Internal (gravity) wave attractors may form in closed containers with boundaries non-parallel and non-normal to the gravity vector. Such attractors have been studied from a theoretical point of view, in laboratory experiments and using linear numerical computations. In the present paper two-dimensional numerical simulations of an internal wave attractor are reported, based upon the nonlinear and non-hydrostatic MIT-gcm numerical code. We first reproduce the laboratory experiment of a wave attractor performed by Hazewinkel et al. (J. Fluid Mech. Vol. 598, 2008 p. 373) and obtain very good agreement with the experimental data. We next propose simple ideas to model the thickness of the attractor. The model predicts that the thickness should scale as the 1/3 power of the non-dimensional parameter measuring the ratio of viscous to buoyancy effects. When the attractor is strongly focusing, the thickness should also scale as the 1/3 power of the spatial coordinate along the attractor. Analysis of the numerical data for two different attractors yields values of the exponent close to 1/3, within 30%. Finally, we study nonlinear effects induced by the attractor.

We investigate the transient ventilation flow within a confined ventilated space, with high- and low-level openings, when the strength of a low-level point source of heat is changed instantaneously. The steady-flow regime in the space involves a turbulent buoyant plume, which rises from the point source to a well-mixed warm upper layer. The steady-state height of the interface between this layer and the lower layer of exterior fluid is independent of the heat flux, but the upper layer becomes progressively warmer with heat flux. New analogue laboratory experiments of the transient adjustment between steady states identify that if the heat flux is increased, the continuing plume propagates to the top of the room forming a new, warmer layer. This layer gradually deepens, and as the turbulent plume entrains fluid from the original warm layer, the original layer is gradually depleted and disappears, and a new steady state is established. In contrast, if the source buoyancy flux is decreased, the continuing plume is cooler than the original plume, so that on reaching the interface it is of intermediate density between the original warm layer and the external fluid. The plume supplies a new intermediate layer, which gradually deepens with the continuing flow. In turn, the original upper layer becomes depleted, both as a result of being vented through the upper opening of the space, but also due to some penetrative entrainment of this layer by the plume, as the plume overshoots the interface before falling back to supply the new intermediate layer. We develop quantitative models which are in good accord with our experimental data, by combining classical plume theory with models of the penetrative entrainment for the case of a decrease in heating. Typically, we find that the effect of penetrative entrainment on the density of the intruding layer is relatively weak, provided the change in source strength is sufficiently large. However, penetrative entrainment measurably increases the rate at which the depth of the draining layer decreases. We conclude with a discussion of the importance of these results for the control of naturally ventilated spaces.

Most rational asymptotic studies of non-rotating Rayleigh–Bénard convection and its cousins have been restricted to the linear or weakly nonlinear regime. An important exception occurs for large Rayleigh-number thermal convection at effectively infinite Prandtl number, i.e. fast but very viscous convection. In this scenario, the temperature field exhibits a layer-like structure surrounding an isothermal core and, crucially, the momentum equation linearizes. These features have been exploited by several authors to obtain semi-analytical nonlinear solutions. At O(1) Prandtl number, the fluid dynamics in the vortex core is dominated by nonlinear inertial rather than linear viscous effects, substantially altering the vortex structure. Here, it is shown that a combination of matched asymptotic analysis and global conservation constraints can be used to obtain a semi-analytic yet strongly nonlinear description of two related flows: (i) Rayleigh–Bénard convection between constant heat-flux boundaries at unit Prandtl number, and (ii) Langmuir circulation (LC), a wind and wave-driven convective flow commonly observed in natural water bodies. A simple analytical prediction is given for the roll-vortex amplitude, which is shown to be independent of the horizontal wavenumber of the convection pattern. In marked contrast to weakly nonlinear convection cells, the fully nonlinear asymptotic solutions exhibit flow features relevant to turbulent convection including the complete vertical redistribution of the basic-state temperature (or, for LC, downwind velocity) field. Comparisons with well-resolved pseudospectral numerical simulations of the full two-dimensional governing equations confirm the accuracy of the asymptotic results.

