The structure of energy-containing turbulence in the outer region of a zero-pressure- gradient boundary layer has been studied using particle image velocimetry (PIV) to measure the instantaneous velocity fields in a streamwise-wall-normal plane. Experiments performed at three Reynolds numbers in the range 930 < Reθ < 6845 show that the boundary layer is densely populated by velocity fields associated with hairpin vortices. (The term ‘hairpin’ is here taken to represent cane, hairpin, horseshoe, or omega-shaped vortices and deformed versions thereof, recognizing these structures are variations of a common basic flow structure at different stages of evolution and with varying size, age, aspect ratio, and symmetry.) The signature pattern of the hairpin consists of a spanwise vortex core located above a region of strong second-quadrant fluctuations (u < 0 and v > 0) that occur on a locus inclined at 30–60° to the wall.

In the outer layer, hairpin vortices occur in streamwise-aligned packets that propagate with small velocity dispersion. Packets that begin in or slightly above the buffer layer are very similar to the packets created by the autogeneration mechanism (Zhou, Adrian & Balachandar 1996). Individual packets grow upwards in the streamwise direction at a mean angle of approximately 12°, and the hairpins in packets are typically spaced several hundred viscous lengthscales apart in the streamwise direction. Within the interior of the envelope the spatial coherence between the velocity fields induced by the individual vortices leads to strongly retarded streamwise momentum, explaining the zones of uniform momentum observed by Meinhart & Adrian (1995). The packets are an important type of organized structure in the wall layer in which relatively small structural units in the form of three-dimensional vortical structures are arranged coherently, i.e. with correlated spatial relationships, to form much longer structures. The formation of packets explains the occurrence of multiple VITA events in turbulent ‘bursts’, and the creation of Townsend's (1958) large-scale inactive motions. These packets share many features of the hairpin models proposed by Smith (1984) and co-workers for the near-wall layer, and by Bandyopadhyay (1980), but they are shown to occur in a hierarchy of scales across most of the boundary layer.

In the logarithmic layer, the coherent vortex packets that originate close to the wall frequently occur within larger, faster moving zones of uniform momentum, which may extend up to the middle of the boundary layer. These larger zones are the induced interior flow of older packets of coherent hairpin vortices that originate upstream and over-run the younger, more recently generated packets. The occurence of small hairpin packets in the environment of larger hairpin packets is a prominent feature of the logarithmic layer. With increasing Reynolds number, the number of hairpins in a packet increases.

We present the results of experimental study of a Couette–Taylor system with superimposed axial flow and an upper free surface, in the high Taylor number regime. At large Taylor numbers, when the rotational speed of the inner cylinder increases, bubbles created near the free surface are distributed throughout the test section and permit the study of the spatial and temporal properties of turbulent flows using visualization techniques. In addition to classic travelling Taylor vortices, intermittent pulses of vortices with higher phase velocities are also observed. These patterns are described in terms of the rotational speed and the intensity of the throughflow.

Consider Stokes flow in the semi-infinite wedge bounded by the sidewalls ϕ = ±α and the endwall z = 0. Viscous fluid fills the region 0 < r < ∞, 0 < z < ∞ bounded by these planes; the motion of the fluid is driven by boundary data given on the endwall z = 0. A consequence of the linearity of the problem is that one can treat the velocity field q(r, ϕ, z) as the sum of a field qa(r, ϕ, z) antisymmetric in ϕ and one symmetric in it, qs(r, ϕ, z). It is shown in each of these cases that there exists a real vector eigenfunction sequence vn(r, ϕ, z) and a complex vector eigenfunction sequence un(r, ϕ, z), each member of which satisfies the sidewall no-slip condition and has a z-behaviour of the form e−kz. It is then shown that one can, for each case, write down a formal representation for the velocity field as an infinite integral over k of the sums of the real and complex eigenfunctions, each multiplied by unknown real and complex scalar functions bn(k) and an(k), respectively, that have to be determined from the endwall boundary conditions. A method of doing this using Laguerre functions and least squares is developed. Flow fields deduced by this method for given boundary data show interesting vortical structures. Assuming that the set of eigenfunctions is complete and that the relevant series are convergent and that they converge to the boundary data, it is shown that, in general, there is an infinite sequence of corner eddies in the neighbourhood of the edge r = 0 in the antisymmetric case but not in the symmetric case. The same conclusion was reached earlier for the infinite wedge by Sano & Hasimoto (1980) and Moffatt & Mak (1999). A difficulty in the symmetric case when 2α = π/2, caused by the merger of two real eigenfunctions, has yet to be resolved.

