We consider a sheet flow in which heavy grains near a packed bed interact with a unidirectional turbulent shear flow of a fluid. We focus on sheet flows in which the particles are supported by their collisional interactions rather than by the velocity fluctuations of the turbulent fluid and introduce what we believe to be the simplest theory for the collisional regime that captures its essential features.

We employ a relatively simple model of the turbulent shearing of the fluid and use kinetic theory for the collisional grain flow to predict profiles of the mean fluid velocity, the mean particle velocity, the particle concentration, and the strength of the particle velocity fluctuations within the sheet. These profiles are obtained as solutions to the equations of balance of fluid and particle momentum and particle fluctuation energy over a range of Shields parameters between 0.5 and 2.5. We compare the predicted thickness of the concentrated region and the predicted features of the profile of the mean fluid velocity with those measured by Sumer et al. (1996). In addition, we calculate the volume flux of particles in the sheet as a function of Shields parameter.

Finally, we apply the theory to sand grains in air for the conditions of a sandstorm and calculate profiles of particle concentration, velocity, and local volume flux.

The breaking of fully nonlinear internal solitary waves of depression shoaling upon a uniformly sloping boundary in a smoothed two-layer density field was investigated using high-resolution two-dimensional simulations. Our simulations were limited to narrow-crested waves, which are more common than broad-crested waves in geophysical flows. The simulations were performed for a wide range of boundary slopes S ∈ [0.01, 0.3] and wave slopes extending the parameter range to weaker slopes than considered in previous laboratory and numerical studies. Over steep slopes (S ≥ 0.1), three distinct breaking processes were observed: surging, plunging and collapsing breakers which are associated with reflection, convective instability and boundary-layer separation, respectively. Over mild slopes (S ≤ 0.05), nonlinearity varies gradually and the wave fissions into a train of waves of elevation as it passes through the turning point where solitary waves reverse polarity. The dynamics of each breaker type were investigated and the predominance of a particular mechanism was associated with a relative developmental time scale. The breaking location was modelled as a function of wave amplitude (a), characteristic wave length and the isopycnal length along the slope. The breaker type was characterized in wave slope (Sw = a/Lw, where Lw is a measure of half of the wavelength) versus S space, and the reflection coefficient (R), modelled as a function of the internal Iribarren number, was in agreement with other studies. The effects of grid resolution and wave Reynolds number (Rew) on R, boundary-layer separation and the evolution of global instability were studied. High Reynolds numbers (Rew ~ 104) were found to trigger a global instability, which modifies the breaking process relative to the lower Rew case, but not necessarily the breaking location, and results in a ~ 10 % increase in R, relative to the Rew ~ 103 case.

The stability of two vortex pairs is analysed as a model for the vortex system generatedby an aircraft in flaps-down configuration. The co-rotating vortices on the starboard andport sides tumble about one another as they propagate downward. This results in atime-periodic basic state for the stability analysis. The dynamics and instability of thetrailing vortices are modelled using thin vortex filaments. Stability equations are derivedby matching the induced velocities from Biot–Savart integrals with kinematic equationsobtained by temporal differentiation of the vortex position vectors. The stability equationsare solved analytically as an eigenvalue problem, using Floquet theory, and numerically asan initial value problem. The instabilities are periodic along the axes of the vortices withwavelengths that are large compared to the size of the vortex cores. The results showsymmetric instabilities that are linked to the long-wavelength Crow instability. Inaddition, new symmetric and antisymmetric instabilities are observed at shorter wavelengths.These instabilities have growth rates 60–100% greater than the Crow instability. The systemof two vortex pairs also exhibits transient growth which can lead to growth factors of 10 or15 in one-fifth of the time required for the same growth due to instability.

