To investigate which component of the anisotropic permeability tensor of porous media influences turbulence over porous walls, direct numerical simulation of anisotropic porous-walled channel flows is performed by the D3Q27 multiple-relaxation-time lattice Boltzmann method. The presently considered anisotropic permeable walls have square pore arrays aligned with the Cartesian axes. Vertical, streamwise and spanwise pore arrays are systematically introduced to the walls to impose anisotropic permeability. Simulations are carried out at a friction Reynolds number of 111 and 230, which is based on the averaged friction velocity of the porous bottom and the smooth top walls. It is found that streamwise and spanwise permeabilities enhance turbulence whilst vertical permeability itself does not. In particular, the enhancement of turbulence is remarkable over porous walls with streamwise permeability. Over streamwise permeable walls, development of high- and low-speed streaks is prevented whilst large-scale intermittent patched patterns of ejection motions are induced. It is revealed by two-point correlation analysis that streamwise permeability allows the development of streamwise large-scale perturbations induced by Kelvin–Helmholtz instability. Spectral analysis reveals that this perturbation contributes to the enhancement of the Reynolds shear stress, leading to significant skin friction of the porous interface. Through the comparison between the two different Reynolds-number cases, it is found that, as the Reynolds number increases, the streamwise perturbation becomes larger and more organized. Consequently, owing to the enhancement of the large-scale perturbation, a significant Reynolds-number dependence of the skin friction of the porous interface can be observed over the streamwise permeable wall. It is also implied that the wavelength of the perturbation can be reasonably scaled by the outer-layer length scale.

A wind tunnel in which an arbitrary negative pressure gradient could be developed has been built for boundary-layer studies. The effects of selected pressure gradients on boundary layers grown on one of the walls of the tunnel were studied. It was possible to obtain equilibrium boundary layers of the type first suggested by Clauser: that is, layers whose non-dimensional velocity-defect distribution is invariant along the direction of flow. The velocity-defect distributions for two such boundary layers were established, corresponding to values of Clauser's dimensionless pressure-gradient parameter β of −0·35 and −0·53. These velocity profiles are compared with the profiles predicted theoretically by Mellor & Gibson (1966). The agreement between the two is very good.

The role of free-stream turbulence (FST) in the hydrodynamic instability mechanisms and transition to turbulence in laminar separation bubbles (LSBs) was investigated using direct numerical simulations (DNS). Towards this end, a set of highly resolved DNS have been carried out, where isotropic FST fluctuations with intensities from 0.1 % to 3 % are introduced to investigate the relevant physical mechanisms governing the interaction of separation and transition in LSBs. For disturbance-free simulations, i.e. without FST, laminar–turbulent transition involves a Kelvin–Helmholtz (KH) instability of the separated shear layer. For LSBs subjected to FST, vortical FST fluctuations penetrate the approaching attached laminar boundary layer upstream of the separation location and induce slowly growing low-frequency disturbances, so-called Klebanoff (K) modes, which cause a spanwise modulation with a distinct spanwise wavelength. Simultaneously, the FST enhances the initial levels of instability waves with frequencies in the frequency range of the KH instability, but at much smaller amplitude levels compared to the K-modes. Results from the calculations based on the linearized Navier–Stokes equations and comparison with DNS results reveal that the K-mode exhibits exponential growth in the separated shear layer until it reaches a peak amplitude. At the same time, two-dimensional (2D) disturbance waves are also exponentially amplified, in fact at larger growth rate compared to the K-mode, due to the primary (convective) shear-layer instability mechanism until they saturate downstream of the peak amplitude associated with the K-mode. Therefore, based on detailed spectral analysis and modal decompositions for the separation bubbles investigated, the transition process is the result of two different mechanisms: (i) strong amplification of high-frequency (order of the shedding frequency), essentially 2D or weakly oblique fluctuating disturbances and (ii) low-frequency, three-dimensional K-modes caused by FST. Depending on the intensity of the FST, one of these mechanisms would dominate the transition process, or both mechanisms act together and contribute simultaneously. The net effect of these two events is an acceleration of transition for an increased level of FST intensity, which in turn leads to a reduction of the extent of the separation bubble in streamwise and wall-normal directions. The ‘roll-up’ into spanwise large-scale vortical structures resulting from the shear-layer instability, and the eventual breakdown of these structures, strongly contribute to the reattachment process. The spanwise coherence of these ‘rollers’ deteriorates due to the presence of large-amplitude K-modes, thus effectively weakening their strength for high levels of FST intensities ($Tu>1\,\%$).