A methodology for deriving a pressure–strain correlation model with variable coefficients is developed. The methodology is based on two important premises: (i) the extreme states of turbulence – the rapid distortion and equilibrium limits – are more amenable to mathematically rigorous modelling because of significant simplifications not possible at other states; and (ii) the models of the extreme states collectively contain all of the relevant physics so that models for any intermediate state can be obtained by suitable interpolation. A pressure–strain model of the standard form is considered and the coefficients are determined from linear analysis in the rapid distortion limit and from a fixed point analysis in the equilibrium limit. The model coefficients, which depend on the mean deformation and turbulence state, vary from flow to flow in a manner consistent with Navier–Stokes physics.

The exact causal relationship between the model coefficients and the model's equilibrium behaviour is established by fixed point analysis performed using representation theory. Then, the equilibrium values of the model coefficients are chosen to yield the observed equilibrium behaviour. The values of the model coefficients in the rapid distortion limit are determined by enforcing consistency with the Crow constraint. The new variable-coefficient model reduces to the traditional constant-coefficient model in strain-dominated turbulent flows near equilibrium. The model performance in bench-mark turbulent flows, in which the traditional models have been calibrated extensively, is preserved intact. The new model is significantly different from the traditional one in mean vorticity-dominated and non-equilibrium turbulence. These two important classes of flows, in which traditional models fail, are successfully captured by the new model.

We have generalized the exact solution of the Riemann problem in special relativistic hydrodynamics (Martí & Müller 1994) for arbitrary tangential flow velocities. The solution is obtained by solving the jump conditions across shocks plus an ordinary differential equation arising from the self-similarity condition along rarefaction waves, in a similar way as in purely normal flow. The dependence of the solution on the tangential velocities is analysed, and the impact of this result on the development of multi-dimensional relativistic hydrodynamic codes (of Godunov type) is discussed.

A comprehensive numerical study on the linear stability of mixed-convection flow in a vertical pipe with constant heat flux is presented with particular emphasis on the instability mechanism and the Prandtl number effect. Three Prandtl numbers representative of different regimes in the Prandtl number spectrum are employed to simulate the stability characteristics of liquid mercury, water and oil. The results suggest that mixed-convection flow in a vertical pipe can become unstable at low Reynolds number and Rayleigh numbers irrespective of the Prandtl number, in contrast to the isothermal case. For water, the calculation predicts critical Rayleigh numbers of 80 and −120 for assisted and opposed flows, which agree very well with experimental values of Rac = 76 and −118 (Scheele & Hanratty 1962). It is found that the first azimuthal mode is always the most unstable, which also agrees with the experimental observation that the unstable pattern is a double spiral flow. Scheele & Hanratty's speculation that the instability in assisted and opposed flows can be attributed to the appearance of inflection points and separation is true only for fluids with O(1) Prandtl number. Our study on the effect of the Prandtl number discloses that it plays an active role in buoyancy-assisted flow and is an indication of the viability of kinematic or thermal disturbances. It profoundly affects the stability of assisted flow and changes the instability mechanism as well. For assisted flow with Prandtl numbers less than 0.3, the thermal–shear instability is dominant. With Prandtl numbers higher than 0.3, the assisted-thermal–buoyant instability becomes responsible. In buoyancy-opposed flow, the effect of the Prandtl number is less significant since the flow is unstably stratified. There are three distinct instability mechanisms at work independent of the Prandtl number. The Rayleigh–Taylor instability is operative when the Reynolds number is extremely low. The opposed-thermal–buoyant instability takes over when the Reynolds number becomes higher. A still higher Reynolds number eventually leads the thermal–shear instability to dominate. While the thermal–buoyant instability is present in both assisted and opposed flows, the mechanism by which it destabilizes the flow is completely different.

