The linear-eddy approach for modelling molecular mixing in turbulent flow involves stochastic simulation on a one-dimensional domain with sufficient resolution to include all physically relevant lengthscales. In each realization, molecular diffusion is implemented deterministically, punctuated by a sequence of instantaneous, statistically independent ‘rearrangement events’ (measure-preserving maps) representing turbulent stirring. These events emulate the effect of compressive strain on the scalar field. An inertial-range similarity law is incorporated.

The model reproduces key features of scalar power spectra, including dependences of spectra! amplitudes and transition wavenumbers on Reynolds and Schmidt numbers. Computed scaling exponents governing scalar power spectra, higher-order fluctuation statistics such as structure functions, and the spatial distribution of scalar level crossings are close to measured exponents. It is inferred that the characterization of stirring as a sequence of independent events (the model analogue of eddies) leads to a useful representation of mixing-field microstructure.

Particle image velocimetry experiments have been performed in a turbulent boundary-layer wind tunnel in order to study the coherent structures taking part in the generation and preservation of wall turbulence. The particular wind tunnel used is suitable for high-resolution experiments ($\delta \gt 0.3$ m) at high Reynolds numbers (up to $R_{\theta} = 19\,000$ in the present results). Eddy structures were identified in instantaneous velocity maps in order to determine their mean characteristics and possible relationships between these structures. In the logarithmic region, the results show that the observed eddy structures appear to organize like elongated vortices, tilted downstream, mainly at an angle of about 45° and having a cane shape. The characteristics of these vortices appear here to be universal in wall units for $R_{\theta}\,{\leq}\,19\,000$. They seem to find their origin at a wall distance of about 25 wall units as quasi-streamwise vortices and to migrate away from the wall while tilting to form a head and a leg. Away from the wall, their radius increases and their vorticity decreases very slowly so that their circulation is nearly constant. Near the wall, the picture obtained is in fair agreement with existing models. The analysis of the results indicates a universality of the buffer-layer mechanism, even at low Reynolds number, and a sensitivity of the logarithmic region to low-Reynolds-number effects.

Since the availability of data from direct numerical simulation (DNS) of turbulence, researchers have utilized the joint PDFs of invariants of the velocity gradient tensor to study the geometry of small-scale motions of turbulence. However, the joint PDFs only give an instantaneous static representation of the properties of fluid particles and dynamical Lagrangian information cannot be extracted. In this paper, the Lagrangian evolution of the invariants of the velocity gradient tensor is studied using conditional mean trajectories (CMT). These CMT are derived using the concept of the conditional mean time rate of change of invariants calculated from a numerical simulation of isotropic turbulence. The study of the CMT in the invariant space (RA, QA) of the velocity-gradient tensor, invariant space (RS, QS) of the rate-of-strain tensor, and invariant space (RW, QW) of the rate-of-rotation tensor show that the mean evolution in the (Σ, QW) phase plane, where Σ is the vortex stretching, is cyclic with a characteristic period similar to that found by Martin et al. (1998) in the cyclic mean evolution of the CMT in the (RA, QA) phase plane. Conditional mean trajectories in the (Σ, QW) phase plane suggest that the initial reduction of QW in regions of high QW is due to viscous diffusion and that vorticity contraction only plays a secondary role subsequent to this initial decay. It is also found that in regions of the flow with small values of QW, the local values of QW do not begin to increase, even in the presence of self-stretching, until a certain self-stretching rate threshold is reached, i.e. when Σ≈0.25 〈QW〉1/2. This study also shows that in regions where the kinematic vorticity number (as defined by Truesdell 1954) is low, the local value of dissipation tends to increase in the mean as observed from a Lagrangian frame of reference. However, in regions where the kinematic vorticity number is high, the local value of enstrophy tends to decrease. From the CMT in the (−QS, RS phase plane, it is also deduced that for large values of dissipation, there is a tendency for fluid particles to evolve towards having a positive local value of the intermediate principal rate of strain.

