Flow around a wall-mounted cube is an example of a turbulent flow around a three-dimensional bluff body attached to a surface. The main experimentally observed feature of this type of flow is the appearance of complex vortical structures, e.g. a horseshoe vortex originating in front of the body and enveloping it. The current paper is a follow-up to Yakhot, Liu & Nikitin (2006) in which we presented results of direct numerical simulation (DNS) of turbulent flow around a cube. Here, it is shown that unsteadiness of the considered flow is caused by inviscid–viscous interaction between the horseshoe vortex and the narrow band of positive vorticity attached to the surface in front of the cube. Details of the spatio-temporal evolution of large-scale vortical structures, including samples of long-term visualization and turbulence statistics, are presented. For the normal-to-the-wall velocity, in the vicinity of the cube's front face, the results reveal an anomalous probability distribution, namely, a bimodal distribution and one with high kurtosis.

The objective of the study is first to examine the optimal transient growth of Görtler–Hämmerlin perturbations in swept Hiemenz flow. This configuration constitutes a model of the flow in the attachment-line boundary layer at the leading-edge of swept wings. The optimal blowing and suction at the wall which minimizes the energy of the optimal perturbations is then determined. An adjoint-based optimization procedure applicable to both problems is devised, which relies on the maximization or minimization of a suitable objective functional. The variational analysis is carried out in the framework of the set of linear partial differential equations governing the chordwise and wall-normal velocity fluctuations. Energy amplifications of up to three orders of magnitude are achieved at low spanwise wavenumbers ($k\,{\sim}\,0.1$) and large sweep Reynolds number ($\hbox{\textit{Re}}\,{\sim}\,2000$). Optimal perturbations consist of spanwise travelling chordwise vortices, with a vorticity distribution which is inclined against the sweep. Transient growth arises from the tilting of the vorticity distribution by the spanwise shear via a two-dimensional Orr mechanism acting in the basic flow dividing plane. Two distinct regimes have been identified: for $k\,{\lesssim}\,0.25$, vortex dipoles are formed which induce large spanwise perturbation velocities; for $k\,{\gtrsim}\,0.25$, dipoles are not observed and only the Orr mechanism remains active. The optimal wall blowing control yields for instance an 80% decrease of the maximum perturbation kinetic energy reached by optimal disturbances at $\hbox{\textit{Re}}\,{=}\,550$ and $k\,{=}\,0.25$. The optimal wall blowing pattern consists of spanwise travelling waves which follow the naturally occurring vortices and qualitatively act in the same manner as a more simple constant gain feedback control strategy.

Experiments are described which measured concentration fields in liquid-phase strong transverse jets over the Reynolds-number range $1.0\,{\times}\,10^3\,{\leq}\,Re_j\,{\leq}\,20\,{\times}\,10^3$. Laser-induced-fluorescence measurements were made of the jet-fluid-concentration fields at a jet-to-freestream velocity ratio of $V_r\,{=}\,10$. The concentration-field data for far-field ($x/d_j\,{=}\,50$) slices of the jet show that turbulent mixing in the transverse jet is Reynolds-number dependent over the range investigated, with a scalar-field PDF that evolves with Reynolds number. A growing peak in the PDF, indicating enhanced spatial homogenization of the jet-fluid concentration field, is found with increasing Reynolds number. Comparisons between transverse jets and jets discharging into quiescent reservoirs show that the transverse jet is an efficient mixer in that it entrains more fluid than the ordinary jet, yet is able to effectively mix and homogenize the additional entrained fluid. Analysis of the structure of the scalar field using distributions of scalar increments shows evidence for well-mixed plateaux separated by sharp cliffs in the jet-fluid concentration field, as previously shown in other flows. Furthermore, the scalar field is found to be anisotropic, even at small length scales. Evidence for local anisotropy is seen in the scalar power spectra, scalar microscales, and PDFs of scalar increments in different directions. The scalar-field anisotropy is shown to be correlated to the vortex-induced large-scale strain field of the transverse jet. These experiments add to the existing evidence that the large and small scales of high-Schmidt-number turbulent mixing flows can be linked, with attendant consequences for the universality of small scales of the scalar field for Reynolds numbers up to at least $Re\,{=}\,20\,{\times}\,10^4$.

