The evolution of a low-amplitude localized disturbance in a laminar boundary layer is considered. Linear inviscid theory illustrates that the disturbance may be divided into two parts: a dispersive wave part, represented by solutions to the Rayleigh equation which travel at their characteristic speeds, and a transient or advective part travelling at the local mean velocity. For a three-dimensional initial disturbance, calculations based on linear inviscid theory indicate that the transient portion of the disturbance does not decay and has the form of an inclined shear layer which elongates as the disturbance propagates downstream. The amplitude of the transient part exceeds by far that of the wave part of the disturbance. Experimental results are presented for a disturbance created by the impulsive motion of a small membrane flush-mounted at the wall. For small amplitudes, the initial evolution of the disturbance is found to be in good qualitative agreement with the inviscid calculations, showing the rapid formation of an inclined shear layer. Further downstream, the transient portion of the disturbance decays owing to viscous effects, leaving a linearly unstable dispersive wave packet. The evolutions of equal and opposite disturbances are compared and it is shown that, despite a weak nonlinearity that develops, the resultant wave packets are equal in structure but of opposite phase.

The purpose of the present paper is to investigate the structure of the laminar–turbulent transition region for the three-dimensional boundary layer along a 30° cone rotating in external axial flow. Spiral vortices, which were assumed as small disturbances in the present stability analysis, are observed experimentally in the transition region. The process of transition to a turbulent boundary layer is visualized in detail. When the ratio of rotational speed to external axial flow is increased, the critical and transition Reynolds numbers decrease remarkably. The spiral angle and the number of vortices appearing on the cone decrease as the rotational speed ratio is increased.

This paper generalizes the linear stability analysis of Pearson for Marangoni instability to the case where surface tension depends on both temperature and solute concentration. The results are expressed in terms of a thermal Marangoni number BT and a solutal Marangoni number BS. It is found when BS > 0 the onset of instability has the form of stationary convection, while when BS < 0 there are circumstances in which the onset of instability is in the form of oscillatory convection.

Bounds on the heat transport in a porous layer are derived using the variational method of Howard (1963) and Busse (1969b). The relatively simple structure of the variational problem in the case of porous convection allows one to formulate the theory more simply and to investigate some of the mathematical questions posed by the earlier work. A precise characterization of the solution with N wavenumbers is given. The variational problem is solved exactly among functions with a single overall wavenumber and this solution is in good agreement with a nonlinear perturbation solution of the governing equations and with experiments. An N-wavenumber solution is constructed for large Nusselt numbers by boundary-layer methods. The asymptotic solution is compared with a numerical solution of the problem for N = 2. The comparison supports the boundary-layer assumptions introduced in the asymptotic analysis.

By establishing that the length scale governing the detachment of a shock wave from a wedge is the distance from the leading edge to the sonic line, and by considering the view of observers with different length scales, it is predicted that the detachment distance increases gradually with wedge angle for relaxing flow and more rapidly in a perfect gas. Both of these features are confirmed by experiments in the free-piston shock tunnel. The influence of other length scales is discussed. The phenomenon is related to a relaxation effect in which a subsonic layer grows from the translational-rotational shock as the wedge inclination is increased beyond the frozen sonic point.

This paper provides the analysis of bursting and transition to turbulence in a Couette flow, based on the growth of nonlinear optimal disturbances. We use a global variational procedure to identify such optimal disturbances, defined as those initial perturbations yielding the largest energy growth at a given target time, for given Reynolds number and initial energy. The nonlinear optimal disturbances are found to be characterized by a basic structure, composed of inclined streamwise vortices along localized regions of low and high momentum. This basic structure closely recalls that found in boundary-layer flow (Cherubini et al., J. Fluid Mech., vol. 689, 2011, pp. 221–253), indicating that this structure may be considered the most ‘energetic’ one at short target times. However, small differences in the shape of these optimal perturbations, due to different levels of the initial energy or target time assigned in the optimization process, may produce remarkable differences in their evolution towards turbulence. In particular, direct numerical simulations have shown that optimal disturbances obtained for large initial energies and target times induce bursting events, whereas for lower values of these parameters the flow is directly attracted towards the turbulent state. For this reason, the optimal disturbances have been classified into two classes, the highly dissipative and the short-path perturbations. Both classes lead the flow to turbulence, skipping the phases of streak formation and secondary instability which are typical of the classical transition scenario for shear flows. The dynamics of this transition scenario exploits three main features of the nonlinear optimal disturbances: (i) the large initial value of the streamwise velocity component; (ii) the streamwise dependence of the disturbance; (iii) the presence of initial inclined streamwise vortices. The short-path perturbations are found to spend a considerable amount of time in the vicinity of the edge state (Schneider et al., Phys. Rev. E, vol. 78, 2008, 037301), whereas the highly dissipative optimal disturbances pass closer to the edge, but they are rapidly repelled away from it, leading the flow to high values of the dissipation rate. After this dissipation peak, the trajectories do not lead towards the turbulent attractor, but they spend some time in the vicinity of an unstable periodic orbit (UPO). This behaviour led us to conjecture that bursting events can be obtained not only as homoclinic orbits approaching the UPO, as recently found by van Veen & Kawahara (Phys. Rev. Lett., vol. 107, 2011, p. 114501), but also as heteroclinic orbits between the equilibrium solution on the edge and the UPO.