The instabilities of barotropic and baroclinic, quasi-geostrophic, f-plane, circular vortices are found using a linearized contour dynamics model. We model the vortex using a circular region of horizontally uniform potential vorticity surrounded by an annulus of uniform, but different, potential vorticity. We concentrate mostly upon isolated vortices with no circulation in the basic state outside the outer radius b. In addition to linear analyses, we also consider weakly nonlinear waves. The amplitude equation has a cubic nonlinearity and, depending upon the sign of the coefficient of the cubic term, may give nonlinear stabilization or nonlinear enhancement of the growth. Barotropic isolated eddies are unstable when the outer annulus is narrow enough; on the other hand, if the scale of the whole vortex is sufficiently small compared to the radius of deformation of a baroclinic mode, the break up may be preferentially to a depth-varying disturbance corresponding to a twisting and tilting of the vortex. As the vortex becomes more baroclinic, we find that large-scale vortices show an elliptical mode baroclinic instability as well which is relatively insensitive to the scale of the outer annulus. When the baroclinic currents in the basic state dominate, the twisting mode disappears, and we see only the instabilities associated with either strong enough shear in the annular region or sufficiently large vortices compared with the deformation radius. The finite amplitude results show that the baroclinic instability mode for large enough vortices is nonlinearly stabilized while in most cases, the other two kinds of instability are nonlinearly destabilized.

The stability of the oscillatory Stokes layers is examined using two quasi-static linear theories and an integration of the full time-dependent linearized disturbance equations. The full theory predicts absolute stability within the investigated range and perhaps for all the Reynolds numbers. A given wavenumber disturbance of a Stokes layer is found to be more stable than that of the motionless state (zero Reynolds number). The quasi-static theories predict strong inflexional instabilities. The failure of the quasi-static theories is discussed.

The classical problem of linear stability of a regular $N$-gon of point vortices to infinitesimal space displacements from an equilibrium of the vortex configuration is generalized to the one for $N$ helical vortices (couple, triplet, etc., $N\,{>}\, 1$) for the first time. As a consequence of this consideration, the analytical form for the stability boundaries has been obtained. This solution allows an efficient analysis to be made of the existence of stable helical vortex arrays, which were repeatedly observed in practice.

Such a stability problem was earlier considered in theory, but only for the case of a plane polygonal array of $N$ point vortices. As for helical vortices, owing to their complexity, intensive study has been mainly on the self-induced motion of the vortex.

The algebraic representation for the velocity of motion of the $N$ helical vortex array was originally obtained as an additional intermediate result. The new formula allows accurate calculations to be made within the whole range of helical pitch variations and has a simpler form than the known asymptotic expressions.

Solution of these two classical problems of vortex dynamics has significance both for theoretic and applied mechanics, as well as for many other areas of natural science, where the rotor (vortex) concept is the basic one.

The integration of oscillating water column (OWC) wave energy converters into a coastal structure (breakwater, jetty, pier, etc.) or, more generally, their installation along the coast is an effective way to increase the accessibility of wave power exploitation. In this paper, a theoretical model is developed based on the linear potential flow theory and eigenfunction matching method to evaluate the hydrodynamic performance of an array of OWCs installed along a vertical straight coast. The chamber of each OWC consists of a hollow vertical circular cylinder, which is half embedded in the wall. The OWC chambers in the theoretical model may have different sizes, i.e. different values of the radius, wall thickness and submergence. At the top of each chamber, a Wells turbine is installed to extract power. The effects of the Wells turbine together with the air compressibility are taken into account as a linear power take-off system. The hydrodynamic and wave power extraction performance of the multiple coast-integrated OWCs is compared with that of a single offshore/coast-integrated OWC and of multiple offshore OWCs. More specifically, we analyse the role of the incident wave direction, chamber size (i.e. radius, wall thickness and submergence), spacing between OWCs and number of OWCs by means of the present theoretical model. It is shown that wave power extraction from the coast-integrated OWCs for a certain range of wave conditions can be significantly enhanced due to both the constructive array effect and the constructive coast effect.

Aerosol particle transport and deposition in vertical and horizontal turbulent duct flows in the presence of different gravity directions are studied. The instantaneous fluid velocity field is generated by the direct numerical simulation of the Navier–Stokes equation via a pseudospectral method. A particle equation of motion including Stokes drag, Brownian diffusion, lift and gravitational forces is used for trajectory analysis. Ensembles of 8192 particle paths are evaluated, compiled, and statistically analysed. The results show that the wall coherent structure plays an important role in the particle deposition process. The simulated deposition velocities under various conditions are compared with the available experimental data and the sublayer model predictions. It is shown that the shear velocity, density ratio, the shear-induced lift force and the flow direction affect the particle deposition rate. The results for vertical ducts show that the particle deposition velocity varies with the direction of gravity, and the effect becomes more significant when the shear velocity is small. For horizontal ducts, the gravitational sedimentation increases the particle deposition rate on the lower wall.

