The sensitivity of the flow around three-dimensional blunt geometry is investigated experimentally at Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}9.2\times 10^4$. Vertical and horizontal control cylinders are used to disturb the natural flow which is the superposition of two reflectional symmetry breaking states (see Part 1 of this study, Grandemange, Gohlke & Cadot, J. Fluid Mech., vol. 722, 2013b, pp. 51–84). When the perturbation breaks the symmetry of the set-up, it can select one of the two asymmetric topologies so that a mean side force is found. When the reflectional symmetry is preserved, some positions of horizontal and vertical control cylinders alter the natural bi-stability of the flow which may result in drag reduction. In addition, it is found that the horizontal perturbation affects the lift force especially when the top and bottom mixing layers are disturbed. The ability of the disturbances to suppress the bi-stable behaviour is discussed and, introducing a formalism of induced drag, a quantification of the impact on the drag of the cross-flow forces observed for the natural bi-stable wake is suggested. Finally, a general concept for a control strategy of separated flows past three-dimensional bluff bodies can be drawn up from these analyses.

We introduce a new boundary layer formalism on the basis of which a class of exact solutions to the Navier–Stokes equations is derived. These solutions describe laminar boundary layer flows past a flat plate under the assumption of one homogeneous direction, such as the classical swept Hiemenz boundary layer (SHBL), the asymptotic suction boundary layer (ASBL) and the oblique impingement boundary layer. The linear stability of these new solutions is investigated, uncovering new results for the SHBL and the ASBL. Previously, each of these flows had been described with its own formalism and coordinate system, such that the solutions could not be transformed into each other. Using a new compound formalism, we are able to show that the ASBL is the physical limit of the SHBL with wall suction when the chordwise velocity component vanishes while the homogeneous sweep velocity is maintained. A corresponding non-dimensionalization is proposed, which allows conversion of the new Reynolds number definition to the classical ones. Linear stability analysis for the new class of solutions reveals a compound neutral surface which contains the classical neutral curves of the SHBL and the ASBL. It is shown that the linearly most unstable Görtler–Hämmerlin modes of the SHBL smoothly transform into Tollmien–Schlichting modes as the chordwise velocity vanishes. These results are useful for transition prediction of the attachment-line instability, especially concerning the use of suction to stabilize boundary layers of swept-wing aircraft.

We study the formation, growth and disintegration of jets following the impact of a drop on a thin film of the same liquid for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{We}<1000$ and $\mathit{Re}<2000$ using a combination of numerical simulations and linear stability theory (Agbaglah, Josserand & Zaleski, Phys. Fluids, vol. 25, 2013, 022103). Our simulations faithfully capture this phenomena and are in good agreement with experimental profiles obtained from high-speed X-ray imaging. We obtain scaling relations from our simulations and use these as inputs to our stability analysis. The resulting predictions for the most unstable wavelength are in excellent agreement with experimental data. Our calculations show that the dominant destabilizing mechanism is a competition between capillarity and inertia but that deceleration of the rim provides an additional boost to growth. We also predict over the entire parameter range of our study the number and timescale for formation of secondary droplets formed during a splash, based on the assumption that the most unstable mode sets the droplet number.

Nonlinear instability in a supersonic boundary layer at Mach 4.8 with two-dimensional roughness is investigated by means of spatial direct numerical simulations (DNS). It was previously found that an important effect of a two-dimensional roughness is to increase significantly the amplitude of two-dimensional waves downstream of the roughness in a certain frequency band through enhanced instability and transient growth, while waves outside this band are damped. Here, we investigate the nonlinear secondary instability induced by a large-amplitude two-dimensional wave, which has received a significant boost in amplitude from this additional roughness-induced amplification. Both subharmonic and fundamental secondary excitation of the oblique secondary waves are considered. We found that even though the growth rate of the secondary perturbations increases compared to their linear amplification, only in some of the cases was a fully resonant state attained by the streamwise end of the domain. A parametric investigation of the amplitude of the primary wave, the phase difference between the primary and the secondary waves, and the spanwise wavenumber has also been performed. The transient growth experienced by the primary wave was found to not influence the secondary instability for most parameter combinations. For unfavourable phase relations between the primary and the secondary waves, the phase speed of the secondary wave decreases significantly, and this hampers its growth. Finally, we also investigated the strongly nonlinear stage, for which both the primary and the subharmonic secondary waves had a comparable, finite amplitude. In this case, the growth of the primary waves was found to vanish downstream of the transient growth region, resulting in a lower amplitude than in the absence of the large-amplitude secondary wave. This feedback also decreases the amplification rate of the secondary wave.

