The Lie group approach developed by Oberlack (1997) is used to derive new scaling laws for high-Reynolds-number turbulent pipe flows. The scaling laws, or, in the methodology of Lie groups, the invariant solutions, are based on the mean and fluctuation momentum equations. For their derivation no assumptions other than similarity of the Navier–Stokes equations have been introduced where the Reynolds decomposition into the mean and fluctuation quantities has been implemented. The set of solutions for the axial mean velocity includes a logarithmic scaling law, which is distinct from the usual law of the wall, and an algebraic scaling law. Furthermore, an algebraic scaling law for the azimuthal mean velocity is obtained. In all scaling laws the origin of the independent coordinate is located on the pipe axis, which is in contrast to the usual wall-based scaling laws. The present scaling laws show good agreement with both experimental and DNS data. As observed in experiments, it is shown that the axial mean velocity normalized with the mean bulk velocity um has a fixed point where the mean velocity equals the bulk velocity independent of the Reynolds number. An approximate location for the fixed point on the pipe radius is also given. All invariant solutions are consistent with all higher-order correlation equations. A large-Reynolds-number asymptotic expansion of the Navier–Stokes equations on the curved wall has been utilized to show that the near-wall scaling laws for at surfaces also apply to the near-wall regions of the turbulent pipe flow.

Enhancement of the temporal growth rate of inviscid three-dimensional instability waves in free shear layers by deformation of the basic flow is studied. The deformation of a two-dimensional mixing layer is assumed to yield a base flow that remains unidirectional, but has a steady spanwise speed variation in addition to the two- dimensional shear. The computed growth rates for hyperbolic tangent base flow, perturbed this way, show enhanced instability in the sense that the neutral waves of the unperturbed flow exhibit positive growth rates. For each imposed spanwise periodicity, an oblique mode is selected that shows maximum growth rate. The results are consistent with related theoretical studies and with qualitative observations in experiments.

This paper describes experiments and numerical simulations of a gravity current flowing down a ramp in a tank stratified in two layers. We study the dynamics of the configuration for different densities of the gravity current and different ramp angles. The experiments show that waves of large amplitude can be generated easily and that, depending on the density of the gravity current, the initial gravity current splits into a gravity current along the interface of the stratified layers, and a gravity current along the bottom of the tank. We also describe numerical simulations which give results in agreement with the results of the experiments and enable us to study three-fluid configurations with wider ranges of density than is possible in the laboratory.

This paper reports on a series of large-eddy simulations of a round jet issuing normally into a crossflow. Simulations were performed at two jet-to-crossflow velocity ratios, 2.0 and 3.3, and two Reynolds numbers, 1050 and 2100, based on crossflow velocity and jet diameter. Mean and turbulent statistics computed from the simulations match experimental measurements reasonably well. Large-scale coherent structures observed in experimental flow visualizations are reproduced by the simulations, and the mechanisms by which these structures form are described. The effects of coherent structures upon the evolution of mean velocities, resolved Reynolds stresses, and turbulent kinetic energy along the centreplane are discussed. In this paper, the ubiquitous far-field counter-rotating vortex pair is shown to originate from a pair of quasi-steady ‘hanging’ vortices. These vortices form in the skewed mixing layer that develops between jet and crossflow fluid on the lateral edges of the jet. Axial flow through the hanging vortex transports vortical fluid from the near-wall boundary layer of the incoming pipe flow to the back side of the jet. There, the hanging vortex encounters an adverse pressure gradient and breaks down. As this breakdown occurs, the vortex diameter expands dramatically, and a weak counter-rotating vortex pair is formed that is aligned with the jet trajectory.

Direct numerical simulations of a turbulent fluid laden with finite-sized particles are performed. The computations, on a 1283 grid along with a maximum of 262 144 particles, incorporated both direct particle interactions via hard-sphere collisions and particle feedback. The ‘reverse’ coupling is accomplished in a manner ensuring correct discrete energy conservation (Sundaram & Collins 1996). A novel two-field formalism (Sundaram & Collins 1994a) is employed to calculate two-point correlations and equivalent spectral densities. An important consideration in these simulations is the initial state of fluid and particles. That is, the initial conditions must be chosen so as to allow a meaningful comparison of the different runs. Using such a carefully initialized set of runs, particle inertia was observed to increase both the viscous and drag dissipations; however, simultaneously, it also caused particle velocities to correlate for longer distances. The combination of effects suggests a mechanism for turbulence enhancement or suppression that depends on the parameter values. Like previous investigators, ‘pivoting’ or crossover of the fluid energy spectra was observed. A possible new scaling for this phenomenon is suggested. Furthermore, investigations of the influence of particle mass and number densities on turbulence modulation are also carried out.