The rapid-distortion model of Hunt & Graham (1978) for the initial distortion of turbulence by a flat boundary is extended to account fully for viscous processes. Two types of boundary are considered: a solid wall and a free surface. The model is shown to be formally valid provided two conditions are satisfied. The first condition is that time is short compared with the decorrelation time of the energy-containing eddies, so that nonlinear processes can be neglected. The second condition is that the viscous layer near the boundary, where tangential motions adjust to the boundary condition, is thin compared with the scales of the smallest eddies. The viscous layer can then be treated using thin-boundary-layer methods. Given these conditions, the distorted turbulence near the boundary is related to the undistorted turbulence, and thence profiles of turbulence dissipation rate near the two types of boundary are calculated and shown to agree extremely well with profiles obtained by Perot & Moin (1993) by direct numerical simulation. The dissipation rates are higher near a solid wall than in the bulk of the flow because the no-slip boundary condition leads to large velocity gradients across the viscous layer. In contrast, the weaker constraint of no stress at a free surface leads to the dissipation rate close to a free surface actually being smaller than in the bulk of the flow. This explains why tangential velocity fluctuations parallel to a free surface are so large. In addition we show that it is the adjustment of the large energy-containing eddies across the viscous layer that controls the dissipation rate, which explains why rapid-distortion theory can give quantitatively accurate values for the dissipation rate. We also find that the dissipation rate obtained from the model evaluated at the time when the model is expected to fail actually yields useful estimates of the dissipation obtained from the direct numerical simulation at times when the nonlinear processes are significant. We conclude that the main role of nonlinear processes is to arrest growth by linear processes of the viscous layer after about one large-eddy turnover time.

We present a numerical solution of the Euler equations of gas dynamics for a weak-shock Mach reflection in a half-space. In our numerical solutions, the incident, reflected, and Mach shocks meet at a triple point, and there is a supersonic patch behind the triple point, as proposed by Guderley. A theoretical analysis supports the existence of an expansion fan at the triple point, in addition to the three shocks. This solution is in complete agreement with the numerical solution of the unsteady transonic small-disturbance equations obtained by Hunter & Brio (2000), which provides an asymptotic description of a weak-shock Mach reflection. The supersonic patch is extremely small, and this work is the first time it has been resolved in a numerical solution of the Euler equations. The numerical solution uses six levels of grid refinement around the triple point. A delicate combination of numerical techniques is required to minimize both the effects of numerical diffusion and the generation of numerical oscillations at grid interfaces and shocks.

The particle tracking (PT) technique is used to study turbulent diffusion of particle pairs in a three-dimensional turbulent flow generated by two oscillating grids. The experimental data show a range where the Richardson–Obukhov law 〈r2〉 = Cεt3 is satisfied, and the Richardson–Obukhov constant is found to be C = 0.5. A number of models predict much larger values. Furthermore, the distance–neighbour function is studied in detail in order to determine its general shape. The results are compared with the predictions of three models: Richardson (1926), Batchelor (1952) and Kraichnan (1966a). These three models predict different behaviours of the distance–neighbour function, and of the three, only Richardson's model is found to be consistent with the measurements. We have corrected a minor error in Kraichnan's (1996a) Lagrangian history direct interaction calculations with the result that we had to increase his theoretical value from C = 2.42 to C = 5.5.

A non-isothermal viscoplastic thin-layer theory is developed to explore the effects of surface cooling, yield stress, and shear thinning on the evolution of non-isothermal domes of lava and laboratory fluids. The fluid is modelled using the Herschel–Bulkley constitutive relations, but modified to have temperature-dependent viscosity and yield stress. The thin-layer equations are solved numerically to furnish models of expanding, axisymmetrical domes. Linear stability theory reveals the possibility of non-axisymmetrical, fingering-like instability in these domes. Finally, the relevance to lava and experiments is discussed.

