Experimental investigations of the boundary layer receptivity, on the sharp leading edge of a at plate, to acoustic waves induced by two-dimensional and three- dimensional perturbers, have been performed for a free-stream Mach number M∞ = 5.92. The fields of controlled free-stream disturbances were studied. It was shown that two-dimensional and three-dimensional perturbers radiate acoustic waves and that these perturbers present a set of harmonic motionless sources and moving sources with constant amplitude. The disturbances excited in the boundary layer were measured. It was found that acoustic waves impinging on the leading edge generate Tollmien–Schlichting waves in the boundary layer. The receptivity coefficients were obtained for several radiation conditions and intensities. It was shown that there is a dependence of receptivity coefficients on the wave inclination angles.

The momentum and mass conservation equations derived in part 1 are used in the limiting case of large solids-to-fluid density ratio. Bubbles generally appear in such fluidized beds and solutions are obtained which describe the motion of the particulate and fluid phases in the presence of a fully developed bubble. The analytical solutions are shown to be in fair agreement with experiment.

The flow in a two-dimensional laminar separation bubble is analyzed by means of finite-difference solutions to the Navier-Stokes equations for incompressible flow. The study was motivated by the need to analyze high-Reynolds-number flow fields having viscous regions in which the boundary-layer assumptions are questionable. The approach adopted in the present study is to analyze the flow in the immediate vicinity of the separation bubble using the Navier-Stokes equations. It is assumed that the resulting solutions can then be patched to the remainder of the flow field, which is analyzed using boundary-layer theory and inviscid-flow analysis. Some of the difficulties associated with patching the numerical solutions to the remainder of the flow field are discussed, and a suggestion for treating boundary conditions is made which would permit a separation bubble to be computed from the Navier-Stokes equations using boundary conditions from inviscid and boundary-layer solutions without accounting for interaction between individual flow regions. Numerical solutions are presented for separation bubbles having Reynolds numbers (based on momentum thickness) of the order of 50. In these numerical solutions, separation was found to occur without any evidence of the singular behaviour at separation found in solutions to the boundary-layer equations. The numerical solutions indicate that predictions of separation by boundary-layer theory are not reliable for this range of Reynolds number. The accuracy and validity of the numerical solutions are briefly examined. Included in this examination are comparisons between the Howarth solution of the boundary-layer equations for a linearly retarded freestream velocity and the corresponding numerical solutions of the Navier-Stokes equations for various Reynolds numbers.

Direct numerical simulations (DNS) of stenotic flows under conditions of steady inlet flow were discussed in Part 1 of this study. DNS of pulsatile flow through the 75% stenosed tube (by area) employed for the computations in Part 1 is examined here. Analogous to the steady flow results, DNS predicts a laminar post-stenotic flow field in the case of pulsatile flow through the axisymmetric stenosis model, in contrast to previous experiments, in which intermittent disturbed flow regions and turbulent breakdown were observed in the downstream region. The introduction of a stenosis eccentricity, that was 5% of the main vessel diameter at the throat, resulted in periodic, localized transition to turbulence. Analysis in this study indicates that the early and mid-acceleration phases of the time period cycle were relatively stable, with no turbulent activity in the post-stenotic region. However, towards the end of acceleration, the starting vortex, formed earlier as the fluid accelerated through the stenosis at the beginning of acceleration, started to break up into elongated streamwise structures. These streamwise vortices broke down at peak flow, forming a turbulent spot in the post-stenotic region. In the early part of deceleration there was intense turbulent activity within this spot. Past the mid-deceleration phase, through to minimum flow, the inlet flow lost its momentum and the flow field began to relaminarize. The start of acceleration in the following cycle saw a recurrence of the entire process of a starting structure undergoing turbulent breakdown and subsequent relaminarization of the post-stenotic flow field. Peak wall shear stress (WSS) levels occurred at the stenosis throat, with the rest of the vessel experiencing much lower levels. Turbulent breakdown at peak flow resulted in a sharp amplification of instantaneous WSS magnitudes across the region corresponding to the turbulent spot, accompanied by large axial and circumferential fluctuations, even while ensemble-averaged axial shear stresses remained mostly low and negative. WSS levels dropped rapidly after the mid-deceleration phase, when the relaminarization process took over, and were almost identical to laminar, axisymmetric shear levels through most of the acceleration phase.

