The stability of the viscometric motion of a viscoelastic fluid held between rotating parallel disks with large radii to small-amplitude perturbations is studied for the Oldroyd-B constitutive model. The disturbances are assumed to be radially localized and are expressed in Fourier form so that a separable eigenvalue problem results; these disturbances describe either axisymmetric or spiral vortices, depending on whether the most dangerous disturbance has zero or non-zero azimuthal wavenumber, respectively. The critical value of the dimensionless radius R* for the onset of the instability is computed as a function of the Deborah number De, a dimensionless time constant of the fluid, the azimuthal and radial wavenumbers, and the ratio of the viscosities of the solvent to the polymer solution. Calculations meant to match the experiments of McKinley et al. (1991) for a Boger fluid show that the most dangerous instabilities are spiral vortices with positive and negative angle that start at the same critical radius and travel outward and inward toward the centre of the disk; the axisymmetric mode also becomes unstable at only slightly greater values of R*, or De for fixed R*. The predicted dependence of the value of De for a fixed R* on the gap between the disks agrees quantitatively with the measurements of McKinley et al., when the longest relaxation time for the fluid at the shear rate corresponding to the maximum value of R* is used to define the time constant in the Oldroyd-B model.

Direct numerical simulations of turbulent flows over riblet-mounted surfaces are performed to educe the mechanism of drag reduction by riblets. The computed drag on the riblet surfaces is in good agreement with the existing experimental data. The mean-velocity profiles show upward and downward shifts in the log–law for drag-decreasing and drag-increasing cases, respectively. Turbulence statistics above the riblets are computed and compared with those above a flat plate. Differences in the mean-velocity profile and turbulence quantities are found to be limited to the inner region of the boundary layer. Velocity and vorticity fluctuations as well as the Reynolds shear stresses above the riblets are reduced in drag-reducing configurations. Quadrant analysis indicates that riblets mitigate the positive Reynolds-shear-stress-producing events in drag-reducing configurations. From examination of the instantaneous flow fields, a drag reduction mechanism by riblets is proposed: riblets with small spacings reduce viscous drag by restricting the location of the streamwise vortices above the wetted surface such that only a limited area of the riblets is exposed to the downwash of high-speed fluid that the vortices induce.

The collapse process of single bubbles in travelling bubble cavitation around two axisymmetric headforms have been studied acoustically to understand the collapse process of a cavitation bubble and characterize the sound emission in travelling bubble cavitation. The bubbles were observed to collapse and then sometimes to rebound and collapse again, resulting in one or two pulses in the acoustic signal from a cavitation event. It was observed that each of the pulses could contain more than one peak. This phenomenon is called multipeaking and is clearly distinct from rebounding. The occurrence of rebounding and multipeaking and their effects on some characteristic measures of the acoustic signal such as power spectra are examined in this paper. Two particular headforms (ITTC and Schiebe) with distinct flow characteristics were investigated.

Both rebounding and multipeaking increased with reduction in the cavitation number for the ITTC headform. Smaller flow velocity, smaller cavitation number and multipeaking delay the rebound. The peak amplitude of the sound emitted from the first collapse was seen to be twice as large as the peak amplitude of sound from the second collapse, suggesting a repeatable process of bubble fission during the collapse process. Multipeaking and rebounding increased the characteristic measures of the acoustic signal such as the acoustic impulse. These characteristic measures have larger magnitudes for smaller flow velocity. Also, the values of these characteristics are larger for the ITTC headform than for Schiebe headform.

Theoretical calculations based on the Rayleigh–Plesset equation were seen to correctly predict the order of magnitude of most of these characteristic measures. However, the distribution of spectral energy is not properly predicted; bubble fission during the collapse is thought to account for this discrepancy. Reduction in the cavitation number and multipeaking are observed to decrease the fraction of spectral energy contained in the high-frequency range (30–80 kHz).

