By making simple assumptions, an analytical theory is deduced for the mean velocity behind a two-dimensional obstacle (of height h) placed on a rigid plane over which flows a turbulent boundary layer (of thickness δ). It is assumed that h [Gt ] δ, and that the wake can be divided into three regions. The velocity deficit − u is greatest in the two regions in which the change in shear stress is important, a wall region (W) close to the wall and a mixing region (M) spreading from the top of the obstacle. Above these is the external region (E) in which the velocity field is an inviscid perturbation on the incident boundary-layer velocity, which is taken to have a power-law profile U(y) = U∞(y − y1)n/δn, where n [Gt ] 1. In (M), assuming that an eddy viscosity (= KhU(h)) can be defined for the perturbed flow in terms of the incident boundary-layer flow and that the velocity is self-preserving, it is found that u(x,y) has the form $\frac{u}{U(h)} = \frac{ C }{Kh^2U^2(h)} \frac{f(n)}{x/h},\;\;\;\; {\rm where}\;\;\;\; \eta = (y/h)/[Kx/h]^{1/(n+2)}$, and the constant which defines the strength of the wake is $C = \int^\infty_0 y^U(y)(u-u_E)dy$, where u = uE(x, y) as y → 0 in region (E).

In region (W), u(y) is proportional to In y. By considering a large control surface enclosing the obstacle it is shown that the constant of the wake flow is not simply related to the drag of the obstacle, but is equal to the sum of the couple on the obstacle and an integral of the pressure field on the surface near the body.

New wind-tunnel measurements of mean and turbulent velocities and Reynolds stresses in the wake behind a two-dimensional rectangular block on a roughened surface are presented. The turbulent boundary layer is artificially developed by well-established methods (Counihan 1969) in such a way that δ = 8h. These measurements are compared with the theory, with other wind-tunnel measurements and also with full-scale measurements of the wind behind windbreaks.

It is found that the theory describes the distribution of mean velocity reasonably well, in particular the (x/h)−1 decay law is well confirmed. The theory gives the correct self-preserving form for the distribution of Reynolds stress and the maximum increase of the mean-square turbulent velocity is found to decay downstream approximately as $ (\frac{x}{h})^{- \frac{3}{2}} $ in accordance with the theory. The theory also suggests that the velocity deficit is affected by the roughness of the terrain (as measured by the roughness length y0) in proportion to In (h/y0), and there seems to be some experimental support for this hypothesis.

Direct numerical simulation (DNS) of a spatially developing turbulent boundary layer (TBL) over a wall roughened with regularly arrayed cubes was performed to investigate the effects of three-dimensional (3-D) surface elements on the properties of the TBL. The cubes were staggered in the downstream direction and periodically arranged in the streamwise and spanwise directions with pitches of px/k = 8 and pz/k = 2, where px and pz are the streamwise and spanwise spacings of the cubes and k is the roughness height. The Reynolds number based on the momentum thickness was varied in the range Reθ = 300−1300, and the roughness height was k = 1.5θin, where θin is the momentum thickness at the inlet, which corresponds to k/δ = 0.052–0.174 from the inlet to the outlet; δ is the boundary layer thickness. The characteristics of the TBL over the 3-D cube-roughened wall were compared with the results from a DNS of the TBL over a two-dimensional (2-D) rod-roughened wall. The introduction of cube roughness affected the turbulent Reynolds stresses not only in the roughness sublayer but also in the outer layer. The present instantaneous flow field and linear stochastic estimations of the conditional averaging showed that the streaky structures in the near-wall region and the low-momentum regions and hairpin packets in the outer layer are dominant features in the TBLs over the 2-D and 3-D rough walls and that these features are significantly affected by the surface roughness throughout the entire boundary layer. In the outer layer, however, it was shown that the large-scale structures over the 2-D and 3-D roughened walls have similar characteristics, which indicates that the dimensional difference between the surfaces with 2-D and 3-D roughness has a negligible effect on the turbulence statistics and coherent structures of the TBLs.

In this paper, we solve numerically the entire set of equations associated with the electro-thermo-convective phenomena that take place in a planar layer of dielectric liquid heated from below and subjected to unipolar injection. For the first time the whole set of coupled equations is solved: Navier–Stokes equations, electrohydrodynamic (EHD) equations and the energy equation. We first validate the numerical simulation by comparing the electro-convection stability criteria with ones obtained with a stability approach. The numerical solution of the electro-thermo-convection problem is then presented entirely with a detailed analysis of stability parameters. In particular, the relation between fluid velocity, non-dimensional electrical parameter T, Rayleigh number Ra and Prandtl number Pr is given. An analytical model is presented in order to understand the flow behaviour at some critical conditions. The way that the onset of motion passes from purely electrical convection to purely thermal convection is, in particular, investigated and explained in detail. Finally, a result on the heat transfer enhancement due to electro-convection is exhibited and compared with data from experimental works available in this field.

