In isotropic ‘box’ turbulence without a mean flow, the Lagrangian frequency spectrum extends to frequencies of order $(\epsilon/\nu)^{\frac{1}{2}}$ (ε is the rate of dissipation of kinetic energy per unit mass and ν is the kinematic viscosity of the fluid). This leads to an estimate that makes the r.m.s. value of du/dt of order $(\epsilon^3/\nu)^{\frac{1}{4}}$. The Eulerian frequency spectrum, however, extends to higher frequencies than its Lagrangian counterpart; this is caused by spectral broadening associated with large-scale advection of dissipative eddies. As a consequence, the r.m.s. value of ∂u/∂t at a fixed observation point is (apart from a numerical factor) $R_{\lambda}^{\frac{1}{2}}$ times as large as the r.m.s. value of du/dt (RΛ is the turbulence Reynolds number based on the Taylor microscale). The results of a theoretical analysis based on these premises agree with data obtained by Comte-Bellot, Shlien and Corrsin. The analysis also suggests that the Eulerian frequency spectrum has a $\omega^{-\frac{5}{3}} $ behaviour in the inertial subrange, and that it is not governed by Kolmogorov similarity.

The problem of closing the Reynolds-stress and dissipation-rate equations at low Reynolds numbers is considered, specific forms being suggested for the direct effects of viscosity on the various transport processes. By noting that the correlation coefficient $\overline{uv^2}/\overline{u^2}\overline{v^2} $ is nearly constant over a considerable portion of the low-Reynolds-number region adjacent to a wall the closure is simplified to one requiring the solution of approximated transport equations for only the turbulent shear stress, the turbulent kinetic energy and the energy dissipation rate. Numerical solutions are presented for turbulent channel flow and sink flows at low Reynolds number as well as a case of a severely accelerated boundary layer in which the turbulent shear stress becomes negligible compared with the viscous stresses. Agreement with experiment is generally encouraging.

Vortex-induced vibrations of bluff bodies occur when the vortex shedding frequency is close to the natural frequency of the structure. Of interest is the prediction of the lift and drag forces on the structure given some limited and scattered information on the velocity field. This is an inverse problem that is not straightforward to solve using standard computational fluid dynamics methods, especially since no information is provided for the pressure. An even greater challenge is to infer the lift and drag forces given some dye or smoke visualizations of the flow field. Here we employ deep neural networks that are extended to encode the incompressible Navier–Stokes equations coupled with the structure’s dynamic motion equation. In the first case, given scattered data in space–time on the velocity field and the structure’s motion, we use four coupled deep neural networks to infer very accurately the structural parameters, the entire time-dependent pressure field (with no prior training data), and reconstruct the velocity vector field and the structure’s dynamic motion. In the second case, given scattered data in space–time on a concentration field only, we use five coupled deep neural networks to infer very accurately the vector velocity field and all other quantities of interest as before. This new paradigm of inference in fluid mechanics for coupled multi-physics problems enables velocity and pressure quantification from flow snapshots in small subdomains and can be exploited for flow control applications and also for system identification.

A new type of solitary wave motion in incompressible fluids of non-uniform density has been investigated experimentally and theoretically. If a fluid is stratified in such a manner that there are two layers of different density joined by a thin region in which the density varies continuously, this type of wave propagates along the density gradient region without change of shape. In contrast to previously known solitary waves, these disturbances can exist even if the fluid depth is infinite. The waves are described by an approximate solution of the inviscid equations of motion. The analysis, which is based on the assumption that the wavelength of the disturbance is large compared with the thickness, L, of the region in which the density is not constant, indicates that the propagation velocity, U, is characterized by the dimensionless group (gL/U2) In (ρ1/ρ2), where g is the gravitational acceleration and ρ is the density. The value of this group, which is dependent on the wave amplitude and the form of the density gradient, is of the order one. Experimentally determined propagation velocities and wave shapes serve to verify the theoretical model.

