The three-dimensional structure and streamwise evolution of two-stream mixing layers at high Reynolds numbers (Reδ ∼ 2.7 × 104) were studied experimentally to determine the effects of mild streamwise curvature ($\delta/ \overline{R}$ < 3%). Mixing layers with velocity ratios of 0.6 and both laminar and turbulent initial boundary layers, were subjected to stabilizing and destabilizing longitudinal curvature (in the Taylor–Görtler sense). The mixing layer is affected by the angular momentum instability when the low-speed stream is on the outside of the curve, and it is stabilized when the streams are reversed so that the high-speed stream is on the outside. In both stable and unstable mixing layers, originating from laminar boundary layers, well-organized spatially stationary streamwise vorticity was generated, which produced significant spanwise variations in the mean velocity and Reynolds stress distributions. These vortical structures appear to result from the amplification of small incoming disturbances (as in the straight mixing layer), rather than through the Taylor–Görtler instability. Although the mean streamwise vorticity decayed with downstream distance in both cases, the rate of decay for the unstable case was lower. With the initial boundary layers on the splitter plate turbulent, spatially stationary streamwise vorticity was not generated in either the stable or unstable mixing layer. Linear growth was achieved for both initial conditions, but the rate of growth for the unstable case was higher than that of the stable case. Correspondingly, the far-field spanwise-averaged peak Reynolds stresses were significantly higher for the destabilized cases than for the stabilized cases, which exhibited levels comparable to, or slightly lower than, those for the straight case. A part of the Reynolds stress increase in the unstable layer is attributed to ‘extra’ production through terms in the transport equations which are activated by the angular momentum instability. Velocity spectra also indicated significant differences in the turbulence structure of the two cases, both in the near- and far-field regions.

Studies of three-dimensional Stokes flow of two Newtonian fluids that converge in a T-type bifurcation have important applications in polymer coextrusion, blood flow through the venous microcirculation, and other problems of science and technology. This flow problem is simulated numerically by means of the finite element method, and the solution demonstrates that the viscosity ratio between the two fluids critically affects flow behaviour. For the parameters investigated, we find that as the viscosity ratio between the side branch and the main branch increases, the interface between the merging fluids bulges away from the side branch. The viscosity ratio also affects the velocity distribution: at the outlet branch, the largest radial gradients of axial velocity appear in the less-viscous fluid. The distribution of wall shear stress is non-axisymmetric in the outlet branch and may be discontinuous at the interface between the fluids.

The nonlinear problem of the steady-state interaction of a closed fluid-filled cylindrical elastic membrane with a slow viscous shear flow has been solved by a series-expansion technique. The problems of successive orders were both formulated and solved by a symbolic manipulation program, and the calculations were carried to sixth order in a dimensionless parameter related to the applied shear rate. Moderately large deformations (aspect ratios approaching 3) fall within the range of this analysis, which yields the dependences of the following global variables on the system parameters: membrane deformation, orientation, and strain, as well as tank-treading frequency, and mean internal pressure. The solution for the flow field around an isolated capsule is also used to calculate the apparent viscosity of a dilute suspension of flexible cylindrical particles, which yields the paradoxical result that the apparent viscosity decreases as the internal viscosity increases.

A theoretical study is presented of three-dimensional turbulent flow provoked in a boundary layer by an array of low-profile vortex generators (VGs) on the surface. The typical VG sits in the logarithmic region of the incident boundary layer, and the turbulence model used seems representative in this region. The governing equations yield a forward-marching three-dimensional vortex-type system, which is solved computationally and analytically for spanwise periodic VG arrays. Streamwise vortex patterns of various strengths are produced downstream, owing to three-dimensional distortion of the original logarithmic profile and to the turbulent stresses present. Predictions are given for certain basic VG shapes, e.g. triangular, with various spanwise spacings, and the predictions are found to agree favourably overall with recent experiments. In addition, the analytical formulae obtained prove useful in suggesting designs for favourable VG distributions, based on three factors: close spanwise packing, increased VG length, and suitably non-smooth spanwise shaping.