The effect of an insoluble surfactant on the stability of two-layer viscous flow in an inclined channel confined by two parallel walls is considered. A lubrication-flow model applicable to long waves and low-Reynolds-number-flow is developed, and pertinent nonlinear evolution equations for the interface position and surfactant concentration are derived. Linear stability analysis based on the lubrication-flow model and the inclusive equations of Stokes flow confirm the recent findings of Frenkel & Halpern (2002) and Halpern & Frenkel (2003) that the presence of an insoluble surfactant induces a Marangoni instability in certain regions of parameter space defined by the layer thickness and viscosity ratios. Numerical simulations based on both approaches show that the interfacial waves may grow and saturate into steep profiles. The lubrication-flow model is adequate in capturing the essential features of the instability for small and moderate wavenumbers.

We investigate the temporal evolution of the geometrical distribution of a passive scalar injected continuously into the far field of a turbulent water jet at a scale d smaller than the local integral scale of the turbulence. The concentration field is studied quantitatively by a laser-induced- fluorescence technique on a plane cut containing the jet axis. Global features such as the scalar dispersion from the source, as well as the fine structure of the scalar field, are analysed. In particular, we define the volume occupied by the regions whose concentration is larger than a given concentration threshold (support of the scalar field) and the surface in which this volume is enclosed (boundary of the support). The volume and surface extents, and their respective fractal dimensions are measured as a function of time t, and the concentration threshold is normalized by the initial concentration Cs/C0 for different injection sizes d. All of these quantities display a clear dependence on t, d and Cs, and their evolutions rescale with the variable ξ = (ut/d)(Cs/C0), the fractal dimension being, in addition, scale dependent. The surface-to-volume ratio and the fractal dimension of both the volume and the surface tend towards unity at large ξ, reflecting the sheet-like structure of the scalar at small scales. These findings suggest an original picture of the kinetics of turbulent mixing.

The influence of dense-gas effects on compressible wall-bounded turbulence is investigated by means of direct numerical simulations of supersonic turbulent channel flows. Results are obtained for PP11, a heavy fluorocarbon representative of dense gases, the thermophysics properties of which are described by using a fifth-order virial equation of state and advanced models for the transport properties. In the dense-gas regime, the speed of sound varies non-monotonically in small perturbations and the dependency of the transport properties on the fluid density (in addition to the temperature) is no longer negligible. A parametric study is carried out by varying the bulk Mach and Reynolds numbers, and results are compared to those obtained for a perfect gas, namely air. Dense-gas flow exhibits almost negligible friction heating effects, since the high specific heat of the fluids leads to a loose coupling between thermal and kinetic fields, even at high Mach numbers. Despite negligible temperature variations across the channel, the mean viscosity tends to decrease from the channel walls to the centreline (liquid-like behaviour), due to its complex dependency on fluid density. On the other hand, strong density fluctuations are present, but due to the non-standard sound speed variation (opposite to the mean density evolution across the channel), the amplitude is maximal close to the channel wall, i.e. in the viscous sublayer instead of the buffer layer like in perfect gases. As a consequence, these fluctuations do not alter the turbulence structure significantly, and Morkovin’s hypothesis is well respected at any Mach number considered in the study. The preceding features make high Mach wall-bounded flows of dense gases similar to incompressible flows with variable properties, despite the significant fluctuations of density and speed of sound. Indeed, the semi-local scaling of Patel et al. (Phys. Fluids, vol. 27 (9), 2015, 095101) or Trettel & Larsson (Phys. Fluids, vol. 28 (2), 2016, 026102) is shown to be well adapted to compare results from existing surveys and with the well-documented incompressible limit. Additionally, for a dense gas the isothermal channel flow is also almost adiabatic, and the Van Driest transformation also performs reasonably well. The present observations open the way to the development of suitable models for dense-gas turbulent flows.