The stability of the flow of an incompressible, viscous fluid through a pipe of circular cross-section, curved about a central axis is investigated in a weakly nonlinear regime. A sinusoidal pressure gradient with zero mean is imposed, acting along the pipe. A WKBJ perturbation solution is constructed, taking into account the need for an inner solution in the vicinity of the outer bend, which is obtained by identifying the saddle point of the Taylor number in the complex plane of the cross-sectional angle coordinate. The equation governing the nonlinear evolution of the leading-order vortex amplitude is thus determined. The stability analysis of this flow to axially periodic disturbances leads to a partial differential system dependent on three variables, and since the differential operators in this system are periodic in time, Floquet theory may be applied to reduce it to a coupled infinite system of ordinary differential equations, together with homogeneous uncoupled boundary conditions. The eigenvalues of this system are calculated numerically to predict a critical Taylor number consistent with the analysis of Papageorgiou (1987). A discussion of how nonlinear effects alter the linear stability analysis is also given. It is found that solutions to the leading-order vortex amplitude equation bifurcate subcritically from the eigenvalues of the linear problem.

Effects of the Lewis number and radiative heat loss on flame bifurcations and extinction of CH4/O2-N2-He flames are investigated numerically with detailed chemistry. Attention is paid to the interaction between radiation heat loss and the Lewis number effect. The Planck mean absorption coefficients of CO, CO2, and H2O are calculated using the statistical narrow-band model and compared with the data given by Tien. The use of Tien's Planck mean absorption coefficients overpredicts radiative heat loss by nearly 30 % in a counter flow configuration. The new Planck mean absorption coefficients are then used to calculate the extinction limits of the planar propagating flame and the counterflow flame when the Lewis number changes from 0.967 to 1.8. The interaction between radiation heat loss and the Lewis number effect greatly enriches the phenomenon of flame bifurcation. The existence of multiple flames is shown to be a physically intrinsic phenomenon of radiating counterflow flames. Eight kinds of typical patterns of flame bifurcation are identified. The competition between radiation heat loss and the Lewis number effect results in two distinct phenomena, depending on if the Lewis number is greater or less than a critical value. Comparisons between the standard limits of the unstrained flames and the ammability limits of the counterflow flames indicate that the ammability limit of the counterflow flame is lower than the standard limit when the Lewis number is less than the critical value and is equal to the standard limit when the Lewis number is higher than this critical value. Finally, a G-shaped curve and a K-shaped curve which respectively represent the ammable regions of the multiple flames for Lewis numbers lower and higher than the critical value are obtained. The G- and K-shaped curves show a clear relationship between the stretched counterflow flame and the unstrained planar flame. The present results provide a good explanation of the physics revealed experimentally in microgravity.

The surface profile histories of gentle spilling breakers generated mechanically with a dispersive focusing technique are studied experimentally. Froude-scaled generation conditions are used to produce waves with three average frequencies: f0=1.42, 1.26, and 1.15 Hz. At each frequency, the strength of the breaker is varied by varying the overall amplitude of the wavemaker motion. It is found that in all cases the beginning of the breaking process is marked by the formation of a bulge in the profile at the crest on the forward face of the wave. The leading edge of this bulge is called the toe. As the breaking process continues, the bulge becomes more pronounced while the toe remains in nearly a fixed position relative to the crest. Capillary waves form ahead of the toe. At a time of about 0.1/f0 after the bulge first becomes visible, the toe begins to move down the face of the wave and very quickly accelerates to a constant velocity which scales with the wave crest speed. During this phase of the breaker evolution, the surface profile between the toe and the crest develops ripples which eventually are left behind the wave crest. It is found that the height of the toe above the mean water level scales with the nominal wavelength λ0=g/(2πf20) of the breaker, while the size and shape of the bulge and the length of the capillary waves ahead of the toe are independent of f0.

