The time-history of the development of the three-dimensional transition features in a nominally two-dimensional flow configuration is established for Reynolds number 220 in a cylinder wake. The identification of the successive stages that evolve very fast during experiments is possible by means of direct numerical simulation. The physical processes related to the creation of streamwise and vertical vorticity components and their impact on the spanwise waviness of the main von Kármán vortex filaments are analysed by means of the Craik–Leibovich shearing instability mechanism and a comparative discussion is given with respect to the elliptic stability theory. This study proves the existence of a further stage in the three-dimensional transition, which substantially modifies the regular spanwise undulation. This is a systematic and repetitive development of natural vortex dislocations in the near wake. The definition of this kind of structure is provided, as well as its properties related to a drastic reduction of the fundamental frequency and to the selection of a lower path in the Strouhal–Reynolds number relation. The induced amplitude modulation of the flow properties along the span is also evaluated. Quantification of these properties is carried out by using wavelet analysis and autoregressive modelling of the time series. The reasons for the development of natural vortex dislocations are analysed and related to specific modulations of the spanwise structure of the longitudinal velocity upstream separation. From this part of the study an optimum shape for the spanwise distribution of this component can be specified, able to trigger the vortex dislocations in wake flows and therefore useful to apply in the context of stability theory analyses and in further DNS studies.

The axisymmetrization of a two-dimensional non-uniform elliptic vortex is studied in terms of the growth of palinstrophy, the squared vorticity gradient. First, it is pointed out that the equation for palinstrophy growth, if written in terms of the strain rate tensor, has a similar form to that of enstrophy growth in three-dimensions – the vortex-stretching equation. Then palinstrophy production is analysed, particularly for non-uniform elliptic vortices. It is shown analytically and verified numerically that a non-uniform elliptic vortex in general has a quadrupole structure for palinstrophy production, and that in the positive production regions, vortex filaments are ejected following the gradient enhancement process for vorticity. Numerical simulations are conducted for two different initial conditions, compact support and Gaussian vorticity distributions. These are characterized by distinctly different features of filament ejection and energy spectra. For both cases, the total palinstrophy production is a good indicator of the development of small-scale vorticity. In particular for the compact support case, a possible intermittency mechanism in the filament ejection process is proposed.

The flow fields at the crests of mechanically generated gentle spilling breakers are studied experimentally with particle image velocimetry (PIV). During the initial development of the breaker, a bulge forms on the forward face of the wave near the crest. The present measurements show that the flow field in this bulge is wavelike in nature and that the maximum horizontal component of the fluid velocity, μmax, is less than the wave crest speed. No vorticity is found during this stage of the breaking process. After a short while, the leading edge (called the toe) of the bulge begins to move rapidly down the wave crest; μmax reaches the speed of the wave crest just as the toe begins to move. Concurrent with the beginning of the toe motion, a vortical region appears at the free surface just downstream of the toe. At about the point in the wave development where the measurements end, the vortical region extends from the toe to the crest.

A new form of dynamic response has been observed in some simple experiments with a lightly damped rotating cylinder in a current. The response is orbital with a period several times the structural natural period and an amplitude ranging up to many diameters. It is mainly dependent on the ratio of current velocity to cylinder surface velocity, α, and the reduced velocity, Vr (the ratio of current velocity to the product of natural frequency in water and diameter). Big orbital responses occur with 0.25 < α < 0.5 and Vr > 5, and are accompanied by the expected large static response. To understand the flow mechanisms causing this response computational simulations have been made for two-dimensional laminar flow and the experimental response characteristics are qualitatively reproduced. Streamline and vorticity contour plots are output through a cycle and are related to instantaneous values of lift and drag coefficients and α (all based on flow relative to the cylinder). The movement of the stagnation point away from and towards the cylinder surface with intermittent wake formation in a cycle causes a large lift variation which is mainly responsible for the dynamic response. The variation of lift coefficient with α (as defined above) shows a generally negative gradient, and a pronounced hysteresis loop when substantial response occurs for α [gsim ] 0.25. The computations show that a small-amplitude, high-frequency response may also be superimposed on the high-amplitude, low-frequency response, most noticeably for α [lsim ] 0.25. This is consistent with a simple potential-flow idealization of the lift force. For α ∼ 0.2 a large dynamic response, not observed in the experiments, was produced in the computations due essentially to attached boundary-layer behaviour.

