A detailed analysis of the ‘turbulent inertia’ (TI) term (the wall-normal gradient of the Reynolds shear stress, $\mathrm{d} \langle -uv\rangle /\mathrm{d} y $), in the axial mean momentum equation is presented for turbulent pipe flows at friction Reynolds numbers $\delta ^{+} \approx 500$, 1000 and 2000 using direct numerical simulation. Two different decompositions for TI are employed to further understand the mean structure of wall turbulence. In the first, the TI term is decomposed into the sum of two velocity–vorticity correlations ($\langle v \omega _z \rangle + \langle - w \omega _y \rangle $) and their co-spectra, which we interpret as an advective transport (vorticity dispersion) contribution and a change-of-scale effect (associated with the mechanism of vorticity stretching and reorientation). In the second decomposition, TI is equivalently represented as the wall-normal gradient of the Reynolds shear stress co-spectra, which serves to clarify the accelerative or decelerative effects associated with turbulent motions at different scales. The results show that the inner-normalised position, $y_m^{+}$, where the TI profile crosses zero, as well as the beginning of the logarithmic region of the wall turbulent flows (where the viscous force is leading order) move outwards in unison with increasing Reynolds number as $y_m^{+} \sim \sqrt{\delta ^{+}}$ because the eddies located close to $y_m^{+}$ are influenced by large-scale accelerating motions of the type $\langle - w \omega _y \rangle $ related to the change-of-scale effect (due to vorticity stretching). These large-scale motions of $O(\delta ^{+})$ gain a spectrum of larger length scales with increasing $\delta ^{+}$ and are related to the emergence of a secondary peak in the $-uv$ co-spectra. With increasing Reynolds number, the influence of the $O(\delta ^{+})$ motions promotes viscosity to act over increasingly longer times, thereby increasing the $y^{+}$ extent over which the mean viscous force retains leading order. Furthermore, the TI decompositions show that the $\langle v \omega _z \rangle $ motions (advective transport and/or dispersion of vorticity) are the dominant mechanism in and above the log region, whereas $\langle - w \omega _y \rangle $ motions (vorticity stretching and/or reorientation) are most significant below the log region. The motions associated with $\langle - w \omega _y \rangle $ predominantly underlie accelerations, whereas $\langle v \omega _z \rangle $ primarily contribute to decelerations. Finally, a description of the structure of wall turbulence deduced from the present analysis and our physical interpretation is presented, and is shown to be consistent with previous flow visualisation studies.

A modification of Homann’s axisymmetric outer potential stagnation-point flow of strain rate is obtained by adding periodic radial and azimuthal velocities of the form and , respectively, where is a shear rate. This leads to the discovery of a new family of asymmetric viscous stagnation-point flows depending on the shear-to-strain-rate ratio that exist over the range . Numerical solutions for the wall shear stress parameters and the displacement thicknesses are given and compared with their large- asymptotic behaviours. Sample similarity velocity profiles are also presented.

The identification of uniform momentum zones in wall-turbulence, introduced by Adrian, Meinhart & Tomkins (J. Fluid Mech., vol. 422, 2000, pp. 1–54) has been applied to turbulent channel flow, revealing a large ‘core’ region having high and uniform velocity magnitude. Examination of the core reveals that it is a region of relatively weak turbulence levels. For channel flow in the range $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}Re_{\tau } = 1000\text {--}4000$, it was found that the ‘core’ is identifiable by regions bounded by the continuous isocontour lines of the streamwise velocity at $0.95U_{CL}$ (95 % of the centreline velocity). A detailed investigation into the properties of the core has revealed it has a large-scale oscillation which is predominantly anti-symmetric with respect to the channel centreline as it moves through the channel, and there is a distinct jump in turbulence statistics as the core boundary is crossed. It is concluded that the edge of the core demarcates a shear layer of relatively intense vorticity such that the interior of the core contains weakly varying, very low-level turbulence (relative to the flow closer to the wall). Although channel flows are generally referred to as ‘fully turbulent’, these findings suggest there exists a relatively large and ‘quiescent’ core region with a boundary qualitatively similar to the turbulent/non-turbulent interface of boundary layers, jets and wakes.