The influence of a viscosity stratification on the interaction between thermal convection and a stable density discontinuity is studied, using laboratory experiments. Initially, two superposed isothermal layers of high-Prandtl-number miscible fluids are suddenly cooled from above and heated from below. By adjusting the concentrations of salt and cellulose, Rayleigh numbers between 300 and 3×107 were achieved for density contrasts between 0.45 % and 5 % and viscosity ratios between 1 and 6.4×104. Heat and mass transfer through the interface were monitored.

Two-layer convection is observed but a steady state is never obtained since penetrative convection occurs. A new interfacial instability is reported, owing to the nonlinear interaction of the unstable thermal and stable chemical density gradients. As a result, the temperature condition at the interface is highly inhomogeneous, driving, on top of the classical small-scale thermal convection, a large-scale flow in each layer which produces cusps at the interface. Entrainment, driven by viscous coupling between the two layers, proceeds through those cusps. The pattern of entrainment is asymmetric: two-dimensional sheets are dragged into the more viscous layer, while three-dimensional conduits are produced in the less viscous layer. A simple entrainment model is proposed and scaling laws for the entrainment rate are derived; they explain the experimental data well.

Experiments and theory show that hydrodynamic instabilities can arise during flow of viscoelastic liquids in curved geometries. A recent study has found that a relatively weak steady transverse flow can delay the onset of instability in the circular Couette geometry until the azimuthal Weissenberg number Weθ is significantly higher than without axial flow. In this work we investigate the effect of superposition of a time-periodic axial Couette flow on the viscoelastic circular Couette and Dean flow instabilities. The analysis, carried out for the upper-convected Maxwell and Oldroyd-B fluids, generally shows increased stability compared to when there is no axial flow. However, we also find that the system shows instability – synchronous resonance – for some values of the axial Weissenberg number, Wez and forcing frequency ω. In particular, instability can be induced not only when ω is of the order of the inverse relaxation time of the fluid but also when it is much smaller. Scaling arguments and numerical results indicate that the high-ω, low-Wez regime is essentially equivalent to Wez=0 in the steady case, implying no stabilization. At high values of ω and Wez, scaling analysis shows that the flow will always be stable. Numerical results are in agreement with these conclusions. Consistent with previous results on parametrically forced systems, we find that the zero-frequency limit is singular. In this limit, the disturbances display quiescent intervals punctuated by periods of large transient growth and subsequent decay.

This study also presents linear and nonlinear stability results for the addition of steady axial Couette and Poiseuille flows to viscoelastic instabilities in azimuthal Dean flows. It is shown that, for high Wez, the qualitative effect of adding a steady axial flow is similar to that in the circular Couette geometry, with a linear relationship between the critical Weθ and Wez. For low Wez, we find that the flow is stabilized, unlike in the circular Couette flow where the critical value of Weθ decreases at low Wez. Further, weakly nonlinear analysis shows that the criticality of the bifurcation depends on the value of Wez and the solvent viscosity, S. Finally, we also show the presence of a codimension-2 Takens–Bogdanov bifurcation point in the linear stability curve of Dean flow. This point represents a transition from one mechanism of instability to another.

This work studies the motion of an expanding or contracting bubble pinned at a submerged tube tip and covered with an insoluble Volmer surfactant. The motion is driven by constant flow rate Q into or out of the tube tip. The purpose is to examine two central assumptions commonly made in the bubble and drop methods for measuring dynamic surface tension, those of uniform surfactant concentration and of purely radial flow. Asymptotic solutions are obtained in the limit of the capillary number Ca→0 with the Reynolds number Re=o(Ca−1, non-zero Gibbs elasticity (G), and arbitrary Bond number (Bo). (Ca=μQ/a2σc, where μ is the liquid viscosity, a is the tube radius, and σc is the clean surface tension.) This limit is relevant to dynamic-tension experiments, and gives M→∞, where M=G/Ca is the Marangoni number. We find that in this limit the deforming bubble at each instant in time takes the static shape. The surfactant distribution is uniform, but its value varies with time as the bubble area changes. To maintain a uniform distribution at all times, a tangential flow is induced, the magnitude of which is more than twice that in the clean case. This is in contrast to the surface-immobilizing effect of surfactant on an isolated translating bubble. These conclusions are confirmed by a boundary integral solution of Stokes flow valid for arbitrary Ca, G and Bo. The uniformity in surfactant distribution validates the first assumption in the bubble and drop methods, but the enhanced tangential flow contradicts the second.

