The equivalents of the classical theorems of hydrodynamic stability are derived for inviscid flow through a flexible tube. An important difference between flows in plane and cylindrical geometries is that the Squire transformation, which states that two-dimensional perturbations in plane parallel flows are always more unstable than three-dimensional perturbations, is not valid for tube flows. Therefore, it is necessary to analyse both axisymmetric and non-axisymmetric perturbations in flows in cylindrical geometries. Perturbations of the form $v_i = v_i {\rm {exp}}[ik(x - ct)+{\rm{i}}n\phi]$ exp[ik(χ –ct) + inϕ] are imposed on a steady axisymmetric mean flow V(r), and the stability of the mean velocity profiles and bounds for the phase velocity of the unstable modes are determined. Here r, ϕ and x are the radial, polar and axial directions, and k and c are the wavenumber and phase velocity. The flexible wall is represented by a standard constitutive equation which contains inertial, elastic and dissipative terms. Results for general velocity profiles are derived in two limiting cases: axisymmetric flows (n = 0) and highly non-axisymmetric flows (n [Gt ] k). The results indicate that axisymmetric perturbations are always stable for (V″ – r−1V′) V < 0 and could be unstable for (V″ – r−1V′) V < 0, while highly non-axisymmetric perturbations are always stable for (V″ + r−1V′) V [ges ] 0 and could be unstable for (V″ + r−1V′) V < 0. In addition, bounds on the real part (cr) and imaginary part (ci) of the phase velocity are also derived. For the practically important case of Hagen–Poiseuille flow, the present analysis indicates that axisymmetric perturbations are always stable, while highly non-axisymmetric perturbations could be unstable. This is in contrast to plane parallel flows where two-dimensional disturbances are always more unstable than three-dimensional ones.

In this study the evolution of initially homogeneous and isotropic turbulence in the presence of a free surface was investigated. The Navier–Stokes equations were solved via direct pseudo-spectral simulation with a resolution of 963. The Reynolds number based on the volume-averaged turbulence kinetic energy and dissipation rate was 147. Periodic boundary conditions were used in two dimensions, and the top and bottom sides of the domain were flat and shear-free. A random, divergence-free velocity field with a prescribed spectrum was used as the initial condition. An ensemble of sixteen separate simulations was used to calculate statistics.

Near the surface, the Reynolds stresses are anisotropic and the anisotropy extends a distance from the surface roughly equal to the turbulent lengthscale. The tangential vorticity fluctuations also vanish near the surface, owing to the no-shear condition, causing a corresponding decrease in the fluctuating enstrophy. The thickness of the region in which the surface affects the vorticity distribution is roughly one-tenth the turbulent lengthscale. The stress anisotropy near the surface appears to be maintained by reduced dissipation for the tangential velocity fluctuations, reduced pressure–strain transfer from the tangential to surface-normal velocity fluctuations, and rapid decay of the surface-normal velocity fluctuations due to dissipation. The turbulence kinetic energy rises in the near-surface region owing to a decrease in dissipation at the surface. This decrease in dissipation results from the local reduction in enstrophy owing to the vanishing of the tangential vorticity fluctuations at the surface. At the free surface, the mean pressure rises. This is also due to the reduction in enstrophy.

While the tangential vorticity must vanish at the free surface, the flow is fully three-dimensional up to the surface and the production of surface-normal vorticity by vortex stretching attains a maximum at the free surface. The contribution to the total enstrophy by the surface-normal vorticity fluctuations remains relatively constant over depth. The production of the surface-normal enstrophy component due to vortex stretching is roughly balanced by turbulent transport of enstrophy away from the surface. Near the surface, there are elevated levels of production of tangential vorticity by both vortex-stretching and fluctuating shear strains.

The turbulent flow development was examined for a two-dimensional boundary layer over a bump. The upstream boundary layer had a momentum-thickness. Reynolds number of approximately 4030. The ratios of upstream boundary layer thickness to bump height and convex radius of curvature were 1.5 and 0.06, respectively. The bump was defined by three tangential circular arcs, which subjected the flow to alternating signs of pressure gradient and surface curvature. The boundary layer grew rapidly on the downstream side of the bump but did not separate. The mean velocity profiles deviated significantly from the law of the wall above the bump. The change from concave to convex surface curvature near the leading edge triggered an internal boundary layer, as shown by knee points in the turbulent stress profiles. The internal layer grew rapidly away from the wall on the downstream side of the bump owing to the adverse pressure gradient. The effect of convex surface curvature was considered small since the flow behaviour was generally explained by the effects due to streamwise pressure gradient. A second internal layer was triggered by the change from convex to concave curvature near the trailing edge. The boundary layer recovered rapidly in the downstream section and approached typical flat-plate boundary layer behaviour at the last measurement location.