Flow visualization, laser Doppler velocimetry and planar and stereoscopic particle image velocimetry were used to investigate the isothermal velocity field along an eccentric annular channel with a diameter ratio of 0.5 and an eccentricity of 0.8 for a Reynolds number of 7300. Observation of the flow development has identified three distinct regions: the entrance region, the fluctuation-growth (FG) region and the rapid-mixing (RM) region. Weak quasi-periodic velocity fluctuations were first detected in the downstream part of the entrance region, and grew into very strong ones, reaching peak-to-peak amplitudes in the narrow gap that were nearly 60 % of the bulk velocity. Two mixing layers were identified on either side of the gap, which generated a street of counter rotating vortices and thorough large-scale mixing of the fluid in the channel.

Numerical simulation results are presented for the linear and nonlinear evolution of disturbances in a flat Stokes layer. The response to a spatially localised impulsive forcing is investigated and it is found that the spatial–temporal development of the flow displays an intriguing family-tree-like structure, which involves the birth of successive generations of distinct wavepacket components. It is shown that some features of this unexpected structure can be predicted using the results of a linear stability analysis based on Floquet theory.

The steady, spanwise-periodic, symmetric (varicose) optimal blowing and suction that maximizes energy amplification in the circular cylinder wake is computed at Reynolds numbers ranging from 50 to 100. It is found that the cylinder wake can sustain large energy amplifications that are associated with the generation by the optimal blowing and suction of streamwise vortices near the cylinder, which then induce the transient spatial growth of high-energy streamwise streaks further downstream. The most amplified perturbations have spanwise wavelengths ranging from five to seven times the cylinder diameter at the Reynolds numbers considered, with the corresponding optimal streaks reaching their maximum amplitude in the near wake, inside the pocket of absolute instability which sustains the global instability. The optimal blowing and suction is shown to stabilize the global linear instability. The most stabilizing spanwise wavelengths are in good agreement with previous findings. The amplitude of optimal blowing and suction required to suppress the global instability decreases when the Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}$ is increased from 75 to 100. This trend reveals the key role played by the non-normal amplification of the streaks in the stabilization process, which is able to overcome the increase of the uncontrolled global growth rate with $\mathit{Re}$. Finally, it is shown that the global instability can be suppressed with control amplitudes smaller than those required by 2-D (spanwise-uniform) control. This result is not what would be expected from first-order sensitivity analyses, which predict a zero sensitivity of the global instability to spanwise-periodic control and, in general, a non-zero sensitivity to spanwise-uniform control.

We study autopropulsion of an interface particle that is driven by the Marangoni stress arising from a self-generated asymmetric temperature or concentration field. We calculate separately the long-range Marangoni flow $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\boldsymbol {v}}^{I}$ due to the stress discontinuity at the interface and the short-range velocity field ${\boldsymbol {v}}^{P}$ imposed by the no-slip condition on the particle surface. Both contributions are evaluated for a spherical floater with temperature monopole and dipole moments. We find that the self-propulsion velocity is given by the amplitude of the ‘source doublet’ that belongs to the short-range contribution ${\boldsymbol {v}}^{P}$. Hydrodynamic interactions, on the other hand, are determined by the long-range Marangoni flow ${\boldsymbol {v}}^{I}$. Its dipolar part results in an asymmetric advection pattern of neighbouring particles, which in turn may perturb the known hexatic lattice or even favour disordered states.

Our concern in this paper is with high-Reynolds-number nonlinear equilibrium solutions of the Navier–Stokes equations for boundary-layer flows. Here we consider the asymptotic suction boundary layer (ASBL) which we take as a prototype parallel boundary layer. Solutions of the equations of motion are obtained using a homotopy continuation from two known types of solutions for plane Couette flow. At high Reynolds numbers, it is shown that the first type of solution takes the form of a vortex–wave interaction (VWI) state, see Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666), and is located in the main part of the boundary layer. On the other hand, here the second type is found to support an equilibrium solution of the unit-Reynolds-number Navier–Stokes equations in a layer located a distance of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}O(\ln \mathit{Re})$ from the wall. Here $\mathit{Re}$ is the Reynolds number based on the free-stream speed and the unperturbed boundary-layer thickness. The streaky field produced by the interaction grows exponentially below the layer and takes its maximum size within the unperturbed boundary layer. The results suggest the possibility of two distinct types of streaky coherent structures existing, possibly simultaneously, in disturbed boundary layers.

The stability of a mixing layer made up of two miscible fluids, with a viscosity-stratified layer between them, is studied. The two fluids are of the same density. It is shown that unlike other viscosity-stratified shear flows, where species diffusivity is a dominant factor determining stability, species diffusivity variations over orders of magnitude do not change the answer to any noticeable degree in this case. Viscosity stratification, however, does matter, and can stabilize or destabilize the flow, depending on whether the layer of varying velocity is located within the less or more viscous fluid. By making an inviscid model flow with a slope change across the ‘viscosity’ interface, we show that viscous and inviscid results are in qualitative agreement. The absolute instability of the flow can also be significantly altered by viscosity stratification.

