A fully developed turbulent channel flow is subjected to a mean strain that approximates that in a spatially developing adverse-pressure-gradient (APG) boundary layer. This is done by applying uniform irrotational temporal deformations to the flow domain of a conventional direct numerical simulation channel code. The velocity difference between the inner and outer layer is also controlled by accelerating the walls in the streamwise plane, in order to duplicate the defining features of both the inner and outer regions of an APG boundary layer. Eventually, the flow reverses at the wall. We address basic physics and modelling issues, and create a database that makes detailed testing of turbulence models easy. As in the corresponding spatial layers, distinct inner- and outer-layer dynamics are observed: a decrease in turbulence intensity near the wall is accompanied by increased energy in the outer layer. The ‘extra strain’ effect associated with the diverging outer-layer streamlines is documented, particularly in the Reynolds-stress budgets.

Laboratory experiments on decaying quasi-two-dimensional turbulence have been performed in stratified fluid inside containers with length-to-width ratios $\delta$ up to 5. The Reynolds number $Re$ of the horizontal flow, based on the r.m.s. velocity of the initial flow field and the half-width $H$ of the container, was typically between 1000 and 3000. The turbulence was generated by towing an array of vertical cylinders through the container which was filled with a two-layer stratified fluid. By varying the grid configuration a different amount of angular momentum could be added to the initial flow. The evolution of the flow was visualized by two-dimensional particle tracking velocimetry. The observed decay has been investigated with the emphasis on the final states as function of $\delta$, $Re$ and the angular momentum initially added to the flow. In addition, numerical simulations were carried out for decaying two-dimensional turbulence on rectangular domains with $\delta\,{=}\,2$ and 3. In these runs zero net angular momentum was added to the initial flow field. The numerical study focused on the final states as a function of $\delta$ and $Re$. The numerically obtained final states appeared to agree with the experimental observations. Furthermore, they indicate a clear difference between the predictions of quasi-stationary final states from statistical-mechanical theories and the final states as found in the numerical simulations.

Langmuir circulation is a convective motion commonly observed in the oceanic mixed layer. Internal waves are a prominent feature of stratified regions, particularly the thermocline bounding the mixed layer. Here, the potential for Langmuir-circulation–internal-wave coupling is investigated using a two-layer ocean model. The density jump across the sharp thermocline confines all rotational motions, including the wind-aligned Langmuir vortices, to the upper (‘mixed’) layer. Linear analysis indicates: (i) that thermocline compliance enhances the onset of Langmuir circulation, and (ii) that the ‘vortex force’ arising from the interaction of surface waves with the wind-driven shear modifies the dynamics of cross-wind propagating internal waves. Weakly nonlinear analysis reveals that resonant cross-wind propagating internal waves can be nonlinearly reflected from stationary Langmuir circulation, a dynamic reminiscent of the ‘Bragg reflection’ of surface waves propagating over sand bars. A key feature of the reflection mechanism is the modification of the linear internal-wave dynamics by the vortex force.

The fully nonlinear theory of global modes in open flows, proposed in recent analyses of amplitude equations, is extended to the case of Navier–Stokes equations using direct numerical simulations. The basic flow under consideration is a parallel wake in a finite domain generated by imposing the wake profile at the inlet boundary and by adding a body force to compensate the basic flow diffusion. The link between the global bifurcation, the absolute or convective nature of the local linear instability, and the theory of speed selection for the front separating an unperturbed domain of the flow from a fully saturated solution is elucidated. In particular, thanks to the parallelism of the flow, the bifurcation scenario and the associated scaling laws for the frequency, the healing length, and the slope at the origin predicted by a previous analysis of amplitude equations are recovered with great precision.

Direct numerical simulations of the motion of 27 three-dimensional deformable buoyant bubbles in periodic domains are presented. The full Navier–Stokes equations are solved by a parallelized finite-difference/front-tracking method that allows a deformable interface between the bubbles and the suspending fluid and the inclusion of surface tension. The Eötvös number is taken as equal to 5, so that the bubbles are ellipsoidal, and the Galileo number is 900, so that the rise Reynolds number of a single bubble in an unbounded flow is about 26. Three values of the void fraction have been investigated: 2%, 6% and 12%. At 6%, a change in the behaviour of the bubbles is observed. The bubbles are initially dispersed homogeneously throughout the flow field and their average rise Reynolds number is 23. After the bubbles have risen by about 90 bubble diameters, they form a vertical stream and accelerate. The microstructure of the bubble suspension is analysed and an explanation is proposed for the formation of these streams. The results for the ellipsoidal bubbles are compared to the results for nearly spherical bubbles, for which the Eötvös number is 1 and the Galileo number is 900. The dispersion of the bubbles and the velocity fluctuations in the liquid phase are analysed.

