A linear stability theory for dune and antidune formation is presented that implements a rotational two-dimensional flow model. As in previous linear theories, the phase-lag between sediment transport and bed elevation remains the main mechanism driving instability. However, it is shown that this phase-lag varies significantly in a neighbourhood of the bed. Moreover, since the layer in which sediments are moving has a finite (though small) thickness, it is assumed that the perturbations of the fluid stress driving bedload transport should be evaluated at the top of the layer itself. It is shown that such an apparently minor modification of the classical approach alters remarkably the balance between stabilizing and destabilizing effects that drives the instability process. This also allows a clarification of the debated role of bedslope on the formation of dunes and antidunes. Following the above ideas, antidunes are shown to form in the absence of suspended sediment load and without any effect associated with sediment inertia. The present analysis ultimately allows a successful unification of the theories of dune and antidune formation.

The shallow-ice approximation is used to analyse the normal-mode stability of long-wavelength, thermally coupled, free-surface flows of glaciers and ice sheets along infinite flow sections. The viscosity of the ice changes with temperature, and the ice is also thermally coupled with the bedrock. An assumption of quasi-uniform flow, where the downstream flux of mass and energy is assumed to be constant over relatively short wavelengths, permits vertical advection due to accumulation to be considered. The assumption allows realistic representation of the vertical velocity distribution and temperature distributions within the modelled flow. Nevertheless, the linearized equations are formally independent of the assumption of quasi-uniform flow. The linearized equations are Fourier-transformed over the horizontal plane and an algebraic eigenvalue problem constructed which considers the Fourier-transforms of the vertically varying temperature and thickness. The dependence of the eigenvalues on wavenumber, both parallel and orthogonal to flow, accumulation rate, geothermal heat flux, surface temperature and slope is computed. Extensive linearly unstable regimes where the base is well below melting point are found, which correspond only approximately to negative slopes in the flux/thickness relationship (supercritical zones). The most unstable parts of these regimes have finite along-flow wavelength, and have very long transverse wavelength. These modes are found to be more unstable in two senses: (i) they bifurcate for lesser thicknesses; (ii) they have greater growth rates. All computed wave velocities of unstable modes were downslope. Less unstable modes with short and medium transverse wavelengths are found which excite the temperature field only. Time constants range between ten and one thousand years. There is no strong evidence for stream-pattern formation through a thermoviscous instability mechanism.

The coalescence of two co-flowing axisymmetric turbulent plumes and the resulting single plume flow is modelled and compared to experiments. The point of coalescence is defined as the location at which only a single peak appears in the horizontal buoyancy profile, and a prediction is made for its height. The model takes into account the drawing together of the two plumes due to their respective entrainment fields. Experiments showed that the model tends to overestimate the coalescence height, though this discrepancy may be partly explained by the sensitivity of the prediction to the entrainment coefficient. A model is then developed to describe the resulting single plume and predict its virtual origin. This prediction and subsequent predictions of flow rate above the merge height compare very well with experimental results.

A model fractal forcing for direct numerical simulations (DNS) of stationary, homogeneous and isotropic turbulence is proposed. The power spectrum of this forcing is a power-law function of wavenumber with a positive exponent that is an increasing function of $D_{f}$, the fractal dimension of the fractal stirrer. The following results are obtained for DNS turbulence subjected to fully self-similar fractal forcing of fractal dimension $D_f$. (i) The Taylor and Kolmogorov microscales are proportional to each other and to the smallest length scale of the fractal forcing. (ii) The integral length scale is much smaller than the size $L_b$ of the DNS box and a decreasing function of the extent of the fractal forcing range because fractal forcing generates very irregular velocity fields. (iii) In qualitative (but not quantitative) agreement with renormalization group (RG) theories of turbulence, higher values of $D_f$ lead to increased energy at the highest wavenumbers. (iv) The energy and energy input rate spectra and the inter-scale energy transfer $T(k)$ all scale with the turbulence r.m.s. velocity $u'$ and the Taylor microscale $\lambda$. (v) More than 80% of the total dissipation occurs in the fractal forcing range of scales which extends from $L_b$ to about one to two times the Taylor microscale. In that range, $T(k)$ is negligible. (vi) $\lambda \,{\sim}\, \nu/u'$ (where $\nu$ is the kinematic viscosity), the kinetic energy dissipation rate per unit mass $\epsilon \,{\sim}\, u'^{3}/\lambda \,{\sim}\,u'^{4}/\nu$ and the velocity derivative skewness $S$ is independent of Reynolds number in the limit where the Reynolds number and the fractal forcing range are increased together. (vii) The intermediate eigenvalue of the strain rate tensor is on average positive, and the negligible values of $T(k)$ in the fractal forcing range are accompanied by lower values of $S$ and a significant reduction in local compression by comparison to turbulence forced only at the large scales. (viii) The geometrical alignments between vorticity, strain rate tensor eigenvectors and vortex stretching vector are qualitatively as in turbulence forced only at the large scales but significantly weakened. (ix) The p.d.f.s of velocity increments are approximately, though not exactly, Gaussian at all scales between the Kolmogorov and integral length scales. A few preliminary DNS results are also given for the case of turbulence generated by a fractal forcing that is discretely, as opposed to fully, self-similar.