We report here the results of experiments on two flows – one on a convex surface and the other on a flat surface – designed to bring out explicitly the influence of streamwise curvature on relaminarization in highly favourable pressure gradients. In both flows, the initial conditions and the streamwise distribution of the Launder pressure-gradient parameter $K$ are virtually identical. The maximum value of $K$ is $6.2\,{\times}\, 10^{-6}$, well above the critical value of about $3.5\,{\times}\, 10^{-6}$ usually advocated for relaminarization. The spatial extent of the acceleration zone is of order 10 initial boundary-layer thicknesses, appreciably shorter than in earlier work in order better to simulate conditions at the leading edge of a typical aircraft wing. The fall in skin friction coefficient is steeper and the rise in shape factor sharper on the convex surface than on the flat surface, indicating that relaminarization on the convex surface is both more rapid and more nearly complete. In the crucial relaminarizing zone, two-layer quasi-laminar theory is found to predict the convex-surface mean-flow parameters more accurately than the flat-surface flow, without any explicit modelling of curvature effects. Thus, experimental results and supporting calculations both indicate that the dominant effect of streamwise convex curvature on the mean flow is to promote more rapid and complete relaminarization in an accelerated turbulent boundary layer, thus enhancing the probability of its occurrence on the leading edge of swept wings where both factors are significantly in operation.

We perform direct numerical simulations (DNS) of homogeneous turbulence subject to periodic shear – $S \,{=}\, S_{\hbox{\scriptsize\it max}} \sin (\omega t)$, where $\omega$ is the forcing frequency and $S_{\hbox{\scriptsize\it max}}$ is the maximum shear. The lattice Boltzmann method (LBM) is employed in our simulations and a periodic body force is introduced to produce the required shear. We find that the turbulence behaviour is a strong function of the forcing frequency. There exists a critical frequency – $\omega_{cr}/S_{\hbox{\scriptsize\it max}} \,{\approx}\, 0.5$ – at which the observed behaviour bifurcates. At lower forcing frequencies ($\omega \,{<}\, \omega_{cr}$), turbulence is sustained and the kinetic energy grows. At higher frequencies, the kinetic energy decays. It is shown that the phase difference between the applied strain and the Reynolds stress decreases monotonically from $\pi$ in the constant shear case to $\pi/2$ in very high frequency shear cases. As a result, the net turbulence production per cycle decreases with increasing frequency. In fact, at $\omega/S_{\hbox{\scriptsize\it max}} \,{\geq}\, 10$, decaying isotropic turbulence results are recovered. The frequency-dependence of anisotropy and Reynolds stress budget are also investigated in detail. It is shown that inviscid rapid distortion theory (RDT) does not capture the observed features: it predicts purely oscillatory behaviour at all forcing frequencies. Second moment closure models do predict growth at low frequencies and decay at high frequencies, but the critical frequency value is underestimated. The challenges posed by this flow to turbulence closure modelling are identified.

Skin-friction drag reduction in turbulent boundary layer flow of inhomogeneous polymer solutions is investigated using direct numerical simulations. A continuum constitutive model (FENE-P) accounting for the effects of polymer microstructure and concentration is used to describe the effect of viscoelasticity. The evolution of wall friction along the streamwise direction is a function of the dynamics of the polymer distribution in the boundary layer. It is observed that polymer transport decreases drag reduction downstream compared to the homogeneous case. The fluctuations of polymer concentration are anti-correlated with those of the streamwise velocity. Concentration is largest in the low-speed streaks. The physical process creating this effect is primarily that of dilution of the high-speed streaks, where due to the local turbulence structure the dispersion of polymer is strongest. Thus, the polymer-induced drag reduction phenomenon is sustained primarily in the vicinity of the low-speed streaks where the injected polymer additive is most effective.

The role of chain-branching cross-over temperatures in shock-induced ignition of reactive materials is studied by numerical simulation, using a three-step chain-branching reaction model. In order to provide insight into shock initiation, the simpler problem of a spatially homogeneous explosion is first considered. It is shown that for ratios of the cross-over temperature to the initial temperature, $T_B$, sufficiently less than unity, the homogeneous explosion can be quantitatively described by a widely used two-step model, while for $T_B$ sufficiently above unity the homogeneous explosion can be effectively described by the standard one-step model. From the matchings between these homogeneous-explosion solutions, the parameters of the reduced models are identified in terms of those of the three-step model. When $T_B$ is close to unity, all the reactions of the three-step model have a leading role, and hence in this case the model cannot be reduced further. In the case of shock initiation, for $T_B$ (which is now the ratio of the cross-over temperature to the initial shock temperature) sufficiently below unity, the three-step solutions are qualitatively described by those of the matched two-step model, but there are quantitative differences due to the assumption in the reduced model that a purely chain-branching explosion occurs instantaneously. For $T_B$ sufficiently above unity, the matched one-step model is found to effectively describe the way in which the heat release and fluid dynamics couple. For $T_B$ close to unity, the competition between chain branching and chain termination is important from the outset. In these cases the speed at which the forward moving explosion wave that emerges from the piston is sensitive to $T_B$, and changes from supersonic to subsonic for a value of $T_B$ just below unity.

