Experiments were done with strong shocks diffracting over steel ramps immersed in argon. Numerical simulations of the experiments were done by integrating the Navier–Stokes equations with a higher-order Godunov finite difference numerical scheme using isothermal non-slip boundary conditions. Adiabatic, slip boundary conditions were also studied to simulate cavity-type diffractions. Some results from an Euler numerical scheme for an ideal gas are presented for comparison. When the ramp angle θ is small enough to cause Mach reflection MR, it is found that real gas effects delay its appearance and that the trajectory of its shock triple point is initially curved; it eventually becomes straight as the MR evolves into a self-similar system. The diffraction is a regular reflection RR in the delayed state, and this is subsequently swept away by a corner signal overtaking the RR and forcing the eruption of the Mach shock. The dynamic transition occurs at, or close to, the ideal gas detachment criterion θe. The passage of the corner signal is marked by large oscillations in the thickness of the viscous boundary layer. With increasing θ, the delay in the onset of MR is increased as the dynamic process slows. Once self-similarity is established the von Neumann criterion is supported. While the evidence for the von Neumann criterion is strong, it is not conclusive because of the numerical expense. The delayed transition causes some experimental data for the trajectory to be subject to a simple parallax error. The adiabatic, slip boundary condition for self-similar flow also supports the von Neumann criterion while θ < θe, but the trajectory angle discontinuously changes to zero at θe, so that θe is supported by the numerics, contrary to experiments.

A partially collapsed lung airway or other flexible tube is modelled as a two-dimensional channel of infinite length. We consider the linear stability of this system conveying a developing flow, analysing the full Orr–Sommerfeld system analytically for long waves and numerically for arbitrary wavelengths. We find a long-wave instability which has not been observed in previous channel studies. This long-wave instability is stabilized by increasing the elastance of the wall, but other wall properties do not affect it except in correction terms. In addition to the long-wave instability, there is the finite wavelength (flutter) instability, which, depending on the parameter values chosen, may be critical at a higher or lower flow speed than the long-wave instability. For special parameter values the long-wave and flutter instabilities are critical at the same flow speed. Comparisons with experiments show that theoretical predictions are in agreement with experimental observations.

An experimental investigation of the binary droplet collision dynamics was conducted, with emphasis on the transition between different collision outcomes. A series of time-resolved photographic images which map all the collision regimes in terms of the collision Weber number and the impact parameter were used to identify the controlling factors for different outcomes. The effects of liquid and gas properties were studied by conducting experiments with both water and hydrocarbon droplets in environments of different gases (air, nitrogen, helium and ethylene) and pressures, the latter ranging from 0.6 to 12 atm. It is shown that, by varying the density of the gas through its pressure and molecular weight, water and hydrocarbon droplets both exhibit five distinct regimes of collision outcomes, namely (I) coalescence after minor deformation, (II) bouncing, (III) coalescence after substantial deformation, (IV) coalescence followed by separation for near head-on collisions, and (V) coalescence followed by separation for off-centre collisions. The present result therefore extends and unifies previous experimental observations, obtained at one atmosphere air, that regimes II and II do not exist for water droplets. Furthermore, it was found that coalescence of the hydrocarbon droplets is promoted in the presence of gaseous hydrocarbons in the environment, suggesting that coalescence is facilitated when the environment contains vapour of the liquid mass. Collision at high-impact inertia was also studied, and the mechanisms for separation of the coalescence are discussed based on time-resolved collision images. A coalescence/separation criterion defining the transition between regimes III and IV for the head-on collisions was derived and found to agree well with the experimental data.

According to classical hydraulic theory, the energy losses within an external bore must occur within the expanding layer. However, the application of this theory to describe the propagation of internal bores leads to contradiction with accepted gravity-current behaviour in the limit as the depth of the expanding layer ahead of the bore becomes small. In seeking an improved expression for the propagation of internal bores, we have rederived the steady front condition for a bore in a two-layer Boussinesq fluid in a channel under the assumption that the energy loss occurs within the contracting layer. The resulting front condition is in good agreement with available laboratory data and numerical simulations, and has the appropriate behaviour in both the linear long-wave and gravity-current limits. Analysis of an idealized internal bore assuming localized turbulent stresses suggests that the energy within the expanding layer should, in fact, increase. Numerical simulations with a two-dimensional non-hydrostatic model also reveal a slight increase of energy within the expanding layer and suggest that the structure of internal bores is fundamentally different from classical external bores, having the opposite circulation and little turbulence in the vicinity of the leading edge. However, if there is strong shear near the interface between layers, the structure and propagation of internal jumps may become similar to their counterparts in classical hydraulic theory. The modified jump conditions for internal bores produce some significant alterations in the traditional Froude-number dependence of Boussinesq shallow-water flow over an obstacle owing to the altered behaviour of the upstream-propagating internal bore.