The flow in a completely filled cylinder driven by a rotating endwall has multiple time-dependent stable states when the endwall rotation exceeds a critical value. These states have been observed experimentally and computed numerically elsewhere. In this article, the linear stability of the basic state, which is a non-trivial axisymmetric flow, is analysed at parameter values where the unsteady solutions exist. We show that the basic state undergoes a succession of Hopf bifurcations and the corresponding eigenvalues and eigenvectors of these excited modes describe most of the characteristics of the observed time-dependent states.

The experimental data of Honkan & Andreopoulos (1997a) have been further analysed and some new statistical results obtained. In the present work, particular emphasis is given to the time-dependent behaviour of the kinematic shear stress, vorticity, enstrophy, dissipation rate, vorticity stretching and several of the matrix invariants of the velocity-gradient tensor, strain-rate tensor and rotation-rate tensor. The invariants are linked with terms appearing in the transport equations of enstrophy and dissipation rate. Indicative of the existence of extremely high fluctuations is that all r.m.s. values are considerably larger than the corresponding mean values. All invariants exhibit a very strong intermittent behaviour, which is characterized by large amplitude of bursts, which may be of the order of 10 times the r.m.s. values. A substantial qualitative agreement is found between the present experimentally obtained statistical properties of the invariants and those obtained from direct numerical simulation data. Patterns with high rates of turbulent kinetic energy dissipation and high enstrophy suggest the existence of strong shear layers in the near-wall region. In many instances, locally high values of the invariants are also associated with peaks in the shear stress. Conditional analysis provides some evidence of the existence of sequences of several vortices during strong vortical activities, with an average frequency of appearance four times higher than the frequency of appearance of hairpin vortices.

A heated or cooled body is positioned in a vertically rising forced flow. This develops both a kinematic and a thermal wake, the latter adding buoyant effects to the otherwise forced flow field. An asymptotic model is developed to treat this mixed convection in both plane and axisymmetric geometry. The model holds for laminar flow in the boundary layer approximation and uses a far-wake expansion for weak buoyant forces. For plane geometry the model is validated against both experiments in water and FEM simulations.

It is found for a heated wake that buoyant forces accelerate the fluid in the thermal wake such that the vertical velocity deficit in the kinematic wake is reduced. For strong heating this may even lead to vertical velocities larger than the forced flow amplitude. In conjunction the entrainment is intensified in a heated wake. The effects in a cooled wake are opposite in that the vertical velocity deficit is increased within the thermal wake and the horizontal flow into the wake is weakened. In a strongly cooled wake the horizontal flow may even invert, going from the wake centre into the ambient. The Prandtl number controls the width of the thermal wake and, thus, the portion of the kinematic wake which is affected by buoyant forces. Large Prandtl numbers superimpose a narrow buoyant plume, small Prandtl numbers a wide buoyant plume, onto the kinematic wake.

This technique solves the two-dimensional Poisson equations in geometries involving cylindrical objects. The method uses three fundamental solutions, corresponding to a line force, a line couple and a pressure gradient, on each cylinder. Superposition of the fundamental solutions due to all the cylinders involved, while approximately satisfying the no-slip condition on each cylinder, yields a mobility matrix relating the various forces and motions of all the cylinders. Any specific problem can be solved by prescribing the motions of the cylinders and solving the matrix. For problems involving few cylinders or with a sufficient degree of symmetry this can be done analytically.

Once constructed, the general method is applied analytically to a series of specific problems. The permeability of an eccentric annulus is derived. The result is numerically indistinguishable from the exact solution to the problem, but unlike the exact solution the present one is obtained in closed form. The drag on two parallel rods moving past one another is also derived and compared to the exact solution. In this case the result is accurate for rod separations down to about 0.2 times the rod diameter. Finally the drag on a rod moving in a triangular array of identical rods is derived. Here it is shown that due to screening it is sufficient to include the six nearest neighbours, regardless of the rod separation. Although the present examples are all worked out analytically, the matrix can also be solved numerically, in which case any two-dimensional arrangement of cylindrical objects can be studied.