Unsteady flow due to an oscillating sphere with a velocity U0cosωt’, in which U0 and ω are the amplitude and frequency of the oscillation and t’ is time, is investigated at finite Reynolds number. The methods used are: (i) Fourier mode expansion in the frequency domain; (ii) a time-dependent finite difference technique in the time domain; and (iii) a matched asymptotic expansion for high-frequency oscillation. The flow fields of the steady streaming component, the second and third harmonic components are obtained with the fundamental component. The dependence of the unsteady drag on ω is examined at small and finite Reynolds numbers. For large Stokes number, ε = (ωa2/2v)½ [Gt ] 1, in which a is the radius of the sphere and v is the kinematic viscosity, the numerical result for the unsteady drag agrees well with the high-frequency asymptotic solution; and the Stokes (1851) solution is valid for finite Re at ε [Gt ] 1. For small Strouhal number, St = ωa/U0 [Lt ] 1, the imaginary component of the unsteady drag (Scaled by 6πU0pfva, in which Pf is the fluid density) behaves as Dml ∼ (h0Stlog St–h1St), m = 1,3,5… This is in direct contrast to an earlier result obtained for an unsteady flow over a stationary sphere with a small-amplitude oscillation in the free-stream velocity (hereinafter referred to as the SA case) in which D1∼ –h1St (Mei, Lawrence & Adrian 1991). Computations for flow over a sphere with a free-stream velocity U0(1–α1+α1cosωt’) at Re = U02a/v = 0.2 and St [Lt ] 1 show that h0 for the first mode varies from 0 (at α1 = 0) to around 0.5 (at α1 = 1) and that the SA case is a degenerated case in which the logarithmic dependence of the drag in St is suppressed by the strong mean uniform flow.

The numerical results for the unsteady drag are used to examine an approximate particle dynamic equation proposed for spherical particles with finite Reynolds number. The equation includes a quasi-steady drag, an added-mass force, and a modified history force. The approximate expression for the history force in the time domain compares very well with the numerical results of the SA case for all frequencies; it compares favourably for the PO case for moderate and high frequencies; it underestimates slightly the history force for the PO case at low frequency. For a solid sphere settling in a stagnant liquid with zero initial velocity, the velocity history is computed using the proposed particle dynamic equation. The results compare very well with experimental data of Moorman (1955) over a large range of Reynolds numbers. The present particle dynamic equation at finite Re performs consistently better than that proposed by Odar & Hamilton (1964) both qualitatively and quantitatively for three different types of spatially uniform unsteady flows.

A general theory which can be used to derive bounds on solutions to the Navier-Stokes equations is presented. The behaviour of the resolvent of the linear operator in the unstable half-plane is used to bound the energy growth of the full nonlinear problem. Plane Couette flow is used as an example. The norm of the resolvent in plane Couette flow in the unstable half-plane is proportional to the square of the Reynolds number (R). This is now used to predict the asymptotic behaviour of the threshold amplitude below which all disturbances eventually decay. A lower bound is found to be R−21/4. Examples, obained through direct numerical simulation, give an upper bound on the threshold curve, and predict a threshold of R−1. The discrepancy is discussed in the light of a model problem.

A low-resolution (643) large-eddy model of forced homogeneous turbulence is numerically simulated using Kraichnan's eddy viscosity. The introduction of a reliable statistical estimate of the ζp exponents allows one to perform a detailed statistical analysis of the velocity field and shows that the probability distribution functions, the structure functions and the power-law exponents ζp agree with previous numerical and experimental results obtained at much higher effective resolution. This result shows how a simple modelling of the energy transfer produces self-similar dynamics extending to the small scales and obtains the right statistical properties of the velocity field.

We study the advection of a passive scalar by a vortex couple in the small-diffusion (i.e. large Péclet number, Pe) limit. The presence of weak diffusion enhances mixing within the couple and allows the gradual escape of the scalar from the couple into the surrounding flow. An averaging technique is applied to obtain an averaged diffusion equation for the concentration inside the dipole which agrees with earlier results of Rhines & Young for large times. At the outer edge of the dipole, a diffusive boundary layer of width O(Pe−½) forms; asymptotic matching to the interior of the dipole yields effective boundary conditions for the averaged diffusion equation. The analysis predicts that first the scalar is homogenized along the streamlines on a timescale O(Pe−$\frac{1}{3}$). The scalar then diffuses across the streamlines on the diffusive timescale, O(Pe). Scalar that diffuses into the boundary layer is swept to the rear stagnation point, and a finite proportion is expelled into the exterior flow. Expulsion occurs on the diffusive timescale at a rate governed by the lowest eigenvalue of the averaged diffusion equation for large times. A split-step particle method is developed and used to verify the asymptotic results. Finally, some speculations are made on the viscous decay of the dipole in which the vorticity plays a role analogous to the passive scalar.

