We present wall-modelled large-eddy simulations (WLES) of oblique shock waves interacting with the turbulent boundary layers (TBLs) (nominal $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\delta _{99}=5.4\ \mathrm{mm}$ and ${\mathit{Re}}_{\theta }\approx 1.4\times 10^4$) developed inside a duct with an almost-square cross-section ($45\ \mathrm{mm}\times 47.5\ \mathrm{mm}$) to investigate three-dimensional effects imposed by the lateral confinement of the flow. Three increasing strengths of the incident shock are considered, for a constant Mach number of the incoming air stream $M\approx 2$, by varying the height (1.1, 3 and 5 mm) of a compression wedge located at a constant streamwise location that spans the top wall of the duct at a 20° angle. Simulation results are first validated with particle image velocimetry (PIV) experimental data obtained at several vertical planes (one near the centre of the duct and three near one of the sidewalls) for the 1.1 and 3 mm-high wedge cases. The instantaneous and time-averaged structure of the flow for the stronger-interaction case (5 mm-high wedge), which shows mean flow reversal, is then investigated. Additional spanwise-periodic simulations are performed to elucidate the influence of the sidewalls, and it is found that the structure and location of the shock system, as well as the size of the separation bubble, are significantly modified by the lateral confinement. A Mach stem at the first reflected interaction is present in the simulation with sidewalls, whereas a regular shock intersection results for the spanwise-periodic case. Low-frequency unsteadiness is observed in all interactions, being stronger for the secondary shock reflections of the shock train developed inside the duct. The downstream evolution of secondary turbulent flows developed near the corners of the duct as they traverse the shock system is also studied.

The early phase of the establishment of the flow past a circular cylinder started impulsively into rotation and translation is investigated by visualizing the flow patterns with solid tracers and by analysing qualitatively (flow topology) and quantitatively (velocity distributions and singular-point trajectories) the corresponding photographs. The range considered corresponds to moderate Reynolds numbers (Re [les ] 1000). The rotating-to-translating-speed ratio α increases from 0 to 3.25 and the motion covers a period during which the cylinder translates 4.5 or even 7 times its diameter. The details of the mechanisms of the near-wake formation are considered in particular and the increase of the flow asymmetry with increase in rotation is pointed out. Thus the existence of two regimes has been confirmed with the creation or non-creation of alternate eddies after an initial one E1 Furthermore, the new phenomena of saddle-point transposition and intermediate-eddy coalescence have been identified in the formation or shedding of respectively the odd and even subsequent eddies Ei (i = 2,3,…) when they exist. The very good agreement between these experimental data and the numerical results of Badr & Dennis (1985), obtained by solving the Navier-Stokes equations and presented in a parallel paper, confirms their respective validity and permits the determination of the flow characteristics not accessible, or accessible only with difficulty, to the present experiments. These flow properties such as drag and vorticity are capable of providing information on the Magnus effect for the former property and on unsteady separated flows for the latter.

The dynamics of layer formation by salt fingers from the uniform temperature and salinity gradients is studied by direct numerical simulations of the two-dimensional Navier–Stokes equations. It is shown that formation of steps in the model is caused by the parametric variation of the flux ratio ($\gamma\,{=}\,{\overline{wT}}/{\overline{wS}}$) as a function of the density ratio ($R$), which leads to an instability of equilibrium with uniform stratification. These unstable large-scale perturbations continuously grow in time until well-defined layers are formed. Subsequent evolution of the numerical staircases is explained by considering the secondary instabilities of a series of salt finger interfaces.