We present an approach combining temporal dynamical systems methods with newly proposed spatial coupling measures – namely, coherence and cross-bicoherence – to identify and quantitatively describe low-dimensional dynamics in transitional open flows. The approach is used to describe a forced mixing layer as a low-dimensional temporal dynamical system and interpret its transitional vortex dynamics.

Experiments were performed in an initially laminar plane mixing layer inside an anechoic chamber using forcing of the fundamental instability only; the forcing frequency and amplitude are used as control parameters. Dynamical invariants calculated show that vortex roll-up and the feedback-driven first two pairing dynamics are well-described by one periodic and at least two low-dimensional chaotic attractors; a phase diagram delineating such dynamical states in the control parameter space is presented. The large spatial extents of these feedback-sustained states (verified using coherence and cross-bicoherence), spanning many instability wavelengths downstream, indicate spatial coupling; feedback has also been experimentally verified. At a fixed forcing frequency, as the forcing amplitude is decreased, the spatially coupled, periodic second pairing dynamics becomes chaotic and spatiotemporal (inferred from decay of coherence and cross-bicoherence); the dynamics in the domain that includes the first pairing, however, remains temporal. This loss of spatial coupling is accompanied by a sudden increase in the attractor dimension, and suggests spatiotemporal chaos. The combination of dynamical systems theory and spatial measures seems to be a promising approach to probe spatiotemporal dynamics in other open flows as well.

High-image-density particle image velocimetry allows characterization of the instantaneous patterns of streamwise vorticity ωx over the cross-section of the near wake of a circular cylinder, and the manner in which they evolve with streamwise distance. Emphasis is on the Reynolds number Re = 10 × 103, for which the Kelvin–Helmholtz (K–H) mode in the separating shear layers has a streamwise wavelength much smaller than that of the Kármán mode. Consequently, the corresponding spanwise wavelength between ωx concentrations increases substantially from its value in the separating shear layer to a larger one in the near wake. This streamwise evolution is defined by spatial correlations of patterns of instantaneous ωx and interpreted with the aid of the quasi-two-dimensional topology of the wake in the base region of the cylinder. The principal features of the phase-averaged topology are foci of the initially formed Kármán vortices and a saddle point between them. Immediately downstream of this saddle, remarkably coherent patterns of ωx concentrations are evident; they have a wavelength approximately equal to the cylinder diameter. Moreover, larger-scale spanwise distortion eventually occurs. This distortion exhibits several modes; the most severe is a nearly discontinuous variation of patterns of ωx.

The entrainment of fluids from two streams into the shear region of an incompressible mixing layer is dominated by the evolution of large coherent structures. However, fine-scale mixing of the entrained fluids mainly occurs at the interfaces of the small-scale turbulence. In this investigation, experiments were conducted to understand the properties of the small scales and to explore a method for controlling the population of the fine-scale turbulence. Furthermore, a dissipation scale, ζ, is found from the zerocrossing of the time derivative of the velocity fluctuations. This scale characterizes the most probable size of fine-scale turbulence, which produces most of the viscous dissipation.

A generalization of the quasi-steady theory is proposed, the aim of which is to model the most important unsteady effects neglected by the conventional quasi-steady assumption. Although this generalized model, referred to as the quasi-unsteady model, can be applied in a vast range of flow-induced vibration problems, including classical aeroelasticity, it was primarily developed to improve the theoretical prediction of the fluidelastic behaviour of a single flexible cylinder positioned in the midst of an array of rigid cylinders. In this context, it is shown that the previous improvement to the quasi-steady theory proposed by Price & Païdoussis can be considered as a particular case of the quasi-unsteady model. Results obtained with the quasi-unsteady model are compared to experimental data and to solutions from the Price & Païdoussis model; both modal parameter variation with flow velocity and stability diagrams are considered. This comparison shows that the quasi-unsteady model is a clear improvement on Price & Païdoussis’ approach, leading to a more reasonable agreement with experimental results and providing refined insights into the physical mechanisms responsible for fluidelastic instability.