In this paper, we present detailed and systematic experimental results on the sedimentation of solid particles in aqueous solutions of polyox and polyacrylamide, and in solutions of polyox in glycerin and water. The tilt angles of long cylinders and flat plates falling in these viscoelastic liquids were measured. The effects of particle length, particle weight, particle shape, liquid properties and liquid temperature were determined. In some experiments, the cylinders fall under gravity in a bed with closely spaced walls. No matter how or where a cylinder is released the axis of the cylinder centres itself between the close walls and falls steadily at a fixed angle of tilt with the horizontal. A discussion of tilt angle may be framed in terms of competition between viscous effects, viscoelastic effects and inertia. When inertia is small, viscoelasticity dominates and the particles settle with their broadside parallel or nearly parallel to the direction of fall. Normal stresses acting at the corners of rectangular plates and squared-off cylinders with flat ends cause shape tilting from the vertical. Cylinders with round ends and cone ends tilt much less in the regime of slow flow. Shape tilting is smaller and is caused by a different mechanism to tilting due to inertia. When inertia is large the particles settle with their broadside perpendicular to the direction of fall. The tilt angle varies continuously from 90° when viscoelasticity dominates to 0° when inertia dominates. The balance between inertia and viscoelasticity was controlled by systematic variation of the weight of the particles and the composition and temperature of the solution. Particles will turn broadside-on when the inertia forces are larger than viscous and viscoelastic forces. This orientation occurred when the Reynolds number Re was greater than some number not much greater than one in any case, and less than 0.1 in Newtonian liquids and very dilute solutions. In principle, a long particle will eventually turn its broadside perpendicular to the stream in a Newtonian liquid for any Re > 0, but in a viscoelastic liquid this turning cannot occur unless Re > 1. Another condition for inertial tilting is that the elastic length λU should be longer than the viscous length ν/U where U is the terminal velocity, ν is the kinematic viscosity and λ = ν/c2 is a relaxation time where c is the shear wave speed measured with the shear wave speed meter (Joseph 1990). The condition M = U/c > 1 is provisionally interpreted as a hyperbolic transition of solutions of the vorticity equation analogous to transonic flow. Strong departure of the tilt angle from θ = 90° begins at about M = 1 and ends with θ = 0° when 1 < M < 4.

An experimental investigation of steep, high-frequency gravity waves (∼ 4 to 5 Hz) and the parasitic capillary waves they generate is reported. Spatial, as well as temporal, non-intrusive surface measurements are made using a new technique. This technique employs cylindrical lenses to magnify the vertical dimension in conjunction with an intensified, high-speed imaging system, facilitating the measurement of the disparate scales with a vertical surface-elevation resolution on the order of 10 μm. Thus, high-frequency parasitic capillary waves and the underlying gravity wave are measured simultaneously and accurately in space and time. Time series of spatial surface-elevation measurements are presented. It is shown that the location of the capillary waves is quasi-stationary in a coordinate system moving with the phase speed of the underlying gravity wave. Amplitudes and wavenumbers of the capillaries are modulated in space; however, they do not propagate with respect to the gravity wave. As capillary amplitudes are seen to decrease significantly and then increase again in a recurrence-like phenomenon, it is conjectured that resonance mechanisms are present. Measured surface profiles are compared to the theories of Longuet-Higgins (1963) and Crapper (1970) and the exact, two-dimensional numerical formulation of Schwartz & Vanden-Broeck (1979). Significant discrepancies are found between experimental and theoretical wavetrains in both amplitude and wavenumber. The theoretical predictions of the capillary wave amplitudes are much smaller than the measured amplitudes when the measured phase speed, amplitude, and wavelength of the gravity wave are used in the Longuet-Higgins model. In addition, this theory predicts larger wavenumbers of the capillaries as compared to experiments. The Crapper model predicts the correct order-of-magnitude capillary wave amplitude on the forward face of the gravity wave, but predicts larger amplitudes on the leeward face in comparison to the experiments. Also, it predicts larger capillary wavenumbers than are experimentally determined. Comparison of the measured profiles to multiple solutions of the stationary, symmetric, periodic solutions determined using the Schwartz & Vanden-Broeck numerical formulation show similar discrepancies. In particular, the assumed symmetry of the waveform about crest and trough in the numerical model precludes a positive comparison with the experiments, whose underlying waves exhibit significantly larger capillaries on their forward face than on their leeward face. Also, the a priori unknown multiplicity of numerical solutions for the same dimensionless surface tension and steepness parameters complicates comparison. Finally, using the temporal periodicity of the wave field, composite images of several successive wavelengths are constructed from which potential energy and surface energy are calculated as a function of distance downstream.