To aid the physical understanding of spiralling-columnar convection emerging in rapidly rotating spheres and spherical shells, two-dimensional thermal convection in a rapidly rotating annulus is investigated through the radial propagation properties of topographic Rossby waves. Two kinds of the boundaries containing the fluid in the axial direction are considered: a convex type modelling a spherical geometry and a concave type for comparison. The linear stability of a basic state with no motion and uniformly unstable stratification is examined and spirally elongated structures of critical convection are obtained for small Prandtl numbers. An analysis of the energy budget shows that a part of the kinetic energy generated in the region with slightly inclined boundaries is dynamically transferred and dissipates through viscosity in the region with strongly inclined boundaries. This indicates that the Rossby waves propagate from the region with slightly inclined boundaries to the region with strongly inclined boundaries. It is presented that the appearance of a spiral structure corresponds to an increase of the local radial wavenumber of the Rossby waves propagating in the radial direction. The flow patterns obtained using the dispersion relation of the Rossby waves coincide with those of the tailing part of the spiral structure obtained numerically. As the Prandtl number increases, the Rossby waves barely propagate because of strong viscous dissipation, and the flow pattern is localized in the region with slightly inclined boundaries. For convex boundaries with unstable stratification concentrating near the outer boundary and concave boundaries with unstable stratification confined near the inner boundary, the flow patterns tilt in the direction inverse to the case of uniform unstable stratification. The tilting direction of the flow pattern is not determined by the curvature of the boundaries considered but instead by the radial propagation direction of the Rossby waves excited by thermal convection.

In this paper we investigate salt transport during the evaporation and upflow of saline groundwater. We describe a model in which a sharp evaporation–precipitation front separates regions of soil saturated with an air–vapour mixture and with saline water. We then consider two idealized problems. We first investigate equilibrium configurations of the freshwater system when the depth of the soil layer is finite, obtaining results for the location of the front and the upflow of water induced by the evaporation. We then develop a solution for a propagating front in a soil layer of infinite depth and investigate the gravitational stability of the salinity profile which develops below the front, obtaining marginal linear stability conditions in terms of a Rayleigh number and a dimensionless salt saturation parameter. Applying our findings to realistic parameter regimes, we predict that salt fingering is unlikely to occur in low-permeability soils, but is likely in high-permeability (sandy) soils under conditions of relatively low evaporative upflow.

This paper is concerned with the nonlinear instability of compressible mixing layers in the regime of small to moderate values of Mach number M, in which subsonic modes play a dominant role. At high Reynolds numbers of practical interest, previous studies have shown that the dominant nonlinear effect controlling the evolution of an instability wave comes from the so-called critical layer. In the incompressible limit (M = 0), the critical-layer dynamics are strongly nonlinear, with the nonlinearity being associated with the logarithmic singularity of the velocity fluctuation (Goldstein & Leib, J. Fluid Mech. vol. 191, 1988, p. 481). In contrast, in the fully compressible regime (M = O(1)), nonlinearity is associated with a simple-pole singularity in the temperature fluctuation and enters in a weakly nonlinear fashion (Goldstein & Leib, J. Fluid Mech. vol. 207, 1989, p. 73). In this paper, we first consider a weakly compressible regime, corresponding to the distinguished scaling M = O(ε1/4), for which the strongly nonlinear structure persists but is affected by compressibility at leading order (where ε ≪ 1 measures the magnitude of the instability mode). A strongly nonlinear system governing the development of the vorticity and temperature perturbation is derived. It is further noted that the strength of the pole singularity is controlled by T′c, the mean temperature gradient at the critical level, and for typical base-flow profiles T′c is small even when M = O(1). By treating T′c as an independent parameter of O(ε1/2), we construct a composite strongly nonlinear theory, from which the weakly nonlinear result for M = O(1) can be derived as an appropriate limiting case. Thus the strongly nonlinear formulation is uniformly valid for O(1) Mach numbers. Numerical solutions show that this theory captures the vortex roll-up process, which remains the most prominent feature of compressible mixing-layer transition. The theory offers an effective tool for investigating the nonlinear instability of mixing layers at high Reynolds numbers.