The instability of viscous capillary jets subject to disturbances consisting of two superposed wavenumbers, and for large disturbance amplitudes is investigated. Disturbances composed of the superposition of a fundamental disturbance (first harmonic) with either its second or third harmonic are used. The influence of the wavenumber of the fundamental disturbance on the jet breakup is studied for a disturbance composed of a first harmonic with an initial non-dimensional amplitude of ε1 = 0.01 and a second harmonic with an initial non-dimensional amplitude of ε2 = 0.05. The influence of the initial amplitudes of the first and second harmonics on the jet breakup is studied for two non-dimensional wavenumbers of the fundamental (first harmonic): k = 0.45 and k = 0.7; the second harmonic is unstable in the former and stable in the latter case. The effect of an added third harmonic is studied only for k = 0.45 but for a wide range of initial amplitudes. All cases are studied for an in-phase and a 180° out-of-phase superposition of the two waves. The nonlinear interaction between the two waves results in the formation of a variety of drop sizes and shapes. The breakup times can be controlled within a wide range using this technique.

This work documents the spatial development of a triad of instability waves consisting of a plane TS mode and a pair of oblique modes with equal-opposite wave angles which are undergoing subharmonic transition in Falkner–Skan boundary layers with adverse pressure gradients. The motivation for this study is that for wings with zero or moderate sweep angles, transition is most likely to occur in the adverse pressure gradient region past the maximum thickness point and, starting with low initial amplitudes, subharmonic mode transition is expected to be the predominant mechanism for the first growth of of three-dimensional modes. The experiment follows that of Corke & Mangano (1989) in which the disturbances to produce the triad of waves are introduced by a spanwise array of heating wires located near Branch I. The initial conditions are carefully controlled. These include the initial amplitudes, frequencies, relative phase and oblique wave angles. The basic flow consisted of a Falkner–Skan (Hartree) boundary layer with a dimensionless pressure gradient parameter in the range -0.06 [les ] βH [les ] -0.09. The frequency of the TS wave was selected to be near the most amplified based on linear theory. The frequency of the oblique waves was the subharmonic of the TS frequency. The oblique wave angles were set to give the largest secondary growth (≈ 60°). Compared to similar conditions in a Blasius boundary layer, the adverse pressure gradient was observed to lead to an extra rapid growth of the two- and three-dimensional modes. In this there was a relatively larger maximum amplitude of the fundamental mode and considerably shortened amplitude saturation region compared to zero pressure gradient cases. Analysis of these results includes frequency spectra, the wall-normal distributions of each mode amplitude, and mean velocity profiles. Finally, the streamwise amplitude development is compared with the amplitude model from the nonlinear critical layer analysis of Goldstein & Lee (1992).

Direct numerical simulation databases have been used to study the effect of compressibility on mixing layers. The simulations cover convective Mach numbers from 0.2 to 1.2 and all contain a fully resolved turbulent energy cascade to small spatial scales. Statistical information is extracted from the databases to determine reasons for the reduced growth rate that is observed as the convective Mach number is increased. It is found that the dilatational contribution to dissipation is negligible even when eddy shocklets are observed in the flow. Also pressure-dilatation is not found to be significant. Using an accurate relation between the momentum thickness growth rate and the production of turbulence kinetic energy together with integrated equations for the Reynolds stress tensor it is shown that reduced pressure fluctuations are responsible for the changes in growth rate via the pressure–strain term. A deterministic model for the required pressure fluctuations is given based on the structure of variable-density vortices and the assumption that the limiting eddies are sonic. Simple anisotropy considerations are used to close the averaged equations. Good agreement with turbulence statistics obtained from the simulations is found.

A direct numerical simulation was carried out of plane turbulent Couette flow at a Reynolds number of 750, based on half the velocity difference between the walls and half the channel width. Particular attention was paid to choosing a computational box that is large enough to accommodate even the largest scales of the turbulence. In the central region of the channel very large elongated structures were observed, in accordance with earlier findings. The study is focused on the properties of these structures, but is also aimed at obtaining accurate turbulence statistics. Terms in the energy budget were evaluated and discussed. Also, the limiting values of various quantities were determined and their relevance in high Reynolds number flows discussed. The large structures were shown to be very sensitive to an imposed system rotation. They could be essentially eliminated with a stabilizing system rotation (around the spanwise axis) small enough for only minor damping of the rest of the scales. Despite the fact that the large structures dominate the appearance of the flow field their energy content was shown to be relatively small, on the order of 10% of the total turbulent kinetic energy.