The interaction between a conventional rectangular (primary) air jet and a co-flowing synthetic jet is investigated experimentally. The nozzles of both jets have the same long dimension but the aspect ratio of the synthetic jet orifice is 25 times larger. Detailed particle image velocimetry (PIV) measurements of the flow in the midspan plane show that primary jet fluid is directed into the synthetic jet orifice and the interaction between the jets leads to the formation of a closed recirculating flow domain. The concomitant formation of a low-pressure region between the jets results in deflection of the primary jet toward the actuator jet despite the absence of an extended control surface (e.g. a diffuser or collar) and is balanced by a force on the primary jet conduit. For a given synthetic jet strength and primary jet speed, the vectoring force depends mainly on the volume flow rate of primary jet fluid that is diverted into the synthetic jet actuator. This flow rate is regulated by restricting the flow of entrained ambient fluid using a small streamwise extension of the synthetic jet orifice that scales with the orifice width. The response of the primary jet to the imposed vectoring is investigated using stepped modulation of the driving signal. The characteristic vectoring time and vectoring angle decrease monotonically with primary jet speed.

The earliest results concerning the turbulence structure in a turbulent boundary layer with very unstable thermal stratification are due to Prandtl (1932). These results were developed further and made more precise by Obukhov (1946, 1960), Monin & Obukhov (1954) and Priestley (1954, 1955, 1956, 1960). All of these authors dealt with a surface layer of the Earth's atmosphere on hot summer days. Such a layer is the most easily accessible example of an unstably stratified boundary layer and it will be the main concern in this paper too. The theoretical predictions by the above-mentioned authors seemed at first to be confirmed by the available experimental data but in the late 1960s it became clear that at least some of the predictions disagreed strongly with the experimental information.

A more elaborate theory was proposed by Betchov & Yaglom (1971) who used a suggestion of Zilitinkevich (1971). According to this theory, within an unstably stratified boundary layer there are three special sublayers where turbulence structure is self-preserving and obeys rather simple power laws. The new theory explained the disagreement between some of the deductions from the old theory and the data. However, the data available in 1971 were insufficient for the confirmation of the new theory and it was even supposed by Betchov & Yaglom (1971) that their theory could not be applied to atmospheric surface layers on hot summer days.

Much new experimental data concerning unstably stratified boundary layers has been obtained in recent years; in particular, extensive experimental information was collected during the summers of 1981–1987 at the Tsimlyansk Field Station of the Moscow Institute of Atmospheric Physics. This paper is a survey of the deductions from the theory by Betchov & Yaglom which concern the mean fields and the one-point fluctuation moments in unstably stratified boundary layers, and a comparison of these deductions with the data available in 1989. It is shown that the data agree more or less satisfactorily with the theoretical predictions and permit one to obtain estimates for a number of coefficients that enter the theoretical equations.

Convection in a porous cavity driven by heating in the horizontal is analysed by a number of different techniques which yield a fairly complete description of the two-dimensional solutions. The solutions are governed by two dimensionless parameters: the Darcy-Rayleigh number R and the cavity aspect ratio A. We first find solutions valid for shallow cavities, A → 0, by using matched asymptotic expansions. These solutions are given up to O(A6R4). For A fixed, we find regular expansions in R by semi-numerical techniques, up to O(R30) in some cases. Series-improvement techniques then enable us to cover the range 0 ≤ R ≤ ∞. A limited result regarding bifurcations is noted. Finally, for R → ∞ with A fixed, we propose a self-consistent boundary-layer theory which extends previous approximate work. The results obtained by these different methods of solution are in good agreement with each other and with experiments.

A theory intended for slow, dense flows of cohesionless granular materials is developed for the case of planar deformations. By considering granular flows on very fine scales, one can conveniently split the individual particle velocities into fluctuating and mean transport components, and employ the notion of granular temperature that plays a central role in rapid granular flows. On somewhat larger scales, one can think of analogous fluctuations in strain rates. Both kinds of fluctuations are utilized in the present paper. Following the standard continuum approach, the conservation equations for mass, momentum and particle translational fluctuation energy are presented. The latter two equations involve constitutive coefficients, whose determination is one of the main concerns of the present paper. We begin with an associated flow rule for the case of a compressible, frictional, plastic continuum. The functional dependence of the flow rule is chosen so that the limiting behaviours of the resulting constitutive relations are consistent with the results of the kinetic theories developed for rapid flow regimes. Following Hibler (1977) and assuming that there are fluctuations in the strain rates that have, for example, a Gaussian distribution function, it is possible to obtain a relationship between the mean stress and the mean strain rate. It turns out, perhaps surprisingly, that this relationship has a viscous-like character. For low shear rates, the constitutive behaviour is similar to that of a liquid in the sense that the effective viscosity decreases with increasing granular temperature, whereas for rapid granular flows, the viscosity increases with increasing granular temperature as in a gas. The rate of energy dissipation can be determined in a manner similar to that used to derive the viscosity coefficients. After assuming that the magnitude of the strain-rate fluctuations can be related to the granular temperature, we obtain a closed system of equations that can be used to solve boundary value problems. The theory is used to consider the case of a simple shear flow. The resulting expressions for the stress components are similar to models previously proposed on a more ad hoc basis in which quasi-static stress contributions were directly patched to rate-dependent stresses. The problem of slow granular flow in rough-walled vertical chutes is then considered and the velocity, concentration and granular temperature profiles are determined. Thin boundary layers next to the vertical sidewalls arise with the concentration boundary layer being thicker than the velocity boundary layer. This kind of behaviour is observed in both laboratory experiments and in granular dynamics simulations of vertical chute flows.