The concept of piecewise constant symmetric vortex in the context of three-dimensional baroclinic balanced geophysical flows is explored. The pressure gradients generated by horizontal cylinders and spherical balls of uniform potential vorticity (PV), or uniform material invariants, are obtained either analytically or numerically, in the general case of Boussinesq and f-plane dynamics as well as under the quasi-geostrophic and semigeostrophic dynamical approximations. Based on the order of magnitude of the different terms in the PV inversion equation, approximated PV equations are deduced. In some of these cases, radial solutions are possible and the interior and exterior solutions are found analytically. In the case of non-radial dependence, exterior solutions can be found numerically. Linear, and upper and lower bound approximations to the full PV inversion equations, and their respective solutions, are also included. However, the general solution for the pressure gradient in the vortex exterior does not have spherical symmetry and remains as an important theoretical challenge. It is suggested that, in order to maintain everywhere the inertial and static stability of the balanced geophysical flows, small balls of finite radius, rather than PV singularities, could become, specially in numerical applications, useful mathematical objects.

The steady propagation of an air finger into a fluid-filled tube of uniform rectangular cross-section is investigated. This paper is primarily focused on the influence of the aspect ratio, α, on the flow properties, but the effects of a transverse gravitational field are also considered. The three-dimensional interfacial problem is solved numerically using the object-oriented multi-physics finite-element library oomph-lib and the results agree with our previous experimental results (de Lózar et al. Phys. Rev. Lett. vol. 99, 2007, article 234501) to within the ±1% experimental error.

At a fixed capillary number Ca (ratio of viscous to surface-tension forces) the pressure drops across the finger tip and relative finger widths decrease with increasing α. The dependence of the wet fraction m (the relative quantity of liquid that remains on the tube walls after the propagation of the finger) is more complicated: m decreases with increasing α for low Ca but it increases with α at high Ca. Our results also indicate that the system is approximately quasi-two-dimensional for α ≥ 8, when we obtain quantitative agreement with McLean & Saffman's two-dimensional model for the relative finger width as a function of the governing parameter 1/B = 12α2Ca. The action of gravity causes an increase in the pressure drops, finger widths and wet fractions at fixed capillary number. In particular, when the Bond number (ratio of gravitational to surface-tension forces) is greater than one the finger lifts off the bottom wall of the tube leading to dramatic increases in the finger width and wet fraction at a given Ca.

For α ≥ 3 a previously unobserved flow regime has been identified in which a small recirculation flow is situated in front of the finger tip, shielding it from any contaminants in the flow. In addition, for α ≳ 2 the capillary number, Cac, above which global recirculation flows disappear has been observed to follow the simple empirical law: Cac2/3α = 1.21.

Taylor's hypothesis of frozen flow has frequently been used to convert temporal experimental measurements into a spatial domain. This technique has led to the discovery of long meandering structures in the log-region of a turbulent boundary layer. There is some contention over whether Taylor's approximation is valid over large distances. This paper presents an experiment that compares velocity fields constructed using Taylor's approximation with those obtained from particle image velocimetry (PIV), i.e. spatial data, obtained in the logarithmic region of a turbulent boundary layer.

A large-eddy-based methodology for the simulation of turbulent sprays is discussed. The transport equations for the spatially filtered gas phase variables, in which source terms accounting for the droplet effects are added, are solved together with a probabilistic description of the liquid phase. The probabilistic approach for the liquid phase is based on the transport equation for the spatially filtered joint probability density function of the variables required in order to describe the state of the liquid phase. In this equation, unclosed terms representing the filtered Lagrangian rates of change of the variables describing the spray are present. General modelling ideas for subgrid-scale (SGS) effects are proposed. The capabilities of the approach and the validity of the closure models, with particular with respect to the SGS dispersion, are investigated through application to a dilute particle-laden turbulent mixing layer. It is demonstrated that the formulation is able to reproduce very closely the measured properties of both the continuous and dispersed phases. The large-eddy simulation (LES) results are also found to be entirely consistent with the experimentally observed characteristics of droplet–gas turbulence interactions. Consistent with direct numerical simulation (DNS) studies of isotropic turbulence laden with particles where the entire turbulence spectrum is found to be modulated by the presence of particles, the present investigation, which comprises the effects of particle transport upon the large-scale vortical structures of a turbulent shear flow, highlights what appears to be a selective behaviour; few large-scale frequencies gain energy whereas the remaining modes are damped.

