Effects of the Lorentz force on near-wall turbulence structures are investigated using the direct numerical simulation technique with the assumption of no induced magnetic field at low magnetic Reynolds number. A uniform magnetic field is applied in the streamwise (x), wall-normal (y) or spanwise (z) direction to turbulent flow in an infinitely long channel with non-conducting walls. The Lorentz force induced from the magnetic field suppresses the dynamically significant coherent structures near the wall. The skin friction decreases with increasing streamwise and spanwise magnetic fields, whereas it increases owing to the Hartmann effect when the strength of the wall-normal magnetic field exceeds a certain value. All the turbulence intensities and the Reynolds shear stress decrease with the wall-normal and spanwise magnetic fields, but the streamwise velocity fluctuations increase with the streamwise magnetic field although all other turbulence intensities decrease. It is also shown that the wall-normal magnetic field is much more effective than the streamwise and spanwise magnetic fields in reducing turbulent fluctuations and suppressing the near-wall streamwise vorticity, even though the wall-normal magnetic field interacts directly with the mean flow and results in drag increase at strong magnetic fields. In the channel with a strong streamwise magnetic field, two-dimensional streamwise velocity fluctuations u(y, z) exist, even after other components of the velocity fluctuations nearly vanish. In the cases of strong wall-normal and spanwise magnetic fields, all turbulence intensities, the Reynolds shear stress and vorticity fluctuations decrease rapidly and become zero. The turbulence structures are markedly elongated in the direction of the applied magnetic field when it is strong enough. It is shown that this elongation of the structures is associated with a rapid decrease of the Joule dissipation in time.

The flow through a compressor passage without and with incoming free-stream grid turbulence is simulated. At moderate Reynolds number, laminar-to-turbulence transition can take place on both sides of the aerofoil, but proceeds in distinctly different manners. The direct numerical simulations (DNS) of this flow reveal the mechanics of breakdown to turbulence on both surfaces of the blade. The pressure surface boundary layer undergoes laminar separation in the absence of free-stream disturbances. When exposed to free-stream forcing, the boundary layer remains attached due to transition to turbulence upstream of the laminar separation point. Three types of breakdowns are observed; they combine characteristics of natural and bypass transition. In particular, instability waves, which trace back to discrete modes of the base flow, can be observed, but their development is not independent of the Klebanoff distortions that are caused by free-stream turbulent forcing. At a higher turbulence intensity, the transition mechanism shifts to a purely bypass scenario. Unlike the pressure side, the suction surface boundary layer separates independent of the free-stream condition, be it laminar or a moderate free-stream turbulence of intensity Tu ~ 3%. Upstream of the separation, the amplification of the Klebanoff distortions is suppressed in the favourable pressure gradient (FPG) region. This suppression is in agreement with simulations of constant pressure gradient boundary layers. FPG is normally stabilizing with respect to bypass transition to turbulence, but is, thereby, unfavourable with respect to separation. Downstream of the FPG section, a strong adverse pressure gradient (APG) on the suction surface of the blade causes the laminar boundary layer to separate. The separation surface is modulated in the instantaneous fields of the Klebanoff distortion inside the shear layer, which consists of forward and backward jet-like perturbations. Separation is followed by breakdown to turbulence and reattachment. As the free-stream turbulence intensity is increased, Tu ~ 6.5%, transitional turbulent patches are initiated, and interact with the downstream separated flow, causing local attachment. The calming effect, or delayed re-establishment of the boundary layer separation, is observed in the wake of the turbulent events.

The classical problem of the motion of a one-dimensional unsteady shock generated by a piston moving with velocity vp = ctn is extended to take into account thermal radiation effects by the similarity method of Taylor and Sedov. Gray gas and local thermodynamic equilibrium are assumed and a modification of the Schuster-Schwarzschild differential equation for the heat flux is adopted. The optical thickness is not restricted to be thin or thick, and the absorption coefficient is assumed to vary with the density and temperature. Numerical results indicate that the pressure and velocity are not affected much by the radiation, but the density, temperature and radiant heat flux are changed considerably.

Normal stress differences are measured in dense suspensions of neutrally buoyant non-Brownian spheres dispersed in a Newtonian fluid. Rotating-rod rheometry is used to characterize the suspension normal stresses which are responsible for a rod-dipping phenomenon. These normal stress differences are seen to strongly increase above a volume fraction of approximately 22 %. During the course of the experiments, a new time-dependent behaviour is also observed: the dip is filled with increasing times. This time evolution is found to be related to particle migration from regions of high shear rate to regions of low shear rate. The behaviour is compared with the predictions of a suspension balance model in which the particle migration flux is related to the normal stresses of the suspension.

