The flow in a channel with its lower wall mounted with streamwise riblets is simulated using a highly efficient spectral element-Fourier method. The range of Reynolds numbers investigated is 500 to 3500, which corresponds to laminar, transitional, and turbulent flow states. A complete study is presented for V-groove riblets; the effect of rounded riblets is also investigated. Our results suggest that in the laminar regime there is no drag reduction, while in the transitional and turbulent regimes drag reduction exists (approximately 6 % at Reynolds number 3500) for the riblet-mounted wall in comparison with the smooth wall of the channel. For the first time, we present detailed turbulent statistics (turbulence intensities, Reynolds shear stresses, skewness and flatness) as well as a temporal analysis using a numerical analog of the VITA technique. The flow structure over the riblet-mounted wall is also analysed in some detail and compared with the corresponding flow over the smooth wall in an attempt to identify the physical mechanisms that cause drag reduction. The accuracy of the computation is established by comparing flow quantities corresponding to the smooth wall with previous direct numerical simulation results as well as with experimental results; on the riblet-mounted wall comparison is made with available experimental results. The agreement is very good for both cases. The current computation is the first direct numerical simulation of turbulence in a complex geometry domain.

The three-dimensional turbulent boundary layer is shown to have a self-consistent two-layer asymptotic structure in the limit of large Reynolds number. In a streamline coordinate system, the streamwise velocity distribution is similar to that in two-dimensional flows, having a defect-function form in the outer layer which is adjusted to zero at the wall through an inner wall layer. An asymptotic expansion accurate to two orders is required for the cross-stream velocity which is shown to exhibit a logarithmic form in the overlap region. The inner wall-layer flow is collateral to leading order but the influence of the pressure gradient, at large but finite Reynolds numbers, is not negligible and can cause substantial skewing of the velocity profile near the wall. Conditions under which the boundary layer achieves self-similarity and the governing set of ordinary differential equations for the outer layer are derived. The calculated solution of these equations is matched asymptotically to an inner wall-layer solution and the composite profiles so formed describe the flow throughout the entire boundary layer. The effects of Reynolds number and cross-stream pressure gradient on the cross-stream velocity profile are discussed and it is shown that the location of the maximum cross-stream velocity is within the overlap region.

We present novel measurements of the primary instabilities of thin liquid films flowing down an incline. A fluorescence imaging method allows accurate measurements of film thickness h(x, y, t) in real time with a sensitivity of several microns, and laser beam deflection yields local measurements with a sensitivity of less than one micron. We locate the instability with good accuracy despite the fact that it occurs (asymptotically) at zero wavenumber, and determine the critical Reynolds number Rc for the onset of waves as a function of angle β. The measurements of Rc(β) are found to be in good agreement with calculations, as are the growth rates and wave velocities. We show experimentally that the initial instability is convective and that the waves are noisesustained. This means that the waveform and its amplitude are strongly affected by external noise at the source. We investigate the role of noise by varying the level of periodic external forcing. The nonlinear evolution of the waves depends strongly on the initial wavenumber (or the frequency f). A new phase boundary f*s(R) is measured, which separates the regimes of saturated finite amplitude waves (at high f) from multipeaked solitary waves (at low f). This boundary probably corresponds approximately to the sign reversal of the third Landau coefficient in weakly nonlinear theory. Finally, we show that periodic waves are unstable over a wide frequency band with respect to a convective subharmonic instability. This instability leads to disordered two-dimensional waves.

At low wind speeds the shortest capillary waves appear to be generated hydrodynamically and not by the wind. This phenomenon is investigated using a Hamiltonian representation of the surface wave dynamics. A perturbation technique of Kolmogorov is used to transform away non-resonant, nonlinear interactions. Resonant interactions are treated by the Hasselmann transport equation, applied to the transformed variables. Calculated spectra show reasonable agreement with the observations of Jähne & Riemer (1990).