The motion of a sphere in the presence of a fluid-fluid interface is studied. First, a solution is derived for a point force near a plane interface. Then the solution is extended to include the higher-order terms which are required to describe the motion of a solid sphere. Singularities of higher orders at the centre of the sphere are obtained by using the method of reflexions. For a fluid–fluid interface with an arbitrary viscosity ratio, the drag force and the hydrodynamic torque are calculated for the special cases of motion of a sphere perpendicular and parallel to the interface. In addition, the rotational motion of a sphere is also investigated.

This paper presents an experimental study devoted to investigating the effects of permeability on wall turbulence. Velocity measurements were performed by means of laser Doppler anemometry in open channel flows over walls characterized by a wide range of permeability. Previous studies proposed that the von Kármán coefficient associated with mean velocity profiles over permeable walls is significantly lower than the standard values reported for flows over smooth and rough walls. Furthermore, it was observed that turbulent flows over permeable walls do not fully respect the widely accepted paradigm of outer-layer similarity. Our data suggest that both anomalies can be explained as an effect of poor inner–outer scale separation if the depth of shear penetration within the permeable wall is considered as the representative length scale of the inner layer. We observed that with increasing permeability, the near-wall structure progressively evolves towards a more organized state until it reaches the condition of a perturbed mixing layer where the shear instability of the inflectional mean velocity profile dictates the scale of the dominant eddies. In our experiments such shear instability eddies were detected only over the wall with the highest permeability. In contrast attached eddies were present over all the other wall conditions. On the basis of these findings, we argue that the near-wall structure of turbulent flows over permeable walls is regulated by a competing mechanism between attached and shear instability eddies. We also argue that the ratio between the shear penetration depth and the boundary layer thickness quantifies the ratio between such eddy scales and, therefore, can be used as a diagnostic parameter to assess which eddy structure dominates the near-wall region for different wall permeability and flow conditions.

Finite difference solutions for the time dependent equations of motion have been carried out in order to extend the range of available data on steady flow around a cylinder to larger Reynolds numbers. At the termination of the calculations for R = 40 and 200, the separation angle, the drag coefficient and the pressure and vorticity distributions around the surface of the cylinder were very close to their steady-state values. For R = 500 the separation angle and drag coefficient were very close to their steady-state values but the pressure distribution and vorticity distribution at the rear of the cylinder were still changing slightly. The results at R = 500 were found to be quite different from those at R = 200 so it is not clear how closely we approximated the steady solution for R → ∞. The forces on the cylinder due to viscous drag and due to pressure drag are found to be smaller for steady flow than for laboratory experiments where the wake is unsteady.

In this paper we present a theoretical model which permits the conservation equations of fluid dynamics to be conditioned in a fashion analogous to the experimentalist's technique of ‘conditioned sampling’. The detailed analysis refers to the best-known sampling condition, outer-edge intermittency; but the model equation may be applicable to other flow situations, wherein condition- ing exposes details of the physical phenomena. The analysis results in predic- tions of the flow variables within the turbulent flow and of the intermittency. Comparison is made with two sets of experimental results for the two-dimensional mixing layer and with a boundary layer.

A general theory for two-dimensional vortex interactions is developed from the observation that, under slowly changing external influences, an individual vortex evolves through a series of equilibrium states until such a state proves unstable. Once an unstable equilibrium state is reached, a relatively fast unsteady evolution ensues, typically involving another nearby vortex. During this fast unsteady evolution, a fraction of the original coherent circulation is lost to filamentary debris, and, remarkably, the flow reorganizes into a set of quasi-steady stable vortices.

The simplifying feature of the proposed theory is its use of adiabatic steadiness and marginal stability to determine the shapes and separation distance of vortices on the brink of an inelastic interaction. As a result, the parameter space for the inelastic interaction of nearby vortices is greatly reduced. In the case of two vortex patches, which is the focus of the present work, inelastic interactions depend only on a single parameter: the area ratio of the two vortices (taking the vorticity magnitude inside each to be equal). Without invoking adiabatic steadiness and marginal stability, one would have to contend with the additional parameters of vortex separation and shape, and the latter is actually an infinitude of parameters.