Measurements of velocity fluctuations u (axial) and v (radial) and temperature fluctuations θ averaged over the turbulent zone have been made in a turbulent heated jet with a co-flowing stream and compared with the conventionally averaged results. The zone-averaged mean temperature and temperature fluctuation intensity appear to be nearly homogeneously distributed in the outer, intermittent region of the jet. This homogeneity does not apply to the u and v fluctuations. The flatness factor of the temperature within the turbulent part of the flow is remarkably constant throughout the intermittent region. Although the skewness of the turbulent θ fluctuations is non-zero, it is smaller than the skewness of the turbulent u and v fluctuations. The average $\overline{\theta v}$ of the heat flux over the turbulent zone increases in the intermittent region whereas the zone-averaged momentum flux $\overline{uv}$ and zone-averaged heat flux $\overline{\theta u}$ continuously decrease. This leads to the Prandtl number of the turbulent fluid being smaller than the conventional Prandtl number in the outer part of the flow. A budget for the square of the temperature fluctuations within the turbulent part of the flow indicates a constant distribution of temperature dissipation. The transport of heat by the large-scale structure of the flow is briefly discussed in the light of available experimental information on other turbulent shear flows.

In this paper the Dean (1928) equations are extended to the case of a helical pipe flow, and it is shown that they depend not only on the Dean number K but also on a new parameter λ/[Rscr ] where λ is the ratio of the torsion τ to the curvature κ of the pipe axis and [Rscr ] the Reynolds number referred in the usual way to the pipe radius a and to the equivalent maximum speed in a straight pipe under the same axial pressure gradient. The fact that the torsion has no first-order effect on the flow is confirmed, but it is shown that this is peculiar to a circular cross-section. In the case of an elliptical cross-section there is a first-order effect of the torsion on the secondary flow, and in the limit λ/[Rscr ] → ∞ (twisted pipes, provided only with torsion), the first-order ‘displacement’ effect of the walls on the secondary flow, analysed in detail by Choi (1988), is recovered.

Different systems of coordinates and different orders of approximations have recently been adopted in the study of the flow in a helical pipe. Thus comparisons between the equations and the results presented in different reports are in some cases difficult and uneasy. In this paper the extended Dean equations for a helical pipe flow recently derived by Kao (1987) are converted to a simpler form by introducing an appropriate modified stream function, and their equivalence with the present set of equations is recovered. Finally, the first-order equivalence of this set of equations with the equations obtained by Murata et al. (1981) is discussed.

Compressibility effects within decaying isotropic turbulence and homogeneous turbulent shear flow have been studied using direct numerical simulation. The objective of this work is to increase our understanding of compressible turbulence and to aid the development of turbulence models for compressible flows. The numerical simulations of compressible isotropic turbulence show that compressibility effects are highly dependent on the initial conditions. The shear flow simulations, on the other hand, show that measures of compressibility evolve to become independent of their initial values and are parameterized by the root mean square Mach number. The growth rate of the turbulence in compressible homogeneous shear flow is reduced compared to that in the incompressible case. The reduced growth rate is the result of an increase in the dissipation rate and energy transfer to internal energy by the pressure–dilatation correlation. Examination of the structure of compressible homogeneous shear flow reveals the presence of eddy shocklets, which are important for the increased dissipation rate of compressible turbulence.

The flow between a shallow rotating cone and a stationary plate has been investigated using flow visualization, hot-film heat-transfer probes, and measurements of the torque required to rotate the cone against the retardation of the viscous fluid that fills the device. Theory appropriate to these experiments is also presented.

An expansion of the Navier–Stokes equations is performed for small values of the single parameter $\tilde{R} = r^2\omega\alpha^2/12\nu $. (Here r is the local radius, ω the angular velocity of the cone, α([Lt ] 1) is the angle between the cone and plate, and v is the fluid kinematic viscosity.) The measurements at low rotational speeds describe a simple linear velocity profile as predicted for the laminar flow of a Newtonian fluid. At larger rotational speeds, strong secondary flows are observed. There is agreement between the laminar theory and the measured streamline angles and shear stresses for values of $\tilde{R} < 0.5$. Turbulence is observed for $\tilde{R} \gtrsim 4$.