The effect of time-periodical blowing through a spanwise slot on the near-wall turbulence characteristics is investigated. The blowing velocity changes in a cyclic manner from 0 to 5 wall units. The frequency of the oscillations is nearly equal to the median frequency of the near-wall turbulence. The measurements of the wall shear stress and the streamwise velocity are reported and discussed. The flow field near the blowing slot is partly relaminarized during the acceleration phase of the injection velocity which extends 40 wall units downstream. The imposed unsteadiness is confined to the buffer layer, and the time-mean structural parameters under unsteady blowing are found to be close to those of isotropic turbulence in this region. The relaminarized phase is unstable and gives way to a coherent spanwise structure that increases the shear from 80 to 300 wall units downstream of the slot in a predictable way. This phenomenon is strongly imposed-frequency dependent.

Transverse stability and instability of solitary waves correspond to a class of perturbations that are travelling in a direction transverse to the direction of the basic solitary wave. In this paper we consider the problem of transverse instability of solitary waves for the water-wave problem, from both the model equation point of view and the full water-wave equations. A new universal geometric condition for transverse instability forms the backbone of the analysis. The theory is first illustrated by application to model PDEs for water waves such as the KP equation, and then it is applied to the full water-wave problem. This is the first theory proposed for transverse instability of solitary waves of the full water-wave problem. The theory suggests the introduction of a new functional for water waves, whose importance is suggested by the mathematical structure. Without explicit calculation, the theory is used to argue that the basic class of solitary waves of the water-wave problem, which bifurcate at Froude number unity, are likely to be stable to transverse perturbations, even at large amplitude.

We investigate the performance of standard stochastic models of single-particle dispersion in two-dimensional turbulence. Owing to the presence of coherent vortices, particle dispersion in two-dimensional turbulence is characterized by a non-Gaussian velocity distribution and a non-exponential velocity autocorrelation, and it cannot be properly captured by either linear or nonlinear stochastic models with a single component process. Based on physical and dynamical considerations, we introduce a family of two-process stochastic models that provide a better parameterization of turbulent dispersion in rotating barotropic flows.

Transition arising from a separated region of flow is quite common and plays an important role in engineering. It is difficult to predict using conventional models and the transition mechanism is still not fully understood. We report the results of a numerical simulation to study the physics of separated boundary-layer transition induced by a change of curvature of the surface. The geometry is a flat plate with a semicircular leading edge. The Reynolds number based on the uniform inlet velocity and the leading-edge diameter is 3450. The simulated mean and turbulence quantities compare well with the available experimental data.

The numerical data have been comprehensively analysed to elucidate the entire transition process leading to breakdown to turbulence. It is evident from the simulation that the primary two-dimensional instability originates from the free shear in the bubble as the free shear layer is inviscidly unstable via the Kelvin–Helmholtz mechanism. These initial two-dimensional instability waves grow downstream with a amplification rate usually larger than that of Tollmien–Schlichting waves. Three-dimensional motions start to develop slowly under any small spanwise disturbance via a secondary instability mechanism associated with distortion of two-dimensional spanwise vortices and the formation of a spanwise peak–valley wave structure. Further downstream the distorted spanwise two-dimensional vortices roll up, leading to streamwise vorticity formation. Significant growth of three-dimensional motions occurs at about half the mean bubble length with hairpin vortices appearing at this stage, leading eventually to full breakdown to turbulence around the mean reattachment point. Vortex shedding from the separated shear layer is also observed and the ‘instantaneous reattachment’ position moves over a distance up to 50% of the mean reattachment length. Following reattachment, a turbulent boundary layer is established very quickly, but it is different from an equilibrium boundary layer.

The factors which affect the sound production of a vortex as it passes through a nozzle are investigated at both low and high Mach numbers using time-accurate inviscid-flow computations. Vortex circulation, initial position, and mean-flow Mach number are shown to be the primary factors which influence the amplitude and phase of the sound produced. Nozzle geometry and distribution of vorticity are also shown to play significant roles in determining the detailed form of the signal. Additionally, it is shown that solution bifurcations are possible at sufficiently large values of vortex circulation. Comparisons are made between sound signals computed directly using a numerical method for the Euler equations and predictions obtained using a compressible vortex-sound analogy coupled with a compact-source assumption for the computation of vorticity dynamics. The results confirm that the latter approach is accurate for a range of problems with low mean-flow Mach numbers. At higher Mach numbers, however, the non-compactness of the source becomes apparent, resulting in significant changes to the character of the signal which cannot be predicted using the analogy-based approach. Implications for the construction of simplified models of vortex sound in solid-rocket nozzles are discussed.