Laser Doppler Velocimetry (LDV) measurements are presented for a nominally two-dimensional constant-density flow over a surface-mounted triangular cylinder. The thickness of the boundary layer approaching the triangular cylinder is much less than the height of the triangle. Momentum and turbulent kinetic energy balances are presented and comparisons are made with other separated and reattaching flows. Also, time domain information is presented in the form of autocorrelations and spectra. From the energy balances, the importance of the pressure transport term at the high-speed edge of the shear layer is seen. Observations of the relationships between the shapes of the spectra and the details of the energy balance are made. For example, the slope of the velocity spectra varies from the free-stream value of −5/3 to a value of −1 in the middle of the recirculation region. Concurrent with this increase in slope is a decrease in the role of shear production in the turbulent kinetic energy balance and an increase in the role of advection and turbulent transport. From the two-component LDV measurements, a very low-frequency unsteadiness is shown to contribute energy preferentially to different components of the velocity fluctuations depending on the location in the flow.

The linear stability of an inviscid two-dimensional liquid sheet falling under gravity in a still gas is studied by analysing the asymptotic behaviour of a localized perturbation (wave-packet solution to the initial value problem). Unlike previous papers the effect of gravity is fully taken into account by introducing a slow length scale which allows the flow to be considered slightly non-parallel. A multiple-scale approach is developed and the dispersion relations for both the sinuous and varicose disturbances are obtained to the zeroth-order approximation. These exhibit a local character as they involve a local Weber number Weη. For sinuous disturbances a critical Weη equal to unity is found below which the sheet is locally absolutely unstable (with algebraic growth of disturbances) and above which it is locally convectively unstable. The transition from absolute to convective instability occurs at a critical location along the vertical direction where the flow Weber number equals the dimensionless sheet thickness. This critical distance, as measured from the nozzle exit section, increases with decreasing the flow Weber number, and hence, for instance, the liquid flow rate per unit length. If the region of absolute instability is relatively small it may be argued that the system behaves as a globally stable one. Beyond a critical size the flow receptivity is enhanced and self-sustained unstable global modes should arise. This agrees with the experimental evidence that the sheet breaks up as the flow rate is reduced. It is conjectured that liquid viscosity may act to remove the algebraic growth, but the time after which this occurs could be not sufficient to avoid possible nonlinear phenomena appearing and breaking up the sheet.

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.

The bifurcation structure of thermohaline-driven flows is studied within one of the simplest zonally averaged models which captures thermohaline transport: a Boussinesq model of surface-forced thermohaline flow in a two-dimensional rectangular basin. Under mixed boundary conditions, i.e. prescribed surface temperature and fresh-water flux, it is shown that symmetry breaking originates from a codimension-two singularity which arises through the intersection of the paths of two symmetry-breaking pitchfork bifurcations. The physical mechanism of symmetry breaking of both the thermally and salinity dominated symmetric solution is described in detail from the perturbation structures near bifurcation. Limit cycles with an oscillation period in the order of the overturning time scale arise through Hopf bifurcations on the branches of asymmetric steady solutions. The physical mechanism of oscillation is described in terms of the most unstable mode just at the Hopf bifurcation. The occurrence of these oscillations is quite sensitive to the shape of the prescribed fresh-water flux. Symmetry breaking still occurs when, instead of a fixed temperature, a Newtonian cooling condition is prescribed at the surface. There is only quantitative sensitivity, i.e. the positions of the bifurcation points shift with the surface heat transfer coefficient. There are no qualitative changes in the bifurcation diagram except in the limit where both the surface heat flux and fresh-water flux are prescribed. The bifurcation structure at large aspect ratio is shown to converge to that obtained by asymptotic theory. The complete structure of symmetric and asymmetric multiple equilibria is shown to originate from a codimension-three bifurcation, which arises through the intersection of a cusp and the codimension-two singularity responsible for symmetry breaking.

The formation and evolution of a diffusive interface in a stable salt-stratified layer cooled from above is studied in a two-dimensional geometry by direct numerical simulation. For a typical example with realistic parameters, the evolution of the flow is computed up to the moment where three layers can be distinguished. Focus is on the development of the first mixed layer. The convective velocity scaling as proposed by Hunt (1984) and previously proposed expressions for the interfacial heat flux (Huppert 1971; Fernando 1989a) are shown to correspond well with the results of the simulation. The evolution of the first layer can be well described by an entrainment relation based on a local balance between kinetic and potential energy with mixing efficiency γ. The new entrainment relation is shown to fit the numerical results well and an interpretation of γ in terms of the overall energy balances of the flow is given.