We consider the dynamics of capillary attraction between an articulated train of rigid rods floating at a liquid–gas interface and a nearby wall. We then explain some of the phenomena that are a result of the strong anisotropy and the extended nature of the system, such as the lining up next to the walling in a ‘zippering’ motion that is observed and compare our results qualitatively with those of experiments.

The flow in a stationary open cylinder driven by the constant rotation of the bottom endwall is unstable to three-dimensional perturbations for sufficiently large rotation rates. The bifurcated state takes the form of a rotating wave. Two distinct physical mechanisms responsible for the symmetry breaking are identified, which depend on whether the fluid depth is sufficiently greater or less than the cylinder radius. For deep systems, the rotating wave results from the instability of the near-wall jet that forms as the boundary layer on the rotating bottom endwall is turned into the interior. In this case the three-dimensional perturbations vanish at the air/water interface. On the other hand, for shallow systems, the fluid at radii less than about half the cylinder radius is in solid-body rotation whereas the fluid at larger radii has a strong meridional circulation. The interface between these two regions of flow is unstable to azimuthal disturbances and the resulting rotating wave persists all the way to the air/water interface. The flow dynamics are explored using three-dimensional Navier–Stokes computations and experimental results obtained via digital particle image velocimetry. The use of a flat stress-free model for the air/water interface reproduces the experimental results in the deep system but fails to capture the primary instability in the shallow system, even though the experimental imperfections, i.e. departures from a perfectly flat and clean air/water interface, are about the same for the deep and the shallow systems. The flat stress-free model boundary conditions impose a parity condition on the numerical solutions, and the consideration of an extended problem which reveals this hidden symmetry provides insight into the symmetry-breaking instabilities.

New scaling laws for turbulent boundary layers recently derived (see Oberlack 2000) using Lie group symmetry methods have been tested against experimental data from the KTH database for zero-pressure-gradient turbulent boundary layers. The most significant new law predicts an exponential variation of the mean velocity defect in the outer (wake) region. It was shown to fit the experimental data very well over a large part of the boundary layer, from the outer part of the overlap region to about half the boundary layer thickness ($\delta_{99}$). In the outermost part of the boundary layer the velocity defect falls more rapidly than predicted by the exponential law. This can partly be attributed to intermittency in that region but the main cause stems from non-parallel effects that are not accounted for in the derivation of the exponential law. The two-point correlation function behaviour in the outer region, where an exponential velocity defect law is observed, was found to be very different from that derived under the assumption of parallel flow. It is found to be plausible that this indeed can be attributed to non-parallel effects. A small modification of the innermost part of the log-layer in the form of an additive constant within the log-function is predicted by the Lie group symmetry method. A qualitative agreement with such a behaviour just below the overlap region was found. The derived scaling law behaviour in the overlap region for the two-point correlation functions was also verified by the experimental data.

Envelope equations are derived for the convection rolls in the Rayleigh–Bénard–Poiseuille system, taking into account both their slow streamwise and transverse variations. At finite $O(1)$ Reynolds numbers, the stability of finite-amplitude longitudinal roll patterns is accessible to analysis in a moving frame of reference and stability is predicted provided a generalized Eckhaus criterion is satisfied. At lower Reynolds numbers, the analysis allows the analytical determination of the Green function for arbitrary orientations of the instability pattern. It clarifies previous results concerning the purely convective nature of all modes of instability except transverse rolls (for which a convective–absolute transition exists), as soon as the Reynolds number is non-zero.

Linear global modes in the Rayleigh–Bénard–Poiseuille system, for the case of two-dimensional non-uniform heating in the form of a single hot spot, are analysed in the framework of the envelope equation formalism. Global mode solutions are sought by means of WKBJ asymptotics. As for the one-dimensional case, an analytical selection criterion for the frequency may be derived from the breakdown of the WKBJ expansion at a two-dimensional double turning point located at the maximum of the local Rayleigh number. The analytical results, including the behaviour of the mode in the vicinity of the turning point, are compared with results obtained from numerical simulations of the envelope equation. Finally, the issue of the selection of the wavevector branches in the WKBJ expansion is discussed.