The evolution of an intense barotropic vortex on the β-plane is analysed for the case of finite Rossby deformation radius. The analysis takes into account conservation of vortex energy and enstrophy, as well as some other quantities, and therefore makes it possible to gain insight into the vortex evolution for longer times than was done in previous studies on this subject. Three characteristic scales play an important role in the evolution: the advective time scale Ta (a typical time required for a fluid particle to move a distance of the order of the vortex size), the wave time scale Tw (the typical time it takes for the vortex to move through its own radius), and the distortion time scale Td (a typical time required for the change in relative vorticity of the vortex to become of the order of the relative vorticity itself). For an intense vortex these scales are well separated, Ta [Lt ] Tw [Lt ] Td, and therefore one can consider the vortex evolution as consisting of three different stages. The first one, t [les ] Tw, is dominated by the development of a near-field dipolar circulation (primary β-gyres) accelerating the vortex. During the second stage, Tw [les ] t [les ] Td, the quadrupole and secondary axisymmetric components are intensified; the vortex decelerates. During the last, third, stage the vortex decays and is destroyed. Our main attention is focused on exploration of the second stage, which has been studied much less than the first stage. To describe the second stage we develop an asymptotic theory for an intense vortex with initially piecewise-constant relative vorticity. The theory allows the calculation of the quadrupole and axisymmetric corrections, and the correction to the vortex translation speed. Using the conservation laws we estimate that the vortex lifetime is directly proportional to the vortex streamfunction amplitude and inversely proportional to the squared group velocity of Rossby waves. For open-ocean eddies a typical lifetime is about 130 days, and for oceanic rings up to 650 days. Analysis of the residual produced by the asymptotic solution explains why this solution is a good approximation for times much longer than the expected formal range of applicability. All our analytical results are in a good qualitative agreement with several numerical experiments carried out for various vortices.

A generalized lubrication theory that is applicable to highly deformable porous layers is developed using an effective-medium approach (Brinkman equation). This theory is valid in the limit where the structure is so compressible that the normal forces generated by elastic compression of the fibres comprising the solid phase are negligible compared to the pressure forces generated within the porous layer. We assume that the deformation of the solid phase is primarily due to boundary compression as opposed to the motion of the fluid phase. A generalized Reynolds equation is derived in which the spatial variation of the Darcy permeability parameter, α = H/√Kp, due to the matrix compression is determined by new local hydrodynamic solutions for the flow through a simplified periodic fibre model for the deformed matrix. Here H is the undeformed layer thickness and Kp the Darcy permeability. This simplified model assumes that the fibres compress linearly with the deformed gap height in the vertical direction, but the fibre spacing in the horizontal plane remains unchanged. The model is thus able to capture the essential nonlinearity that results from large-amplitude deformations of the matrix layer.

The new theory shows that there is an unexpected striking similarity between the gliding motion of a red cell moving over the endothelial glycocalyx that lines our microvessels and a human skier or snowboarder skiing on compressed powder. In both cases one observes an order-of-magnitude compression of the matrix layer when the motion is arrested and predicts values of α that are of order 100. In this large-α limit one finds that the pressure and lift forces generated within the compressed matrix are four orders-of-magnitude greater than classical lubrication theory. In the case of the red cell these repulsive forces may explain why red cells do not experience constant adhesive molecular interactions with the endothelial plasmalemma, whereas in the case of the skier or snowboarder the theory explains why a 70 kg human can glide through compressed powder without sinking to the base as would occur if the motion is arrested. The principal difference between the tightly fitting red cell and the snowboarder is the lateral leakage of the excess pressure at the edges of the snowboard which greatly diminishes the lift force. A simplified axisymmetric model is presented for the red cell to explain the striking pop out phenomenon in which a red cell that starts from rest will quickly lift off the surface and then glide near the edge of the glycocalyx and also for the unexpectedly large apparent viscosity measured by Pries et al. (1994) in vivo.

Despite extensive study, there remain significant questions about the Reynolds-number scaling of the zero-pressure-gradient flat-plate turbulent boundary layer. While the mean flow is generally accepted to follow the law of the wall, there is little consensus about the scaling of the Reynolds normal stresses, except that there are Reynolds-number effects even very close to the wall. Using a low-speed, high-Reynolds-number facility and a high-resolution laser-Doppler anemometer, we have measured Reynolds stresses for a flat-plate turbulent boundary layer from Reθ = 1430 to 31 000. Profiles of u′2, v′2, and u′v′ show reasonably good collapse with Reynolds number: u′2 in a new scaling, and v′2 and u′v′ in classic inner scaling. The log law provides a reasonably accurate universal profile for the mean velocity in the inner region.