Buoyancy-driven instabilities of a horizontal interface between two miscible solutions in the gravity field are theoretically studied in porous media and Hele-Shaw cells (two glass plates separated by a thin gap). Beyond the classical Rayleigh–Taylor (RT) and double diffusive (DD) instabilities that can affect such two-layer stratifications right at the initial time of contact, diffusive-layer convection (DLC) as well as delayed-double diffusive (DDD) instabilities can set in at a later time when differential diffusion effects act upon the evolving density profile starting from an initial step-function profile between the two miscible solutions. The conditions for these instabilities to occur can therefore be obtained only by considering time evolving base-state profiles. To do so, we perform a linear stability analysis based on a quasi-steady-state approximation (QSSA) as well as nonlinear simulations of a diffusion–convection model to classify and analyse all possible buoyancy-driven instabilities of a stratification of a solution of a given solute A on top of another miscible solution of a species B. Our theoretical model couples Darcy's law to evolution equations for the concentration of species A and B ruling the density of the miscible solutions. The parameters of the problem are a buoyancy ratio R quantifying the ratio of the relative contribution of B and A to the density as well as δ, the ratio of diffusion coefficients of these two species. We classify the region of RT, DD, DDD and DLC instabilities in the (R, δ) plane as a function of the elapsed time and show that, asymptotically, the unstable domain is much larger than the one captured on the basis of linear base-state profiles which can only obtain stability thresholds for the RT and DD instabilities. In addition the QSSA allows one to determine the critical time at which an initially stable stratification of A above B can become unstable with regard to a DDD or DLC mechanism when starting from initial step function profiles. Nonlinear dynamics are also analysed by a numerical integration of the full nonlinear model in order to understand the influence of R and δ on the dynamics.

In the traditional hybrid RANS/LES approaches for the simulation of wall-bounded fluid turbulence, such as detached-eddy simulation (DES), the whole flow domain is divided into an inner layer and an outer layer. Typically the Reynolds-averaged Navier–Stokes (RANS) equations are used for the inner layer, while large-eddy simulation (LES) is used for the outer layer. The transition from the inner-layer solution to the outer-layer solution is often problematic due to the lack of small-scale dynamics in the RANS region. In this paper, we propose to simulate the whole flow domain by large-eddy simulation while enforcing a Reynolds-stress constraint on the subgrid-scale (SGS) stress model in the inner layer. Both the algebraic eddy-viscosity model and the one-equation Spalart–Allmaras (SA) model have been used to constrain the Reynolds stress in the inner layer. In this way, we improve the LES methodology by allowing the mean flow of the inner layer to satisfy the RANS solution while small-scale dynamics is included. We validate the Reynolds-stress-constrained large-eddy simulation (RSC-LES) model by simulating three-dimensional turbulent channel flow and flow past a circular cylinder. Our model is able to predict mean velocity, turbulent stress and skin-friction coefficients more accurately in turbulent channel flow and to estimate the pressure coefficient after separation more precisely in flow past a circular cylinder compared with the pure dynamic Smagorinsky model (DSM) and DES using the same grid resolution. Furthermore, the computational cost of the RSC-LES is almost the same as that of DES.

The influence of different wing kinematic models on the aerodynamic performance of a hovering insect is investigated by means of two-dimensional time-dependent Navier–Stokes simulations. For this, simplified models are compared with averaged representations of the hovering fruit fly wing kinematics. With increasing complexity, a harmonic model, a Robofly model and two more-realistic fruit fly models are considered, all dynamically scaled at Re = 110. To facilitate the comparison, the parameters of the models were selected such that their mean quasi-steady lift coefficients were matched. Details of the vortex dynamics, as well as the resulting lift and drag forces, were studied.

The simulation results reveal that the fruit fly wing kinematics result in forces that differ significantly from those resulting from the simplified wing kinematic models. In addition, light is shed on the effect of different characteristic features of the insect wing motion. The angle of attack variation used by fruit flies increases aerodynamic performance, whereas the deviation is probably used for levelling the forces over the cycle.