The three-dimensional flow transition behind a heated cylinder subjected to a horizontal flow (water is used as the working fluid; $Pr\,{\simeq}\,7$) at a Reynolds number $\hbox{\it Re}\,{=}\,85$ and a Richardson number $\hbox{\it Ri}\,{=}\,1.0$, manifests itself in the far wake as escaping mushroom-type structures from the upper vortices. The origin of the escaping mushroom-type structures lies in the generation of streamwise vorticity in the near wake, which is described as a cyclic process. In the presence of a spanwise temperature gradient in the near wake, streamwise vorticity is generated, which results from baroclinic vorticity production. Owing to these streamwise vorticity regions, low-speed flow will move upwards at so-called in-plume positions resulting in high- and low-speed streaks in the upper half of the wake. Next, ‘transverse’ vorticity is generated by the spanwise gradients in the streamwise velocity component, resulting in counter-rotating vortices directly behind the cylinder. These vortices lead to high- and low-temperature regions in the spanwise direction and the process repeats itself.

Direct numerical simulations have been performed to study the dynamics of isolated turbulent spots in compressible isothermal-wall boundary layers. Results of a bypass transition scenario at Mach 2, 4 and 6 are presented. At all Mach numbers the evolved spots have a leading-edge overhang, followed by a turbulent core and a calmed region at the rear interface. The spots have an upstream-pointing arrowhead shape when visualized by near-wall slices, but a downstream-pointing arrowhead in slices away from the wall. The lateral spreading of the spot decreases substantially with the Mach number, consistent with a growth mechanism based on the instability of lateral shear layers. Evidence for a supersonic (Mack) mode substructure is found in the Mach 6 case, where coherent spanwise structures are observed under the spot overhang region.

The problem of low-gravity isothermal capillary flow along interior corners that are rounded is revisited analytically in this work. By careful selection of geometric length scales and through the introduction of a new geometric scaling parameter $\overline T_c$, the Navier–Stokes equation is reduced to a convenient$\,{\sim}\, O(1)$ form for both analytic and numeric solutions for all values of corner half-angle $\alpha$ and corner roundedness ratio $\lambda$ for perfectly wetting fluids. The scaling and analysis of the problem captures much of the intricate geometric dependence of the viscous resistance and significantly reduces the reliance on numerical data compared with several previous solution methods and the numerous subsequent studies that cite them. In general, three asymptotic regimes may be identified from the large second-order nonlinear evolution equation: (I) the ‘sharp-corner’ regime, (II) the narrow-corner ‘rectangular section’ regime, and (III) the ‘thin film’ regime. Flows are observed to undergo transition between regimes, or they may exist essentially in a single regime depending on the system. Perhaps surprisingly, for the case of imbibition in tubes or pores with rounded interior corners similarity solutions are possible to the full equation, which is readily solved numerically. Approximate analytical solutions are also possible under the constraints of the three regimes, which are clearly identified. The general analysis enables analytic solutions to many rounded-corner flows, and example solutions for steady flows, perturbed infinite columns, and imbibing flows over initially dry and prewetted surfaces are provided.

We present measurements, over a wide range of Reynolds numbers ($40\leq R_{\lambda} \leq 470$), of grid-generated turbulence subjected to axisymmetric strain, and of the subsequent evolution of the turbulence after the strain is released. The Reynolds number was varied by the use of both passive and active grids and the strain was produced by a 4:1 area-change axisymmetric contraction placed at various distances from the grid. The time scale ratio of the turbulence to that of the mean strain varied from approximately 10 to 30. The results show reasonable agreement with (linear) rapid distortion theory (RDT) for the velocity variances but, contrary to linear theory, the strained longitudinal, $u_{1}$, spectrum peaked at significantly higher wavenumber than the transverse, $u_{2}$, spectrum. The mismatch in peaks increased with increasing $R_{\lambda}$ and at the highest Reynolds number ($R_{\lambda} = 470$) the peak of the strained $u_{1}$-spectrum occurred at a wavenumber 200 times greater than that of the $u_{2}$-spectrum. As the flow relaxed toward isotropy after the contraction, further evidence of the non-locality in the flow field became apparent, with a second peak in the $u_{2}$-spectrum emerging at a similar wavenumber to the high-frequency peak in the $u_{1}$-spectrum. The strain also caused the longitudinal derivative skewness to change sign but as the flow evolved after the contraction the derivative skewness returned to its typical value of ${-}$0.4. We also show that single-point turbulence models are inadequate to describe the relaxation of the turbulence towards an isotropic state in the postcontraction region.