The effect of the steady second-order velocities on the drift forces and moments acting on marine structures in waves and a (small) current is considered. The second-order velocities are found to arise due to first-order evanescent modes and linear body responses. Their contributions to the horizontal drift forces and yaw moment, obtained by pressure integration at the body, and to the yaw drift moment, obtained by integrating the angular momentum flux in the far field, are expressed entirely in terms of the linear first-order solution. The second-order velocities may considerably increase the forward speed part of the mean yaw moment on realistic marine structures, with the most important contribution occurring where the wave spectrum often has its maximal value. The contribution to the horizontal forces obtained by pressure integration is, however, always found to be small. The horizontal drift forces obtained by the linear momentum flux in the far field are independent of the second-order velocities, provided that there is no velocity circulation in the fluid.

The dynamical behaviour of stretchable, orientable microstructure suspended in a general three-dimensional fluid flow is investigated. Model equations given by Olbricht, Rallison & Leal (1982) are examined in the case of microstructure travelling through arbitrarily complicated flows of the carrier fluid. As in the two-dimensional analysis of Szeri, Wiggins & Leal (1991), one must first treat the orientation dynamics problem; only then can the equation for stretch of the microstructure be analyzed rationally. In three-dimensional flows that are steady in the Lagrangian frame, attractors for the orientation dynamics are shown to be equilibria or limit cycles; this asymptotic behaviour was first deduced by Bretherton (1962). In three-dimensional flows that are time periodic in the Lagrangian frame (e.g. recirculating flows), the orientation dynamics may be characterized by periodic or quasi-periodic attractors. Thus, robust (generic) behaviour in these cases is always characterized by a single global attractor; there is no asymptotic dependence of orientation dynamics on the initial orientation. The type of asymptotic orientation dynamics – steady, periodic, or quasi-periodic - is signified by a simple criterion. Details of the relevant bifurcations, as well as history-dependent strong flow criteria are developed. Examples which illustrate the various types of behaviour are given.

The linear, nonlinear and breakdown stages in the transition of localized disturbances in plane Poiseuille flow is studied by direct numerical simulations and analysis of the linearized Navier–Stokes equations. Three-dimensionality plays a key role and allows for algebraic growth of the normal vorticity through the linear lift-up mechanism. This growth primarily generates elongated structures in the streamwise direction since it is largest at low streamwise wavenumbers. For finite-amplitude disturbances such structures will be generated essentially independent of the details of the initial disturbance, since the preferred nonlinear interactions transfer energy to low streamwise wavenumbers. The nonlinear interactions also give a decrease in the spanwise scales. For the stronger initial disturbances the streamwise vorticity associated with the slightly inclined streaks was found to roll up into distinct streamwise vortices in the vicinity of which breakdown occurred. The breakdown starts with a local rapid growth of the normal velocity bringing low-speed fluid out from the wall. This phenomenon is similar to the low-velocity spikes previously observed in transition experiments. Soon thereafter a small turbulent spot is formed. This scenario represents a bypass of the regular Tollmien–Schlichting, secondary instability process. The simulations have been carried out with a sufficient spatial resolution to ensure an accurate description of all stages of the breakdown and spot formation processes. The generality of the observed processes is substantiated by use of different types of initial disturbances and by Blasius boundary-layer simulations. The present results point in the direction of universality of the observed transition mechanisms for localized disturbances in wall-bounded shear flows.

We examine thermal convection in a rotating spherical shell with a spatially non-uniformly heated outer surface, concentrating on three distinct heating modes: first, with wavelength and symmetry corresponding to the most unstable mode of the uniformly heated problem; secondly, with the critical wavelength but opposite equatorial symmetry; and thirdly, with wavelength much larger than that of the most unstable mode. Analysis is focused on boundary-locked convection, the associated spatial resonance phenomena, the stability properties of the resonance solution, and time-dependent secondary convection. A number of new forms of instability and convection are found: the most interesting is perhaps the saddle-node bifurcation, which is the first to be found for realistic fluid systems governed by partial differential equations. An analogous Landau amplitude equation is also analysed, providing an important mathematical framework for understanding the complicated numerical solutions.