With the aim to characterize the near-wall flow structures and their interaction with large-scale motions in the log-law region, time-resolved planar and volumetric flow field measurements were performed in the near-wall and log-law region of an adverse pressure gradient turbulent boundary layer following a zero pressure gradient turbulent boundary layer at a friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}=5000$. Due to the high spatial and temporal resolution of the measurements, it was possible to resolve and identify uniform-momentum zones in the region $z/\unicode[STIX]{x1D6FF}<0.15$ or $z^{+}<350$ and to relate them with well known coherent flow motions near the wall. The space–time results confirm that the turbulent superstructures have a strong impact even on the very near-wall flow motion and also their alternating appearance in time and intensity could be quantified over long time sequences. Using the time record of the velocity field, rare localized separation events appearing in the viscous sublayer were also analysed. By means of volumetric particle tracking velocimetry their three-dimensional topology and dynamics could be resolved. Based on the results, a conceptual model was deduced that explains their rare occurrence, topology and dynamics by means of a complex interaction process between low-momentum turbulent superstructures, near-wall low-speed streaks and tilted longitudinal and spanwise vortices located in the near-wall region.

When an acoustic wave impinges on an aerofoil cascade, a convective vorticity mode is generated giving rise to transverse velocity perturbations. This mode conversion process is investigated to explain the flow dynamics observed when swirlers are submitted to incident acoustic disturbances. The phenomenon is first studied in the case of a two-dimensional aerofoil cascade using a model derived from an actuator disk theory. The model is simplified to deal with low-Mach-number flows. The velocity field on the downstream side of the cascade features two components, an axial perturbation associated with the transmitted acoustic wave and a transverse disturbance corresponding to the vorticity wave generated at the cascade trailing edge. The model provides the amplitude of both components and defines their phase shift. Numerical simulations are carried out in a second stage to validate this model in the case of a cascade operating at a low Reynolds number Rec = 2700 based on the chord length. Space–time diagrams of velocity perturbations deduced from these simulations are used to retrieve the two types of modes. Experiments are then carried out in the case of an axial swirler placed in a cylindrical duct and submitted to plane acoustic waves emitted on the upstream side of the swirler. The amplitude and phase of the two velocity components measured in the axial and azimuthal directions are found to be in good agreement with theoretical estimates and with numerical calculations. This analysis is motivated by combustion dynamics observed in flames stabilized by aerodynamic swirlers in continuous combustors.

It is well established that when a turbulent flow is subjected to a non-uniform body force, the turbulence may be significantly suppressed in comparison with that of the flow of the same flow rate and hence the flow is said to be laminarised. This is the situation in buoyancy-aided mixed convection when severe heat transfer deterioration may occur. Here we report results of direct numerical simulations of flow with a linear or a step-change profile of body force. In contrast to the conventional view, we show that applying a body force to a turbulent flow while keeping the pressure force unchanged causes little changes to the key characteristics of the turbulence. In particular, the mixing characteristics of the turbulence represented by the turbulent viscosity remain largely unaffected. The so-called flow laminarisation due to a body force is in effect a reduction in the apparent Reynolds number of the flow, based on an apparent friction velocity associated with only the pressure force of the flow (i.e. excluding the contribution of the body force). The new understanding allows the level of the flow ‘laminarisation’ and when the full laminarisation occurs to be readily predicted. In terms of the near-wall turbulence structure, the numbers of ejections and sweeps are little influenced by the imposition of the body force, whereas the strength of each event may be enhanced if the coverage of the body force extends significantly away from the wall. The streamwise turbulent stress is usually increased in accordance with the observation of more and stronger elongated streaks, but the wall-normal and the circumferential turbulent stresses are largely unchanged.

Quantitative measurements of the structure of ejections from the wall region have been made using conditional-sampling techniques. Discrete ejections from a burst event were identified using fluorescent-dye flow visualization simultaneously with streamwise, u, and normal, v, velocity measurements. These velocity measurements, at y+ = 15, were conditionally sampled based on different phases of the ejection event. Features of the ejection which were educed from the conditional sampling were found to be very sensitive to the phase alignment. Results showed that ejections were characterized by a rapid deceleration at the leading edge followed by a strong positive velocity gradient at the trailing edge. An intense second-quadrant uv spike occurred immediately following the leading edge. This uv peak was highly correlated with a positive peak in the v velocity. The first ejection which occurred in a burst was found to be significantly more intense than the following ejections. Many characteristics of bursts which have been obtained from previous conditional-sampling studies were found to correspond to different phases of the ejection event.

We report on a novel method to generate fast transient microjets and study their characteristics. The simple device consists of two electrodes on a substrate with a hole in between. The side of the substrate with the electrodes is submerged in a liquid. Two separate microjets exit through the tapered hole after an electrical discharge is induced between the electrodes. They are formed during the expansion and collapse of a single cavitation bubble. The cavitation bubble dynamics as well as the jets were studied with high-speed photography at up to 500 000 f.p.s. With increasing jet velocity they become unstable and spray formation is observed. The jet created during expansion (first jet) is in most cases slower than the jet created during bubble collapse, which can reach up to $400~\text{m}~\text{s}^{-1}$. The spray exiting the orifice is at least in part due to the presence of cavitation in the microchannel as observed by high-speed recording. The effect of viscosity was tested using silicone oil of 10, 50 and 100 cSt. Interestingly, for all liquids the transition from a stable to an unstable jet occurs at $We\sim 4600$. We demonstrate that these microjets can penetrate into soft material; thus they can be potentially used as a needleless drug delivery device.