The first in a new series of European research conferences in mechanics was held in Berlin on 5 and 6 April 1965. The subject was the Coanda effect, or boundary layers and jets on highly curved walls. Participation was restricted to about 40 people, invited on the basis of their active interest in the subject. These Colloquia are intended to have an informal, workshop-like, character, and formal papers are not normally available, nor will any full proceedings be published. The first author was the chairman of the Colloquium. The following account of the scientific developments of the Colloquium has been prepared to make them widely available. The references quoted give further details of the work discussed at the Colloquium and of related previous work.

A theoretical decomposition of mean skin friction generation into physical phenomena across the whole profile of the incompressible zero-pressure-gradient smooth-flat-plate boundary layer is derived from a mean streamwise kinetic-energy budget in an absolute reference frame (in which the undisturbed fluid is not moving). The Reynolds-number dependences in the laminar and turbulent cases are investigated from direct numerical simulation datasets and Reynolds-averaged Navier–Stokes simulations, and the asymptotic trends are consistently predicted by theory. The generation of the difference between the mean friction in the turbulent and laminar cases is identified with the total production of turbulent kinetic energy (TKE) in the boundary layer, represented by the second term of the proposed decomposition of the mean skin friction coefficient. In contrast, the analysis introduced by Fukagata et al. (Phys. Fluids, vol. 14 (11), 2002, pp. 73–76), based on a streamwise momentum budget in the wall reference frame, relates the turbulence-induced excess friction to the Reynolds shear stress weighted by a linear function of the wall distance. The wall-normal distribution of the linearly-weighted Reynolds shear stress differs from the distribution of TKE production involved in the present discussion, which consequently draws different conclusions on the contribution of each layer to the mean skin friction coefficient. At low Reynolds numbers, the importance of the buffer-layer dynamics is confirmed. At high Reynolds numbers, the present decomposition quantitatively shows for the first time that the generation of the turbulence-induced excess friction is dominated by the logarithmic layer. This is caused by the well-known decay of the relative contributions of the buffer layer and wake region to TKE production with increasing Reynolds numbers. This result on mean skin friction, with a physical interpretation relying on an energy budget, is consistent with the well-established general importance of the logarithmic layer at high Reynolds numbers, contrary to the friction breakdown obtained from the approach of Fukagata et al. (Phys. Fluids, vol. 14 (11), 2002, pp. 73–76), essentially based on a momentum budget. The new decomposition suggests that it may be worth investigating new drag reduction strategies focusing on TKE production and on the nature of the logarithmic layer dynamics. The decomposition is finally extended to the pressure-gradient case and to channel and pipe flows.

A numerical method employing an upwind finite-difference technique is adopted for an investigation of peristaltic pumping in circular cylindrical tubes. such as some organs in the living body. Various peristaltic flows are calculated under conditions of finite wave amplitudes, finite wavelengths and finite Reynolds numbers, and the influence of the magnitude of these quantities on the flow is investigated. The fluid mechanics of peristaltic mixing and transport are studied in detail by analysing the reflux and the trapping phenomena. The mechanical efficiency of peristaltic pumping is also discussed, with reference to engineering and physiological applications. It is shown that quantitative differences are observed between the results obtained for flows in a circular cylindrical tube and a two-dimensional plane channel. However, for both cases the appearance of peristaltic reflux depends upon the Reynolds number and the wavenumber (mean tube radius/wavelength). Much greater peristaltic mixing and transport are realized in a circular tube than in a plane channel.

The existing model equations governing the accelerated motion of a spherical particle are examined and their predictions compared with the results of the numerical solution of the full Navier–Stokes equations for unsteady, axisymmetric flow around a freely moving sphere injected into an initially stationary or oscillating fluid. The comparison for the particle Reynolds number in the range of 2 to 150 and the particle to fluid density ratio in the range of 5 to 200 indicates that the existing equations deviate considerably from the Navier–Stokes equations. As a result, we propose a new equation for the particle motion and demonstrate its superiority to the existing equations over a range of Reynolds numbers (from 2 to 150) and particle to fluid density ratios (from 5 to 200). The history terms in the new equation account for the effects of large relative acceleration or deceleration of the particle and the initial relative velocity between the fluid and the particle. We also examine the temporal structure of the near wake of the unsteady, axisymmetric flow around a freely moving sphere injected into an initially stagnant fluid. As the sphere decelerates, the recirculation eddy size grows monotonically even though the instantaneous Reynolds number of the sphere decreases.