This paper describes numerical and asymptotic solutions of the steady two-dimensional boundary-layer equations governing buoyant flow on a horizontal, thermally insulated surface. The class of flows considered is one for which there is a uniform external stream at constant temperature but for which conditions upstream lead to a statically stable temperature field within the boundary layer. This has the effect of generating an adverse pressure gradient which, if sufficiently strong, causes the boundary-layer solution to terminate in a singularity. Results are obtained for a range of Prandtl numbers.

New experimental results are reported for the motion of a liquid surface caused by the heat released from a hot wire below the surface. Starting from a base state with steady convection and steady deformation of the free surface caused by variations in surface tension and heat transport to the surface, the system loses its stability through a supercritical Hopf bifurcation occurring on a curve f(ΔT*, d) = 0 in which d is the distance between hot wire and surface and ΔT* a critical temperature difference. These experiments are a model for more complex laser heating experiments in which chaotic motions may occur. Some emphasis is placed on the characterization of propagating waves produced on the surface after the occurrence of the bifurcation.

An experimental and theoretical study of the mixing produced by a plume rising in a confined stratified environment is presented. As a result of the pre-existing stable stratification, the plume penetrates only part way into the region; at an intermediate level it intrudes laterally forming a horizontal layer. As time evolves, this layer of mixed fluid is observed to increase in thickness. The bottom front advects downward in a way analogous to the first front in the filling box of Baines & Turner (1969), while the lateral spreading of the plume occurs at an ever-increasing level and an ascending top front results. We develop a model of this stratified filling box; the model predicts the rate at which the two fronts advance into the environment.

It is found that stratification in the environment, when smooth, has no significant influence on the dynamics of the descending front. We show that the rate of rise of the ascending front is determined by the turbulent mixing occurring at the spreading level. Entrainment of environmental fluid from above into the overshooting plume is significant; as a result, a density interface develops at this level. Asymptotically, the system reaches a state in which a bottom convecting layer, with an almost homogeneous density, deepens in a stratified background. The model proposed for this large-time behaviour is based on the simple energetic formulation that a constant fraction of the kinetic energy supplied by the plume, for mixing across the interface, is converted into potential energy of the convective layer. Our experimental results suggest an efficiency of approximately 50 % for this conversion.

We discuss our results in the light of previous studies on turbulent penetrative convection and conclude that the theory developed should be valid for an intermediate range of values of the Richardson number characterizing the dynamic conditions at the interface. The model is applied quantitatively to the process of cooling of a room wherein stratification is relevant. The geological problem of replenishment of a magma chamber by a light input of magma is also analysed.

Results of dynamic simulations of bubbles rising through a liquid are presented. The Reynolds number of the flow based on the radius and the terminal speed of bubbles is large compared to unity, and the Weber number, which is the ratio of inertial to surface tension forces, is small. It is assumed that the bubbles do not coalesce when they approach each other but rather bounce instantaneously, conserving the momentum and the kinetic energy of the system. The flow of the liquid is assumed to be irrotational and is determined by solving the many-bubble interaction problem exactly. The viscous force on the bubbles is estimated from the rate of viscous energy dissipation. It is shown that the random state of bubbly liquids under these conditions is unstable and that the bubbles form aggregates in planes transverse to gravity. These aggregates form even when the size distribution of the bubbles is non-uniform. While the instability results primarily from the nature of inertial interaction among pairs of bubbles, which causes them to be attracted toward each other when they are aligned in the plane perpendicular to gravity, it is shown that the presence of viscous forces facilitates the process.

