the gravity and the density gradient in the direction of the shear, as was proposed by Candelier (J. Fluid Mech., vol. 685, pp. 191–201) for a tilted stratified jet. This picture is completely different for a strongly stratified wake since a new unstable mode appears, creating a von Kármán vortex street with a smaller Strouhal number. This surprising result is due to the presence of tilted vortices with no vertical velocity, i.e. with horizontal elliptic streamlines. This mode occurs in a band of Froude numbers which becomes smaller and smaller when the tilt angle increases, and eventually disappears for a horizontal cylinder. The presence of the tilt has thus a large impact on the structure of the wake at small Froude numbers and might need to be taken into account in geophysical flows.

A two-dimensional model describing river morphodynamic processes under mixed-size sediment conditions is analysed with respect to its well posedness. Well posedness guarantees the existence of a unique solution continuously depending on the problem data. When a model becomes ill posed, infinitesimal perturbations to a solution grow infinitely fast. Apart from the fact that this behaviour cannot represent a physical process, numerical simulations of an ill-posed model continue to change as the grid is refined. For this reason, ill-posed models cannot be used as predictive tools. One source of ill posedness is due to the simplified description of the processes related to vertical mixing of sediment. The current analysis reveals the existence of two additional mechanisms that lead to model ill posedness: secondary flow due to the flow curvature and the effect of gravity on the sediment transport direction. When parametrising secondary flow, accounting for diffusion in the transport of secondary flow intensity is a requirement for obtaining a well-posed model. When considering the theoretical amount of diffusion, the model predicts instability of perturbations that are incompatible with the shallow water assumption. The effect of gravity on the sediment transport direction is a necessary mechanism to yield a well-posed model, but not all closure relations to account for this mechanism are valid under mixed-size sediment conditions. Numerical simulations of idealised situations confirm the results of the stability analysis and highlight the consequences of ill posedness.

We calculate the leading-order correction to the time period of rotation of a neutrally buoyant spheroid of arbitrary aspect ratio, in a simple shear flow ($\boldsymbol{u}^{\infty }=\dot{\unicode[STIX]{x1D6FE}}y\mathbf{1}_{1}$; $\mathbf{1}_{1}$ is the unit vector in the flow direction, $y$ being the coordinate along the gradient direction), in its long-time orbit set up by the weak fluid inertial drift at $O(Re)$. Here, $Re$ is the microscale Reynolds number, a dimensionless measure of the fluid inertial effects on the length scale of the spheroid, and is defined as $Re=\dot{\unicode[STIX]{x1D6FE}}L^{2}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D707}$, where $L$ is the semimajor axis of the spheroid, $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70C}$ are respectively the viscosity and density of the fluid, and $\dot{\unicode[STIX]{x1D6FE}}$ is the shear rate. This long-time orbit is the tumbling orbit for prolate spheroids; for oblate spheroids, it is the spinning orbit for aspect ratios greater than $0.137$, and can be either the tumbling or the spinning orbit for oblate spheroids of aspect ratios less than $0.137$. We also calculate the leading-order correction to the time period of rotation of a neutrally buoyant triaxial ellipsoid in a simple shear flow, rotating with its intermediate principal axis aligned along the vorticity of the flow; the latter calculation is in light of recent evidence, by way of numerical simulations (Rosen, PhD dissertation, 2016, Stockholm), of the aforementioned rotation being stabilized by weak inertia. The correction to the time period for arbitrary $Re$ is expressed as a volume integral using a generalized reciprocal theorem formulation. For $Re\ll 1$, it is shown that the correction at $O(Re)$ is zero for spheroids (with aspect ratios of order unity) as well as triaxial ellipsoids in their long-time orbits. The first correction to the time period therefore occurs at $O(Re^{3/2})$, and has a singular origin, arising from fluid inertial effects in the outer region (distances from the spheroid or triaxial ellipsoid of the order of the inertial screening length of $O(LRe^{-1/2})$), where the leading-order Stokes approximation ceases to be valid. Since the correction comes from the effects of inertia in the far field, the rotating spheroid (triaxial ellipsoid) is approximated as a time-dependent point-force-dipole singularity, allowing for the reciprocal theorem integral to be evaluated in Fourier space. It is shown for all relevant cases that fluid inertia at $O(Re^{3/2})$ leads to an increase in the time period of rotation compared with that in the Stokes limit, consistent with the results of recent numerical simulations at finite $Re$. Finally, combination of the $O(Re^{3/2})$ correction derived here with the $O(Re)$ correction derived earlier by Dabade et al. (J. Fluid Mech., vol. 791, 2016, 631703) yields a uniformly valid description of the first effects of inertia for spheroids of all aspect ratios, including prediction of the arrest of rotation for extreme-aspect-ratio spheroids.