A physical model for compressible flows with phase transition is described, in which all the processes of phase transition, i.e. nucleation, droplet growth, droplet evaporation and de-nucleation, are incorporated. The model is focused on dilute mixtures of vapour and droplets in a carrier gas with typical maximum liquid mass fraction smaller than 0.02. The new model is based on a reinterpretation of Hill's method of moments of the droplet size distribution function. Starting from the general dynamic equation, it is emphasized that nucleation or de-nucleation correspond to the rates at which droplets enter or leave droplet size space, respectively. Nucleation and de-nucleation have to be treated differently in agreement with their differences in physical nature. Attention is given to the droplet growth model that takes into account Knudsen effects and temperature differences between droplets and gas. The new phase transition model is then combined with the Euler equations and results in a new numerical method: ASCE2D. The numerical method is first applied to the problem of shock/expansion wave formation in a closed shock tube with humid nitrogen as a driver gas. Nucleation and droplet growth are induced by the expansion wave, and in turn affect the structure of the expansion wave. When the main shock, reflected from the end wall of the low-pressure section, passes the condensation zone, evaporation and de-nucleation occur. As a second example, the problem of the flow of humid nitrogen in a pulse-expansion wave tube, designed to study nucleation and droplet growth in monodisperse clouds, is investigated experimentally and numerically.

The mean wall-normal gradients of the Reynolds shear stress and the turbulent kinetic energy have direct connections to the transport mechanisms of turbulent-boundary-layer flow. According to the Stokes–Helmholtz decomposition, these gradients can be expressed in terms of velocity–vorticity products. Physical experiments were conducted to explore the statistical properties of some of the relevant velocity–vorticity products. The high-Reynolds-number data (Rθ ≃ O(106), where θ is the momentum thickness) were acquired in the near neutrally stable atmospheric-surface-layer flow over a salt playa under both smooth- and rough-wall conditions. The low-Rθ data were from a database acquired in a large-scale laboratory facility at 1000 > Rθ > 5000. Corresponding to a companion study of the Reynolds stresses (Priyadarshana & Klewicki, Phys. Fluids, vol. 16, 2004, p. 4586), comparisons of low- and high-Rθ as well as smooth- and rough-wall boundary-layer results were made at the approximate wall-normal locations yp/2 and 2yp, where yp is the wall-normal location of the peak of the Reynolds shear stress, at each Reynolds number. In this paper, the properties of the vωz, wωy and uωz products are analysed through their statistics and cospectra over a three-decade variation in Reynolds number. Here u, v and w are the fluctuating streamwise, wall-normal and spanwise velocity components and ωy and ωz are the fluctuating wall-normal and spanwise vorticity components. It is observed that v–ωz statistics and spectral behaviours exhibit considerable sensitivity to Reynolds number as well as to wall roughness. More broadly, the correlations between the v and ω fields are seen to arise from a ‘scale selection’ near the peak in the associated vorticity spectra and, in some cases, near the peak in the associated velocity spectra as well.

Thiria, Goujon-Durand & Wesfreid (J. Fluid Mech. vol. 560, 2006, p. 123), it was shown that vortex shedding from a rotationally oscillating cylinder at moderate Reynolds number can be characterized by the spatial coexistence of two distinct patterns, one of which is related to the forcing frequency in the near wake and the other to a frequency close to the natural one for the unforced case downstream of this locked region. The existence and the modification of these wake characteristics were found to be strongly affected by the frequency and the amplitude of the cylinder oscillation. In this paper, a linear stability analysis of these forced regimes is performed, and shows that the stability characteristics of such flows are governed by a strong mean flow correction which is a function of the oscillation parameters. We also present experiments on the spatial properties of the global mode and on the selection of the vortex shedding frequency as a function of the forcing conditions for Re = 150. Finally, we elucidate a diagram of locked and non-locked states, for a large range of frequencies and amplitudes of the oscillation.