A numerical study of the three-dimensional fluid dynamics inside a model left ventricle during diastole is presented. The ventricle is modelled as a portion of a prolate spheroid with a moving wall, whose dynamics is externally forced to agree with a simplified waveform of the entering flow. The flow equations are written in the meridian body-fitted system of coordinates, and expanded in the azimuthal direction using the Fourier representation. The harmonics of the dependent variables are normalized in such a way that they automatically satisfy the high-order regularity conditions of the solution at the singular axis of the system of coordinates. The resulting equations are solved numerically using a mixed spectral–finite differences technique. The flow dynamics is analysed by varying the governing parameters, in order to understand the main fluid phenomena in an expanding ventricle, and to obtain some insight into the physiological pattern commonly detected. The flow is characterized by a well-defined structure of vorticity that is found to be the same for all values of the parameters, until, at low values of the Strouhal number, the flow develops weak turbulence.

A new criterion for the direct initiation of cylindrical or spherical detonations by a localized energy source is presented. The analysis is based on nonlinear curvature effects on the detonation structure. These effects are first studied in a quasi-steady-state approximation valid for a characteristic timescale of evolution much larger than the reaction timescale. Analytical results for the square-wave model and numerical results for an Arrhenius law of the quasi-steady equations exhibit two branches of solutions with a C-shaped curve and a critical radius below which generalized Chapman–Jouguet (CJ) solutions cannot exist. For a sufficiently large activation energy this critical radius is much larger than the thickness of the planar CJ detonation front (typically 300 times larger at ordinary conditions) which is the only intrinsic lengthscale in the problem. Then, the initiation of gaseous detonations by an ideal point energy source is investigated in cylindrical and spherical geometries for a one-step irreversible reaction. Direct numerical simulations show that the upper branch of quasi-steady solutions acts as an attractor of the unsteady blast waves originating from the energy source. The critical source energy, which is associated with the critical point of the quasi-steady solutions, corresponds approximately to the boundary of the basin of attraction. For initiation energy smaller than the critical value, the detonation initiation fails, the strong detonation which is initially formed decays to a weak shock wave. A successful initiation of the detonation requires a larger energy source. Transient phenomena which are associated with the intrinsic instability of the quasi-steady detonations branch develop in the induction timescale and may induce additional mechanisms close to the critical condition. In conditions of stable or weakly unstable planar detonations, these unsteady phenomena are important only in the vicinity of the critical conditions. The criterion of initiation derived in this paper works to a good approximation and exhibits the huge numerical factor, 106–108, which has been experimentally observed in the critical value of the initiation energy.

The conjecture of Vainshtein & Zel'dovich (1972) concerning the existence of a fast dynamo (i.e. one whose growth rate is independent of magnetic diffusivity η in the limit η → 0) is discussed with particular reference to (i) the stretch–twist–fold cycle which can double the strength of a magnetic flux tube, and (ii) the space-periodic Beltrami flow of maximal helicity, which has been shown to be capable of space-periodic dynamo action with the same period as the velocity field, by Arnold & Korkina (1983) and by Galloway & Frisch (1984). The topological constraint associated with conservation of magnetic helicity is shown to preclude fast dynamo action unless the scale of the magnetic field is almost everywhere of order η½ as η → 0; in this case, the field structure is severely singular in the limit. A steady incompressible velocity field, quadratic in the space variables, is shown to mimic the action of the stretch–twist–fold cycle, and is proposed as a plausible candidate for fast dynamo action.

The transverse migration of both clean and fully contaminated spherical bubbles rising near a plane vertical wall in a quiescent viscous liquid is studied experimentally using an optical technique. Knowing the bubble radius $R$, rising speed $U$ and separation distance $L$ between the bubble centre and the wall as a function of time, the transverse or lift component of the hydrodynamic force is determined as a function of $L/R$ for Reynolds numbers $Re\,{=}\,2 UR/\nu $ less than 100 ($\nu $ is the kinematic viscosity). At low Reynolds number, the lift force is directed away from the wall for both clean and contaminated bubbles and its magnitude is found to be in good agreement with available analytical solutions. For $Re\,{>}\,1$, contaminated bubbles are still repelled from the wall, but the magnitude of the lift force is always larger than predicted by the low-$Re$ theory, the difference being an increasing function of $Re$. The behaviour of clean bubbles is markedly different as the experiments reveal that the lift force is directed away from the wall for $Re\,{<}\,35$ and toward it for higher $Re$. The differences found in the evolution of the lift force for clean and contaminated bubbles are analysed by considering the relative strength of two hydrodynamical mechanisms of wall interaction, one being due to the vorticity generated at the bubble surface while the other is due to the irrotational dipole associated with the bubble. Empirical correlations based on the strength of these two mechanisms are derived to obtain practical expressions of the lift force as a function of $Re$ and $L/R$. At high enough $Re$, clean bubbles rising sufficiently close to the wall are found to bounce very close to it. Based on present and past experimental results, we suggest that this bouncing process is essentially due to the competition between inertial and viscous effects, rather than to bubble deformation as previously believed, and we propose that observations may be explained by considering the effect of the vorticity present in the wall boundary layer. Evaluation of the various contributions involved in the lateral force balance of bouncing bubbles reveals the central role of history effects due to unsteady diffusion of vorticity from the bubble surface.