The connection between fluid flow and optical flow is explored in typical flow visualizations to provide a rational foundation for application of the optical flow method to image-based fluid velocity measurements. The projected-motion equations are derived, and the physics-based optical flow equation is given. In general, the optical flow is proportional to the path-averaged velocity of fluid or particles weighted with a relevant field quantity. The variational formulation and the corresponding Euler–Lagrange equation are given for optical flow computation. An error analysis for optical flow computation is provided, which is quantitatively examined by simulations on synthetic grid images. Direct comparisons between the optical flow method and the correlation-based method are made in simulations on synthetic particle images and experiments in a strongly excited turbulent jet.

We present experiments showing vertical jetting from the apex of a viscous drop which impacts onto a pool of lower viscosity liquid. This jet is produced by the ejecta sheet which emerges from the free surface of the pool, and moves up and wraps around the surface of the drop. When this sheet of liquid converges and collides at the top apex of the drop it produces a thin upward jet at velocities of more than 10 times the drop impact velocity. This jetting occurs for a limited range of impact conditions, where the ejecta speed is sufficient for the sheet to travel around the entire drop periphery, but not so fast that it separates from the drop surface. The lower bound for the jetting region is thereby set by a minimal Reynolds number, but the upper bounds are subject to a maximum-Weber-number criterion. The strongest observed jets appear for viscous drops impacting onto liquid pools with the lowest viscosity as well as lowest surface tension, such as acetone and methanol. Jetting has also been observed for drops which are immiscible with the pool liquid, under a different range of impact conditions. However, jetting is never observed for pools of water, as the surface tension is then significantly larger than that of the drop. We believe that Marangoni stresses act in this case to promote separation of the sheet to prevent the jetting. A movie is available with the online version of the paper.

We derive a relation for the growth of the mean square of vertical displacements, δz, of fluid particles of stratified turbulence. In the case of freely decaying turbulence, we find that for large times 〈δz2〉 goes to a constant value 2(EP(0) + aE(0))/N2, where EP(0) and E(0) are the initial mean potential and total turbulent energy per unit mass, respectively, a < 1 and N is the Brunt–Väisälä frequency. In the case of stationary turbulence, we find that 〈δz2〉 = 〈δb2〉/N2 + 2εPt/N2, where εP is the mean dissipation of turbulent potential energy per unit mass and 〈δb2〉 is the Lagrangian structure function of normalized buoyancy fluctuations. The first term is the same as that obtained in the case of adiabatic fluid particle dispersion. This term goes to the finite limit 4EP/N2 as t → ∞. Assuming that the second term represents irreversible mixing, we show that the Osborn & Cox model for vertical diffusion is retained. In the case where the motion is dominated by a turbulent cascade with an eddy turnover time T ≫ N−1, rather than linear gravity waves, we suggest that there is a range of time scales, t, between N−1 and T, where 〈δb2〉 = 2πCPLεPt, where CPL is a constant of the order of unity. This means that for such motion the ratio between the adiabatic and the diabatic mean-square displacement is universal and equal to πCPL in this range. Comparing this result with observations, we make the estimate CPL ≈ 3.