Gas flow over a flat-plate airfoil at very-low Reynolds number is investigated in order to understand the aerodynamic issues related to micro air vehicle design and performance. Studies have shown that such low Reynolds number flow exhibits rarefied phenomena and a flat plate having a thickness ratio of 5% has better aerodynamic performance than conventional streamlined airfoils. This paper simulates air flows over a 5% flat plate using a hybrid continuum–particle approach for flows having a Mach number of 0.2 and a Reynolds number varying between 1 and 200. Investigation shows that low Reynolds number flows are viscous and compressible, and rarefied effects increase when the Reynolds number decreases. It is also found that there is a minimum lift slope for the plate airfoil at a Reynolds number near 10 and the drag coefficient monotonically increases with decreasing Reynolds number.

It is well-known that oceanic vortices exist for years, whereas almost all theoretical studies indicate that they must be unstable. A rare exception is the work by Dewar & Killworth (1995), who demonstrated that a Gaussian vortex in the upper layer of a two-layer ocean becomes stable if accompanied by a weak co-rotating circulation in the lower layer. Note that this paper assumed the lower-layer circulation to have the same profile as the main (upper-layer) vortex. The present paper considers the case where the profile of the circulation in the lower layer is determined by the condition that potential vorticity (PV) there is constant – which models a vortex surrounded by water of a different origin. Given that most oceanic vortices are shed by frontal currents, such model appears to be more realistic than any ad hoc choice. The stability of vortices with uniform lower-layer PV is examined for both quasigeostrophic and ageostrophic cases, numerically and asymptotically, assuming that the upper layer of the ocean is thin. It is shown that such vortices are stable for a wide range of parameters. The effect of vortex stabilization is interpreted through representation of the unstable disturbance by two phase-locked Rossby waves, rotating around the vortex in the upper and lower layers. Then, if the lower-layer PV gradient is zero, it cannot support the corresponding wave, which inhibits the instability.

The self-similar evolution to turbulence of a multi-mode Rayleigh–Taylor mix at small density differences ($A_{t} \sim 7.5 \times 10^{ - 4}$), is investigated through particle image velocimetry (PIV), and high-resolution thermocouple measurements. The density difference has been achieved through a temperature difference in the fluid. Cold fluid enters above the hot in a closed channel to form an unstable interface. This buoyancy-driven mixing experiment allows for long data collection times, short transients, and is statistically steady. First-, second-, and third-order statistics with spectra of velocity and temperature fields are presented. Analysis of the measurements has shed light on the structure of mixing as it develops to a self-similar regime in this flow. The onset of self-similarity is marked by the development of a self-preserving form of the temperature spectra, and the collapse of velocity profiles expressed in self-similar units. Vertical velocity fluctuations dominate horizontal velocity fluctuations in this experiment, with a ratio approaching 2:1 in the self-similar regime. This anisotropy extends to the Taylor microscales that undergo differential straining in the direction of gravity. Up to two decades of velocity spectra development, and four decades of temperature spectra, have been captured from the experiment. The velocity spectra consist of an inertial range comprised of anisotropic vertical and horizontal velocity fluctuations, and a more isotropic dissipative range. Buoyancy forcing occurs across the spectrum of velocity and temperature scales, but was not found to affect the structure of the spectra, resulting in a $-5/3$ slope, similar to other canonical turbulent flows. A scaling argument is presented to explain this observation. The net kinetic energy dissipation, as the flow evolves from an initial state to a final self-similar state was measured to be 49% of the accompanying loss in potential energy, and is in close agreement with values obtained from three-dimensional numerical simulations.

In this paper, the effects of adiabatic and isothermal conditions on the statistics in compressible turbulent channel flow are investigated using direct numerical simulation (DNS). DNS of two compressible turbulent channel flows (Cases 1 and 2) are performed using a mixed Fourier Galerkin and B-spline collocation method. Case 1 is compressible turbulent channel flow between isothermal walls, which corresponds to DNS performed by Coleman et al. (1995). Case 2 is the flow between adiabatic and isothermal walls. The flow of Case 2 can be a very useful framework for the present objective, since it is the simplest turbulent channel flow with an adiabatic wall and provides ideal information for modelling the compressible turbulent flow near the adiabatic wall. Note that compressible turbulent channel flow between adiabatic walls is not stationary if there is no sink of heat. In Cases 1 and 2, the Mach number based on the bulk velocity and sound speed at the isothermal wall is 1.5, and the Reynolds number based on the bulk density, bulk velocity, channel half-width, and viscosity at the isothermal wall is 3000.