An experimental investigation has been made of the effect of ejecting nitrogen and helium coolant gases at the nose of a bluff body at a nominal Mach number of 5·8. The gases were ejected with ‘swirl’, to encourage them to flow tangentially to the model surface at ejection, and also straight out. Measurements were made of the pressure and temperature on the surface of the model at incidences of 0, 4, 8°, and for a range of coolant gas flows. From these measurements the flow patterns and distributions of heat flux were deduced.

It was found that ejection with swirl did not in fact lead to an easing of the heating problem because the high tangential velocity with which the coolant was injected into the boundary layer increased the wall shear stress, and hence, by the Reynolds analogy, the heat flux, so that it predominated over the reduced driving temperature difference associated with the cooled boundary layer.

With straight-out ejection it was found that the heat alleviation capabilities of the ejected coolant were reduced considerably if the flow rate was sufficiently large that the bow shock wave was bulged out. However, provided that the external flow is not disturbed, straight-out ejection provides an effective way of reducing the heat flux.

Droplet dynamics on a solid substrate is significantly influenced by surfactants. It remains a challenging task to model and simulate the moving contact line dynamics with soluble surfactants. In this work, we present a derivation of the phase-field moving contact line model with soluble surfactants through the first law of thermodynamics, associated thermodynamic relations and the Onsager variational principle. The derived thermodynamically consistent model consists of two Cahn–Hilliard type of equations governing the evolution of interface and surfactant concentration, the incompressible Navier–Stokes equations and the generalized Navier boundary condition for the moving contact line. With chemical potentials derived from the free energy functional, we analytically obtain certain equilibrium properties of surfactant adsorption, including equilibrium profiles for phase-field variables, the Langmuir isotherm and the equilibrium equation of state. A classical droplet spread case is used to numerically validate the moving contact line model and equilibrium properties of surfactant adsorption. The influence of surfactants on the contact line dynamics observed in our simulations is consistent with the results obtained using sharp interface models. Using the proposed model, we investigate the droplet dynamics with soluble surfactants on a chemically patterned surface. It is observed that droplets will form three typical flow states as a result of different surfactant bulk concentrations and defect strengths, specifically the coalescence mode, the non-coalescence mode and the detachment mode. In addition, a phase diagram for the three flow states is presented. Finally, we study the unbalanced Young stress acting on triple-phase contact points. The unbalanced Young stress could be a driving or resistance force, which is determined by the critical defect strength.

When a paintbrush is dipped into a pot of paint and pulled out, surface tension forces cause the individual hairs in the brush to coalesce even as the brush becomes impregnated with paint. We study a simple model of this elastocapillary interaction in the context of the surface-tension-driven vertical rise of a liquid between two long flexible hydrophilic sheets that are held a small distance apart at one end. We provide an analytic theory for the static shapes of the sheets as well as the liquid rise height which is different from that of the classical law of Jurin, and show that our experiments are quantitatively consistent with the theory.

Results of direct numerical simulations of stably stratified, freely evolving, homogeneous turbulence are presented. An examination of initial data designed to give insight into laboratory flows suggests that the numerical simulations have a satisfactory degree of realism, insofar as statistical parameters such as total energy and length scales are concerned. The motion is decomposed into a stratified turbulence (vortical) component and a wave component. For initial-value problems similar to laboratory studies of stratified flows, the vortical component decays at a rate virtually identical to that of the non-buoyant case up to t = 6N−1 (N is the Brunt-Väisälä frequency). The decay rate decreases after this time, suggesting a kind of turbulence ‘collapse’. The temperature structure that emerges clearly shows the development of the collapse stage of the flow, which is also diagnosed by the behaviour of parameters such as the Thorpe scale.

We next examine the very small-Froude-number regime in order to understand possible universal aspects of the flow. An examination of various initial conditions with different proportions of stratified and wave components indicates a lack of universality. For initial data containing only vortical motion (motions derived from the vertical vorticity field), the vortical field tends to dominate, in subsequent evolution, at strong stratification. However, contrary to two-dimensional turbulence, the flow is more strongly dissipative than two-dimensional flows due to the frictional effect associated with layering. Other quantities examined are frequency spectra, and the probability distribution for vertical shear. The frequency spectra exhibit some features in common with spectra extracted from oceanographic data.