A separated boundary-layer flow at the rear of a bump is considered. Two-dimensional equilibrium stationary states of the Navier–Stokes equations are determined using a nonlinear continuation procedure varying the bump height as well as the Reynolds number. A global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The onset of instability is shown to be characterized by a family of modes with localized structures around the reattachment point becoming almost simultaneously unstable. The optimal perturbation analysis, by projecting the initial disturbance on the set of temporal eigenmodes, reveals that the non-normal modes are able to describe localized initial perturbations associated with the large transient energy growth. At larger time a global low-frequency oscillation is found, accompanied by a periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier–Stokes time integration and is shown to trigger the nonlinear ‘flapping’ typical of separation bubbles. It is possible to follow the stationary equilibrium state on increasing the Reynolds number far beyond instability, ruling out for the present flow case the hypothesis of some authors that topological flow changes are responsible for the ‘flapping’.

This theoretical investigation of steady fluid flow through a rigid three-dimensional branching geometry is motivated by applications to haemodynamics in the brain especially, while the flow through a tube with a blockage or through a collapsed tube provides another motivation with a biomedical background. Three-dimensional motion without symmetry is addressed through one mother vessel to two or several daughters. A comparatively long axial length scale of the geometry leads to a longitudinal vortex system providing a slender-flow model for the complete mother-and-daughters flow response. Computational studies and subsequent analysis, along with comparisons, are presented. The relative flow rate varies in terms of an effective Reynolds number dependence, allowing a wide range of flow rates to be examined theoretically; also any rigid cross-sectional shape and ratio of cross-sectional area expansion or contraction from the mother vessel to the daughters can be accommodated in principle in both the computations and the analysis. Swirl production with substantial crossflows is found. The analysis shows that close to any carina (the ridge separating daughter vessels) or carinas at a branch junction either forward or reversed motion can be observed locally at the saddle point even though the bulk of the motion is driven forward into the daughters. The local forward or reversed motion is controlled, however, by global properties of the geometry and incident conditions, a feature which applies to any of the flow rates examined.

We present a new formulation of the motion of a flexible body with a vortex-sheet wake and use it to study propulsive forces generated by a flexible body pitched periodically at the leading edge in the small-amplitude regime. We find that the thrust power generated by the body has a series of resonant peaks with respect to rigidity, the highest of which corresponds to a body flexed upwards at the trailing edge in an approximately one-quarter-wavelength mode of deflection. The optimal efficiency approaches 1 as rigidity becomes small and decreases to 30–50% (depending on pitch frequency) as rigidity becomes large. The optimal rigidity for thrust power increases from approximately 60 for large pitching frequency to ∞ for pitching frequency 0.27. Subsequent peaks in response have power-law scalings with respect to rigidity and correspond to higher-wavenumber modes of the body. We derive the power-law scalings by analysing the fin as a damped resonant system. In the limit of small driving frequency, solutions are self-similar at the leading edge. In the limit of large driving frequency, we find that the distribution of resonant rigidities ~k−5, corresponding to fin shapes with wavenumber k. The input power and output power are proportional to rigidity (for small-to-moderate rigidity) and to pitching frequency (for moderate-to-large frequency). We compare these results with the range of rigidity and flapping frequency for the hawkmoth forewing and the bluegill sunfish pectoral fin.

This paper examines the evolution of a two-dimensional vortex which initially consists of an axisymmetric monopole vortex with a perturbation of azimuthal wavenumber m = 2 added to it. If the perturbation is weak, then the vortex returns to an axisymmetric state and the non-zero Fourier harmonics generated by the perturbation decay to zero. However, if a finite perturbation threshold is exceeded, then a persistent nonlinear vortex structure is formed. This structure consists of a coherent vortex core with two satellites rotating around it.

The paper considers the formation of these satellites by taking an asymptotic limit in which a compact vortex is surrounded by a weak skirt of vorticity. The resulting equations match the behaviour of a normal mode riding on the vortex with the evolution of fine-scale vorticity in a critical layer inside the skirt. Three estimates of inviscid thresholds for the formation of satellites are computed and compared: two estimates use qualitative diagnostics, the appearance of an inflection point or neutral mode in the mean profile. The other is determined quantitatively by solving the normal mode/critical-layer equations numerically. These calculations are supported by simulations of the full Navier–Stokes equations using a family of profiles based on the tanh function.