To compare compressible and incompressible turbulent flows, DNS of two incompressible turbulent channel flows with passive scalar transport (Cases A and B) are performed using a mixed Fourier Galerkin and Chebyshev tau method. The wall boundary conditions of Cases A and B correspond to those of Cases 1 and 2, respectively. Case A corresponds to the DNS of Kim & Moin (1989). In Cases A and B, the Reynolds number based on the friction velocity, the channel half-width, and the kinematic viscosity is 150.

The mean velocity and temperature near adiabatic and isothermal walls for compressible turbulent channel flow can be explained using the non-dimensional heat flux and the friction Mach number. It is found that Morkovin's hypothesis is not applicable to the near-wall asymptotic behaviour of the wall-normal turbulence intensity even if the variable property effect is taken into account. The mechanism of the energy transfers among the internal energy, mean and turbulent kinetic energiesis investigated, and the difference between the energy transfers near isothermal and adiabatic walls is revealed. Morkovin's hypothesis is not applicable to the correlation coeffcient between velocity and temperature fluctuations near the adiabatic wall.

The breakup of immiscible fluid particles in a prototypical turbulent flow has been investigated. Dispersed fluids of varying density, viscosity and interfacial tension with water were injected continuously on the centreline in the fully developed region of a turbulent water jet. Digital image-processing techniques were used to track the particle size distributions as the initial globules of the dispersed fluid were broken into smaller particles and convected downstream in the jet. Particle breakup frequencies were calculated from the evolution of the measured particle size distributions using a simplified version of the Boltzmann equation. The results of these calculations indicate that the breakup frequency of fluid particles at low Weber numbers scales with the passage frequency of the large-scale turbulent features of the flow, approximated as $u'/L$, where $u'$ is the r.m.s. value of turbulent velocity fluctuations and $L$ is the local integral length scale. High-speed video images corroborate this result. Prior to breakup, dispersed fluid particles with initial diameters within the inertial subrange of the background flow stretch to lengths comparable to the local integral scale. These elongated particles subsequently break owing to capillary effects resulting from differences in the radius of curvature along their length. The breakup time of these particles scales with the capillary time $t_d = \mu_dD/\sigma$, where $\mu_d$ is the dispersed fluid viscosity, $D$ is the undeformed particle diameter, and $\sigma$ is the interfacial tension between the dispersed fluid and water. These results are analogous to the breakup mechanisms observed by several investigators in low-Reynolds-number flows; however, they contradict the classical theory for turbulent particle breakup, which suggests that fragmentation results from isolated interactions with turbulent velocity fluctuations over distances comparable to or smaller than the undeformed dispersed particle diameter.

This study deals with the nonlinear cyclo-geostrophic adjustment of a circular density front in a two-layer fluid. Laboratory experiments have been performed to investigate the dynamical evolution of a fixed volume of buoyant water, initially confined within a bottomless cylinder, which is quickly released in a dense rotating fluid. This configuration corresponds to a rapid input of potential energy in a geostrophic fluid layer and reproduces some dynamical processes which occur during oceanic upwelling or stratospheric warming events. We focus our efforts on the visualization techniques in order to have simultaneous and independent measurements of both the horizontal velocity field and the vertical density field. We thus obtained, for the first time, quantitative measurements of the potential vorticity and the flow balance after a geostrophic adjustment process. The density profile of the mean adjusted state observed in the experiment is in good agreement with the prediction of the standard adjustment theory based on Lagrangian conservation of potential vorticity except in the frontal region. There, strong three-dimensional motions (plume structures, shocks and rapid transient instabilities) take place during the early stage of adjustment. These transient three-dimensional motions could dissipate up to 50% of the initial energy of the system, especially when the size of the initial density anomaly is close to or larger than the deformation radius. Therefore, it significantly changes the velocity and the energy budget predicted by the standard Rossby adjustment. Both the kinetic energy of the mean adjusted state and the energy transferred to inertia–gravity wave modes are reduced by these transient dissipative processes.

The process of the formation and early evolution of a condensation trail (‘contrail’) in the near field of an aircraft wake was numerically studied by means of a mixed Eulerian/Lagrangian two-phase flow approach. Large-eddy simulations were used for the carrier phase, while, for the dispersed phase, a Lagrangian particle tracking method was used, coupled with a microphysics model to account for ice nucleation. The basic configuration was an exhaust engine jet loaded with soot particles and water vapour and interacting with a wing-tip trailing vortex. The thermodynamic conditions for contrail formation were identified by tracking the spatial distribution of supersaturation around particles. A strong mass coupling between the two phases was demonstrated by the simulations: the condensation of water vapour over soot particles, induced by exhaust dispersion into cold ambient air, leads to the formation of ice crystals whose size grows until thermodynamic equilibrium between the two phases is reached. Finally, local vapour depletion causes significant deviation from the classical mixing line theory and is also responsible for polydispersion of particle radii.