Data from the near-wall-turbulent region of the high-Reynolds-number atmospheric surface layer are used to analyse the attached-eddy model of wall turbulence. All data were acquired during near-neutral conditions at the Surface Layer Turbulence and Environmental Science Test (SLTEST) facility located in the western Utah Great Salt Lake Desert. Instantaneous streamwise and wall-normal components of velocity were collected with a wall-normal array of two-component hot wires within the first 2 m above the surface of the salt flats. Streamwise and wall-normal turbulence intensities and spectra are directly compared to corresponding laboratory data and similarity formulations hypothesized from the attached-eddy model of wall turbulence. This affords the opportunity to compare results with Reynolds numbers varying over three orders of magnitude. The wall-normal turbulence-intensity similarity formulation is extended. The results show good support for the similarity arguments forwarded by the attached-eddy model as well as Townsend's (1956) Reynolds-number similarity hypothesis and lack of the ‘inactive’ motion influence on the wall-normal velocity component. The effects of wall roughness and the spread in the convection velocity due to this roughness are also discussed.

Laboratory experiments have been conducted to study the shoaling of internal solitary waves of depression in a two-layer system on a uniform slope. The shoaling of a single solitary wave results in wave breaking and the production of multiple turbulent surges, or boluses, which propagate up the slope. Significant vertical mixing occurs everywhere inshore of the breaking location. The kinematics of the breaking and bolus runup are described and a breaking criterion is found. The energetics of the breaking are investigated. Over the range of parameters examined, 15 (±5) % of the energy lost from first-mode wave motion inshore of the break point goes into vertical mixing.

A two-dimensional stability analysis is presented of flow of low Froude number over an erodible bed. Particular regard is given to the modelling of the turbulent flow close to the bed. In contrast to previous theories that use a constant eddy-viscosity approach the present theory predicts the occurrence of two separate modes of instability, with wavelengths related to the roughness of the bed and the depth of the flow. It is postulated that these two modes correspond to the formation of ripples and dunes respectively. The results are strongly dependent on the two parameters z0, the roughness length of the bed, and β, the effect of the local bed slope on the bed-load transport. Using physically plausible estimates for these parameters the results of the analysis are in good agreement with observations for both ripples and dunes.

We analyse the dilute, steady, fully developed flow of relatively massive particles in a turbulent gas in the context of a vertical pipe. The idea is that the exchange of momentum in collisions between the grains and between the grains and the wall plays a significant role in the balance of forces in the particle phase. Consequently, the particle phase is considered to be a dilute system of colliding grains, in which the velocity fluctuations are produced by collisions rather than by the gas turbulence. The balance equations for rapid granular flow are modified to incorporate the drag force from the gas, and boundary conditions, based on collisional exchanges of momentum and energy at the wall, are employed. The turbulence of the gas is treated using a one-equation closure. A numerical solution of the resulting governing equations provides velocity and turbulent energy profiles in agreement with the measurements of Tsuji et al. (1984).

Upon review of past experimental results and theoretical efforts it is apparent that the mechanism by which combustion noise is generated is not well understood. A theory of combustion noise is developed in this paper which follows rigorously from the principles of fluid mechanics. Lighthill's approach, used in his studies of aerodynamic noise, is closely followed in the present work. The sound radiated from open, turbulent flames is found to depend strongly upon the structure of such flames; at present their structure is not well known. However, meaningful bounds and scaling rules for the sound power output and spectral content are derived based upon the present limited knowledge. A framework is developed which explains past experimental work and the origin of combustion noise.

We perform direct numerical simulations of a rigid sphere translating parallel to a flat wall in an otherwise quiescent ambient fluid. A spectral element method is employed to perform the simulations with high accuracy. For $Re\,{<}\,100$, we observe the lift coefficient to decrease with both Reynolds number and distance from the wall. In this regime the present results are in good agreement with the low-Reynolds-number theory of Vasseur & Cox (1977), with the recent experiments of Takemura & Magnaudet (2003) and with the simulations of Kim et al. (1993). The most surprising result from the present simulations is that the wall-induced lift coefficient increases dramatically with increasing $Re$ above about 100. Detailed analysis of the flow field around the sphere suggests that this increase is due to an imperfect bifurcation resulting in the formation of a double-threaded wake vortical structure. In addition to a non-rotating sphere, we also simulate a freely rotating sphere in order to assess the importance of free rotation on the translational motion of the sphere. We observe the effect of sphere rotation on lift and drag forces to be small. We also explore the effect of the wall on the onset of unsteadiness.