This paper is concerned with convective and absolute instabilities in the boundary-layer flow over the outer surface of a sphere rotating in an otherwise still fluid. Viscous and streamline-curvature effects are included and the analysis is conducted between latitudes of 10° and 80° from the axis of rotation. Both convective and absolute instabilities are found at each latitude within specific parameter spaces. The results of the convective instability analysis show that a crossflow instability mode is the most dangerous below θ = 66°. Above this latitude a streamline-curvature mode is found to be the most dangerous, which coincides with the appearance of reverse flow in the radial component of the mean flow. At low latitudes the disturbances are considered to be stationary, but at higher latitudes they are taken to rotate at 76% of the sphere surface speed, as observed in experimental studies. Our predictions of the Reynolds number and vortex angle at the onset of convective instability are consistent with existing experimental measurements. Results are also presented that suggest that the occurrence of the slowly rotating vortices is associated with the dominance of the streamline-curvature mode at θ = 66°. The local Reynolds number at the predicted onset of absolute instability matches experimental data well for the onset of turbulence at θ = 30°; beyond this latitude the discrepancy increases but remains relatively small below θ = 70°. It is suggested that this absolute instability may cause the onset of transition below θ = 70°. Close to the pole the predictions of each stability analysis are seen to approach those of existing rotating disk investigations.

The effect of a temperature-dependent solute diffusion coefficient on a model of unidirectional solidification of a binary melt with a quasi-equilibrium mushy (two-phase) zone is studied. The Soret effect (thermodiffusion) is also included in the analysis. The concentration field in the liquid, solid and mushy phases, as well as the rate of solidification and mushy zone thickness, are found analytically as functions of all thermophysical parameters. The role of the nonlinear solute transport is detailed in the analysis. On the basis of analytical solutions, the regime of solidification with a quasi-equilibrium mushy zone is replaced by an equivalent discontinuity surface (frontal) regime with new boundary conditions.

A fully developed Mach 3 turbulent boundary layer subjected to four expansion regions (centred and gradual expansions of 7° and 14°) was investigated with laser Doppler velocimetry. Measurements were acquired in the incoming flat-plate boundary layer and to s/δ≃20 downstream of the expansions. While mean velocity profiles exhibit significant progress towards recovery by the most downstream measurements, the turbulence structure remains far from equilibrium. Comparisons of computed (method of characteristics) and measured velocity profiles indicate that the post-expansion flow evolution is largely inviscid for approximately 10δ. Turbulence levels decrease across the expansion, and the reductions increase in severity as the wall is approached. Downstream of the 14° expansions, the reductions are more severe and reverse transition is indicated by sharp reductions in turbulent kinetic energy levels and a change in sign of the Reynolds shear stress. Dimensionless parameters such as anisotropy and shear stress correlation coefficient highlight the complex evolution of the post-expansion boundary layer. An examination of the compressible vorticity transport equation and estimates of the perturbation impulses attributable to streamline curvature, acceleration, and dilatation both confirm dilatation to be the primary stabilizer. However, the dilatation impulse increases only slightly for the 14° expansions, so the dramatic differences downstream of the 7° and 14° expansions indicate nonlinear boundary layer response. Differences attributable to the varied radii of surface curvature are fleeting for the 7° expansions, but persist through the spatial extent of the measurements for the 14° expansions.

The paper investigates experimentally the global wake dynamics of a simplified three-dimensional ground vehicle at a Reynolds number of $Re\simeq 4.0\times 10^{5}$. The after-body has a blunt rectangular trailing edge leading to a massive flow separation. Both the inclination (yaw and pitch angles) and the distance to the ground (ground clearance) are accurately adjustable. Two different aspect ratios of the rectangular base are considered; wider than it is tall (minor axis perpendicular to the ground) and taller than it is wide (major axis perpendicular to the ground). Measurements of the spatial distribution of the pressure at the base and velocity fields in the wake are used as topological indicators of the flow. Sensitivity analyses of the base pressure gradient expressed in polar form (modulus and phase) varying ground clearance, yaw and pitch are performed. Above a critical ground clearance and whatever the inclination is, the modulus is always found to be large due to the permanent static symmetry-breaking instability, and slightly smaller when aligned with the minor axis of the base rather than when aligned with the major axis. The instability can be characterized with a unique wake mode, quantified by this modulus (asymmetry strength) and a phase (wake orientation) which is the key ingredient of the global wake dynamics. An additional deep rear cavity that suppresses the static instability allows a basic flow to be characterized. It is shown that both the inclination and the ground clearance constrain the phase dynamics of the unstable wake in such way that the component of the pressure gradient aligned with the minor axis of the rectangular base equals that of the basic flow. Meanwhile, the other component related to the major axis adjusts to preserve the large modulus imposed by the instability. In most cases, the dynamics explores only two possible opposite values of the component along the major axis. Their respective probability depends on the geometrical environment of the wake: base shape, body inclination, ground proximity and body supports. An expression for the lateral force coefficients taking into account the wake instability is proposed.