Numerical simulations are used to investigate the effect of aspect ratio on the wake topology and hydrodynamic performance of thin ellipsoidal flapping foils. The study is motivated by the quest to understand the hydrodynamics of fish pectoral fins. The simulations employ an immersed boundary method that allows us to simulate flows with complex moving boundaries on fixed Cartesian grids. A detailed analysis of the vortex topology shows that the wake of low-aspect-ratio flapping foils is dominated by two sets of interconnected vortex loops that evolve into distinct vortex rings as they convect downstream. The flow downstream of these flapping foils is characterized by two oblique jets and the implications of this characteristic on the hydrodynamic performance are examined. Simulations are also used to examine the thrust and propulsive efficiency of these foils over a range of Strouhal and Reynolds numbers as well as pitch-bias angles.

The injection of liquid from a central well into a partially saturated aquifer of finite thickness is described using similarity solutions. The solutions illustrate that injection leads to a growing zone around the source in which the fluid fills the whole depth of the aquifer. Beyond this zone, the current adjusts to the depth in the far field as the fluid slumps under gravity. The radial extent of the zone in which the aquifer is fully flooded depends on the ratio of the buoyancy-driven flow speed to the pressure-driven flow speed associated with the injection. New laboratory experiments, using a model porous medium, support the model predictions in the case of an initially unsaturated layer. The analysis is then developed to allow for a fully saturated aquifer, containing fluid of lower density than the injectate, and a further class of similarity solutions is developed. Again, these are shown to be consistent with new laboratory experiments. In concluding, we briefly consider how the results may be combined, to explore the self-similar dynamics of a relatively dense fluid injected into an aquifer which is partially saturated with fluid of smaller density.

The interaction between the wall and the core region of turbulent channels is studied using direct numerical simulations at friction Reynolds number ${\hbox{\it Re}}_{\tau} \approx 630$. In these simulations the near-wall energy cycle is effectively removed, replacing the smooth-walled boundary conditions by prescribed velocity disturbances with non-zero Reynolds stress at the walls. The profiles of the first- and second-order moments of the velocity are similar to those over rough surfaces, and the effect of the boundary condition on the mean velocity profile is described using the equivalent sand roughness. Other effects of the disturbances on the flow are essentially limited to a layer near the wall whose height is proportional to a length scale defined in terms of the additional Reynolds stress. The spectra in this roughness sublayer are dominated by the wavenumber of the velocity disturbances and by its harmonics. The wall forcing extracts energy from the flow, while the normal equilibrium between turbulent energy production and dissipation is restored in the overlap region. It is shown that the structure and the dynamics of the turbulence outside the roughness sublayer remain virtually unchanged, regardless of the nature of the wall. The detached eddies of the core region only depend on the mean shear, which is not modified beyond the roughness sublayer by the wall disturbances. On the other hand, the large scales that are correlated across the whole channel scale with $U_{\hbox{\scriptsize{\it LOG}}}=u_{\tau}\kappa^{-1}\log({\hbox{\it Re}}_\tau)$, both in smooth- and in rough-walled flows. This velocity scale can be interpreted as a measure of the velocity difference across the log layer, and it is used to modify the scaling proposed and validated by del Álamo et al. (J. Fluid Mech., vol. 500, 2004, p. 135) for smooth-walled flows.

Direct numerical simulation (DNS) and a linear analysis of the global instability of a buoyancy layer have been performed. The spatially developing basic flow under consideration is induced by a vertical heated flat plate immersed in a thermally stratified medium. It is revealed numerically that, depending on the modified Grashof number, the disturbed flat-plate boundary-layer flows may not relax to the basic state but instead oscillate with an intrinsic frequency. The front of globally unstable waves in numerical simulations agrees very well with the position of marginal absolute instability, and the dominant frequencies in the oscillating region are identical and tuned to the marginal absolute frequency derived from the local linear dispersion relation based on the unperturbed basic state. The front of the nonlinear global modes is thus of a pulled type in this buoyancy-driven flow system.