Based on a previous linear theory (van Duin & Janssen 1992), turbulent air flow over a surface gravity wave of finite amplitude is studied analytically by the methods of matched asymptotic expansions and multiple-scale analysis. In particular, an initial-value problem for weakly nonlinear waves is solved, where the initial conditions are prescribed by a single Stokes wave, displacing the water surface. The water is inviscid and incompressible, and there is no mean shear current. Wave–wave interactions are not taken into account. The validity of the theory is restricted to slow waves and small drag coefficient.

We investigate in detail the change of the mean air flow with the evolution of the wave, with a prescribed order of magnitude of the initial wave slope. The rate of change of this flow is fully determined by an evolution equation for the wave slope, which is obtained from the continuity condition for the normal stress at the air-water interface. This equation also determines the amplitude-dependent rate of growth or damping of the wave, for which a closed-form expression is derived. It turns out that nonlinear effects reduce the rate of energy transfer from the mean air flow to the growing wave, which implies that nonlinearity has a stabilizing effect.

For sufficienty large time scales, the slope of the growing wave becomes so large that the original evolution equation, which is approximately a Landau equation, ceases to be valid. For such relatively large wave slope, an alternative evolution equation is derived, which presumably describes the further evolution of the wave until the occurrence of wave breaking. The relative effects of nonlinearity, which can be characterized by a single parameter, increase with increasing wave slope and decreasing wave frequency.

A description of the behaviour of the second-order moments of concentration for a variety of source types is derived within the context of the classical phenomenology of isotropic turbulence. The sources considered include instantaneous area, line and point sources and can also be interpreted as relating to plumes from continuous point sources and continuous crosswind line sources in a strong uniform mean flow. A large number of different regimes are identified corresponding to different relative sizes of the many length scales involved. Perhaps the most interesting result is the identification of an ‘Inertial–meander’ subrange when the inertial-subrange eddies contribute to the meandering of the plume.

Using an active grid devised by Makita (1991), shearless decaying turbulence is studied for the Taylor-microscale Reynolds number, Rλ, varying from 50 to 473 in a small (40 × 40 cm2 cross-section) wind tunnel. The turbulence generator consists of grid bars with triangular wings that rotate and flap in a random way. The value of Rλ is determined by the mean speed of the air (varied from 3 to 14 m s–1) as it passes the rotating grid, and to a lesser extent by the randomness and rotation rate of the grid bars. Our main findings are as follows. A weak, not particularly well-defined scaling range (i.e. a power-law dependence of both the longitudinal (u) and transverse (v) spectra, F11(k1) and F22(k1) respectively, on wavenumber k1) first appears at Rλ ∼ 50, with a slope, n1, (for the u spectrum) of approximately 1.3. As Rλ was increased, n1 increased rapidly until Rλ ∼ 200 where n ∼ 1.5. From there on the increase in n1 was slow, and even by Rλ = 473 it was still significantly below the Kolmogorov value of 1.67. Over the entire range, 50 [les ] Rλ [les ] 473, the data were well described by the empirical fit: $n_1 = \frac{5}{3}(1-3.15R_\lambda^{-2/3})$. Using a modified form of the Kolmogorov similarity law: F11(k1) = C1*ε2/3k1–5/3(k1η)5/3–n1 where ε is the turbulence energy dissipation rate and η is the Kolmogorov microscale, we determined a linear dependence between n1 and C1*: C1* = 4.5 – 2.4n1. Thus for n1 = 5/3 (which extrapolation of our results suggests will occur in this flow for Rλ ∼ 104), C1* = 0.5, the accepted high-Reynolds-number value of the Kolmogorov constant. Analysis of the p.d.f. of velocity differences Δu(r) and Δv(r) where r is an inertial subrange interval, conditional dissipation, and other statistics showed that there was a qualitative difference between the turbulence for Rλ < 100 (which we call weak turbulence) and that for Rλ > 200 (strong turbulence). For the latter, the p.d.f.s of Δu(r) and Δv(r) had super Gaussian tails and the dissipation (both of the u and v components) conditioned on Δu(r) and Δv(r) was a strong function of the velocity difference. For Rλ < 100, p.d.f.s of Δu(r) and Δv(r) were Gaussian and conditional dissipation statistics were weak. Our results for Rλ > 200 are consistent with the predictions of the Kolmogorov refined similarity hypothesis (and make a distinction between the dynamical and kinematical contributions to the conditional statistics). They have much in common with similar statistics done in shear flows at much higher Rλ, with which they are compared.