The flow of a viscous fluid from a point or line source on an inclined plane is analysed using the equations of lubrication theory in which surface tension is neglected. At short times, when the gradient of the interfacial thickness is much greater than that of the plane, the fluid is shown to spread symmetrically from the source, as on a horizontal plane. At long times, the flow is predominantly downslope, with some cross-slope spreading for the case of a point source. Similarity solutions for the long-time behaviour of the governing nonlinear partial differential equations are found for the case in which the volume of fluid increases with time like tα, where α is a constant. The two-dimensional equations appropriate to a line source are hyperbolic in the self-similar regime and the similarity profile is found analytically to end abruptly at a downslope position which increases like t(2α+1)/3. Inclusion of higher-order terms in the analysis resolves this frontal shock into a boundary-layer structure of width comparable to the thickness of the current. Owing to the term representing cross-slope spreading, the mathematical structure of the equations is considerably more complex for flow from a point source and the similarity form is found numerically in this case. Though the downslope and cross-slope extents of the current again increase with time according to a power-law if α > 0, they also depend on a power of In t if α = 0. The leading-order near-source structure is shown to be that of steady flow from a constant-flux source of strength given by the instantaneous flow rate. For sources with α > 1, the contact line advances at all points on the perimeter of the flow and the entire plane is eventually covered by the flow; for sources with 0 < α < 1, only a portion of the contact line is advancing at any time and only that part of the plane with |y| [les ] cx3α/(4α+3) is eventually covered, where x and y are the downslope and cross-slope coordinates and c is a constant. The theoretical spreading relationships and planforms are found to be in good agreement with experimental measurements of constant-volume and constant-flux flows of viscous fluids from a point source on a plane. At very long times, however, the experimental flows are observed to be unstable to the formation of a capillary rivulet at the nose of the current.

We present the results of a combined experimental and theoretical investigation of millimetric droplets bouncing on a vertically vibrating fluid bath. We first characterize the system experimentally, deducing the dependence of the droplet dynamics on the system parameters, specifically the drop size, driving acceleration and driving frequency. As the driving acceleration is increased, depending on drop size, we observe the transition from coalescing to vibrating or bouncing states, then period-doubling events that may culminate in either walking drops or chaotic bouncing states. The drop’s vertical dynamics depends critically on the ratio of the forcing frequency to the drop’s natural oscillation frequency. For example, when the data describing the coalescence–bouncing threshold and period-doubling thresholds are described in terms of this ratio, they collapse onto a single curve. We observe and rationalize the coexistence of two non-coalescing states, bouncing and vibrating, for identical system parameters. In the former state, the contact time is prescribed by the drop dynamics; in the latter, by the driving frequency. The bouncing states are described by theoretical models of increasing complexity whose predictions are tested against experimental data. We first model the drop–bath interaction in terms of a linear spring, then develop a logarithmic spring model that better captures the drop dynamics over a wider range of parameter space. While the linear spring model provides a faster, less accurate option, the logarithmic spring model is found to be more accurate and consistent with all existing data.

This paper analyses the development, according to the Navier-Stokes equations, of the two-dimensional flow in the neighbourhood of the rear stagnation point on a cylinder which is set in motion impulsively with constant velocity. The local flow is idealized to the extent that the cylindrical boundary is taken to be an infinite plane bounding a semi-infinite domain of fluid. The velocity field is taken to be a linear function of the co-ordinate measured parallel to the boundary, and the initial flow is taken to be the (unique) irrotational form of such a field, namely, inviscid flow away from a stagnation point. Thereafter this irrotational flow is maintained as the outer boundary condition at a large distance from the boundary.

It is suggested that, outside a viscous layer on the boundary, the asymptotic flow for large times is described by a similarity solution of the inviscid form of the governing equation, with a length scale normal to the boundary which increases exponentially with time. This inviscid solution has a steady velocity of slip along the boundary which is equal but opposite to that of the initial flow, so that the flow in the viscous layer ultimately becomes the well-known stagnation flow towards a boundary. The suggestion is supported by a numerical solution of the initial-value problem.

Particle image velocimetry (PIV) measurements were performed in a wave tank under water waves propagating and breaking on a 1/15 sloping beach. The wave transformation occurred in the surf zone over a large domain covering several wavelengths from incipient breaking to swash zone beyond the shoreline. PIV spatial interrogation windows must be small enough to obtain accurate velocities, and one window covers only a small part of the domain. To overcome this problem and to measure the instantaneous velocity field over the whole surf zone area, we have split the full field into 14 overlapping smaller windows of the same size. Local measurements were synchronized with each other using pulsed TTL triggers and wave gauge data. The full velocity field was then reconstructed at every time step by gathering the 14 PIV fields. We then measured the complete space–time evolution of the velocity field over the whole surf zone. We determined also the ensemble-period-average and phase-average components of the flow with their associated fluctuating parts. We used the PIV images and velocity measurements to estimate the void fraction in each point of the surf zone. Special attention was given to the calculation of the spatial derivatives in order to obtain relevant information on vorticity and on the physical terms that appear in the fluctuating kinetic energy transport equation.