Previously, two rival mechanisms have been proposed that determine the final thickness of the first layer (Turner 1968; Fernando 1987). One of the distinguishing features of both mechanisms is whether a transition in entrainment regime – as the first layer develops – is a necessary condition for the mixed layer to stop growing. Another is the presence of a buoyancy jump over the interface before substantial convection in the second layer occurs. From the numerical results, we find a significant buoyancy jump even before the thermal boundary layer ahead of the first layer becomes unstable. Moreover, the convective activity in the second layer is too small to be able to stop the growth of the first layer. We therefore favour the view proposed by Fernando (1987) that a transition in entrainment regime determines the thickness of the first layer. Following this, a new one-dimensional model of layer formation is proposed. Important expressions within this model are verified using the results of the numerical simulation. The model contains two constants which are determined from the numerical results. The results of the new model fit experimental results quite well and the parameter dependence of the thickness of the first layer is not sensitive to the values of the two constants.

The two-dimensional wake–shear layer forming behind a rectangular-based forebody with independent ambient streams on either side of the forebody is examined by direct numerical simulation. Theoretical aspects of global modes and frequency selection criteria based on local and global stability arguments are tested by computing local stability properties using local, time-averaged velocity profiles obtained from the numerical simulations and making the parallel-flow approximation. The theoretical results based on the assumption of a slightly non-parallel, spatially developing flow are shown to provide a firm basis for the frequency selection of vortex shedding and for defining the conditions for its onset. Distributed suction or blowing applied at the base of the forebody is used as a means of wake flow modification. The critical suction velocity to suppress vortex shedding is calculated. It is shown that local directional control (i.e. vectoring) of the near-wake flow is possible, but only when all global modes are suppressed.

Bifurcation characteristics of stably stratified plane Poiseuille flow have been investigated on a weakly nonlinear basis. It is found that the results are sensitive to the value of the Prandtl number, in that subcritical bifurcation persists for most values of the Prandtl number but is replaced by supercritical bifurcation over a range of small values of the Prandtl number. This range includes values characteristic of some liquid metals. The bifurcation becomes degenerate at a particular parameter set where the real part of the cubic nonlinear coefficient in the Stuart–Landau equation vanishes at criticality, and the situation is discussed by including higher-order terms in the manner of Eckhaus & Iooss (1989). An exact hyper-degenerate situation is also found to be possible for which the cubic and the quintic nonlinear coefficients lose their real parts simultaneously; this case is also analysed. For large values of the Prandtl number, stable stratification tends to promote subcritical instability.

The free surface of a viscous fluid is a source of convective flow (Marangoni convection) if its surface tension is distributed non-uniformly. Such non-uniformity arises from the dependence of the surface tension on a scalar quantity, either surfactant concentration or temperature. The surface-tension-induced velocity redistributes the scalar forming a closed-loop interaction. It is shown that under the assumptions of (i) small Reynolds number and (ii) vanishing diffusivity this nonlinear process is described by a single self-consistent two-dimensional evolution equation for the scalar field at the free surface that can be derived from the three-dimensional basic equations without approximation. The formulation of this equation for a particular system requires only the knowledge of the closure law, which expresses the surface velocity as a linear functional of the active scalar at the free surface. We explicitly derive these closure laws for various systems with a planar non-deflecting surface and infinite horizontal extent, including an infinitely deep fluid, a fluid with finite depth, a rotating fluid, and an electrically conducting fluid under the influence of a magnetic field. For the canonical problem of an infinitely deep layer we demonstrate that the dynamics of singular (point-like) surfactant or temperature distributions can be further reduced to a system of ordinary differential equations, equivalent to point-vortex dynamics in two-dimensional perfect fluids. We further show, using numerical simulations, that the dynamical evolution of initially smooth scalar fields leads in general to a finite-time singularity. The present theory provides a rational framework for a simplified modelling of strongly nonlinear Marangoni convection in high-Prandtl-number fluids or systems with high Schmidt number.

Statistical properties of multidimensional Burgers' turbulence evolving in the presence of a force field with random potential, which is delta-correlated in time and smooth in space, are studied in the inviscid limit and at the physical level of rigorousness. The solution algorithm reduces to finding multistream fields describing the motion of an auxiliary gas of interacting particles in a force field. Consequently, the statistical description of forced Burgers' turbulence is obtained by finding the largest possible value of the least action for the auxiliary gas. The exponential growth of the number of streams is found to be a necessary condition for the existence of stationary regimes.