We analyse the density evolution of fluid within a confined ventilated space resulting from the action of a dense turbulent plume originating at the top of the space with finite source volume flux, $Q_0$, and initial source buoyancy flux, $B_0$. The space is ventilated through upper and lower openings of areas $A_u$ and $A_l$ respectively, which are separated by a vertical distance $H$. We show that if $Q_0^3\,{<}\, 2 B_0 H c_l^2 A_l^2$ (where $c_l$ is an empirically determined discharge coefficient) then a two-layer steady stratification becomes established in the room, with outflow through the lower opening and inflow through the upper opening. The interface location depends not only on the geometry of the openings, but also the source conditions. We show that as $Q_0$ increases for fixed $B_0$, the height of the interface, which equals the depth of the lower layer of relatively dense fluid, increases. Eventually, when the source volume flux has a value greater than $Q_m\,{=}\,(c_l A_l)^{2/3}(2B_0 H)^{1/3}$, the natural exchange flow becomes blocked and a steady outflow through both of the openings develops. As a result, the density of the fluid throughout the room gradually evolves towards the density of the incoming dense fluid. We compare our theoretical predictions with a series of laboratory experiments, and discuss the implications of our model for the design of ventilation systems.

It is shown that each structure function $S_{n,m}(r)$, of order $n+m$, in strong turbulence is characterized by its own dissipation scale $\eta_{n,m}$. In the limit $n\rightarrow\infty$, the dissipation scale $\eta_{n,0}= O(\hbox{\it Re}^{-1})$, which is much smaller than the Kolmogorov scale $\eta=O(\hbox{\it Re}^{-{3}/{4}})$, $\hbox{\it Re}$ being the large-scale Reynolds number. This result has implications for the resolution requirements of direct simulations of turbulence. A new rigorous dynamic constraint relating scaling exponents of the structure functions to the codimension of the most singular features of turbulence is derived. A modification of the model by Gotoh & Nakano (2003) for the pressure–velocity correlations, based on the Bernoulli equation, is proposed. This proposal leads to an analytic expression for the scaling exponents of velocity structure functions.

Numerical calculations of the resonant interactions of three-dimensional short-crested waves very near their two-dimensional standing wave limit are performed for water of uniform depth. A detailed study of the properties of the solutions indicates that both classes of waves admit multiple solutions that are connected to each other through turning points. It is also shown that the solutions match each other at the limit. Then a study on the superharmonic instabilities (resonant interactions) of short-crested waves was performed in the vicinity of the standing wave limit. The matching allowed extrapolation of the short-crested wave stability results to standing waves. The results are that for resonant waves, superharmonic instabilities associated with harmonic resonance are dominant. The possible jumps from one solution to another may lead to a drastic change of the wave itself. Since the superharmonic instability enhances this property one may conclude that this class of waves can be considered non-stationary. By contrast, non-resonant waves are weakly unstable or stable and are the only waves that are likely to exist. Thus, this class of waves can be considered as quasi-permanent.

Sound transmission through ducts of constant cross-section with a uniform inviscid mean flow and a constant acoustic lining (impedance wall) is classically described by a modal expansion, where the modes are eigenfunctions of the corresponding Laplace eigenvalue problem along a duct cross-section. A natural extension for ducts with cross-section and wall impedance that are varying slowly (compared to a typical acoustic wavelength and a typical duct radius) in the axial direction is a multiple-scales solution. This has been done for the simpler problem of circular ducts with homentropic irrotational flow. In the present paper, this solution is generalized to the problem of ducts of arbitrary cross-section. It is shown that the multiple-scales problem allows an exact solution, given the cross-sectional Laplace eigensolutions. The formulation includes both hollow and annular geometries. In addition, the turning point analysis is given for a single hard-wall cut-on, cut-off transition. This appears to yield the same reflection and transmission coefficients as in the circular duct problem.