The effects of stable stratification on turbulence structure have been experimentally investigated in stratified open-channel flow and a theoretical spectral-equation model has been applied to the stably stratified flow. The measurements were made in the outer layer of open-channel flow with strongly stable density gradient, where the wall effect was small. Velocity and temperature fluctuations were simultaneously measured by a laser-Doppler velocimeter and a cold-film probe. Measurements include turbulent intensities, correlation coefficients of turbulent fluxes and coherence–phase relationships. These turbulent quantities were correlated with the local gradient Richardson number and compared with the values calculated using a spectral-equation model and with other laboratory measurements. In stable conditions, turbulent motions approach wavelike motions, and negative heat and momentum transfer against the mean temperature and velocity gradient occurs in strongly stable stratification.

When a ferromagnetic fluid with a horizontal free surface is subjected to a uniform vertical applied magnetic field B0, it is known (Cowley & Rosensweig 1967) that the surface may be unstable when the field strength exceeds a certain critical value Bc. In this paper we consider, by means of an energy minimization principle, the possible forms that the surface may then take. Under the assumption that |μ − 1| [Lt ] 1 (where μ is the magnetic permeability of the fluid), it is shown that when B0 is near to Bc there are three equilibrium configurations for the surface: (i) flat surface, (ii) stationary hexagonal pattern, (iii) stationary square pattern. Configuration (i) is stable for B0 < Bc, (ii) is stable for B0 > Bc and B0−Bc sufficiently small, and (iii) is stable for some higher values of B0. In each configuration the fluid is static, and the surface is in equilibrium under the joint action of gravity, surface tension, and magnetic forces. The amplitude of the surface perturbation in cases (ii) and (iii) is calculated, and hysteresis effects associated with increase and decrease of B0 are discussed.

An experimental investigation of the flow development of an axisymmetric jet exhausting into a moving air stream is made for two values of the ratio of jet velocity to external air velocity. The u-component turbulence intensity and Reynolds shear stress measurements together with the dissipation length scales inferred from measured u-component spectra suggest that the turbulence similarity assumptions are incorrect for the present flow situation. A discussion of the turbulence structure of the flow indicates that self-preservation does not apply for this situation and that the flow far downstream depends strongly on the complete past history.

Numerical simulations of Navier–Stokes equations are performed to study the flow originated by an oscillating pressure gradient close to a wall characterized by small imperfections. The scenario of transition from the laminar to the turbulent regime is investigated and the results are interpreted in the light of existing analytical theories. The ‘disturbed-laminar’ and the ‘intermittently turbulent’ regimes detected experimentally are reproduced by the present simulations. Moreover it is found that imperfections of the wall are of fundamental importance in causing the growth of two-dimensional disturbances which in turn trigger turbulence in the Stokes boundary layer. Finally, in the intermittently turbulent regime, a description is given of the temporal development of turbulence characteristics.

We present a parametric experimental study of convective electrokinetic instability (EKI) in an isotropically etched, cross-shaped microchannel using quantitative epifluorescence imaging. The base state is a three-inlet, one-outlet electrokinetic focusing flow configuration where the centre sample stream and sheath flows have mismatched ionic conductivities. Electrokinetic flows with conductivity gradients become unstable when the electroviscous stretching and folding of conductivity interfaces grows faster than the dissipative effect of molecular diffusion. Scalar images, critical applied fields required for instability, and temporal and spatial scalar energy are presented for flows with a wide range of applied d.c. electric field and centre-to-sheath conductivity ratios. These parameters impose variations of the electric Rayleigh number across four orders of magnitude. We introduce a scaling for charge density in the bulk fluid as a function of local maximum conductivity gradients in the flow. This scaling shows that the flow becomes unstable at a critical electric Rayleigh number ($Ra_{e, \ell} \,{=}\, 205$) and applies to a wide range of applied field and centre-to-sheath conductivity ratios. This work is relevant to on-chip electrokinetic flows with conductivity gradients such as field amplified sample stacking, flow at the intersections of multi-dimensional assays, electrokinetic control and separation of sample streams with poorly specified chemistry, and low-Reynolds number micromixing.