Gravity currents created by the release of a fixed volume of a suspension into a lighter ambient fluid are studied theoretically and experimentally. The greater density of the current and the buoyancy force driving its motion arise primarily from dense particles suspended in the interstitial fluid of the current. The dynamics of the current are assumed to be dominated by a balance between inertial and buoyancy forces; viscous forces are assumed negligible. The currents considered are two-dimensional and flow over a rigid horizontal surface. The flow is modelled by either the single- or the two-layer shallow-water equations, the two-layer equations being necessary to include the effects of the overlying fluid, which are important when the depth of the current is comparable to the depth of the overlying fluid. Because the local density of the gravity current depends on the concentration of particles, the buoyancy contribution to the momentum balance depends on the variation of the particle concentration. A transport equation for the particle concentration is derived by assuming that the particles are vertically well-mixed by the turbulence in the current, are advected by the mean flow and settle out through the viscous sublayer at the bottom of the current. The boundary condition at the moving front of the current relates the velocity and the pressure head at that point. The resulting equations are solved numerically, which reveals that two types of shock can occur in the current. In the late stages of all particle-driven gravity currents, an internal bore develops that separates a particle-free jet-like flow in the rear from a dense gravity-current flow near the front. The second type of bore occurs if the initial height of the current is comparable to the depth of the ambient fluid. This bore develops during the early lock-exchange flow between the two fluids and strongly changes the structure of the current and its transport of particles from those of a current in very deep surroundings. To test the theory, several experiments were performed to measure the length of particle-driven gravity currents as a function of time and their deposition patterns for a variety of particle sizes and initial masses of sediment. The comparison between the theoretical predictions, which have no adjustable parameters, and the experimental results are very good.

Stokes flow in a two-dimensional cavity of rectangular section, induced by the motion of one of the walls, is considered. A direct, efficient calculational procedure, based on an eigenfunction expansion, is used to study the eddy structure in the cavity. It is shown that some of the results of earlier studies are quantitatively in error. More importantly, two interesting questions, namely the extent of the symmetry of the corner eddies and their relationship to the large-eddy structure are settled. By carefully examining the rather sudden change in the main eddy structure for cavities of depth around 1.629, it is shown that the main eddies are formed by the merger of the primary corner eddies; the secondary corner eddies then become the primary corner eddies and so on. Thus, in the evolution of the large-eddy structure the corner eddies, in some sense, play the role of progenitors. This explicit prediction should be experimentally verifiable.

The disturbances on the free surface of dielectric fluids resulting from intense laser heating of their boundary layer are studied theoretically and experimentally. The heating is accompanied by pronounced evaporation from the surface and thereby leads to a recoil pressure momentum applied to the surface. For small values of total momentum transferred to the fluid, the low-amplitude initially hollow-like displacement of the surface in the impact zone decays to produce linear gravity–capillary waves (GCW) spreading out on the surface. This regime is treated analytically and the results obtained are compared with experiments involving weakly viscous (water, ethanol) and highly viscous (glycerol) liquids. An experimental arrangement for remote generation and subsequent detection of probe GCW-packets is given. The evolution of broadband GCW-disturbances on clean and surfactant-contaminated water surfaces are described. Results of GCW-attenuation spectrum measurements on clean water surfaces and on film-covered surfaces are presented.

High total recoil momentum values give rise to substantially nonlinear surface motion: after a short transient stage the surface takes the shape of a hemisphere expanding into the liquid, and later the liquid above the hemisphere closes up to form a cavity and slow down the expansion. For this regime the dynamics of the hemisphere expansion are determined and satisfactory agreement with experimental data obtained with the shadowgraph technique is established. Consistency of theory and experiment allowed the determination of the total recoil pressure momentum and its surface distribution.

In the intermediate case of moderate values of recoil momentum, the nonlinear evolution of broadband GCW-packets on clean and surfactant-contaminated water surfaces is investigated experimentally.

Stably stratified and non-stratified flows past a touching pair of vortices with continuous velocity are considered. An asymptotic solution for the very long eddies is determined. Numerical results cover the whole range of subcritical stratification and eddy length.