We obtain an analytic solution for the generation of internal gravity waves by tidal flow past a vertical barrier of height $b$ in a uniformly stratified ocean of depth $h\,{>}\,b$ and buoyancy frequency $N$. The radiated power (watts per metre of barrier) is $$\textstyle\frac{1}{4} \pi \rho_0 b^2 U^2 N \sqrt{1-(f/\omega)^2}M(b/h),$$ where $\rho_0$ is the mean density of seawater, $U \cos (\omega t)$ the tidal velocity, and $f$ the Coriolis frequency. The function $M(b/h)$ increases monotonically with $M(0)\,{=}\,1$, $M(0.92)\,{=}\,2$ and $M(1)\,{=}\,\infty$. As $b/h \,{\to}\, 1$, $M$ diverges logarithmically and consequently the radiated power grows as $\ln[(h-b)/b]$. We also calculate the conversion in a realistically stratified ocean with strongly non-uniform buoyancy frequency, $N(z)$. A rough approximation to the radiated power in this case is $$\textstyle\frac{1}{4} \pi \rho_0 b^2 U^2 N(b) \sqrt{1- (f/\omega)^2} M(B/\pi),$$ where $N(b)$ is the buoyancy frequency at the tip of the ridge and $B$ is the height of the ridge in WKB coordinates. (The WKB coordinate is normalized so that the total depth of the ocean is $\pi$.) The approximation above is an over-estimate of the actual radiation by as much as 20% when $B/\pi \,{\approx}\, 0.8$. But the formula correctly indicates the strong dependence of conversion on stratification through the factor $N(b)$.

This paper presents a two-dimensional depth-integrated theory for the gravity-driven free-surface flow of a granular avalanche over an arbitrarily but gently curved and twisted topography which is an important extension of the original Savage & Hutter theory. In contrast to previous extensions the present coordinate system is based on a reference curve with curvature and torsion. Its derivation was necessary because real avalanches are often guided by rather curved and twisted valleys or more general slopes. The aim is to gain fundamental insight into the effects of non-uniform curvature and torsion, using an orthogonal coordinate system that rotates with torsion, and find an analytic description of flow avalanches. We present a set of model equations which comprises nonlinear hyperbolic partial differential equations for the space and time evolution of the granular pile height and the depth-averaged streamwise velocity distribution of a finite mass of granulates. The emerging theory is believed to be capable of predicting the flow of dense granular materials over moderately curved and twisted channels of general type.

An experimental investigation is made into the active control of the near-wall region of a turbulent boundary layer ($Re_\theta \,{=}\, 1960$) using a linear active control scheme. System identification in the boundary layer provides optimal transfer functions that predict the downstream characteristics of the streamwise velocity fluctuations. Enhanced detection techniques isolate the large-scale turbulent motion and improve the downstream correlations, resulting in greater controllability. The control is applied using a spanwise array of resonant synthetic jet actuators that introduce pairs of streamwise vortices into the flow. Control results show that a maximum reduction of 30% in the streamwise velocity fluctuations is achieved. This reduction is greatest at the point of optimization but spans a few hundred viscous lengths downstream of the actuator, about 50 viscous lengths in the wall-normal direction and 150 viscous lengths in the spanwise direction. The wall pressure fluctuation and the mean wall shear stress (measured approximately using mean velocity profiles near the wall) were reduced by 15% and 7% respectively. The bursting frequency, based on VITA event detection was also reduced by up to 23%.

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 viscous and steady flow about two distinct parallel infinite rotating circular cylinders is theoretically investigated. Because any inner steady Stokes flow is not quiescent far from the cylinders, a strictly steady analysis requires matching an inner steady Stokes approximation with an outer solution of the steady Navier–Stokes equations. However, except for the case of identical cylinders of equal angular velocities, it is impossible to determine this outer solution. In the same spirit as Nakanishi et al. (1997) and Ueda et al. (2001), the present work therefore first addresses the unsteady viscous flow induced by cylinders impulsively set into both steady rotation and translation ${\bm W}.$ Using integral representations of the stream function and the vorticity, the resulting long-time flow is approximated in the limit of large viscosity. Letting time tend to infinity for ${\bm W}$ non-zero extends Lee & Leal (1986) and agrees with Watson (1996), whereas the required steady flow is obtained by making ${\bm W}$ vanish before letting time go to infinity. At the obtained leading-order approximation, the ‘lift’ and ‘drag’ forces on each cylinder (parallel and normal to the line of centres) are respectively zero and independent of the Reynolds number. The drag experienced by each body is plotted versus the gap between the cylinders for several values of the rotation, both for identical and non-identical cylinders.

The aim of this paper is the derivation of a multiphase model of compressible fluids. Each fluid has a different average translational velocity, density, pressure, internal energy as well as the energies related to rotation and vibration. The main difficulty is the description of these various translational, rotational and vibrational motions in the context of a one-dimensional model. The second difficulty is the determination of closure relations for such a system: the ‘drag’ force between inviscid fluids, pressure relaxation rate, vibration and rotation creation rates, etc. The rotation creation rate is particularly important for turbulent flows with shock waves. In order to derive the one-dimensional multiphase model, two different approaches are used. The first one is based on the Hamilton principle. This method gives thermodynamically consistent equations with a clear mathematical structure, coupling the various motions: translation, rotation and vibration. However the relaxation effects have to be added phenomenologically. In order to achieve the closure of the system and its numerical resolution we use the second approach, in which the pure fluid equations are discretized at the microscopic level and then averaged. In this context, the flow is considered to be the annular flow of two turbulent fluids. We also derive the continuous limit of this model which provides explicit formulae for the closure laws. The structure of this system of partial differential equations is the same as the one obtained by the Hamilton principle. The final issue is to determine the rate of energy (or entropy) rotation. We assume that all the entropy creation related to the various relaxation effects after the passage of the shock wave is converted to rotational motion. The one-dimensional model is validated by comparing its predictions with averaged two-dimensional direct numerical results. The problem on which this model is tested is the interaction of a shock wave propagating in a heavy gas with a light gas bubble. The results obtained by the one-dimensional multiphase model are in a very good agreement with the two-dimensional averaged results.