The flow of a Boussinesq density-stratified fluid of large depth past the algebraic mountain (‘Witch of Agnesi’) is studied in the hydrostatic limit using the asymptotic theory of Kantzios & Akylas (1993). The upstream conditions are those of constant velocity and Brunt–Väisälä frequency. On the further assumptions that the flow is steady and there is no permanent alteration of the upstream flow conditions (no upstream influence), Long's model (Long 1953) predicts a critical amplitude of the mountain (ε = 0.85) above which local density inversions occur, leading to convective overturning. Linear stability analysis demonstrates that Long's steady flow is in fact unstable to infinitesimal modulations at topography amplitudes below this critical value, 0.65 [lsim ] ε < 0.85. This instability grows at the expense of the mean flow and may be attributed to a discrete spectrum of modes that become trapped over the mountain in the streamwise direction. The transient problem is also solved numerically, mimicking impulsive startup conditions. In the absence of instability, Long's steady flow is reached. For topography amplitudes in the unstable range 0.65 [lsim ] ε < 0.85, however, the flow fluctuates about Long's steady state over a long timescale; there is no significant upstream influence and no evidence of transient wave breaking is found for ε [les ] 0.75.

The shearing motion of monodisperse suspensions of two-dimensional deformable liquid drops with uniform interfacial tension is studied by means of numerical simulations. In the theoretical model, the drops are distributed randomly within a square that is repeated periodically in two directions yielding a doubly periodic flow. Under the assumption that inertial effects are negligible and the viscosity of the drops is equal to that of the suspending fluid, the motion is investigated as a function of the area fraction of the suspended drops and of the capillary number. The evolution of the suspension from an initial configuration with randomly distributed circular drops is computed using an improved implementation of the method of interfacial dynamics which is based on the standard boundary integral formulation for Stokes flow. The numerical procedure incorporates the method of multipole expansions to account for far-drop interactions, and interpolation through tables for computing the doubly periodic Green's function; the latter allows considerable savings in the cost of the computations. Dynamic simulations are carried out for suspensions with up to 49 drops within each periodic cell, for an extended period of time up to kt = 60, where k is the shear rate. Comparisons with previous numerical results for solid particles reveal that particle deformability and interfacial mobility play an important role in the character of the motion. The effects of particle area fraction and capillary number on the effective rheological properties of the suspension are discussed, and the statistics of the drop motion is analysed with reference to the drop-centre pair distribution function and probability density functions of drop aspect ratio and inclination. It is found that the effective rheo-logical properties may be predicted with remarkable accuracy from a knowledge of the instantaneous mean drop deformation and orientation alone, even at high area fractions. Cluster formation is not as important as in suspension of solid particles. The apparent random motion of the individual drops, when viewed at a sequence of time intervals that are large compared to the inverse shear rate, is described in terms of an effective non-isotropic long-time diffusivity tensor, and the transverse component of this tensor is computed from the results of the simulations with some uncertainty.

The second-order wave force is analysed for diffraction of monochromatic water waves by a vertical cylinder. The force is evaluated directly from pressure integration over the cylinder, and the second-order potential is derived by Weber transformation of the corresponding forcing function on the free surface. This forcing function is reduced to a form which involves a simple factor inversely proportional to the radial coordinate plus an oscillatory function which decays more rapidly in the far field. This feature alleviates the slow rate of convergence involved in capturing the far-field effect. Benchmark computations are obtained and compared with other works. Asymptotic approximations are derived for long and short wavelengths. The analysis and results are primarily for the case of infinite fluid depth, but the finite-depth case is also considered to facilitate comparison with other computations and to illustrate the importance of finite-depth effects in the long-wavelength asymptotic regime.