The Lagrangian transport due to non-breaking and breaking focusing wave packets is examined. We present direct numerical simulations of the two-phase air–water Navier–Stokes equations describing focusing wave packets, investigating the Lagrangian drift by tracking tracer particles in the water before, during and after the breaking event. The net horizontal transport for non-breaking focusing packets is well described by the classical Stokes drift, both at the surface and in the bulk of the fluid, where the e-folding scale of the evanescent vertical profile is given by the characteristic wavenumber. For focusing wave packets that lead to breaking, we observe an added drift that can be ten times larger than the classical Stokes drift for a non-breaking packet at the surface, while the initial depth of the broken fluid scales with the wave height at breaking. We find that the breaking induced Lagrangian transport scales with the breaking strength. A simple scaling argument is proposed to describe this added drift and is found to be consistent with the direct numerical simulations. Applications to upper ocean processes are discussed.

A lubrication analysis is presented for the near-contact axisymmetric motion of spherical drops covered with an insoluble non-diffusing surfactant. Detailed results are presented for the surfactant distribution, the interfacial velocity, and the gap width between the drop surfaces. The effect of surfactant is characterized by a dimensionless force parameter: the external force normalized by Marangoni stresses. Critical values of the force parameter have been established for drop coalescence and separation. Surfactant-covered drops are stable to rapid coalescence for external forces less than 4πkTac0, where c0 is the surfactant concentration at the edge of the near-contact region and a is the reduced drop radius.

For subcritical forces, the behaviour of surfactant-covered drops is described by two time scales: a fast time scale characteristic of near-contact motion between drops with clean interfaces and a slow time scale associated with rigid particles. The surfactant distribution evolves on the short time scale until Marangoni stresses approximately balance the external force. Supercritical values of the external force cannot be balanced; coalescence and separation occur on the fast time scale. The coalescence time normalized by the result for drops with clean interfaces is independent of the viscosity ratio and initial gap width.

Under subcritical force conditions, a universal long-time behaviour is attained on the slow time scale. At long times, the surfactant distribution scales with the near-contact region and the surface velocity is directed inward which impedes the drop approach and accelerates their separation compared to rigid particles. For drops pressed together with a sufficiently large subcritical force, a shrinking surfactant-free clean spot forms.

Surfactant-covered drops exhibit an elastic response to unsteady external forces because of energy stored in the surfactant distribution.

An experimental investigation on the initial instability of nonlinear deep-water wave trains including wind effects is reported. The experiment was conducted at the Ocean Engineering Laboratory wind-wave facility (50 m long, 4.2 m wide, 2.1 m deep), with a fully computer-controlled mechanical wave generator to explore the parameter space: steepness; sideband frequency; wind speed. The estimated growth rates of the Benjamin–Feir instability from seeded wind-free experiments agreed well with the theoretical prediction derived from Krasitskii's four-wave reduced equation as computed here. Wind was added to the same wave system; the growth rates of the sidebands were reduced for weak, and enhanced for strong wind forcing. Experiments with naturally selected sidebands, i.e. unseeded, were conducted as well; measurements showed that wind did not inhibit the growth of sidebands in the case of either two-dimensional or three-dimensional instabilities. A comparison of the results with earlier work suggests that there are two independent effects of wind: first, the alteration of the inviscid growth for a given modulational frequency as shown by comparison with the seeded experiments without wind; second, a change in the natural modulational frequency appearing in the presence of wind which is a function of the wave age, as observed in unseeded experiments. Both effects combined will determine whether the modulational instability is enhanced or suppressed; comparison of experimental results with theoretical predictions suggests that the effect of wind on the natural selection of the modulational frequency is the dominant effect. It was shown that for moderate to old waves, the net effect of wind on the modulational instability is small. For all the experiments except a few unseeded cases with weak breakers, the modulation was small and no breaking was observed within the tank.