The conditions are examined in which stationary hydraulic jumps may occur in a continuously stratified layer of fluid of finite thickness moving over a horizontal boundary at z = 0 and beneath a deep static layer of uniform density. The velocity and density in the flowing layer are modified by turbulent mixing in the transition region. Entrainment of fluid from the overlying static layer is possible. Results are expressed in terms of a Froude number, Fr, characterizing the flow upstream of a transition. A Froude number, Fr*, is found that must be exceeded if conditions for the conservation of volume, mass and momentum fluxes across a hydraulic transition are satisfied. The condition Fr > Fr* is satisfied if the kinetic energy (KE) per unit area is greater than the potential energy per unit area, or if ∫0h [u2(z) − z2N2(z)]dz > 0, in a flow of speed u(z), in a layer of thickness h, with buoyancy frequency N(z). In the particular case (referred to as an ‘η profile’) of a flow with velocity and density that are constant if z ≤ ηh, decrease linearly if ηh ≤ z ≤ h, and in which u(z) = 0 and density is constant when z ≥ h, long linear internal waves can propagate upstream, ahead of a stationary hydraulic jump, for Fr in a range Fr* < Fr < Frc; here Frc is the largest Fr at which long waves, and wave energy, can propagate upstream in a flow with specified η. (Profiles other than the η profile exhibit similar properties.) It is concluded that whilst, in general, Fr > Fr* is a necessary condition for a hydraulic jump to occur, a more stringent condition may apply in cases where Fr* < Frc, i.e. that Fr > Frc.

Physical constraints are imposed on the form of the flow downstream of the hydraulic jump or transition that relate, for example, to its static and dynamic stability and its stability against a further hydraulic jump. A further condition is imposed that relates the rate of dissipation of turbulent KE within a transition to the loss in energy flux of the flow in passing through the transition from the upstream side to the downstream. The constraints restrict solutions for the downstream flow to those in which the flux of energy carried downstream by internal waves is negligible and in which dissipation of energy occurs within the transition region.

Although the problem is formulated in general terms, particular examples are given for η profiles, specifically when η = 0 and 0.4. The jump amplitude, the entrainment rate, the loss of energy flux and the shape of the velocity and density profiles in the flow downstream of a transition are determined when Fr > Fr* (and extending to those with Fr > Frc) in a number of extreme conditions: when the loss of energy flux in transitions is maximized, when the entrainment is maximized, when the jump amplitude is least and when loss of energy flux is maximized subject to the entrainment into the transitions being made zero. The ratio of the layer thickness downstream and upstream of a transition, the jump amplitude, is typically at least 1.4 when jumps are just possible (i.e. when Fr ~ Frc). The amplitude, entrainment and non-dimensionalized loss in energy flux increase with Fr in each of the extreme conditions, and the maximum loss in energy flux is close to that when the entrainment is greatest. The magnitude of the advective and diffusive fluxes across isopycnal surfaces, i.e. the diapycnal fluxes characterizing turbulent mixing in the transition region, also increase with Fr. Results are compared to those in which the velocity and density profiles downstream of the transition are similar to those upstream, and comparisons are also made with equivalent two-layer profiles and with a cosine-shaped profile with continuous gradients of velocity and density. If Fr is larger than a certain value (about 7 and > Frc, if η = 0.4), no solutions for flows downstream of a transition are found if there is no entrainment, implying that fluid must be entrained if a transition is to occur in flows with large Fr. Although the extreme conditions of loss of energy flux, jump amplitude or entrainment provide limits that it must satisfy, in general no unique downstream flow is found for a given flow upstream of a jump.

We consider the analytic structure of interfaces in several families of steady and unsteady two-dimensional Stokes flows, focusing on the formation of corners and cusps. Previous experimental and theoretical studies have suggested that, without surface tension, the interfaces spontaneously develop such singular points. We investigate whether and how corners and cusps actually develop in a time-dependent flow, and assess the stability of stationary cusped shapes predicted by previous authors. The motion of the interfaces is computed with high resolution using a boundary integral method for three families of flows. In the case of a bubble that is subjected to the family of straining flows devised by Antanovskii, we find that a stationary cusped shape is not likely to occur as the asymptotic limit of a transient deformation. Instead, the pointed ends of the bubble disintegrate in a process that is reminiscent of tip streaming. In the case of the flow due to an array of point-source dipoles immersed beneath a free surface, which is the periodic version of a flow proposed by Jeong & Moffatt, we find evidence that a cusped shape indeed arises as the result of a transient deformation. In the third part of the numerical study, we show that, under certain conditions, the free surface of a liquid film that is levelling under the action of gravity on a horizontal or slightly inclined surface develops an evolving corner or cusp. In certain cases, the film engulfs a small air bubble of ambient fluid to obtain a composite shape. The structure of a corner or a cusp in an unsteady flow does not have a unique shape, as it does at steady state. In all cases, a small amount of surface tension is able to prevent the formation of a singularity, but replacing the inviscid gas with a viscous liquid does not have a smoothing effect. The ability of the thin-film lubrication equation to produce mathematical singularities at the free surface of a levelling film is also discussed.