The fine scale three-dimensional structures usually associated with streamwise vortices in the near wake of a circular cylinder have been studied at Reynolds numbers ranging from 170 to 2200. Spatially continuous velocity measurements along lines parallel to the cylinder axis were obtained with a scanning laser anemometer. To detect the streamwise vortices in the amplitude modulated velocity field, it was necessary to develop a spatial decomposition technique to split the total flow into a primary flow component and a secondary flow component. The primary flow is comprised of the mean flow and Strouhal vortices, while the secondary flow is the result of the three-dimensional streamwise vortices that are the essence of transition to turbulence. The three-dimensional flow amplitude increases in the primary vortex formation region, then saturates shortly after the maximum amplitude in the primary flow is reached. In the near-wake region the wavelength decreases approximately like Re−0.5, but increases with downstream distance. A discontinuous increase in wavelength occurs below Re = 300 suggesting a fundamental change in the character of the three-dimensional flow. At downstream distances (x/D = 10-20), the spanwise wavelength decreases from 1.42D to 1.03D as the Reynolds number increases from 300 to 1200.

Associated with Taylor’s classical analysis of scalar solute dispersion in the laminar flow of a solvent in a straight pipe, this work explores the approach towards transverse uniformity of concentration distribution. Mei’s homogenization technique is extended to find solutions for the concentration transport. Chatwin’s result for the approach to longitudinal normality is recovered in terms of the mean concentration over the cross-section. The asymmetrical structure of the concentration cloud and the transverse variation of the concentration distribution are concretely illustrated for the initial stage. The rate of approach to uniformity is shown to be much slower than that to normality. When the longitudinal normality of mean concentration is well established, the maximum transverse concentration difference remains near one-half of the centroid concentration of the cloud. A time scale up to $10 R^2/D$ ($R$ is the radius of the pipe and $D$ is the molecular diffusivity) is suggested to characterize the transition to transverse uniformity, in contrast to the time scale of $0.1 R^2/D$ estimated by Taylor for the initial stage of dispersion, and that of $1.0 R^2/D$ by Chatwin for longitudinal normality.

Self-diffusion in a monodisperse suspension of non-Brownian particles in simple shear flow is studied using accelerated Stokesian dynamics (ASD) simulation. The availability of a much faster computational algorithm allows the study of large systems (typically of 1000 particles) and the extraction of accurate results for the complete shear-induced self-diffusivity tensor. The finite, and often large, autocorrelation time requires the mean-square displacements to be followed for very long times, which is now possible with ASD. The self-diffusivities compare favourably with the available experimental measurements when allowance is made for the finite strains sampled in the experiments. The relationship between the mean-square displacements and the diffusivities appearing in a Fokker–Planck equation when advection couples to diffusion is discussed.

Measurements of sectional and total forces and the spanwise correlation of vortex shedding are presented for a circular cylinder in planar oscillatory flow at Keulegan-Carpenter numbers, KC, in the range from about 4 to 55. The viscous parameter β is in the range from around 100 to 1665. Circulation measurements around a circuit close to and enclosing the cylinder, are also presented. A mode-averaging technique was used for both sectional forces and circulation measurements and this gave, for typical modes of vortex shedding, time histories over an average cycle. The transverse force and the circulation tend to fluctuate in sympathy with each other, except around the instant of flow reversal when the force changes sign but the circulation remains high. Values of the strength of shed vortices, estimated from the measured circulation, are found to be comparable with steady-flow results. For KC [lsim ] 30, modes of vortex shedding occur over distinct ranges of KC with spanwise correlation high at the centre of a KC-range for a particular mode of shedding but low at the boundaries. Above KC ≈ 30 the correlation is no longer very sensitive to KC and the correlation length is estimated to be equal to 4.65 cylinder diameters. In the transverse vortex-street regime (8 [lsim ] KC [lsim ] 15) the cylinder was found to experience a steady transverse force, the coefficient of which is estimated to be about 0.5 at KC = 14.