This study investigates the stability and transition of Görtler vortices in a hypersonic boundary layer using linear stability theory and direct numerical simulations. In the simulations, Görtler vortices are separately excited by wall blowing and suction with spanwise wavelengths of 3, 6 and 9 mm. In addition to primary streaks with the same wavelength as the blowing and suction, secondary streaks with half the wavelength also emerge in the 6 and 9 mm cases. The streaks develop into mushroom structures before breaking down. The breakdown processes of the three cases are dominated by a sinuous-mode instability, a varicose-mode instability and a combination of the two, respectively. Both fundamental and subharmonic instabilities are relevant in all cases. Multiple modes are identified in the secondary-instability stage, some of which originate from the primary instabilities (first and second Mack modes). We demonstrate that the first Mack mode can be destabilized to either a varicose-mode or sinuous-mode streak instability depending on its frequency and wavelength, whereas the second Mack mode undergoes a stabilizing stage before turning into a varicose mode in the 6 and 9 mm cases. An energy analysis reveals the stabilizing and destabilizing mechanisms of the primary instabilities under the influence of Görtler vortices, highlighting the role played by the spanwise production based on the spanwise gradient of the streamwise velocity in both varicose and sinuous modes. The effects introduced by the secondary streaks are examined by filtering the secondary streaks in two new simulations with nominally identical conditions to those of the 6 and 9 mm cases. Remarkably, the secondary streaks can destabilize the Görtler vortices, therefore advancing the transition. The stability theory results are in good agreement with those from direct numerical simulations.

We present a model for the relative velocity of inertial particles in turbulent flows that provides new physical insight into this problem. Our general formulation shows that the relative velocity has contributions from two terms, referred to as the ‘generalized acceleration’ and ‘generalized shear’, because they reduce to the well-known acceleration and shear terms in the Saffman–Turner limit. The generalized shear term represents particles' memory of the flow velocity difference along their trajectories and depends on the inertial particle pair dispersion backward in time. The importance of this backward dispersion in determining the particle relative velocity is emphasized. We find that our model with a two-phase separation behaviour, an early ballistic phase and a later tracer-like phase, as found by recent simulations for the forward (in time) dispersion of inertial particle pairs, gives good fits to the measured relative speeds from simulations at low Reynolds numbers. In the monodisperse case with identical particles, the generalized acceleration term vanishes and the relative velocity is determined by the generalized shear term. At large Reynolds numbers, our model gives a St1/2-dependence of the relative velocity on the Stokes number St in the inertial range for both the ballistic behaviour and the Richardson separation law. This leads to the same inertial-range scaling for the two-phase separation that well fits the simulation results. Our calculations for the bidisperse case show that, with the friction timescale of one particle fixed, the relative speed as a function of the other particle's friction time has a dip when the two timescales are similar. This indicates that similar-size particles tend to have stronger velocity correlation than different ones. We find that the primary contribution at the dip, i.e. for similar particles, is from the generalized shear term, while the generalized acceleration term is dominant for particles of very different sizes. Future numerical studies are motivated to check the accuracy of the assumptions made in our model and to investigate the backward-in-time dispersion of inertial particle pairs in turbulent flows.

Details of flow visualization, aerodynamic forces and pitching moment, static pressure and skin friction measurements have been carried out on a symmetrical aerofoil at fixed angle of incidence in longitudinal oscillations parallel to the uniform airstream of a wind-tunnel.

This investigation shows weak unsteady effects at incidences below that of static stall. For higher incidences, strong unsteady effects appear and depend on the frequency and amplitude of the oscillations. The measurements indicate an overshoot of the instantaneous lift and drag which is explained by a strong vortex shedding process occurring during the dynamic stall encountered by the aerofoil in decelerated motion, as observed for profiles oscillating in pitch through stall. When the aerofoil is going forward in accelerated motion dynamic reattachment may be observed at very high incidence over a short part of the period of oscillation.