Turbulent channel flow simulations are performed using second- and fourth-order finite difference codes. A systematic comparison of the large-eddy simulation (LES) results for different grid resolutions, finite difference schemes, and several turbulence closure models is performed. The use of explicit filtering to reduce numerical errors is compared to results from the traditional LES approach. Filter functions that are smooth in spectral space are used, as the findings of this investigation are intended for application of LES to complex domains. Explicit filtering introduces resolved subfilter-scale (RSFS) as well as subgrid-scale (SGS) turbulence terms. The former can be theoretically reconstructed; the latter must be modelled. The dynamic Smagorinsky model, the dynamic mixed model, and the new dynamic reconstruction model are all studied. It is found that for explicit filtering, increasing the reconstruction levels for the RSFS stress improves the mean velocity as well as the turbulence intensities. When compared to LES without explicit filtering, the difference in the mean velocity profiles is not large; however the turbulence intensities are improved for the explicit filtering case.

A turbulent Ekman layer created by a steady wind near the water surface is investigated using the numerical method of large-eddy simulations. The classical case of a flow unaffected by density stratification and surface waves is revisited to understand the internal structure of the flow and implications of the traditional assumptions of constant effective viscosity and the ‘$f$-plane’ approximation. A series of numerical experiments reveals that the Ekman solution needs correcting even in this case. The examination of the effective viscosity hypothesis confirms its validity but shows that the viscosity varies strongly with depth. It increases in the subsurface layer of thickness about 1/4 the turbulent length scale and decreases below this level. A Bessel function solution is proposed that corresponds to the approximate effective viscosity profile and matches with the LES results. Strong flow dependence on the latitude and wind direction is detected and explained by the effects of redistribution of turbulent kinetic energy between the velocity components and modification of the vertical transfer of turbulent momentum.

A three-dimensional boundary-layer model of shallow-water flows assuming hydrostatic pressure with negligible numerical diffusion and wave damping has been extended to turbulent flow. A standard two-layer mixing-length model determines vertical length scales. The horizontal mixing length is made a multiple $\beta$ of the vertical value and $\beta$ is determined from comparison with experiment. Eddy viscosity is of a general three-dimensional form where, for example, the horizontal mixing length and associated strain rates determine the magnitude of eddy viscosity and hence vertical mixing (and vice versa). Direct comparison is made with previous experiments for subcritical flow around a conical island of small side slope which exhibits the transition from a vigorous vortex-shedding wake to a steady recirculating wake as the stability parameter, $St$, is increased. The value of $\beta$ influences wake structure, particularly for stability parameters close to the critical (the value at which the wake becomes steady or stable). The critical value in the experiments was 0.4 and this was reproduced in the model with $\beta \,{=}\,6$. Vortex shedding patterns with $\hbox{\it St} \,{=}\, 0.26$ and 0.36 were qualitatively reproduced. The flows were subcritical with an onset Froude number of about 0.2, with values approaching 0.6–0.7 in areas where depth-averaged vorticity magnitude was also greatest, at a small distance from the wet/dry intersection. At this intersection, depth-averaged vorticity approached zero while potential vorticity (depth-averaged vorticity/depth) was at a maximum, indicating the importance of the intersection as an origin for vorticity.

Finite-amplitude capillary waves, which can accompany the axisymmetric flow of a thin viscous film over a rotating disk, are considered. A system of approximate evolution equations for the film thickness and volumetric flow rates in the radial and azimuthal directions is derived, which contains two similarity parameters. In order to inspire confidence in this model, its steady solutions and their linear stability characteristics are compared to those of the full Navier–Stokes equations. Localized equations, which account for the presence of inertial, capillary, centrifugal and Coriolis forces, are obtained via truncation of the approximate system. Periodic solutions of these equations are then determined and found to be similar to those observed experimentally. Our results suggest that the steady quasi-periodic waves with largest amplitude compare well with experimentally observed wave profiles.

Journal of Fluid Mechanics, vol. 471 (2002), pp. 203–217