An experimental and numerical investigation of the unsteady phenomena induced in a hydrogen-fuelled scramjet combustor under high-equivalence-ratio conditions is carried out, focusing on the processes leading up to unstart. The configuration for the study is the fuelled flow path of the HyShot II flight experiment. Experiments are performed in the HEG reflected-shock wind tunnel, and results are compared with those obtained from unsteady numerical simulations. High-speed schlieren and OH∗ chemiluminescence visualization, together with time-resolved surface pressure measurements, allow links to be drawn between the experimentally observed flow and combustion features. The transient flow structures signalling the onset of unstart are observed to take the form of an upstream-propagating shock train. Both the speed of propagation and the downstream location at which the shock train originates depend strongly on the equivalence ratio. The physical nature of the incipient shock system, however, appears to be similar for different equivalence ratios. Both experiments and computations indicate that the primary mechanism responsible for the transient behaviour is thermal choking, though localized boundary-layer separation is observed to accompany the shock system as it moves upstream. In the numerical simulations, the global choking behaviour is dictated by the limited region of maximum heat release around the shear layer between the injected hydrogen and the incoming air flow. This leads to the idea of ‘local’ thermal choking and results in a lower choking limit than is predicted by a simple integral analysis. Such localized choking makes it possible for new quasi-steady flow topologies to arise, and these are observed in both experiments and simulation. Finally, a quasi-unsteady one-dimensional analytical model is proposed to explain elements of the shock-propagation behaviour.

The instantaneous velocity distribution in trailing vortices generated by lifting hydrofoils has been measured in the Low Turbulence Water Tunnel at the California Institute of Technology. Two different rectangular planform hydrofoils with small aspect ratios were tested. Double-pulsed holography of injected microbubbles, which act much as Lagrangian flow tracers, was used to determine instantaneous axial and tangential velocities. Measurements were made at various free-stream velocities, angles of attack, and downstream distances. The vortex core mean axial velocity is consistently greater than the free-stream velocity near the hydrofoil trailing edge, and decreases with downstream distance. The mean axial velocity is strongly Reynolds-number dependent.

Axial flow in the trailing vortex is highly unsteady for all the flow conditions studied; peak-to-peak fluctuations on the centreline as large as the free-stream velocity have been observed. The amplitude of these fluctuations falls rapidly with increasing distance from the centreline. For an angle of attack of 10° the fluctuations consist of both ‘fast’ and ‘slow’ components, whereas for α = 5° only ‘fast’ fluctuations have been observed. Peak decelerations of the centreline fluid occur with amplitude comparable to the maximum centripetal acceleration around the centreline. Certain unusual structures of the vortex core - regions in which the flow direction quickly diverges from the free-stream direction, and then equally quickly recovers - have been labelled ‘vortex kinks.’

In this paper we study the fluidization of 1204 spheres at Reynolds numbers in the thousands using the method of distributed Lagrange multipliers. The results of the simulation are compared with an experiment. This is the first direct numerical simulation of a fluidized bed at the finite Reynolds numbers encountered in applications. The simulations are processed to give straight lines in log–log plots leading to power laws as in the celebrated experimental correlations of Richardson & Zaki (1954). The numerical method allows the first direct calculation of the slip velocity and other averaged values used in two-fluid continuum models. The computation and the experiment show that a single particle may be in balance with respect to weight and drag for an interval of fluidizing velocities; the expectation that the fluidizing velocity is unique is not realized. The numerical method reveals that the dynamic pressure decreases slowly with the fluidizing velocity. Tentative interpretations of these new results are discussed.

The formation of vortex streets behind stationary cylinders is found to be caused by an absolute instability in the wake immediately behind the cylinder. The inviscid Orr–Sommerfeld equation is used together with measured profiles at Reynolds numbers of (a) Re = 56 when the absolute instability provides a Strouhal number of 0.13; and (b) Re = 140000 providing a Strouhal number of 0.21, both in agreement with experimental values. At the subcritical Re = 34 the instability is of the convective type; i.e. the disturbance decays, being convected away once the external disturbance is removed, in agreement with experimental observations. Finally, the instability of the mode which causes a symmetric array of vortices is shown to be always of the convective type.

This paper describes analytical and numerical solutions for the flow of a Bingham plastic in an eccentric annulus. Analytical solutions are obtained by expanding in powers of δ, the ratio of the difference in radii of the bounding cylinders to their mean. The solution over most of the annulus is similar to that in a slot of uniform width, containing a central plug-like region over which the velocity is independent of the radial variable. However, unlike the uniform-slot solution, the velocity in the plug varies around the annulus and the stress exceeds the yield stress. This simple structure is supplemented by true plugs (over which the velocity is constant and the stress is below the yield stress) at the widest and, in some cases, the narrowest parts of the annulus. A simple criterion is given for conditions under which the fluid ceases to flow on the narrow side and bounds are obtained for the extent of the motionless region and for the true plugs.

The predictions of the theory have been compared to numerical results over a wide range of eccentricities, radius ratios, fluid properties and flow parameters. Good quantitative agreement has been reached for radius ratios in excess of about 0.7. In particular the extent and location of pseudo-plugs and true plugs are confirmed.