A theoretical model is developed giving a moderately detailed description of the flow in a turbulent bore, the velocity profiles, the shear stresses, the energy dissipation, etc. An analysis of the flow conditions at the toe of the turbulent front indicates significant differences from the usual description based on the finite-amplitude shallow-water equations, and it is shown that the present model gives a closer description of the actual physical conditions. Finally, numerical results are presented that illustrate how the model works, and test its validity on an example with known properties.

We consider flow in a thin film generated by partially submerging an inclined rigid plate in a reservoir of ethanol– or methanol–water solution and wetting its surface. Evaporation leads to concentration and surface tension gradients that drive flow up the plate. An experimental study indicates that the climbing film is subject to two distinct instabilities. The first is a convective instability characterized by flattened convection rolls aligned in the direction of flow and accompanied by free-surface deformations; in the meniscus region, this instability gives rise to pronounced ridge structures aligned with the mean flow. The second instability, evident when the plate is nearly vertical, takes the form of transverse surface waves propagating up the plate.

We demonstrate that the observed longitudinal rolls are driven by the combined influence of surface deformations and alcohol concentration gradients. Guided by the observation that the rolls are flattened, we develop a quasi-two-dimensional theoretical model for the instability of the film, based on lubrication theory, which includes the effects of gravity, capillarity and Marangoni stresses at the surface. We develop stability criteria for the film which are in qualitative agreement with our experimental observations. Our analysis yields an equation for the shape of the interface which is solved numerically and reproduces the salient features of the observed flows, including the slow lateral drift and merging of the ridges.

This paper discusses the theory of the generation of sound which occurs when a frozen turbulent eddy is convected in a mean flow past an airfoil or a semi-infinite plate, with and without the application of a Kutta condition and with and without the presence of a mean vortex sheet in the wake. A sequence of two-dimensional mathematical problems involving a prototype eddy in the form of a line vortex is examined, it being argued that this constitutes the simplest realistic model. Important effects of convection are deduced which hitherto have not been revealed by analyses which assume quadrupole sources to be at rest relative to the plate or airfoil. It is concluded that, to the order of approximation to which the sound from convected turbulence near a scattering body is usually estimated, the imposition of a Kutta condition at the trailing edge leads to a complete cancellation of the sound generated when frozen turbulence convects past a semi-infinite plate, and to the cancellation of the diffraction field produced by the trailing edge in the case of an airfoil of compact chord.

Observations made in a well-developed, thermally stratified, horizontal, flat- plate boundary layer are used to study the effects of buoyancy on the mean flow and turbulence structure. These are represented in a similarity framework obtained from the concept of local equilibrium in a fully developed turbulent flow. Mean velocity and temperature profiles in both the inner and outer layers are strongly dependent on the thermal stratification, the former suggesting an increase in the thickness of the viscous sublayer with increasing stability. The coefficients of skin friction and heat transfer, on the other hand, decrease with increasing stability.

Normalized turbulent intensities, fluxes and their correlation coefficients also vary with buoyancy. In stable conditions, turbulence becomes rapidly suppressed with increasing stability as more and more energy has to be expended in over- coming buoyancy forces. The buoyancy effects are found to be more dominant in the stress budget than in the turbulent energy budget. The horizontal heat flux is much greater than the vertical heat flux and their ratio increases with stability. The ratio of the eddy diffusivities of heat and momentum, on the other hand, decreases with increasing stability. The spectra of velocity and temperature fluctuations indicate no buoyancy subrange, but the wavenumber corresponding to peak energy is found to increase with increasing stability.

We consider the problem of a thin liquid layer falling down an inclined plate that is subjected to non-uniform heating. The plate temperature is assumed to be linearly distributed and both directions of the temperature gradient with respect to the flow are investigated. The film flow is not only influenced by gravity and mean surface tension, but in addition by the thermocapillary force acting along the free surface. The coupling of thermocapillary instability and surface-wave instabilities is studied for two-dimensional disturbances. Applying the long-wave theory, a nonlinear evolution equation is derived. When the plate temperature is decreasing in the downstream direction, linear stability analysis exhibits a film stabilization, compared to a uniformly heated film. In contrast, for increasing temperature along the plate, the film becomes less stable. Numerical solution of the evolution equation indicates the existence of permanent finite-amplitude waves of different kinds. The shape of the waves depends mainly on the mean flow and the mean surface tension, but their amplitudes and phase speeds are influenced by thermocapillarity.