The von Kármán (1921) rotating disk problem is extended to the case of flow started impulsively from rest; also, the steady-state problem is solved to a higher degree of accuracy than previously by a simple analytical-numerical method which avoids the matching difficulties in Cochran's (1934) well-known solution. Exact representations of the non-steady velocity field and pressure are given by suitable power-series expansions in the angle of rotation, Ωt, with coefficients that are functions of a similarity variable. The first four equations for velocity coefficient functions are solved exactly in closed form, and the next six by numerical integration. This gives four terms in the series for the primary flow and three terms in each series for the secondary flow.

The results indicate that the asymptotic steady state is approached after about 2 radians of the disk's motion and that it can be approximately obtained from the initial-value, time-dependent analysis. Furthermore, the non-steady flow has three phases, the first two of which are accurately and fully described with the terms computed. During the first-half radian (phase 1), the velocity field is essentially similar in time, with boundary-layer thickening the only significant effect. For 0·5 [lsim ] Ωt [lsim ] 1·5 (phase 2), boundary-layer growth continues at a slower rate, but simultaneously the velocity profiles adjust towards the shape of the ultimate steady-state profiles. At about Ωt = 1·5, some flow quantities overshoot the steady-state values by small amounts. In analogy with the ‘Greenspan-Howard problem’ (1963) it is believed that the third phase (Ωt > 1·5) consists of a small amplitude decaying oscillation about the steady-state solution.

Measurements of nearly isotropic turbulence downstream of an active grid are performed as a high-Reynolds-number ($Re_{\lambda} \approx 720$) update of the Comte-Bellot & Corrsin (1971) data set. Energy spectra at four downstream distances from the grid, ranging from $x/M=20$ to $x/M=48$, are measured and documented for subsequent initialization of, and comparison with, large-eddy simulations (LES). Data are recorded using an array of four X-wire probes which enables measurement of filtered velocities, filtered in the streamwise (using Taylor's hypothesis) and cross-stream directions. Different filter sizes are considered by varying the separation between the four probes. Higher-order statistics of filtered velocity are quantified by measuring probability density functions, and hyper-flatness and skewness coefficients of two-point velocity increments. The data can be used to study the ability of LES to reproduce both spectral and higher-order statistics of the resolved velocity field. In this study, the Smagorinsky, dynamic Smagorinsky, and dynamic mixed nonlinear models are considered. They are implemented in simulations of decaying isotropic turbulence using a pseudospectral code with initial conditions that match the measured energy spectra at $x/M=20$. Overall, it is found that the various LES models predict accurate low-order statistics of resolved scales in isotropic turbulence during the decay. For the spectral cutoff filter, the dynamic Smagorinsky model simulates the energy spectrum more closely at smaller wavenumber, and the dynamic mixed nonlinear model has closer agreement at large wavenumber. For the graded physical-space (Gaussian) filter, the dynamic mixed nonlinear model provides the best spectral results. The three models considered here underpredict the intermittency of longitudinal velocity increments at small distances. For transverse velocity increments, the models' predictions are closer to the measured values, but differ among themselves, with the mixed nonlinear model predicting reduced intermittency. Comparisons of probability density functions of subgrid-scale dissipation and stresses from simulations and experimental data reveal pronounced differences among the different models.

The behaviour of a turbulent boundary layer on a flat surface as it encounters the nose of a cylindrical wing mounted normal to that surface is being investigated. A three-component laser anemometer has been developed to measure this highly turbulent three-dimensional flow. Measurements of all the non-zero mean-velocity and Reynolds-stress components have been made with this instrument in the plane of symmetry upstream of the wing. These data have been used to estimate some of the component terms of the turbulence kinetic energy equation. Histograms of velocity fluctuations and short-time cross-correlations between the laser anemometer and a hot-wire probe have also been measured in the plane of symmetry. In all, these results reveal much of the time-dependent and time-averaged turbulence structure of the flow here.