A detailed investigation of fully developed transient flow in a pipe has been undertaken using water as the working fluid. Linearly increasing or decreasing excursions of flow rate were imposed between steady initial and final values. A three-beam, two-component laser Doppler anemometer was used to make simultaneous measurements of either axial and radial, or axial and circumferential, components of local velocity. Values of ensemble-averaged mean velocity, root-mean-square velocity fluctuation and turbulent shear stress were found from the measurements.

Being the first really detailed study of ramp-type transient turbulent flow, the present investigation has yielded new information and valuable insight into certain fundamental aspects of turbulence dynamics. Some striking features are evident in the response of the turbulence field to the imposed excursions of flow rate. Three different delays have been identified: a delay in the response of turbulence production; a delay in turbulence energy redistribution among its three components; and a delay associated with the propagation of turbulence radially. The last of these is the most pronounced under the conditions of the present study. A dimensionless delay parameter τ+[= √2τUτ0/D] is proposed to describe it. The first response of turbulence is found to occur in the region near the wall where turbulence production peaks. The axial component of turbulence responds earlier than the other two components and builds up faster. The response propagates towards the centre of the pipe through the action of turbulent diffusion at a speed which depends on the Reynolds number at the start of the excursion. In the core region, the three components of turbulence energy respond in a similar manner. Turbulence intensity is reduced in the case of accelerating flow and increased in decelerating flow. This is mainly as a result of the delayed response of turbulence. A dimensionless ramp rate parameter γ[= (dUb/dt) (1/Ub0)(D/Uτ0] is proposed, which determines the extent to which the turbulence energy differs from that of pseudo-steady flow as a result of the delay in the propagation of turbulence.

This paper examines how increasing the value of the Reynolds number $Re$ affects the ability of spanwise-forcing techniques to yield turbulent skin-friction drag reduction. The considered forcing is based on the streamwise-travelling waves of spanwise-wall velocity (Quadrio et al., J. Fluid Mech., vol. 627, 2009, pp. 161–178). The study builds upon an extensive drag-reduction database created via direct numerical simulation of a turbulent channel flow for two fivefold separated values of $Re$, namely $Re_{\unicode[STIX]{x1D70F}}=200$ and $Re_{\unicode[STIX]{x1D70F}}=1000$. The sheer size of the database, which for the first time systematically addresses the amplitude of the forcing, allows a comprehensive view of the drag-reducing characteristics of the travelling waves, and enables a detailed description of the changes occurring when $Re$ increases. The effect of using a viscous scaling based on the friction velocity of either the non-controlled flow or the drag-reduced flow is described. In analogy with other wall-based drag-reduction techniques, like riblets for example, the performance of the travelling waves is well described by a vertical shift of the logarithmic portion of the mean streamwise velocity profile. Except when $Re$ is very low, this shift remains constant with $Re$, at odds with the percentage reduction of the friction coefficient, which is known to present a mild, logarithmic decline. Our new data agree with the available literature, which is however mostly based on low-$Re$ information and hence predicts a quick drop of maximum drag reduction with $Re$. The present study supports a more optimistic scenario, where for an airplane at flight Reynolds numbers a drag reduction of nearly 30 % would still be possible thanks to the travelling waves.

Two-dimensional numerical simulations are used to study the coupled fluid–structure–electric interaction of a simple model of an inverted piezoelectric flag, and to investigate the dynamics of the flow–structure interaction of this configuration as well as its energy harvesting performance. In particular, the dynamic response of the inverted flag as well as the associated flow patterns are examined for a range of inertia, bending stiffness and Reynolds numbers, and categorized into distinct vibrational states based on the symmetry of the motion as well as the amplitude. Simulations indicate that large-amplitude vibrations can be achieved over a large range of parameters and there is also evidence of lock-on between the flag flutter and the intrinsic wake shedding phenomenon. The initial inclination of the flag to the prevailing flow is found to significantly affect the flutter performance for inclination angles exceeding $15^{\circ }$. The state with large symmetric flutter is identified as being most promising for energy harvesting, and the effect of piezoelectric material parameters on the energy harvesting performance of this flutter state is examined in detail. The maximum energy efficiency of the flags is found to be approximately $7\,\%$, and the maximum occurs when there is a match between the time scales of flutter and the intrinsic time scale of the piezoelectric circuit. The simulations are used to examine a simple scaling law that could be used to predict the energy harvesting performance of such devices.