We present a study of Eulerian and Lagrangian statistics from a high-resolution numerical simulation of isotropic and homogeneous turbulence using the FLASH code, with an estimated Taylor microscale Reynolds number of around 600. Statistics are evaluated over a data set with 18563 spatial grid points and with 2563 = 16.8 million particles, followed for about one large-scale eddy turnover time. We present data for the Eulerian and Lagrangian structure functions up to the tenth order. We analyze the local scaling properties in the inertial range. The Eulerian velocity field results show good agreement with previous data and confirm the puzzling differences previously found between the scaling of the transverse and the longitudinal structure functions. On the other hand, accurate measurements of sixth-and-higher-order Lagrangian structure functions allow us to highlight some discrepancies from earlier experimental and numerical results. We interpret this result in terms of a possible contamination from the viscous scale, which may have affected estimates of the scaling properties in previous studies. We show that a simple bridge relation based on a multifractal theory is able to connect scaling properties of both Eulerian and Lagrangian observables, provided that the small differences between intermittency of transverse and longitudinal Eulerian structure functions are properly considered.

Creeping flow of a two-layer system with a monolayer of an insoluble surfactant on the interface is considered. The linear-stability theory of plane Couette–Poiseuille flow is developed in the Stokes approximation. To isolate the Marangoni effect, gravity is excluded. The shear-flow instability due to the interfacial surfactant, uncovered earlier for long waves only (Frenkel & Halpern 2002), is studied with inclusion of all wavelengths, and over the entire parameter space of the Marangoni number $M$, the viscosity ratio $m$, the interfacial velocity shear $s$, and the thickness ratio $n$ (${\ge}\,1$). The complex wave speed of normal modes solves a quadratic equation, and the growth rate function is continuous at all wavenumbers and all parameter values. If $M\,{>}\,0$, $s\,{\ne}\,0$, $m\,{<}\,n^2$, and $n\,{>}\,1$, the small disturbances grow provided they are sufficiently long wave. However, the instability is not long wave in the following sense: the unstable waves are not necessarily much longer than the smaller of the two layer thicknesses. On the other hand, there are parametric regimes for which the instability has a mid-wave character, the flow being stable at both sufficiently large and small wavelengths and unstable in between. The critical (instability-onset) manifold in the parameter space is investigated. Also, it is shown that for certain parametric limits the convergence of the dispersion function is non-uniform with respect to the wavenumber. This is used to explain the parametric discontinuities of the long-wave growth-rate exponents found earlier.

The flow field of an airfoil oscillated periodically over a reduced frequency range, 0 ≤ k ≤ 1.6, is studied experimentally at chord Reynolds numbers of Rc = 22000 and 44000. For most of the data, the NACA0012 airfoil is pitched sinusoidally about one quarter chord between angles of attack α of 5° and 25°. The cyclic variation of the near wake flow field is documented through flow visualization and phase-averaged vorticity measurements. In addition to the familiar dynamic stall vortex (DSV), an intense vortex of opposite sign is observed to originate from the trailing edge just when the DSV is shed. The two together take the shape of the cross-section of a large ‘mushroom’ while being convected away from the airfoil. The phase delay in the shedding of the DSV with increasing k, as observed by previous researchers, is documented for the full range of k. It is observed that the sum of the absolute values of all vorticity convected into the wake over a cycle is nearly constant and is independent of the reduced frequency and amplitude of oscillation but dependent on the mean α. The time varying component of the lift is estimated in a novel way from the shed vorticity flux. The analytical foundation of the method and the various approximations are discussed. The estimated lift hysteresis loops are found to be in reasonable agreement with available data from the literature as well as with limited force balance measurements. Comparison of the lift hysteresis loops with the corresponding vorticity fields clearly shows that major features of the lift variation are directly linked to the evolution of the large-scale vortical structures and the phase delay phenomenon.

The principle and basic physics of a new type of liquid-in-air jet are described. This jet is generated by a primary fluid jet that emerges from a nozzle below the surface of a stationary (secondary) fluid. After breaking the surface, the jet consists of the central primary jet surrounded by a sheath of secondary fluid which has been entrained by the primary jet during its passage through the secondary fluid. Normally the flow in this compound jet is laminar, and it breaks up into drops due to capillary instability.

In this paper the generation of compound jets is discussed, and the flow pattern in the jet is studied experimentally both in stable and unstable conditions. Three different types of instabilities can be elicited on the jet. The jet mechanism is described quantitatively, and expressions for the jet velocity and some other parameters are derived.