The problem of subgrid modelling, that is, of representing energy transfers from large to small eddies in terms of the large eddies only, must arise in any large eddy simulation, whether the equations of motion are open or direct (unaveraged) or closed (averaged). Models for closed calculations are derived from classical closures, and these are used to determine the effect of filter shape, grid-scale spectrum and grid-scale anisotropy on the effective eddy viscosity: the Leonard or resolvable-scale stress is calculated separately and is found to account for 14% of the total drain in a typical high Reynolds number case.

The validity of using these eddy viscosities in an open calculation is considered. It is concluded that this is not unreasonable, but that the simulation would be much improved if the gross drain could be separated into net drain and backscatter.

Fast liquid jets, called micro-jets, are produced within cavitation bubbles experiencing an aspherical collapse. Here we review micro-jets of different origins, scales and appearances, and propose a unified framework to describe their dynamics by using an anisotropy parameter $\unicode[STIX]{x1D701}\geqslant 0$, representing a dimensionless measure of the liquid momentum at the collapse point (Kelvin impulse). This parameter is rigorously defined for various jet drivers, including gravity and nearby boundaries. Combining theoretical considerations with hundreds of high-speed visualisations of bubbles collapsing near a rigid surface, near a free surface or in variable gravity, we classify the jets into three distinct regimes: weak, intermediate and strong. Weak jets ($\unicode[STIX]{x1D701}<10^{-3}$) hardly pierce the bubble, but remain within it throughout the collapse and rebound. Intermediate jets ($10^{-3}<\unicode[STIX]{x1D701}<0.1$) pierce the opposite bubble wall close to the last collapse phase and clearly emerge during the rebound. Strong jets ($\unicode[STIX]{x1D701}>0.1$) pierce the bubble early during the collapse. The dynamics of the jets is analysed through key observables, such as the jet impact time, jet speed, bubble displacement, bubble volume at jet impact and vapour-jet volume. We find that, upon normalising these observables to dimensionless jet parameters, they all reduce to straightforward functions of $\unicode[STIX]{x1D701}$, which we can reproduce numerically using potential flow theory. An interesting consequence of this result is that a measurement of a single observable, such as the bubble displacement, suffices to estimate any other parameter, such as the jet speed. Remarkably, the dimensionless parameters of intermediate and weak jets ($\unicode[STIX]{x1D701}<0.1$) depend only on $\unicode[STIX]{x1D701}$, not on the jet driver (i.e. gravity or boundaries). In the same regime, the jet parameters are found to be well approximated by power laws of $\unicode[STIX]{x1D701}$, which we explain through analytical arguments.

The average thickness of the wetting film left behind during the slow passage of an air bubble in a water-filled capillary tube of circular cross-section has been determined experimentally as a function of bubble speed and bubble length. For bubbles of length many times the tube radius, the ratio of film thickness to tube radius is found to be a function of the capillary number only, in agreement with previous experimental studies. As has been found previously, the asymptotic result of Bretherton (1961) significantly underpredicts the film thickness, the discrepancy being greatest at the lowest speeds. For bubbles of length less than about 20 tube radii, on the other hand, good agreement with the Bretherton theory is obtained over two orders of magnitude in bubble speed. The theoretical profile of long bubbles is shown to be unstable; however the explanation of the observed behaviour is, as yet, incomplete.

To study the waves and runup/rundown generated by a sliding mass, a numerical simulation model, based on the large-eddy-simulation (LES) approach, was developed. The Smagorinsky subgrid scale model was employed to provide turbulence dissipation and the volume of fluid (VOF) method was used to track the free surface and shoreline movements. A numerical algorithm for describing the motion of the sliding mass was also implemented.