This paper presents three-dimensional (3D) direct numerical simulations (DNS) of flow past a circular cylinder over a range of Reynolds number ($Re$) up to 300. The gradual wake transition process from mode A* (i.e. mode A with large-scale vortex dislocations) to mode B is well captured over a range of $Re$ from 230 to 260. The mode swapping process is investigated in detail with the aid of numerical flow visualization. It is found that the mode B structures in the transition process are developed based on the streamwise vortices of mode A or A* which destabilize the braid shear layer region. For each case within the transition range, the transient mode swapping process consists of dislocation and non-dislocation cycles. With the increase of $Re$, it becomes more difficult to trigger dislocations from the pure mode A structure and form a dislocation cycle, and each dislocation stage becomes shorter in duration, resulting in a continuous decrease in the probability of occurrence of mode A* and a continuous increase in the probability of occurrence of mode B. The occurrence of mode A* results in a relatively strong flow three-dimensionality. A critical condition is confirmed at approximately $Re=265{-}270$, where the weakest flow three-dimensionality is observed, marking a transition from the disappearance of mode A* to the emergence of increasingly disordered mode B structures.

In this paper, the nonlinear dynamics of an axisymmetric liquid bridge held captive between two coaxial, circular, solid disks that are separated at a constant velocity are considered. As the disks are continuously pulled apart, the bridge deforms and ultimately breaks when its length attains a limiting value, producing two drops that are supported on the two disks. The evolution in time of the bridge shape and the rupture of the interface are investigated theoretically and experimentally to quantitatively probe the influence of physical and geometrical parameters on the dynamics. In the computations, a one-dimensional model that is based on the slender jet approximation is used to simulate the dynamic response of the bridge to the continuous uniaxial stretching. The governing system of nonlinear, time-dependent equations is solved numerically by a method of lines that uses the Galerkin/finite element method for discretization in space and an adaptive, implicit finite difference technique for discretization in time. In order to verify the model and computational results, extensive experiments are performed by using an ultra-high-speed video system to monitor the dynamics of liquid bridges with a time resolution of 1/12 th of a millisecond. The computational and experimental results show that as the importance of the inertial force – most easily changed in experiments by changing the stretching velocity – relative to the surface tension force increases but does not become too large and the importance of the viscous force – most easily changed by changing liquid viscosity – relative to the surface tension force increases, the limiting length that a liquid bridge is able to attain before breaking increases. By contrast, increasing the gravitational force – most readily controlled by varying disk radius or liquid density – relative to the surface tension force reduces the limiting bridge length at breakup. Moreover, the manner in which the bridge volume is partitioned between the pendant and sessile drops that result upon breakup is strongly influenced by the magnitudes of viscous, inertial, and gravitational forces relative to surface tension ones. Attention is also paid here to the dynamics of the liquid thread that connects the two portions of the bridge liquid that are pendant from the top moving rod and sessile on the lower stationary rod because the manner in which the thread evolves in time and breaks has important implications for the closely related problem of drop formation from a capillary. Reassuringly, the computations and the experimental measurements are shown to agree well with one another.

A swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity and referred to as a squirmer. The centre of mass of the sphere may be displaced from the geometric centre, and the effects of inertia and Brownian motion are neglected. The well-known Stokesian dynamics method is modified in order to simulate squirmer motions in a concentrated suspension. The movement of 216 identical squirmers in a concentrated suspension without any imposed flow is simulated in a cubic domain with periodic boundary conditions, and the coherent structures within the suspension are investigated. The results show that (a) a weak aggregation of cells appears as a result of the hydrodynamic interaction between cells; (b) the cells generate collective motions by the hydrodynamic interaction between themselves; and (c) the range and duration of the collective motions depend on the volume fraction and the squirmers' stresslet strengths. These tendencies show good qualitative agreement with previous experiments.