Flow in a rapidly rotating, precessing spherical shell is studied with and without an applied magnetic dipole field in order to model the Earth's core. The primary response of the fluid to precessional forcing is a solid body rotation about an axis other than the rotation axis of the shell. The orientation and energy of that flow is predicted well by an asymptotic theory. Ekman layers at the boundaries of the shell break down at critical latitudes and spawn internal shear layers. The limit of small precession rate is investigated in particular: at zero magnetic field, the strongest shear layers are inclined at 30° with respect to the rotation axis of the shell and erupt at 30° latitude from the inner core. When a magnetic dipole field with its dipole oriented along the rotation axis of the shell is applied, shear zones develop additional structure and change position and orientation. At an Elsasser number of 10, most flow structures tend to align with the rotation axis of the shell.

The Stokes flow occurring within a non-neutrally buoyant spherical droplet translating by buoyancy through an immiscible liquid which is undergoing simple shear is shown to be chaotic under many circumstances for which the droplet translates by buoyancy through the entraining fluid. This flow is easily produced, for example, when the droplet rises (or falls) through the annular space of a vertical concentric-cylinder Couette viscometer or through a vertical Poiseuille flow. The parameters studied include: (i) droplet/bulk fluid viscosity ratio; (ii) shear strength/bubble rise velocity ratio; and (iii) the angle between the translational bubble velocity vector and the vorticity vector characterizing the undisturbed shear. Streamlines existing within a droplet that translates perpendicular to this vorticity vector are shown to be non-chaotic for all choices of physical parameters. Other relative orientations frequently contain chaotic trajectories. When solute initially dissolved within the droplet is extracted into the bulk fluid, the resulting overall mass-transfer coefficient (calculated via generalized Taylor dispersion theory) quantifying the extraction rate at asymptotically long times is shown to be significantly higher in the chaotic flow case.

Following the investigation of the long-time limit of the impulse response of an incompressible swept boundary layer (Taylor & Peake 1998), we now consider the corresponding behaviour of two representative sets of compressible swept-wing profiles, one set in subsonic flow and the other in supersonic flow. The key feature of the incompressible analysis was the occurrence of modal pinch points in the cross-flow wavenumber plane, and in this paper the existence of such pinches over a wide portion of space in high-speed flow is confirmed. We also show that close to the attachment line, no unstable pinches in the chordwise wavenumber plane can be found for these realistic wing profiles, contrary to predictions made previously for incompressible flow with simple Falker–Skan–Cooke profiles (Lingwood 1997). A method for searching for absolute instabilities is described and applied to the compressible boundary layers, and we are able to confirm that these profiles are not absolutely unstable. The pinch point property of the compressible boundary layers is used here to predict the maximum local growth rate achieved by waves in a wavepacket in any given direction. By determining the direction of maximum amplification, we are able to derive upper bounds on the amplification rate of the wavepacket over the wing, and initial comparison with experimental data shows that the resulting N-factors are more consistent than might be expected from existing conventional methods.

The evolution of artificially generated localized disturbances in the shape of hairpin vortices, in laminar axisymmetric rotating shear flows, is investigated experimentally. The results are compared with the predictions of a theoretical model (Levinski & Cohen 1995) with respect to the growth of such disturbances. Hairpin vortices were generated at the surface of the inner cylinder of an axisymmetric Couette apparatus, employing an injection–suction technique. The flow field was analysed from flow visualization using top and side views and by measurements of the mean and instantaneous velocity fields, carried out using laser Doppler anemometry and particle image velocimetry. An instability domain, within the range of base flow parameters where the flow is known to be linearly stable, was found. The marginal ratio between the angular velocities of the inner and outer cylinders beyond which the flow is stable to finite-amplitude localized disturbances agrees with the theoretical prediction based on the measured mean flow in the region of the disturbance. The dependence of the hairpin's inclination angle on the ratio between the two angular velocities is fairly well predicted by the theoretical model.