This paper extends the resolvent analysis of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to consider flow control techniques that employ linear control laws, focusing on opposition control (Choi, Moin & Kim, J. Fluid Mech., vol. 262, 1994, pp. 75–110) as an example. Under this formulation, the velocity field for turbulent pipe flow is decomposed into a series of highly amplified (rank-1) response modes, identified from a gain analysis of the Fourier-transformed Navier–Stokes equations. These rank-1 velocity responses represent propagating structures of given streamwise/spanwise wavelength and temporal frequency, whose wall-normal footprint depends on the phase speed of the mode. Opposition control, introduced via the boundary condition on wall-normal velocity, affects the amplification characteristics (and wall-normal structure) of these response modes; a decrease in gain indicates mode suppression, which leads to a decrease in the drag contribution from that mode. With basic assumptions, this rank-1 model reproduces trends observed in previous direct numerical simulation and large eddy simulation, without requiring high-performance computing facilities. Further, a wavenumber–frequency breakdown of control explains the deterioration of opposition control performance with increasing sensor elevation and Reynolds number. It is shown that slower-moving modes localized near the wall (i.e. attached modes) are suppressed by opposition control. Faster-moving detached modes, which are more energetic at higher Reynolds number and more likely to be detected by sensors far from the wall, are further amplified. These faster-moving modes require a phase lag between sensor and actuator velocity for suppression. Thus, the effectiveness of opposition control is determined by a trade-off between the modes detected by the sensor. However, it may be possible to develop control strategies optimized for individual modes. A brief exploration of such mode-optimized control suggests the potential for significant performance improvement.

The bulk rheology of bidisperse mixtures of granular materials is examined under homogeneous shear flow conditions using the event-driven simulation method. The granular material is modelled as a system of smooth inelastic disks, interacting via the hard-core potential. In order to understand the effect of size and mass disparities, two cases were examined separately, namely, a mixture of different sized particles with particles having either the same mass or the same material density. The relevant macroscopic quantities are the pressure, the shear viscosity, the granular energy (fluctuating kinetic energy) and the first normal stress difference.

Numerical results for pressure, viscosity and granular energy are compared with a kinetic-theory constitutive model with excellent agreement in the low dissipation limit even at large size disparities. Systematic quantitative deviations occur for stronger dissipations. Mixtures with equal-mass particles show a stronger shear resistance than an equivalent monodisperse system; in contrast, however, mixtures with equal-density particles show a reduced shear resistance. The granular energies of the two species are unequal, implying that the equi-partition principle assumed in most of the constitutive models does not hold. Inelasticity is responsible for the onset of energy non-equipartition, but mass disparity significantly enhances its magnitude. This lack of energy equipartition can lead to interesting non-monotonic variations of the pressure, viscosity and granular energy with the mass ratio if the size ratio is held fixed, while the model predictions (with the equipartition assumption) suggest a monotonic behaviour in the same limit. In general, the granular fluid is non-Newtonian with a measurable first normal stress difference (which is positive if the stress is defined in the compressive sense), and the effect of bidispersity is to increase the normal stress difference, thus enhancing the non-Newtonian character of the fluid.

The coalescence of two co-flowing axisymmetric turbulent plumes and the resulting single plume flow is modelled and compared to experiments. The point of coalescence is defined as the location at which only a single peak appears in the horizontal buoyancy profile, and a prediction is made for its height. The model takes into account the drawing together of the two plumes due to their respective entrainment fields. Experiments showed that the model tends to overestimate the coalescence height, though this discrepancy may be partly explained by the sensitivity of the prediction to the entrainment coefficient. A model is then developed to describe the resulting single plume and predict its virtual origin. This prediction and subsequent predictions of flow rate above the merge height compare very well with experimental results.

Three-dimensional numerical solutions are obtained describing convection with a square lattice in a layer heated from below with no-slip top and bottom boundaries. The limit of infinite Prandtl number and a linear dependence of the viscosity on temperature are assumed. The stability of the three-dimensional solutions with respect to disturbances fitting the square lattice is analysed. It is shown that convection in the form of two-dimensional rolls is stable for low variations of viscosity, while square-pattern convection becomes stable when the viscosity contrast between upper and lower parts of the fluid layer is sufficiently strong. The theoretical results are in qualitative agreement with experimental observations.