An experimental and numerical investigation of the density distribution produced in a container by a negatively buoyant jet has been undertaken to evaluate the effect of the forced vertical motion of the environment. Vertical motion results from inflows and exhausts above and below the jet. Three distinct cases were identified. In the first, a velocity in the environment opposed the jet and produced a steady flow. This configuration was used to measure the entrainment flux along the length of the fountain. This configuration is similar to a jet impinging on an interface for which the entrainment depends on the local Froude number. The experiments covered a wider range of local Froude numbers than previously published and have produced results which are different from those in the literature. In the second case, the environment was at rest except for the motion induced by the fountain. An interface formed at the base of the fountain and moved quickly to the top. Once there, it advanced slowly due to entrainment through the end of the fountain and the length of the fountain increased. The final case is a co-flowing environment. No interface formed if the environment velocity was greater than the advance velocity of the end of the fountain. However, one formed for a smaller environment velocity and the end of the fountain was observed to undergo a quasi-periodic jump phenomenon. The top of the fountain would advance with the environment particles for a short time and then snap back to the elevation of a fountain in an infinite environment. A new interface formed and the cycle repeated.

The shape of a two-dimensional liquid curtain issuing from a slot and falling under the influence of gravity is predicted theoretically and verified experimentally for cases where a pressure is applied to the curtain. A set of approximate equations is derived which governs the location of the curtain for a liquid having surface tension σ, density ρ, volumetric flow per unit width Q, and local free-fall velocity V. These equations possess a singularity at the point where the local Weber number, We = ρQV/2σ, is equal to 1. Despite the fact that previous work on the stability of two-dimensional curtains shows that curtains having locations where We < 1 are unstable to small disturbances, our experiments show that these curtains can exist over a wide range of flow conditions. Thus, it is necessary to consider how the singularity is resolved when a pressure is applied.

It is found that the singularity can be eliminated from the governing equations if the curtain assumes a definite direction as it leaves the slot. By contrast, if the curtain leaves the slot such that We > 1, there is no such restriction, and experimentally it is found that the curtain leaves parallel to the slot walls. The theoretical predictions of the curtain shapes are in agreement with those measured experimentally for all Weber numbers investigated.

A series of experiments has been performed on a laminar flat-plate boundary layer undergoing transition to turbulence. Reproducible disturbances were introduced via a loudspeaker embedded at some upstream location. Time series of the velocity fluctuations were obtained at a sequence of downstream locations using hot-wire anemometry and the phase portraits were reconstructed at each position. A new technique has been used to estimate the number of independent modes. Nonlinear maps were then fitted that transform the portrait at one streamwise location onto the portrait at the neighbouring downstream position. In this way the spatial evolution of disturbances is modelled explicitly. These maps agree with classical linear stability theory for small disturbances, and appear to give rise to ‘Smale horse-shoe’-like behaviour for larger amplitude disturbances. This may provide a mechanism for generating sensitive dependence on initial conditions, and illustrates a possible role for low-dimensional chaos in boundary-layer transition.

The three-dimensional, parabolized form of the coupled equations of motion and energy are solved to study the development of mixed-convection heat transfer in the entrance region of horizontal square ducts. The specific problem considered here is that of axially uniform heat flux and peripherally uniform temperature with parabolic inlet profile and uniform inlet temperature in a square cross-section. Previous studies have been confined mostly to two-dimensional flows in the fully developed region. Recent two-dimensional calculations have indicated the existence of multiple steady-state solutions in the fully developed region with two- and four-cell flow structure. Present three-dimensional calculations, carried over the whole entrance length and the full cross-section, indicate the evolutionary path that lead to such two-dimensional flows. Furthermore they shed new light on the nature of flows in regions where all known two-dimensional solutions become unstable in some manner. For low Grashof numbers and Pr = 0.73 the secondary velocities develop into an axially invariant state with two counter-rotating vortices. For Grashof numbers above 2.2 × 105 the inlet profiles evolve into a state with a four-cell secondary flow structure. As in a related problem of flow in a curved channel (Winters 1987; Bara et al. 1992), the two-dimensional, four-cell solutions are found to be unstable to asymmetric perturbations. Such perturbations trigger a new, streamwise-periodic mode which is sustained over long lengths in the flow direction. For Grashof numbers above 5.6 × 105 axially invariant two-cell solutions reappear. Some of the three-dimensional solutions corresponding to the streamwise periodic mode lack the reflective symmetry about the vertical centreline and hence these flows occur with multiplicity of two.

The motion of two cavitation bubbles near a rigid boundary is observed experimentally using a high-speed camera and compared against numerical solutions obtained using a boundary integral method. The comparisons are favourable with regard to both bubble shape history and centroid motion. The bubbles show a range of responses depending on the experimental configuration. Elongated bubbles, jets directed towards or away from the rigid boundary and bubble splitting phenomena are all observed and predicted for the given parameters. It is clear that nearby bubbles are equally as important as the presence of a rigid boundary in determining the behaviour of bubbles.