Dynamic stall and dynamic reattachment contribute to a favourable effect of unsteadiness on the mean lift coefficient, which increases as compared to the steady state one, and which is expressed through an empirical formula involving incidence, frequency and amplitude of oscillations. At given incidence, optimization of this feature is achieved by matching the frequency and the amplitude of oscillation, respectively with the frequency linked with the highest peak of energy in the wake, and with the distance between two consecutive vortices in the mean wake when modelled as a von Kármán's vortex street.

A new system of approximating equations is derived for three-dimensional steady viscous compressible flows in which a primary flow direction is present, but in which both transverse velocity components can be large. If the transverse velocity vector which corrects a given potential flow is first decomposed into ‘potential’ and ‘rotational’ vector components, then a re-examination of three-dimensional boundary-layer theory shows that both components (vϕ, wϕ) of the potential-velocity vector may be assumed small, whereas both components (vψ, wψ) of the rotational-velocity vector and hence of the composite secondary flow (v, w) can remain of order unity. An assumption of small scalar potential then leads to a system of governing equations whose characteristic polynomial has a non-elliptic form for arbitrary Mach number, without introducing any direct approximation of either streamwise or transverse pressure gradient terms. These non-elliptic equations can be solved very economically as a well-posed initial/boundary-value problem. Computed results for laminar subsonic flow in a curved square duct confirm the small scalar-potential approximation for both large (R/d = 100) and small (R/d = 2) radius of curvature. Other computations for R/d = 2.3 are in good agreement with the measurements of Taylor, Whitelaw & Yianneskis (1980).

A circular shear zone is created in a thin layer of fluid. The Kelvin-Helmholtz instability induces regular, steady patterns of m vortices. The experimental conditions are such that neither the centrifugal nor the Coriolis forces play a role in the motion. The state of the flow is defined by a Reynolds number, the value of which is controlled by the imposed velocities. The pattern of vortices can be characterized by its wavevector k or by m, the order of its symmetry. As k is quantized, its evolution, due to an increase or a decrease of the controlled stress, leads to transitions between patterns of different m. The transitory states between different symmetries are investigated. The experiments are performed with a soap film which provides a new type of visualization of an air flow.

The boundary layer generated by the harmonic oscillations of a wavy wall in a fluid otherwise at rest is studied. First the wall waviness is assumed to be of small amplitude and large values of the Reynolds number are considered. The results obtained by means of a linear analysis, where the time variable appears only as a parameter, show that resonance may occur. Indeed it is found that when the Reynolds number is larger than a critical value, an instant within the decelerating part of the cycle exists such that a waviness of infinitesimal amplitude induces unbounded perturbations of the flow in the Stokes layer. The passage through resonance is then studied by means of a multiple-timescale approach, taking into account the damping effect of local acceleration within a small time range around resonance. The asymptotic approach fails beyond a threshold value of the Reynolds number, because the damping effect of the local acceleration terms spreads over the whole cycle. The problem is then tackled by means of an approach that takes into account the above damping effect throughout the whole cycle. Finally, a numerical procedure is used that also allows the inclusion of nonlinear terms and the study of the interactions among forced and free modes. The numerical approach reveals that, even for relatively large values of the amplitude of the wall waviness, nonlinear effects are negligible and the damping of resonance is mainly due to local acceleration effects. The relevance of the results to the understanding of transition to turbulence in Stokes layers is discussed.

Convex transverse curvature effects in wall-bounded turbulent flows are significant if the boundary-layer thickness is large compared to the radius of curvature (large γ = δ/a). The curvature affects the inner part of the flow if a+, the cylinder radius in wall units, is small.

Two direct numerical simulations of a model problem approximating axial flow boundary layers on long cylinders were performed for γ = 5 (a+ ≈ 43) and γ = 11 (a+ ≈ 21). Statistical and structural data were extracted from the computed flow fields. The effects of the transverse curvature were identified by comparing the present results with those of the plane channel simulation of Kim, Moin & Moser (1987), performed at a similar Reynolds number. As the curvature increases, the skin friction increases, the slope of the logarithmic region decreases and turbulence intensities are reduced. Several turbulence statistics are found to scale with a curvature dependent velocity scale derived from the mean momentum equation. Near the wall, the flow is more anisotropic than in the plane channel with a larger percentage of the turbulent kinetic energy resulting from the streamwise velocity fluctuations. As the curvature increases, regions of strong normal vorticity develop near the wall.