We explore the nature of inertial equilibration of equatorial flows in the presence of mean meridional and vertical shears of the basic state, with oceanic applications in mind. The study is motivated by the observational evidence that the subthermocline equatorial mean circulation displays nearly zero Ertel potential vorticity away from the equator, when taking into account the non-traditional horizontal component of the Earth rotation. This observed state precisely verifies the marginal condition for inertial instability: a linear analysis for the equatorial β-plane confirms that the usual condition of instability, namely that Ertel potential vorticity should be of opposite sign to the vertical Coriolis parameter, remains valid even when the traditional approximation is relaxed. Analytical linear normal modes reveal that a meridional shear of the basic state leads to a vertical stacking of equatorially-trapped zonal flows of alternate signs, with a new centre of symmetry located at the dynamical equator. A vertical shear of the basic state causes a meridional stacking of extra-equatorial zonal flows.

In an inviscid framework, a two-dimensional formulation is ill-posed and we resort to non-hydrostatic viscous simulations to determine the nonlinear normal forms of the system. The influence of a small-scale eddy diffusivity and a large-scale Rayleigh damping on the equilibrated vertical scale is determined numerically. The nonlinear equilibration occurs through a steady-state bifurcation from a basic state without jets to another steady state with secondary jets of alternate signs. The final state corresponds to eastward jets located on the geographic equator, while westward jets are located near the dynamical equator. These results are consistent with in situ observations of equatorial deep jets.

The analogy between the equatorial meridional shear flow and the cylindrical Couette–Taylor flow with an axial density stratification is detailed. There is a strong similarity in the general symmetries and nonlinear normal forms of the two problems. Similarly to the homogeneous Couette–Taylor flow, the gap width between the two cylinders is important for determining the axial scale of the secondary flow through the Reynolds number. For the equatorial problem, an upper bound for the height scale of inertial jets is such that the corresponding equatorial radius of deformation times √2 fits between the geographic and dynamic equators.

One of our main conclusions is that the raisond’être of the observed region of zero Ertel potential vorticity is to facilitate angular momentum exchanges between the two hemispheres and inertial deep jets are the byproducts of this angular momentum mixing.

A thin-film approximation is used to study the effects of surface tension on a thin liquid layer lining the interior of a cylindrical tube, where the tube has radius a and a centreline with weak, uniform curvature δ/a. Centreline curvature induces a pressure gradient in the fluid layer, analogous to that due to a weak gravitational field, that drives fluid from the inner to the outer wall of the tube, i.e. away from the centre of centreline curvature. The resulting draining flow is computed numerically under the assumption of axial uniformity, and the large-time asymptotic draining regimes and flow structures are identified. In the absence of destabilizing intermolecular interactions, the inner wall remains wet, covered with a vanishingly thin fluid layer, while a near-equilibrium lobe forms on the outer wall. The stability of this quasi-static lobe to axial variations is then investigated by using numerical and perturbation methods to solve the linearized Young–Laplace equation, prescribing zero contact angle at the lobe's free boundary. Conditions on δ, the fluid volume a3V and the tube length aL are identified separating axially uniform lobes (which are stable for low V/(δL) or small L), wavy lobes (some with a solitary structure) and localized fluid droplets (which exist for sufficiently large V/δ and L). Hysteresis is demonstrated between multiple equilibria, the topology of which can change dramatically as parameters are varied. The application of these results to lung airways is discussed.

This paper is concerned with the theoretical behaviour of the laminar Ekman layer and the family of related rotating problems that includes both the Bödewadt and the von Kármán boundary-layer flows. Results from inviscid and viscous analyses are presented. In both cases, within specific regions of the parameter space, it is shown that the flows are absolutely unstable in the radial direction, i.e. disturbances grow in time at every radial location within these regions. Outside these regions, the flows are convectively unstable or stable. The absolute or convective nature of the flows is determined by examining the branch-point singularities of the dispersion relation. The onset of absolute instability is consistent with available experimental observations of the onset of laminar–turbulent transition in these flows.

For two-dimensional periodic water waves or sound waves, the kinetic energy per wavelength is ½mdc2, and the momentum per wavelength is ±mdc, where c is the wave velocity, and md is the drift mass per wavelength. These results also hold for three-dimensional periodic waves, for which the kinetic energy, momentum, and drift mass are all for one wave cell, the area of which is the product of the wavelengths in two perpendicular directions.

The results obtained are rigorous, and not restricted to linear waves or even to nonlinear symmetric waves. For linear water waves, in particular, the kinetic energy can be shown to be equal to the sum of the potential energy and the surface energy (due to surface tension), so that the total energy E is twice the kinetic energy, and

formula here

McIntyre's (1981) contention that wave momentum is a myth is discussed at length for both water waves and sound waves.