The paper proposes a one-dimensional model for predicting gas entrainment into a non-evaporating full cone steady spray injected into a stagnant gas at uniform pressure. The main outcome is a law relating the entrained mass flow rate to injected mass flow rate, gas properties, mean droplet diameter and axial distance from the nozzle. A comparison with available experimental data is presented. The model allows the apparent inconsistency of the experimental results obtained under different conditions to be eliminated by identifying two new non-dimensional parameters.

Effects of the Lorentz force on near-wall turbulence structures are investigated using the direct numerical simulation technique with the assumption of no induced magnetic field at low magnetic Reynolds number. A uniform magnetic field is applied in the streamwise (x), wall-normal (y) or spanwise (z) direction to turbulent flow in an infinitely long channel with non-conducting walls. The Lorentz force induced from the magnetic field suppresses the dynamically significant coherent structures near the wall. The skin friction decreases with increasing streamwise and spanwise magnetic fields, whereas it increases owing to the Hartmann effect when the strength of the wall-normal magnetic field exceeds a certain value. All the turbulence intensities and the Reynolds shear stress decrease with the wall-normal and spanwise magnetic fields, but the streamwise velocity fluctuations increase with the streamwise magnetic field although all other turbulence intensities decrease. It is also shown that the wall-normal magnetic field is much more effective than the streamwise and spanwise magnetic fields in reducing turbulent fluctuations and suppressing the near-wall streamwise vorticity, even though the wall-normal magnetic field interacts directly with the mean flow and results in drag increase at strong magnetic fields. In the channel with a strong streamwise magnetic field, two-dimensional streamwise velocity fluctuations u(y, z) exist, even after other components of the velocity fluctuations nearly vanish. In the cases of strong wall-normal and spanwise magnetic fields, all turbulence intensities, the Reynolds shear stress and vorticity fluctuations decrease rapidly and become zero. The turbulence structures are markedly elongated in the direction of the applied magnetic field when it is strong enough. It is shown that this elongation of the structures is associated with a rapid decrease of the Joule dissipation in time.

In experiments recently performed at Melbourne, Pitot-tube mean velocity profiles in a boundary layer disagreed with those obtained with hot wires. The standard MacMillan (1956) correction for the probe displacement effect and a correction for turbulence intensity were both required for obtaining agreement between the two sets of mean velocity data. We were thus motivated to reanalyse the Princeton superpipe data using the same two corrections. The result is a plausible conclusion that the superpipe is rough at the higher Reynolds numbers and its data follow the Colebrook (1939) formula for commercial pipes rather well. It also appears that the logarithmic law of the wall is valid, with a Kármán constant close to that found recently by Österlund (1999) from boundary layer measurements with a hot wire. The smooth regime in the pipe gave almost the same additive constant for the log-law as Österlund's. A comparison between the superpipe data and the pipe data of Perry, Henbest & Chong (1997) suggests that the conventional velocity defect law may be valid down to lower Reynolds numbers than concluded by Zagarola & Smits (1998).

We first demonstrate that, if the contributions of higher-order mean flow are ignored, the parabolized stability equations (Bertolotti et al. 1992) and the ‘full’ non-parallel equation of Govindarajan & Narasimha (1995, hereafter GN95) are both equivalent to order R−1 in the local Reynolds number R to Gaster's (1974) equation for the stability of spatially developing boundary layers. It is therefore of some concern that a detailed comparison between Gaster (1974) and GN95 reveals a small difference in the computed amplitude ratios. Although this difference is not significant in practical terms in Blasius flow, it is traced here to the approximation, in Gaster's method, of neglecting the change in eigenfunction shape due to flow non-parallelism. This approximation is not justified in the critical and the wall layers, where the neglected term is respectively O(R−2/3) and O(R−1) compared to the largest term. The excellent agreement of GN95 with exact numerical simulations, on the other hand, suggests that the effect of change in eigenfunction is accurately taken into account in that paper.