Measurements of mean velocity, mean flow direction, normal turbulent stress in the direction of flow, and mean static pressure are reported for the subsonic flow field generated by identical twin jets of air issuing from parallel slot nozzles in a common wall and mixing turbulently with ambient room air. At the low nozzle velocity employed (72ft./sec), the two-dimensional plano-symmetric flow was effectively incompressible. Since the end walls prevented interjet air entrainment from the surroundings, a region of highly convergent flow was formed near the nozzles. In this region, contour maps clearly reveal (1) the sub-atmospheric static pressure through that accounts for the jet convergence, (2) a free stagnation point on the plane of symmetry, (3) stable symmetrical contrarotary vortices which recycle air on the concave side of each converging jet, and (4) the super-atmospheric static pressure mound that redirects the merging jet streams in a common downstream direction. Comparisons are made between the development of the flow, in both the region of jet convergence and the region of combined jet flow, and that of the single-jet counterpart which was previously reported.

Further evidence of a universal maximum drag reduction asymptote is presented. In the elastic sublayer model, inferred therefrom, the mean velocity profile during drag reduction is approximated by three zones: the usual viscous sublayer, an elastic sublayer where the mixing-length constant is derived from the maximum drag reduction asymptote, and an outermost region with Newtonian mixinglength constant. Upon integration the model yields a friction factor relation, parametric in elastic sublayer thickness, which properly reproduces the known features of turbulent dilute polymer solution flow. The dependence of elastic sublayer thickness upon flow and polymeric parameters is inferred from experimental data revealing two hitherto unknown relationships: namely that on Prandtl co-ordinates, 1/f½vs. log Re f½ the difference in slope between a polymer solution and solvent is proportional to the square root of molar concentration and to the three-halves power of backbone chain links in the macromolecule. The proportionality constant in the preceding relationship is approximately the same for several different polymer species of carbon-carbon or similar skeletal structure in various thin solvents; there is an indication that this constant further depends upon the product of solvent viscosity times the cube of the effective bond length per chain link of the polymer species. Some recent results regarding the onset of drag reduction are also summarized.

The unsteady flow past a circular cylinder which starts translating and rotating impulsively from rest in a viscous fluid is investigated both theoretically and experimentally in the Reynolds number range 103 [les ] R [les ] 104 and for rotational to translational surface speed ratios between 0.5 and 3. The theoretical study is based on numerical solutions of the two-dimensional unsteady Navier–Stokes equations while the experimental investigation is based on visualization of the flow using very fine suspended particles. The object of the study is to examine the effect of increase of rotation on the flow structure. There is excellent agreement between the numerical and experimental results for all speed ratios considered, except in the case of the highest rotation rate. Here three-dimensional effects become more pronounced in the experiments and the laminar flow breaks down, while the calculated flow starts to approach a steady state. For lower rotation rates a periodic structure of vortex evolution and shedding develops in the calculations which is repeated exactly as time advances. Another feature of the calculations is the discrepancy in the lift and drag forces at high Reynolds numbers resulting from solving the boundary-layer limit of the equations of motion rather than the full Navier–Stokes equations. Typical results are given for selected values of the Reynolds number and rotation rate.

In many industrial processes granular materials are mixed together in partially filled slowly rotating drums. In this paper a general theoretical framework is developed for the quasi-two-dimensional motion of granular material in a rotating drum. The key assumption is that the body can be divided into a fluid-like and a solid-like region, that are separated by a non-material singular surface at which discontinuities occur. Experiments show that close to the free surface there is a thin rapidly moving fluid-like avalanche that flows downslope, and beneath it there is a large region of slowly rotating solid-like material. The solid region provides a net transport of material upslope and there is strong mass transfer between the two regions. In the theory the avalanche is treated as a shallow incompressible Mohr–Coulomb or inviscid material sliding on a moving bed at which there is erosion and deposition. The solid is treated as a rigid rotating body, and the two regions are coupled together using a mass jump condition. The theory has the potential to model time-dependent intermittent flow with shock waves, as well as steady-state continuous flow. An exact solution for the case of steady continuous flow is presented. This demonstrates that when the base of the avalanche lies above the axis of revolution a solid core develops in the centre of the drum. Experiments are presented to show how a mono-disperse granular material mixes in the drum, and the results are compared with the predictions using the exact solution.