The primary instability of a falling film selectively amplifies two-dimensional noise down-stream over three-dimensional modes with transverse variation. If the initial three-dimensional noise is weak or if it has short wavelengths such that they are effectively damped by the capillary mechanism of the primary instability, our earlier study (Chang et al. 1993a) showed that the primary instability leads to a weakly nonlinear, nearly sinusoidal γ1 stationary wave which then undergoes a secondary transition to a strongly nonlinear γ2 wave with a solitary wave structure. We show here that the primary transition remains in the presence of significant three-dimensional noise but the secondary transition can be replaced by a selective excitation of oblique triad waves which can even include stable primary disturbances. The resulting secondary checkerboard pattern is associated with a subharmonic mode in the streamwise direction. If the initial transverse noise level is low, a secondary transition to a two-dimensional γ2 solitary wave is followed by a tertiary ‘phase instability’ dominated by transverse wave crest modulations.

The fine scale three-dimensional structures usually associated with streamwise vortices in the near wake of a circular cylinder have been studied at Reynolds numbers ranging from 170 to 2200. Spatially continuous velocity measurements along lines parallel to the cylinder axis were obtained with a scanning laser anemometer. To detect the streamwise vortices in the amplitude modulated velocity field, it was necessary to develop a spatial decomposition technique to split the total flow into a primary flow component and a secondary flow component. The primary flow is comprised of the mean flow and Strouhal vortices, while the secondary flow is the result of the three-dimensional streamwise vortices that are the essence of transition to turbulence. The three-dimensional flow amplitude increases in the primary vortex formation region, then saturates shortly after the maximum amplitude in the primary flow is reached. In the near-wake region the wavelength decreases approximately like Re−0.5, but increases with downstream distance. A discontinuous increase in wavelength occurs below Re = 300 suggesting a fundamental change in the character of the three-dimensional flow. At downstream distances (x/D = 10-20), the spanwise wavelength decreases from 1.42D to 1.03D as the Reynolds number increases from 300 to 1200.

A global, three-dimensional stability analysis of the steady and the periodic cylinder wake is carried out employing a low-dimensional Galerkin method. The steady flow is found to be asymptotically stable with respect to all perturbations for Re < 54. The onset of periodicity is confirmed to be a supercritical Hopf bifurcation which can be modelled by the Landau equations. The periodic solution is observed to be only neutrally stable for 54 < Re < 170. While two-dimensional perturbations of the vortex street rapidly decay, three-dimensional perturbations with long spanwise wavelengths neither grow nor decay. The periodic solution becomes unstable at Re = 170 by a perturbation with the spanwise wavelength of 1.8 diameters. This instability is shown to be a supercritical Hopf bifurcation in the spanwise coordinate and leads to a three-dimensional periodic flow. Finally the transition scenario for higher Reynolds numbers is discussed.

The evolution of three-dimensional disturbances in an incompressible three-dimensional stagnation-point flow in an inviscid fluid is investigated. Since it is not possible to apply classical normal-mode analysis to the disturbance equations for the fully three-dimensional stagnation-point flow to obtain solutions, an initial-value problem is solved instead. The evolution of the disturbances provides the necessary information to determine stability and indeed the complete transient as well. It is found that when considering the disturbance energy, the planar stagnation-point flow, which is independent of one of the transverse coordinates, represents a neutrally stable flow whereas the fully three-dimensional flow is either stable or unstable, depending on whether the flow is away from or towards the stagnation point in the transverse direction that is neglected in the planar stagnation point.

A kinematic description is presented of how turbulent diffusion from a continuous source varies with the sampling time in stationary, homogeneous turbulence. Unlike most previous theories, the sampling is assumed to take place at fixed downstream distances from the source. It is shown that the sampling-time effects depend on two-particle velocity statistics. Thus, time-average diffusion at fixed downstream distances is more akin to relative diffusion than to absolute diffusion. Two simple diffusion models are developed from the kinematic equations. These models are in fairly good agreement with diffusion data obtained both in a wind tunnel and in the field. Moreover, these models have significant practical implications. For example, the models indicate that care must be taken when using band-pass spectral filtering as a paradigm for turbulent diffusion. Also, the models show that the mean flow speed U has an important influence on the sampling-time effects. To account for U properly, diffusion measurements with differing sampling times Λ should be compared using the product UΛ, and not just Λ.