The nonlinear evolution of perturbed equilibrium configurations of constant-vorticity vortices is calculated. To illustrate a variety of nonlinear behaviour, we consider the following relatively simple configurations: the corotating configurations of N vortices whose linear stability has been treated in a previous study; the elliptical vortex; and the annular vortex. Our calculations test for nonlinear stability as well as categorize the possible forms of stability and instability. The energy ideas announced in the previous study are found to greatly constrain vortex evolution. In particular, we show that two vortices and an elliptical vortex may evolve into each other, and that an annular vortex may break cleanly into five co-rotating vortices.

The existence of fast dynamos caused by steady motion of an electrically conducting fluid is established by consideration of a two-dimensional spatially periodic flow: the velocity, which is independent of the vertical coordinate z, is finite and continuous everywhere but the vorticity is infinite at the X-type stagnation points. A mean-field model is developed using boundary-layer methods valid in the limit of large magnetic Reynolds number R. The magnetic field is confined to sheets, width of order R−½. The mean magnetic field lies and is uniform on horizontal planes: its direction is independent of time but rotates once about the vertical axis over a short distance 2πl, where l−1 = R½β and β is a vertical stretched wavenumber independent of R. Its alternating direction gives it a rope-like structure within the sheets. An α-effect is calculated for the model, whose strength for a given flow is a function of β and R. Two sources of α-effect are isolated whose relative importance depends critically on the size of β. When the vorticity is finite everywhere and β [Lt ] 1, the dynamo is ‘almost’ fast with growth rates of order (ln R)−1. The maximum growth rate ln (ln R)/ln R occurs when, correct to leading order, β is (ln R)−½. The asymptotic results valid for large R compare excellently with Roberts (1972) modal analysis for finite R.

This article reports an analysis of the first detailed chemistry direct numerical simulation (DNS) of a high Karlovitz number laboratory premixed flame. The DNS results are first compared with those from laser-based diagnostics with good agreement. The subsequent analysis focuses on a detailed investigation of the flame area, its local thickness and their rates of change in isosurface following reference frames, quantities that are intimately connected. The net flame stretch is demonstrated to be a small residual of large competing terms: the positive tangential strain term and the negative curvature stretch term. The latter is found to be driven by flame speed–curvature correlations and dominated in net by low probability highly curved regions. Flame thickening is demonstrated to be substantial on average, while local regions of flame thinning are also observed. The rate of change of the flame thickness (as measured by the scalar gradient magnitude) is demonstrated, analogously to flame stretch, to be a competition between straining tending to increase gradients and flame speed variations in the normal direction tending to decrease them. The flame stretch and flame thickness analyses are connected by the observation that high positive tangential strain rate regions generally correspond with low curvature regions; these regions tend to be positively stretched in net and are relatively thinner compared with other regions. High curvature magnitude regions (both positive and negative) generally correspond with lower tangential strain; these regions are in net negatively stretched and thickened substantially.

In a previous work, two-dimensional film flows were modelled using a weighted-residual approach that led to a four-equation model consistent at order $\epsilon^2$. A two-equation model resulted from a subsequent simplification but at the cost of lowering the degree of the approximation to order $\epsilon$ only. A Padé approximant technique is applied here to derive a refined two-equation model consistent at order $\epsilon^2$. This model, formulated in terms of coupled evolution equations for the film thickness $h$ and the flow rate $q$, accounts for inertia effects due to the deviations of the velocity profile from the parabolic shape, and closely follows the asymptotic long-wave expansion in the appropriate limit. Comparisons of two-dimensional wave properties with experiments and direct numerical simulations show good agreement for the range of parameters in which a two-dimensional wavy motion is reported in experiments.

The stability of two-dimensional travelling waves to three-dimensional pertur-bations is investigated based on the extension of the models to include spanwise dependence. The secondary instability is found to be not very selective, which explains the widespread presence of the synchronous instability observed in the experiments by Liu et al.(1995) whereas Floquet analysis predicts a subharmonic scenario in most cases. Three-dimensional wave patterns are computed next assuming periodic boundary conditions. Transition from two- to three-dimensional flows is shown to be strongly dependent on initial conditions. The herringbone patterns, the synchronously deformed fronts and the three-dimensional solitary waves observed in experiments are recovered using our regularized model, which is found to be an excellent compromise between the complete model, which has seven equations, and the simplified model, which does not include the second-order inertia corrections. Those corrections are found to play a role in the selection of the type of secondary instability as well as of the spanwise wavelength of the emerging pattern.