To validate the numerical model, we conducted a set of large-scale experiments in a wave tank of 104m long, 3.7m wide and 4.6m deep with a plane slope (1:2) located at one end of the tank. A freely sliding wedge with two orientations and a hemisphere were used to represent landslides. Their initial positions ranged from totally aerial to fully submerged, and the slide mass was also varied over a wide range. The slides were instrumented to provide position and velocity time histories. The time-histories of water surface and the runup at a number of locations were measured.

Comparisons between the numerical results and experimental data are presented only for wedge shape slides. Very good agreement is shown for the time histories of runup and generated waves. The detailed three-dimensional complex flow patterns, free surface and shoreline deformations are further illustrated by the numerical results. The maximum runup heights are presented as a function of the initial elevation and the specific weight of the slide. The effects of the wave tank width on the maximum runup are also discussed.

The dynamical properties of a fluid, occupying the space between two concentric rotating spheres, are considered, attention being focused on the case where the angular velocities of the spheres are only slightly different and the Reynolds number R of the flow is large. It is found that the flow properties differ inside and outside a cylinder [Cscr ], circumscribing the inner sphere and having its generators parallel to the axis of rotation. Outside [Cscr ] the fluid rotates as if rigid with the angular velocity of the outer sphere. Inside [Cscr ] the fluid rotates with an angular velocity intermediate to the angular velocities of the two spheres and determined by the condition that the flux of fluid into the boundary layer of the faster-rotating sphere is equal to the flux out of the boundary layer of the slower-rotating sphere at the same distance from the axis. The return of fluid is effected by a shear layer near [Cscr ] and we show that it has a complicated structure for it can be divided into three separate layers, two outer ones, of thickness $\sim R^{-\frac{2}{7}}$ and ∼R−¼, and an inner layer of thickness $\sim R^{-\frac{1}{3}}$.

This work uses direct numerical simulations of time evolving annular mixing layers, which correspond to the early development of round jets, to study compressibility effects on turbulence in free shear flows. Nine cases were considered with convective Mach numbers ranging from Mc = 0.1 to 1.8 and turbulence Mach numbers reaching as high as Mt = 0.8.

Growth rates of the simulated mixing layers are suppressed with increasing Mach number as observed experimentally. Also in accord with experiments, the mean velocity difference across the layer is found to be inadequate for scaling most turbulence statistics. An alternative scaling based on the mean velocity difference across a typical large eddy, whose dimension is determined by two-point spatial correlations, is proposed and validated. Analysis of the budget of the streamwise component of Reynolds stress shows how the new scaling is linked to the observed growth rate suppression. Dilatational contributions to the budget of turbulent kinetic energy are found to increase rapidly with Mach number, but remain small even at Mc = 1.8 despite the fact that shocklets are found at high Mach numbers. Flow visualizations show that at low Mach numbers the mixing region is dominated by large azimuthally correlated rollers whereas at high Mach numbers the flow is dominated by small streamwise oriented structures. An acoustic timescale limitation for supersonically deforming eddies is found to be consistent with the observations and scalings and is offered as a possible explanation for the decrease in transverse lengthscale.

Experimental measurements of the advancing head of a gravity current are described. The head, of depth d in mean cross-section and advancing at mean speed U, has an elevated nose whose mean height h satisfies h/d = 0·61 Re-0·23±0·01 through the range 300 < Re < 10000 (Re = Ud/ν). A shifting pattern of lobes and clefts ranges across the head, in which the mean lobe width $\overline{b}$ is found experimentally to satisfy $\overline{b}/d = 7.4\,Re^{-0.39\pm 0.02}$. By moving the floor in the direction of the current, or by introducing a thin dense layer beneath it, the lobes and clefts can be suppressed and a purely two-dimensional flow obtained. It is concluded that the lower boundary plays an essential role in determining the substructure of the head and that this originates in convective instability as the head rides over less dense fluid.