Since 1904, when Prandtl formulated the boundary-layer equations, it has been presumed that due to the viscous nature of the boundary layers the solution of the Prandtl equations should be sought in the class of continuous functions. However, there are clear mathematical reasons for discontinuous solutions to exist. Moreover, under certain conditions they represent the only possible solutions of the boundary-layer equations.

In this paper we consider, as an example, an unsteady analogue of the laminar jet problem first studied by Schlichting in 1933. In Schlichting's formulation the jet emerges from a narrow slit in a flat barrier and penetrates into a semi-infinite region filled with fluid which would remain at rest if the slit were closed. Assuming the flow steady, Schlichting was able to demonstrate that the corresponding solution to the Prandtl equations may be written in an explicit analytic form. Here our concern will be with unsteady flow that is initiated when the slit is opened and the jet starts penetrating into the stagnant fluid. To study this process we begin with the numerical solution of the unsteady boundary-layer equations. Since discontinuities were expected, the equations were written in conservative form before finite differencing. The solution shows that the jet has a well-established front representing a discontinuity in the velocity field, similar to the shock waves that form in supersonic gas flows.

Then, in order to reveal the ‘internal structure’ of the shock we turn to the analysis of the flow in a small region surrounding the discontinuity. With Re denoting the Reynolds number, the size of the inner region is estimated as an order Re−1/2 quantity in both longitudinal and lateral directions. We found that the fluid motion in this region is predominantly inviscid and may be treated as quasi-steady if considered in the coordinate frame moving with the jet front. These simplifications allow a simple formula for the front speed to be deduced, which proved to be in close agreement with experimental observation of Turner (J. Fluid Mech. vol. 13 (1962), p. 356).

Laminar-to-turbulent transition of pipe flows occurs, for sufficiently high Reynolds numbers, in the form of slugs. These are initiated by disturbances in the entrance region of a pipe flow, and grow in length in the axial direction as they move downstream. Sequences of slugs merge at some distance from the pipe inlet to finally form the state of fully developed turbulent pipe flow. This formation process is generally known, but the randomness in time of naturally occurring slug formation does not permit detailed study of slug flows. For this reason, a special test facility was developed and built for detailed investigation of deterministically generated slugs in pipe flows. It is also employed to generate the puff flows at lower Reynolds numbers. The results reveal a high degree of reproducibility with which the triggering device is able to produce puffs. With increasing Reynolds number, ‘puff splitting’ is observed and the split puffs develop into slugs. Thereafter, the laminar-to-turbulent transition occurs in the same way as found for slug flows. The ring-type obstacle height, h, required to trigger fully developed laminar flows to form first slugs or puffs is determined to show its dependence on the Reynolds number, Re = DU/ν (where D is the pipe diameter, U is the mean velocity in the axial direction and ν is the kinematic viscosity of the fluid). When correctly normalized, h+ turns out to be independent of Reτ (where h+ = hUτ/ν, Reτ = DUτ/ν and ; τw is the wall shear stress and ρ is the density of the fluid).

Integral models proposed to simulate positively buoyant jets are used to model jets with negative or reversing buoyancy issuing into a calm, homogeneous or density-stratified environment. On the basis of the self-similarity assumption, ‘top hat’ and Gaussian cross-sectional distributions are employed for concentration and velocity. The entrainment coefficient is considered to vary with the local Richardson number, between the asymptotic values for simple jets and plumes, estimated from earlier experiments in positively buoyant jets. Top-hat and Gaussian distribution models are employed in a wide range of experimental data on negatively buoyant jets, issuing vertically or at an angle into a calm homogeneous ambient, and on jets with reversing buoyancy, discharging into a calm, density-stratified fluid. It is found that geometrical characteristics such as the terminal (steady state) height of rise, the spreading elevation in stratified ambient and the distance to the point of impingement are considerably underestimated, resulting in lower dilution rates at the point of impingement, especially when the Gaussian formulation is applied. Reduction of the entrainment coefficient in the jet-like flow regime improves model predictions, indicating that the negative buoyancy reduces the entrainment in momentum-driven, negatively buoyant jets.