This work focuses on the direct numerical simulation of the pulsatile flow through a bileaflet mechanical heart valve under physiological conditions and in a realistic aortic root geometry. The motion of the valve leaflets has been computed from the forces exerted by the fluid on the structure both being considered as a single dynamical system. To this purpose the immersed boundary method, combined with a fluid–structure interaction algorithm, has shown to be an inexpensive and accurate technique for such complex flows. Several complete flow cycles have been simulated in order to collect enough phase-averaged statistics, and the results are in good agreement with experimental data obtained for a similar configuration. The flow analysis, strongly relying on the data accessibility provided by the numerical simulation, shows how some features of the leaflets motion depend on the flow dynamics and that the criteria for the red cell damages caused by the valve need to be formulated using very detailed analysis. In particular, it is shown that the standard Eulerian computation of the Reynolds stresses, usually employed to assess the risk of haemolysis, might not be adequate on several counts: (i) Reynolds stresses are only one part of the solicitation, the other part being the viscous stresses, (ii) the characteristic scales of the two solicitations are very different and the Reynolds stresses act on lengths much larger than the red cells diameter and (iii) the Eulerian zonal assessment of the stresses completely misses the information of time exposure to the solicitation which is a fundamental ingredient for the phenomenon of haemolysis. Accordingly, the trajectories of several fluid particles have been tracked in a Lagrangian way and the pointwise instantaneous viscous stress tensor has been computed along the paths. The tensor has been then reduced to an equivalent scalar using the von Mises criterion, and the blood damage index has been evaluated following Grigioni et al. (Biomech. Model Mechanobiol., vol. 4, 2005, p. 249).

The steady motion of a liquid drop in another liquid of comparable density and viscosity is studied theoretically. Both inside and outside the drop, the Reynolds number is taken to be large enough for boundary-layer theory to hold, but small enough for surface tension to keep the drop nearly spherical. Surface-active impurities are assumed absent. We investigate the boundary layers associated with the inviscid first approximation to the flow, which is shown to be Hill's spherical vortex inside, and potential flow outside. The boundary layers are shown to perturb the velocity field only slightly at high Reynolds numbers, and to obey linear equations which are used to find first and second approximations to the drag coefficient and the rate of internal circulation.

Drag coefficients calculated from the theory agree quite well with experimental values for liquids which satisfy the conditions of the theory. There appear to be no experimental results available to test our prediction of the internal circulation.

A control-volume analysis of a hydraulic jump is used to obtain the mean vorticity downstream of the jump as a function of the Froude number. To do this it is necessary to include the conservation of angular momentum. The mean vorticity increases from zero as the cube of Froude number minus one, and, in dimensionless form, approaches a constant at large Froude number. Digital particle imaging velocimetry was applied to travelling hydraulic jumps giving centre-plane velocity field images at a frequency of 15 Hz over a Froude number range of 2–6. The mean vorticity determined from these images confirms the control-volume prediction to within the accuracy of the experiment. The flow field measurements show that a strong shear layer is formed at the toe of the wave, and extends almost horizontally downstream, separating from the free surface at the toe. Various vorticity generation mechanisms are discussed.

Similarity rules are constructed for the load distributions induced on a thin two-dimensional wing at subsonic speeds by sinusoidal gusts whose wave fronts are at an angle to the leading edge of the wing. It is shown that these rules divide into two groups according to the value of a parameter dependent on the Mach number and the angle between the gust front and the wing. The similarity rules for each group relate all the members of the group to a simpler problem whose solution can be found by existing methods. The similarity between the two groups is also discussed in terms of the two methods of solution available and it is shown that each method of solution is applicable in all cases.

The paper considers the problem of calculating the statistical characteristics of a passive scalar dispersed by a homogeneous turbulence field. In many turbulent shear flows the time-scale for the evolution of the scalar field is intrinsically related to that of the turbulent velocity field. This is by no means always the case, however, and it is at this more general situation that the present work is aimed. An approximate transport equation for the rate of dissipation of scalar variance is proposed which, it is argued, must contain (at least) two sink terms one of which responds to the time scale of the velocity field while the other reflects that of the scalar field itself. The model has been applied to the limited number of homogeneous scalar flows for which data are available and achieves satisfactory agreement as judged by the evolution of the mean-square scalar variance.