This paper is a study of the flow of a viscous incompressible fluid in the immediate neighbourhood of a saddle point of attachment, near which the external flow is irrotational with components {ax, by, −(a + b)z}, where a > 0 > b. It is shown that the flow is of a boundary-layer character, and that part of the boundary-layer flow is reversed when b/a < − 0·4294.

On the assumption that such flows are physically plausible, the problem may be solved for all values of b/a [ges ] −1. Even in the limiting case b/a = −1, an effect of the boundary layer is everywhere to draw fluid towards the wall, so that vorticity is still convected towards the wall.

Numerical solutions have been computed, and some of the results are presented in the tables and diagrams.

The flow rate of liquid metals is commonly measured by electromagnetic flowmeters. In these the fluid moves through a region of transverse magnetic field, inducing a potential difference between two electrodes on the walls of the pipe. The ratio of signal to flow rate is dependent on the velocity profile, and this is affected by electromagnetic forces.

In this paper the ultimate steady velocity profile and its associated pressure gradient and induced potential are calculated for the case of laminar flow in a circular pipe whose walls are conducting but without contact resistance. Laminar flow is encouraged by a transverse field. When the fluid conductivity and field strength are sufficiently high, boundary layers occur with a thickness inversely proportional to normal field intensity. The induced potential difference is then 0·926 of the value corresponding to the case of uniform velocity if the walls are non-conducting.

The distance the fluid must travel after entering the transverse field before the steady state is reached is next estimated by a Rayleigh approximation. The inlet velocity is taken to be uniform and effects which occur at the edge of the field are neglected. The process falls into two stages, first a boundary-layer growth and then an adjustment of the velocity away from the walls, occupying a much greater length of pipe. The entry length is shorter than it is in the case of flow in a rectangular pipe, but is still too long for appreciable distortion of the velocity profile to occur within practical flowmeters except at low flow rates. The pressure drop associated with the adjustment of the velocity profile is found to be independent of field strength, if this is high, and about one-eighth of the drop which occurs in the non-conducting case.

Experiments are described in which steady-state pressure gradients and induced potential differences were measured in mercury flowing along Perspex pipes of 0·5 and 0·25 in. bore in transverse fields up to 14500 gauss. The results confirmed the steady-state theory within the limitations of experimental accuracy and the assumption in the theory of high conductivity and an intense field. The experiments also covered the entry region in many cases, and showed that the theoretical entry lengths were correct in order of magnitude but over-estimated. However, the exact entry condition was uncertain, and steady readings were difficult to obtain in the entry region.

Frost heave is the process by which the freezing of water-saturated soil causes the deformation and upward thrust of the ground surface. We describe the fundamental interactions between phase change and fluid flow in partially frozen, saturated porous media (soils) that are responsible for frost heave. Water remains only partially frozen in a porous medium at temperatures below $0\,^\circ$C owing both to the depression of the freezing temperature at curved phase boundaries and to interfacial premelting caused by long-range intermolecular forces. We show that while the former contributes to the geometry of fluid pathways, it is solely the latter effect that generates the forces necessary for frost heave. We develop a simple model describing the formation and evolution of the ice lenses (layers of ice devoid of soil particles) that drive heave, based on integral force balances. We determine conditions under which either (i) a single ice lens propagates with no leading frozen fringe, or (ii) a single, propagating ice lens is separated from unfrozen soil by a partially frozen fringe, or (iii) multiple ice lenses form.

A model problem is analysed to study the microscopic flow near the surface of two-dimensional porous media. In the idealized problem we consider axial flow through infinite and semi-infinite lattices of cylindrical inclusions. The effect of lattice geometry and inclusion shape on the permeability and surface flow are examined. Calculations show that the definition of a slip coefficient for a porous medium is meaningful only for extremely dilute systems. Brinkman's equation gives reasonable predictions for the rate of decay of the mean velocity for certain simple geometries, but fails for to predict the correct behaviour for media anisotropic in the plane normal to the flow direction.