The flow structure on low-aspect-ratio wings arising from pitch-up motion is addressed via a technique of particle image velocimetry. The objectives are to: determine the onset and evolution of the three-dimensional leading-edge vortex; provide complementary interpretations of the vortex structure in terms of streamlines, projections of spanwise and surface-normal vorticity, and surfaces of constant values of the second invariant of the velocity gradient tensor (iso- surfaces); and to characterize the effect of wing planform (rectangular versus elliptical) on this vortex structure. The pitch-up motion of the wing (plate) is from 0 to over a time span corresponding to four convective time scales, and the Reynolds number based on chord is 10 000. Volumes of constant magnitude of the second invariant of the velocity gradient tensor are interpreted in conjunction with three-dimensional streamline patterns and vorticity projections in orthogonal directions. The wing motion gives rise to ordered vortical structures along its wing surface. In contrast to development of the classical two-dimensional leading-edge vortex, the flow pattern evolves to a strongly three-dimensional form at high angle of attack. The state of the vortex system, after attainment of maximum angle of attack, has a similar form for extreme configurations of wing planform. Near the plane of symmetry, a large-scale region of predominantly spanwise vorticity dominates. Away from the plane of symmetry, the flow is dominated by two extensive regions of surface-normal vorticity, i.e. swirl patterns parallel to the wing surface. This similar state of the vortex structure is, however, preceded by different sequences of events that depend on the magnitude of the spanwise velocity within the developing vortex from the leading edge of the wing. Spanwise velocity of the order of one-half the free stream velocity, which is oriented towards the plane of symmetry of the wing, results in regions of surface-normal vorticity. In contrast, if negligible spanwise velocity occurs within the developing leading-edge vortex, onset of the regions of surface-normal vorticity occurs near the tips of the wing. These extremes of large and insignificant spanwise velocity within the leading-edge vortex are induced respectively on rectangular and elliptical planforms.

A nonlinear equation based on the hydrodynamic equations is solved analytically using perturbation expansions to calculate the flow field of a steady isothermal, compressible and laminar gas flow in either a circular or a planar microchannel. The solution takes into account slip-flow effects explicitly by utilizing the classical velocity-slip boundary condition, assuming the gas properties are known. Consistent expansions provide not only the cross-stream but also the streamwise evolution of the various flow parameters of interest, such as pressure, density and Mach number. The slip-flow effect enters the solution explicitly as a zero-order correction comparable to, though smaller than, the compressible effect. The theoretical calculations are verified in an experimental study of pressure-driven gas flow in a long microchannel of sub-micron height. Standard micromachining techniques were utilized to fabricate the microchannel, with integral pressure microsensors based on the piezoresistivity principle of operation. The integrated microsystem allows accurate measurements of mass flow rates and pressure distributions along the microchannel. Nitrogen, helium and argon were used as the working fluids forced through the microchannel. The experimental results support the theoretical calculations in finding that acceleration and non-parabolic velocity profile effects were found to be negligible. A detailed error analysis is also carried out in an attempt to expose the challenges in conducting accurate measurements in microsystems.

Axial and azimuthal turbulence intensities in the rod-gap region are shown, for developed turbulent flow through parallel rod arrays, to increase strongly with decreasing rod spacing. Two array geometries are reported: one was constructed from a rectangular cross-section duct containing four rods and spaced at five pitch-to-diameter or wall-to-diameter ratios; the second was a test section containing six rods set in a regular square-pitch array to represent the interior flow region of a large array.

Measurements were made of the mean axial velocity, wall-shear-stress variation, axial-pressure distribution and Reynolds stresses. Techniques for resolving secondary-flow velocities to within ±1% of the local axial velocity failed to detect significant non-zero mean secondary-flow components. Analysis of the turbulent flow structure showed an energetic azimuthal turbulent-velocity component in the open rod gap for both geometries. The axial turbulent velocity has a coupled large-scale semiperiodic structure, with an antiphase relationship across the rod-gap centre or subchannel boundary. This structure is considered to be generated by an incompressible-flow parallel-channel instability, and, for closely spaced rod arrays, is the dominant process for intersubchannel mass, heat or momentum exchange.

A comprehensive study on confined swirling flows in an operational gas-turbine injector was performed by means of large-eddy simulations. The formulation was based on the Favre-filtered conservation equations and a modified Smagorinsky treatment of subgrid-scale motions. The model was then numerically solved by means of a preconditioned density-based finite-volume approach. Calculated mean velocities and turbulence properties show good agreement with experimental data obtained from the laser-Doppler velocimetry measurements. Various aspects of the swirling flow development (such as the central recirculating flow, precessing vortex core and Kelvin–Helmholtz instability) were explored in detail. Both co- and counter-rotating configurations were considered, and the effects of swirl direction on flow characteristics were examined. The flow evolution inside the injector is dictated mainly by the air delivered through the primary swirler. The impact of the secondary swirler appears to be limited.