A nonlinear analysis of Bénard–Marangoni convection in a horizontal fluid layer of infinite extent is proposed. The nonlinear equations describing the fields of temperature and velocity are solved by using the Gorkov–Malkus–Veronis technique, which consists of developing the steady solution in terms of a small parameter measuring the deviation from the marginal state. This work generalizes an earlier paper by Schlüter, Lortz & Busse wherein only buoyancy-driven instabilities were handled. In the present work both buoyancy and temperature-dependent surface-tension effects are considered. The band of allowed steady states of convection near the onset of convection is determined as a function of the Marangoni number and the wavenumber. The influence of various dimensionless quantities like Rayleigh, Prandtl and Biot numbers is examined. Supercritical as well as subcritical zones of instability are displayed. It is found that hexagons are allowable flow patterns.

An experiment is described in which a turbulent boundary layer was generated along a sloping wall of a laboratory tank containing salt-stratified fluid. Initially, the pycnocline separating the upper fresh-water layer from the lower saline layer was relatively thin; it thickened in response to the mixing as the experiment proceeded. Two types of mean circulation developed as a result of the boundary mixing. In the boundary layer, counterflowing mean streams were observed that augmented the diffusion of salt up- and downslope. This augmented dispersion in turn tended to spread the salt beyond the depth range of the pycnocline in the ambient fluid, producing a mean convergence in the boundary layer and intrusions from the layer into the ambient fluid.

The evolving density structure in the ambient fluid was measured by conductivity-probe traverses, from which the net buoyancy flux (or salt flux) and volume flux in the boundary layer were determined. The level of zero volume flux was found to coincide closely with the level of maximum stability frequency N, so that the buoyancy transport across this level was entirely the result of turbulent dispersion in the boundary layer. At other levels, the convective transports up and down contribute significantly. A simple theory provides scaling in terms of the laboratory parameters; in terms of the inferred overall turbulent viscosity νe, the buoyancy transport resulting from boundary-layer dispersion was found to be \[ F_{\rm B} = 0.60 \left\{\frac{\nu_{\rm e}N(z)}{\sin\theta}\right\}^{\frac{3}{2}}\cos^2\theta, \] and the intrusion velocity into the ambient fluid is \[ v_1 = 0.42\frac{\nu_{\rm e}^{\frac{3}{2}}}{N^{\frac{1}{2}}h^2}\frac{\cos^2\theta}{(\sin\theta)^{\frac{3}{2}}}, \] where θ is the angle of the sloping bed. These expressions must become invalid when θ becomes vanishingly small; their application to estuarine and continental slope flows is discussed briefly.

The role of large vortex structures in the evolution of a two-dimensional shear layer is studied numerically. The motion of up to 4096 vortices is followed on a 256 × 256 grid using the cloud-in-cell algorithm. The scaling predictions of self-preservation theory are confirmed for low-order velocity correlations, although the existence of vortex structures produces large fluctuations even in a simulation of this size. The simple picture of the shear layer as a line of vortex blobs, that merge pairwise thus thickening the layer, is not seen. On the contrary, the layer seems to thicken by the scattering of vortex structures of roughly fixed size about the midline. The size of the vortex structures does not scale with the layer thickness. A study of the entrainment of a passive marker shows that flow visualization experiments may have overestimated the size of the vortex structures. It appears that the finite area vortices have time to equilibrate between mergings, and the consequences of applying equilibrium statistical mechanics to their internal structure are explored. A simple model is presented which demonstrates how the size and separation of vortex structures may lock into a fixed ratio. This is precisely the type of mechanism that is needed to produce simple scaling in a flow that has initially several distinct length scales. A number of consistency checks on the numerical results are performed. In particular, the evolution of the same vortex configuration on two grids of different size is compared. This test showed that, although errors on subgrid scales do propagate to small wavenumbers, the dominant wavenumber of vorticity cascades back ahead of the peak in the error spectrum.

In this paper calculations are made of the two-dimensional flow field of an incompressible viscous fluid in a long parallel-sided channel whose walls pulsate in a prescribed way. The study covers all values of the unsteadiness parameter α and the steady-streaming Reynolds number. The wall motion is, in general, assumed to be of small amplitude and sinusoidal. Particular attention is given to the steady component of the flow at second order in the amplitude parameter ε. The results for the corresponding problem in axisymmetric geometry are given in an appendix.

Next the following problem is considered: the calculation of the wall motion which will result, in response to prescribed unsteady pressures imposed at the ends of the channel and outside its walls, if the walls are assumed to respond elastically to variations in transmural pressure. It is found that the system has a natural frequency of oscillation, and that resonance will occur if this frequency is close to a multiple of the frequency of the external pressure fluctuations. Finally the preceding work is applied in a discussion of blood flow in the coronary arteries of large mammals.