Wave formation on a falling film is an intriguing hydrodynamic phenomenon involving transitions among a rich variety of spatial and temporal structures. Immediately beyond an inception region, short, near-sinusoidal capillary waves are observed. Further downstream, long, near-solitary waves with large tear-drop humps preceded by short, front-running capillary waves appear. Both kinds of waves evolve slowly downstream such that over about ten wavelengths, they resemble stationary waves which propagate at constant speeds and shapes. We exploit this quasi-steady property here to study wave evolution and selection on a vertically falling film. All finite-amplitude stationary waves with the same average thickness as the Nusselt flat film are constructed numerically from a boundary-layer approximation of the equations of motion. As is consistent with earlier near-critical analyses, two travelling wave families are found, each parameterized by the wavelength or the speed. One family γ1 travels slower than infinitesimally small waves of the same wavelength while the other family γ2 and its hybrids travel faster. Stability analyses of these waves involving three-dimensional disturbances of arbitrary wavelength indicate that there exists a unique nearly sinusoidal wave on the slow family γ1 with wavenumber αs (or α2) that has the lowest growth rate. This wave is slightly shorter than the fastest growing linear mode with wavenumber αm and approaches the wave on γ1 with the highest flow rate at low Reynolds numbers. On the fast γ2 family, however, multiple bands of near-solitary waves bounded below by αf are found to be stable to two-dimensional disturbances. This multiplicity of stable bands can be interpreted as a result of favourable interaction among solitary-wave-like coherent structures to form a periodic train. (All waves are unstable to three-dimensional disturbances with small growth rates.) The suggested selection mechanism is consistent with literature data and our numerical experiments that indicate waves slow down immediately beyond inception as they approach the short capillary wave with wavenumber α2 of the slow γ1 family. They then approach the long stable waves on the γ2 family further downstream and hence accelerate and develop into the unique solitary wave shapes, before they succumb to the slowly evolving transverse disturbances.

In an experimental study of the vortex-induced oscillations of a long flexible circular cylinder, the observed stationary amplitudes describe an hysteresis loop partially different from earlier studies. Each branch of the loop is associated with a vortex shedding mode and, as a jump from one branch to the other occurs, the phase difference between the cylinder displacement and the vortex shedding undergoes an abrupt change. The critical flow velocities at which the jump occurs concur with the flow visualization observations of Williamson & Roshko (1988) on the vortex shedding modes near the fundamental synchronization region. Impulsive regimes, obtained at a given flow velocity with the cylinder initially at rest or pre-excited, and progressive regimes resulting from a variation of the flow velocity, are examined. The occurrence of bifurcations is detected for a flow velocity range in the case of the impulsive regimes. The coordinates of the bifurcations define a boundary between two vortex shedding modes, a boundary that verifies the critical curve obtained by Williamson & Roshko (1988). The experimental set-up of this study simulates half the wavelength of a vibrating cable, eliminates the end effects present in oscillating rigid cylinder set-up and has one of the lowest damping ratios reported for the study of this phenomenon.

The constant Cμ and the near-wall damping function fμ in the eddy-viscosity relation of the k–ε model are evaluated from direct numerical simulation (DNS) data for developed channel and boundary-layer flow, each at two Reynolds numbers. Various existing fμ model functions are compared with the DNS data, and a new function is fitted to the high-Reynolds-number channel flow data. The ε-budget is computed for the fully developed channel flow. The relative magnitude of the terms in the ε-equation is analysed with the aid of scaling arguments, and the parameter governing this magnitude is established. Models for the sum of all source and sink terms in the ε-equation are tested against the DNS data, and an improved model is proposed.

Parallel cold wires were used to measure the temperature derivative, in each of the three spatial directions, in the self-preserving region of a turbulent round jet. The temperature derivative variances were inferred from the correlation method and from the temperature derivative spectra after correcting these for the effect of wire separation. Both methods yielded fully consistent results for the components of the average temperature dissipation: the radial and azimuthal values are nearly equal and only slightly larger than the axial component. The resulting departure from isotropy of the temperature dissipation is small, especially when compared with results in other free shear flows. The high-wavenumber behaviour of the corrected temperature derivative spectra conforms closely with isotropy on the jet axis but small departures occur away from the axis. Conditional averages, based on spatially coherent temperature jumps, indicate that, while the organized motion makes a significant contribution to the temperature variance, its contribution to the temperature-derivative variances is small.