An investigation into the magnitude and direction of the convective velocity in a plane air jet was performed. Convective velocities were obtained from cross-correlation measurements. They are defined as the ratio of the spacing between two hot-wire probes and the time delay between their signals to reach maximum correlation. These velocities were larger in magnitude than the local mean velocities for lateral distances greater than the half-width of the jet. Frequency analysis of the convective velocity indicates that the large-scale eddies move slower than the mean flow while the small scales move faster. Based on the convective velocity vector, broadband ‘convection lines’ were defined and found to point outward with respect to the streamlines for all values of y/b [Gt ] 0·5. Likewise, frequency investigation indicates that ‘convection lines’ point outward for all y/b [Lt ] 1·3 and then inward for larger values of y/b.

The small-amplitude free vibrations of a spherical drop immersed in an outer immiscible fluid and in partial contact with a solid support are considered when both fluids are assumed to be viscous and incompressible, while gravity effects are neglected. Using the normal-mode decomposition and the Green-function method, the solution of the linearized Navier-Stokes equations is reduced to the solution of an eigenvalue problem. The model includes as particular cases the viscous model for a free drop proposed by Prosperetti (1980) and the inviscid model for a supported drop previously proposed by the authors.

The influence of the viscosity and of the support size are analysed both for the bubble and for the drop. At large values of the viscosity, the free drop shows significant differences with respect to the unsupported drop and a singular behaviour of the eigenvalue problem as the support size tends to zero.

The comparison with the available experimental data shows a quite satisfactory agreement for both the vibration frequency and the damping constant, provided that the support angle is not too large.

Swimming micro-organisms often have to propel themselves in complex non-Newtonian fluids. We carry out experiments with self-propelling helical swimmers driven by an externally rotating magnetic field in shear-thinning inelastic fluids. Similarly to swimming in a Newtonian fluid, we obtain for each fluid a locomotion speed that scales linearly with the rotation frequency of the swimmer, but with a prefactor that depends on the power index of the fluid. The fluid is seen to always increase the swimming speed of the helix, up to 50 % faster, and thus the strongest of such type reported to date. The maximum relative increase is for a fluid power index of approximately 0.6. Using simple scalings, we argue that the speed increase is not due directly to the local decrease of the flow viscosity around the helical filament, but hypothesise instead that it originates from confinement-like effect due to viscosity stratification around the swimmer.

Steady Couette and pressure-driven turbulent channel flows have large regions in which the gradients of the viscous and Reynolds stresses are approximately in balance (stress gradient balance regions). In the case of Couette flow, this region occupies the entire channel. Moreover, the relevant features of pressure-driven channel flow throughout the channel can be obtained from those of Couette flow by a simple transformation. It is shown that stress gradient balance regions are characterized by an intrinsic hierarchy of ‘scaling layers’ (analogous to the inner and outer domains), filling out the stress gradient balance region except for locations near the wall. The spatial extent of each scaling layer is found asymptotically to be proportional to its distance from the wall.

There is a rigorous connection between the scaling hierarchy and the mean velocity profile. This connection is through a certain function $A(y^+)$ defined in terms of the hierarchy, which remains $O(1)$ for all $y^+$. The mean velocity satisfies an exact logarithmic growth law in an interval of the hierarchy if and only if $A$ is constant. Although $A$ is generally not constant in any such interval, it is arguably almost constant under certain circumstances in some regions. These results are obtained completely independently of classical inner/outer/overlap scaling arguments, which require more restrictive assumptions.

The possible physical implications of these theoretical results are discussed.

A linear stability analysis for dune and ripple formation is presented that implements a rotational two-dimensional flow model valid in the smooth as well as in the transitional and rough flow regimes. Sediment is assumed to be transported as bedload, disregarding the role of suspension. Therefore, the main mechanism driving instability, for both ripples and dunes, is the phase lag between bed shear stress and bed elevation. Ripples are shown to be confined to relatively low values of the Shields parameter and of the particle Reynolds number. For higher values of the Shields parameter and of the particle Reynolds number (and thus of the Froude number and of the roughness Reynolds number), ripples are replaced by dunes. The present analysis ultimately allows for a successful unification of the theories of dune and ripple formation and for a clarification of the debated role of ripples on the formation of dunes. A good agreement between predicted and observed wavelengths for both ripples and dunes is found.