This paper examines the streamwise dispersion of passive contaminant molecules released in a time-dependent laminar flow through a tube in the presence of boundary absorption or a catalytic wall reaction, which causes a depletion of contaminant in the flow. A finite-difference implicit scheme has been used to solve the unsteady convective–diffusion equation for all time. Here it is shown how the mixing of the cross-sectionally integrated concentration of contaminant molecules is influenced by the frequency of pressure pulsation and the heterogeneous reaction at the boundary. The behaviour of the dispersion coefficient due to the shear effects of steady, oscillatory, and the combined action of steady and periodic currents have been examined separately. The comparison reveals that for all cases the dispersion coefficient asymptotically reaches a stationary state after a certain time and it decreases with the absorption parameter. The increased wall absorption causes negatively skewed deviations from Gaussianity.

Using numerical simulations of rapidly rotating Boussinesq convection in a Cartesian box, we study the formation of long-lived, large-scale, depth-invariant coherent structures. These structures, which consist of concentrated cyclones, grow to the horizontal scale of the box, with velocities significantly larger than the convective motions. We vary the rotation rate, the thermal driving and the aspect ratio in order to determine the domain of existence of these large-scale vortices (LSV). We find that two conditions are required for their formation. First, the Rayleigh number, a measure of the thermal driving, must be several times its value at the linear onset of convection; this corresponds to Reynolds numbers, based on the convective velocity and the box depth, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\gtrsim }100$. Second, the rotational constraint on the convective structures must be strong. This requires that the local Rossby number, based on the convective velocity and the horizontal convective scale, ${\lesssim }0.15$. Simulations in which certain wavenumbers are artificially suppressed in spectral space suggest that the LSV are produced by the interactions of small-scale, depth-dependent convective motions. The presence of LSV significantly reduces the efficiency of the convective heat transport.

A subgrid model is presented for a passive scalar advected by a randomly prescribed frozen velocity field. In addition to an eddy diffusivity it has a term which describes the injection of noise from the subgrid scales, and a term which implicitly describes the coupling between the large eddies and the eddies just below the limit of resolution of the finite-difference grid. The construction of the model is based upon an iterative procedure which models away first the dissipation-range eddies and then the eddies of a slightly larger size, etc.

We observed a gradual change in the structure of a shock wave passing through a long tube of bubbly liquid, which we attribute to the motion of the bubbles relative to the liquid. We show that the effect of the motion on the structure of a shock wave is like that of thermal relaxation on gasdynamic shock waves: the pertinent relaxation time is the time viscous forces in the fluid take to alter the velocity of a bubble to that of the fluid. Our theory predicts certain changes in the speed of the shock wave and in its structure. We could not verify the prediction as to wave speed: in dilute mixtures it is too small to be measured. But we report experiments on the structure of the wave, which support our theoretical conclusion that the observed changes are due to the relative motion.

Moore's asymptotic analysis of vortex-sheet motion predicts that the Kelvin–Helmholtz instability leads to the formation of a weak singularity in the sheet profile at a finite time. The numerical studies of Meiron. Baker & Orszag, and of Krasny, provide only a partial validation of his analysis. In this work, the motion of periodic vortex sheets is computed using a new, spectrally accurate approximation to the Birkhoff–Rott integral. As advocated by Krasny, the catastrophic effect of round-off error is suppressed by application of a Fourier filter, which itself operates near the level of the round-off. It is found that to capture the correct asymptotic behaviour of the spectrum, the calculations must be performed in very high precision, and second-order terms must be included in the Ansatz to the spectrum. The numerical calculations proceed from the initial conditions first considered by Meiron, Baker & Orszag. For the range of amplitudes considered here, the results indicate that Moore's analysis is valid only at times well before the singularity time. Near the singularity time the form of the singularity departs away from that predicted by Moore, with the real and imaginary parts of the solutions becoming differentiated in their behaviour; the real part behaves in accordance with Moore's prediction, while the singularity in the imaginary part weakens. In addition, the form of the singularity apparently depends upon the initial amplitude of the disturbance, with the results suggesting that either Moore's analysis gives the complete form of the singularity only in the zero amplitude limit, or that the initial data considered here is not yet sufficiently small for the behaviour to be properly described by the asymptotic analysis. Convergence of the numerical solution beyond the singularity time is not observed.