Scaling analyses describe the evolution of an extrusion of viscous or plastic fluid in the presence of surface cooling and solidification, under the assumption that the flows consist of two components: an isothermal interior and a surface crust. The ‘crust’ is a complex thermal and rheological boundary layer which we model using a viscous, plastic or brittle rheology. These models are thought to be relevant to some types of lava flows and address the effects of cooling on their morphology and rate of advance of the flow front. They show that effects of crust strength will dominate over both viscous and yield stresses in the interior when the ratio of crust thickness to flow length exceeds the ratio of effective yield stress of the crust to basal shear stress exerted on the bulk of the flow, a condition that appears likely to be met by many lava flows and small outgrowths on large flows. Similarity solutions are compared with measurements on the spreading of extrusions of wax beneath cold water in the laboratory, where the extruded liquid is viscous but develops a solid crust. Crust strength provides the dominant retarding force for the wax flows in cases where surface solidification occurs rapidly compared with lateral advection. These conditions give flows topped by sheets of solid that buckle or rift apart, or extrusions that enlarge by small bulbous outgrowths (analogous to ‘pillows’ on submarine lavas and ‘toes’ on some sub-aerial basalt flows). A comparison of the models with data for the growth of lava domes in the craters of Mount St Helens and Soufrière of St Vincent volcanoes reveals that spreading of those domes was not controlled by stresses in the flow interior. Instead the data are consistent with a balance between gravity and yield stresses in a thin crustal layer over most of the growth period.

Experiments are performed to demonstrate the occurrence and explore the characteristics of a secondary flow superposed upon the natural convection main flow on an inclined plate. A flow visualization technique is employed whereby the flow pattern is made visible by local changes of colour of the fluid itself, the colour change being brought about by a change in pH. The secondary flow consists of longitudinal vortices or rolls distributed more or less periodically across the width of the plate. The number of such vortices increases with the temperature difference between the surface and the ambient fluid, but appears to be relatively insensitive to the inclination angle of the plate. The secondary flow results from the destabilizing effect of the buoyancy force component, which acts normal to the plate surface. The longitudinal vortices are the first stage of the laminarturbulent transition process. This is in contrast to the case of natural convection on a vertical plate, where the first stage of transition is Tollmien-Schlichting waves.

Oscillations of a cavity shear layer, involving a downstream-travelling wave and associated vortex formation, its impingement upon the cavity corner, and upstream influence of this vortex-corner interaction are the subject of this experimental investigation.

Spectral analysis of the downstream-travelling wave reveals low-frequency components having substantial amplitudes relative to that of the fundamental (instability) frequency component; using bicoherence analysis it is shown that the lowest-frequency component can interact with the fundamental either to reinforce itself or to produce an additional (weaker) low-frequency component. In both cases, all frequency components exhibit an overall phase difference of almost 2kπ(k = 1, 2,…) between separation and impingement. Furthermore, the low-frequency and fundamental components have approximately the same amplitude growth rates and phase speeds; this suggests that the instability wave is amplitude-modulated at the low frequency, as confirmed by the form of instantaneous velocity traces.

At the downstream corner of the cavity, successive vortices, arising from the amplified instability wave, undergo organized variations in (transverse) impingement location, producing a low-frequency component(s) of corner pressure. The spectral content and instantaneous trace of this impingement pressure are consistent with those of velocity fluctuations near the (upstream) shear-layer separation edge, giving evidence of the strong upstream influence of the corner region.

This paper modifies and refines earlier work of Palm (1960) concerning the finiteamplitude steady state of cellular convective motion attained when a horizontal layer of fluid becomes unstable as a result of being heated from below. The two non-linear ordinary differential equations to which the problem was reduced by Palm (under certain conditions) are given in a corrected form, and are then analysed in some detail. The principal conclusions are that, for the model considered, hexagonal convection cell may be the stable equilibrium state only if the variation of kinematic viscosity with temperature is sufficiently great. Under the same circumstances a two-dimensional roll cell is also possible, the initial conditions determining which state actually occurs. Although further work is indicated, it seems probable that in an actual experiment with sufficiently large kinematic-viscosity variation, the hexagonal cells are more likely to appear. The analysis enables conclusions to be drawn concerning the flow direction at the cell centre, and also shows that a disturbance of sufficient magnitude may grow even though the situation is a stable one by linearized theory. Comparison with experiment is discussed.