This work aims to improve understanding of the damping induced by an axial flow on a rigid cylinder undergoing small lateral oscillations within the framework of the quasistatic assumption. The study focuses on the normal force exerted on the cylinder for a Reynolds number of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=24\, 000$ (based on the cylinder diameter and axial flow velocity). Both dynamic and static approaches are investigated. With the static approach, fluid forces, pressure distributions and velocity fields are measured for different yaw angles and cylinder lengths in a wind tunnel. It is found that for yaw angles smaller than $5{^\circ }$, the normal force varies linearly with the angle and is fully dominated by its lift component. The lift originates from the high pressure coefficient at the front of the cylinder, which is found to depend linearly on the angle, and from a base pressure coefficient that remains close to zero independent of the yaw angle. At the base, a flow deficit and two counter-rotating vortices are observed. A numerical simulation using a $k\mbox{--}\omega $ shear stress transport turbulence model confirms the static experimental results. A dynamic experiment conducted in a water tunnel brings out damping-rate values during free oscillations of the cylinder. As expected from the linear dependence of the normal force on the yaw angle observed with the static approach, the damping rate increases linearly with the axial flow velocity. Satisfactory agreement is found between the two approaches.

Wicking flow inside capillary tubes can attain considerable momentum so as to produce a liquid jet at the end of the tube. Auto-ejection refers to the formation of droplets at the tip of such a jet. Experimental observations suggest that a tapering nozzle at the end of the capillary tube is necessary for auto-ejection; it has never been reported for a straight tube. Besides, most experimental realizations require microgravity, although it is possible under normal gravity if the nozzle has a sufficiently sharp contraction. This computational study focuses on two related issues: the critical condition for auto-ejection, and the hydrodynamics of the liquid meniscus as affected by geometric parameters. We adopt a diffuse-interface Cahn–Hilliard model for the moving contact line, and allow the dynamic contact angle to deviate from the static one through wall energy relaxation. From analyzing the dynamics of the meniscus in the straight tube and the nozzle, we establish a critical condition for the onset of auto-ejection based on a Weber number defined at the exit of the nozzle and an effective length that encompasses the geometric features of the tube–nozzle combination. In particular, this shows that capillary ejection is not possible in straight tubes. With steeper contraction in the nozzle, we predict two additional regimes of interfacial rupture: rapid ejection of multiple droplets and air bubble entrapment. The numerical results are in general agreement with available experiments.

We present a continuum approach to predicting the electrophoretic mobility of a charged dielectric colloidal particle in a concentrated multivalent electrolyte. Our model takes into account steric (excluded volume) hindrance between ions via Bikerman’s approach (Philos. Mag., vol. 33, 1942, p. 384) and ion–ion electrostatic (Coulombic) correlations via the work of Bazant et al. (Phys. Rev. Lett., vol. 106, 2011, 046102). The latter can result in the prediction of an electrophoretic mobility reversal, that is, the migration velocity of a particle switches direction with increasing ion concentration. Our model predictions compare favourably with experiments that observe mobility reversals in multivalent electrolytes.

A model of the total volume flux and entrainment occurring in two coalescing axisymmetric turbulent plumes is developed and compared with laboratory experiments. The dynamical evolution of the two plumes is divided into three regions. In region 1, where the plumes are separate, the entrainment in each plume is unaffected by the other plume, although the two plumes are drawn together due to the entrainment of ambient fluid between them. In region 2 the two plumes touch each other but are not yet merged. In this region the total entrainment is a function of both the dynamics of the touching plumes and the reduced surface area through which entrainment occurs. In region 3 the two plumes are merged and the entrainment is equivalent to that in a single plume. We find that the total volume flux after the two plumes touch and before they merge increases linearly with distance from the sources, and can be expressed as a function of the known total volume fluxes at the touching and merging heights. Finally, we define an ‘effective’ entrainment constant, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\alpha _{eff}$, as the value of $\alpha $ needed to obtain the same total volume flux in two independent plumes as that occurring in two coalescing plumes. The definition of $\alpha _{eff}$ allows us to find a single expression for the development of the total volume flux in the three different dynamical regions. This single expression will simplify the representation of coalescing plumes in more complex models, such as in large-scale geophysical convection, in which plume dynamics are not resolved. Experiments show that the model provides an accurate measure of the total volume flux in the two coalescing plumes as they evolve through the three regions.