In an experimental and theoretical study, we model a phenomenon observed in the summer Arctic, where a fresh-water layer at a temperature of 0°C floats both over a sea-water layer at its freezing point and under an ice layer. Our results show that the ice growth in this system takes place in three phases. First, because the fresh-water density decreases upon supercooling, the rapid diffusion of heat relative to salt from the fresh to the salt water causes a density inversion and thereby generates a high Rayleigh number convection in the fresh water. In this convection, supercooled water rises to the ice layer, where it nucleates into thin vertical interlocking ice crystals. When these sheets grow down to the interface, supercooling ceases. Second, the presence of the vertical ice sheets both constrains the temperature T and salinity s to lie on the freezing curve and allows them to diffuse in the vertical. In the interfacial region, the combination of these processes generates a lateral crystal growth, which continues until a horizontal ice sheet forms. Third, because of the T and s gradients in the sea water below this ice sheet, the horizontal sheet both migrates upwards and increases in thickness. From one-dimensional theoretical models of the first two phases, we find that the heat-transfer rates are 5–10 times those calculated for classic thermal diffusion.

In this paper, the low-Reynolds number (Re = 80) flow around a row of nine square cylinders placed normal to the oncoming flow is investigated using the lattice-Boltzmann method. The effects of the cylinder spacing on the flow are studied for spacing to diameter ratios of 0.3 to 12. No significant interaction between the wakes is observed with spacings greater than six times the diameter. At smaller spacings, the flow regimes as revealed by vorticity field and drag coefficient signal are: synchronized, quasi-periodic and chaotic. These regimes are shown to result from the interaction between primary (vortex shedding) and secondary (cylinder interaction) frequencies; the strength of the latter frequency in turn depends on the cylinder spacing. The secondary frequency is also related to transition between narrow and wide wakes behind a cylinder.

The mean drag coefficient and Strouhal number are found to increase rapidly with a decrease in spacing; correlations of these parameters with spacing are proposed. The Strouhal number based on gap velocity becomes approximately constant for a large range of spacings, highlighting the significance of gap velocity for this class of flows. It is also possible to analyse the vortex pattern in the synchronized and quasi-periodic regimes with the help of vorticity dynamics. These results, most of which have been obtained for the first time, are of fundamental significance.

When drag-reducing additives are confined entirely to the linear sublayer of a turbulent channel flow of water, both the spanwise spacing and bursting rate of the wall-layer structure are the same as those for a water flow and there is no evidence of drag reduction. Drag reduction is measured downstream of the location where the additives injected into the sublayer begin to mix in significant quantities with the buffer region (10 < y+ < 100)The superscript + denotes a dimensionless quantity scaled with the kinematic viscosity ν and the wall shear velocity v* = (τw/ρ)½. of the channel flow. At streamwise locations where drag reduction does occur and where the injected fluid is not yet uniformly mixed with the channel flow, the dimensionless spanwise streak spacing increases and the average bursting rate decreases. The decrease in bursting rate is larger than the corresponding increase in streak spacing. The wall-layer structure is like the structure in the flow of a homogeneous, uniformly mixed, drag-reducing solution. Thus, the additives have a direct effect on the flow processes in the buffer region and the linear sublayer appears to have a passive role in the interaction of the inner and outer portions of a turbulent wall layer.

Three-dimensional particle tracking velocimetry (PTV) and phase-sensitive constant temperature anemometry in pseudo-turbulence – i.e. flow solely driven by rising bubbles – were performed to investigate bubble clustering and to obtain the mean bubble rise velocity, distributions of bubble velocities and energy spectra at dilute gas concentrations (α ≤ 2.2 %). To characterize the clustering the pair correlation function G(r, θ) was calculated. The deformable bubbles with equivalent bubble diameter db = 4–5 mm were found to cluster within a radial distance of a few bubble radii with a preferred vertical orientation. This vertical alignment was present at both small and large scales. For small distances also some horizontal clustering was found. The large number of data points and the non-intrusiveness of PTV allowed well-converged probability density functions (PDFs) of the bubble velocity to be obtained. The PDFs had a non-Gaussian form for all velocity components and intermittency effects could be observed. The energy spectrum of the liquid velocity fluctuations decayed with a power law of −3.2, different from the ≈ −5/3 found for homogeneous isotropic turbulence, but close to the prediction −3 by Lance & Bataille (J. Fluid Mech., vol. 222, 1991, p. 95) for pseudo-turbulence.