We examine the effects of a fluid yield stress on the classical Rayleigh–Bénard instability between heated parallel plates. The focus is on a qualitative characterization of these flows, by theoretical and computational means. In contrast to Newtonian fluids, we show that these flows are linearly stable at all Rayleigh numbers, ${\hbox{{\it Ra}}$, although the usual linear modal stability analysis cannot be performed. Below the critical Rayleigh number for energy stability of a Newtonian fluid, ${\hbox{{\it Ra}}}_E$, the Bingham fluid is also globally asymptotically stable. Above ${\hbox{{\it Ra}}}_E$, we provide stability bounds that are conditional on ${\hbox{{\it Ra}}}\,{-}\,{\hbox{\it Ra}}_E$, as well as on the Bingham number $B$, the Prandtl number $\hbox{\it Pr}$, and the magnitude of the initial perturbation. The stability characteristics therefore differ considerably from those for a Newtonian fluid. A second important way in which the yield stress affects the flow is that when the flow is asymptotically stable, the velocity perturbation decays to zero in a finite time. We are able to provide estimates for the stopping time for the various types of stability. A consequence of the finite time decay is that the temperature perturbation decays on two distinctly different time scales, i.e. before/after natural convection stops. The two decay time scales are clearly observed in our computational results.

We are also able to determine approximate marginal stability parameters via computation, when in the conditional stability regime, although computation is not ideal for this purpose. When just above the marginal stability limits, perturbations grow into a self-sustained cellular motion that appears to resemble closely the Newtonian secondary motion, i.e. Rayleigh–Bénard cells. When stable, however, the decaying flow pattern is distinctly different to that of a Newtonian perturbation. As $t\rightarrow\infty$, a stable Newtonian perturbation decays exponentially and asymptotically resembles the least stable eigenfunction of the linearized problem. By contrast, as $t$ approaches its stopping value, the Bingham fluid is characterized by growth of a slowly rotating (almost) unyielded core within each convection cell, with fully yielded fluid contained in a progressively narrow layer surrounding the core. Finally, preliminary analyses and remarks are made concerning extension of our results to inclined channels, stability of three-dimensional flows and the inclusion of residual stresses in the analysis.

An ultra-fine liquid atomization procedure combining the advantages of electrospray and flow focusing is presented. Both techniques are known to produce strikingly small and steady liquid micro-jets issuing from menisci held by capillary forces. Such menisci take the form of a cusp-like drop attached to the feeding tube (flow focusing: FF) or a Taylor cone (electrospray: ES). The issuing micro-jets are forced or ‘sucked’ from the parent meniscus either by pressure or electrohydrodynamic forces. Subsequent capillary breakup of the jet leads to fine sprays of remarkable quality. Here we describe the joint effect of pressurization and electrification in a flow focusing device, and the subsequent coupling of both ES and FF phenomena. For any given liquid and flow rate, the combined procedure gives rise to significantly smaller droplet sizes than observed in any of the source techniques. The co-flowing gas stream removes space charges; in addition, the perforated plate facing the feed tube provides an electric barrier, shielding the jet-meniscus or ‘production’ area from the spray or ‘product’ area. As a result, space charges and electrified droplets are removed from the production area, thus avoiding the ambient electric saturation which becomes a limiting factor in ES-spraying: a significantly enhanced spraying stability ensues, with a much wider operation range than FF or ES. Other unexpected outcomes from the combination are also shown. A theoretical model is developed to predict the emitted droplet size: a first integral of the momentum equation yielding a generalized Bernoulli equation, and an explicit approximation for the jet diameter and droplet size, accurate within a broad parametrical band.

The existence and magnitude of slip velocities between deionized water and a smooth glass surface is studied experimentally. Sub-micron fluorescent particles are suspended in water and imaged using total internal reflection velocimetry (TIRV). For water flowing over a hydrophilic surface, the measurements are in agreement with previous experiments and indicate that slip, if present, is minimal at low shear rates, but increases slightly as the shear rate increases. Surface hydrophobicity is observed to induce a small slip velocity, with the slip length reaching a maximum of 96 nm at a shear rate of 1800$\,{\rm s}^{-1}$. Issues associated with the experimental technique and the interpretation of results are also discussed.

The motion of a spherical particle suspended in gravity-driven film flow down an inclined plane is considered in the limit of vanishing Reynolds and Bond numbers where the free-surface deformation is infinitesimal. Taking advantage of the axially symmetry of the boundaries of the flow with respect to the axis that is normal to the wall and free surface and passes through the particle centre, the problem is formulated as a system of one-dimensional integral equations for the first Fourier coefficients of the unknown traction and velocity along the boundary contours in a meridional plane. It is found that the particle translational velocity scaled by the unperturbed velocity evaluated at the particle centre increases monotonically as the particle approaches the free-surface, whereas the corresponding angular velocity of rotation scaled by the unperturbed vorticity evaluated at the particle centre reaches a maximum at a certain intermediate position. The free-surface velocity vector field and deformation are displayed, the force and torque exerted on a spherical particle adhering to the wall are tabulated, and the associated flow pattern is discussed.