Separation occurs in the plane of symmetry because of the adverse pressure gradient imposed by the wing. In the time mean the resulting separated flow consists of two fairly distinct regions: a thin upstream region characterized by low mean backflow velocities and a relatively thick downstream region dominated by the intense recirculation of the mean junction vortex. In the upstream region the turbulence stresses develop in a manner qualitatively similar to those of a two-dimensional boundary layer separating in an adverse pressure gradient. In the vicinity of the junction vortex, though, the turbulence stresses are much greater and reach’ values many times larger than those normally observed in turbulent flows. These large stresses are associated with bimodal (double-peaked) histograms of velocity fluctuations produced by a velocity variation that is bistable. These observations are consistent with large-scale low-frequency unsteadiness of the instantaneous flow structure associated with the junction vortex. This unsteadiness seems to be produced by fluctuations in the momentum and vorticity of fluid from the outer part of the boundary layer which is recirculated as it impinges on the leading edge of the wing. Though we would expect these fluctuations to be produced by coherent structures in the boundary layer, frequencies of the large-scale unsteadiness are substantially lower than the passage frequency of such structures. It therefore seems that only a fraction of the turbulent structures are recirculated in this way.

Building on our recent work on induced-charge electro-osmosis (ICEO) and electrophoresis (ICEP), as well as the Russian literature on spherical metal colloids, we examine the rich consequences of broken geometric and field symmetries upon the ICEO flow around conducting bodies. Through a variety of paradigmatic examples involving ideally polarizable (e.g. metal) bodies with thin double layers in weak fields, we demonstrate that spatial asymmetry generally leads to a net pumping of fluid past the body by ICEO, or, in the case of a freely suspended colloidal particle, translation and/or rotation by ICEP. We have chosen model systems that are simple enough to admit analysis, yet which contain the most important broken symmetries. Specifically, we consider (i) symmetrically shaped bodies with inhomogeneous surface properties, (ii) ‘nearly symmetric’ shapes (using a boundary perturbation scheme), (iii) highly asymmetric bodies composed of two symmetric bodies tethered together, (iv) symmetric conductors in electric-field gradients, and (v) arbitrarily shaped conductors in general non-uniform fields in two dimensions (using complex analysis). In non-uniform fields, ICEO flow and ICEP motion exist in addition to the more familiar dielectrophoretic forces and torques on the bodies (which also vary with the square of the electric field). We treat all of these problems in two and three dimensions, so our study has relevence for both colloids and microfluidics. In the colloidal context, we describe principles to ‘design’ polarizable particles which rotate to orient themselves and translate steadily in a desired direction in a DC or AC electric field. We also describe ‘ICEO spinners’ that rotate continuously in AC fields of arbitrary direction, although we show that ‘near spheres’ with small helical perturbations do not rotate, to leading order in the shape perturbation. In the microfluidic context, strong and steady flows can be driven by small AC potentials applied to systems containing asymmetric structures, which holds promise for portable or implantable self-powered devices. These results build upon and generalize recent studies in AC electro-osmosis (ACEO). Unlike ACEO, however, the inducing surfaces in ICEO can be physically distinct from the driving electrodes, increasing the frequency range and geometries available.

Direct numerical simulations of fully developed pressure-driven turbulent flow in a rotating channel have been performed. The unsteady Navier–Stokes equations were written for flow in a constantly rotating frame of reference and solved numerically by means of a finite-difference technique on a 128 × 128 × 128 computational mesh. The Reynolds number, based on the bulk mean velocity Um and the channel half-width h, was about 2900, while the rotation number Ro = 2|Ω|h/Um varied from 0 to 0.5. Without system rotation, results of the simulation were in good agreement with the accurate reference simulation of Kim, Moin & Moser (1987) and available experimental data. The simulated flow fields subject to rotation revealed fascinating effects exerted by the Coriolis force on channel flow turbulence. With weak rotation (Ro = 0.01) the turbulence statistics across the channel varied only slightly compared with the nonrotating case, and opposite effects were observed near the pressure and suction sides of the channel. With increasing rotation the augmentation and damping of the turbulence along the pressure and suction sides, respectively, became more significant, resulting in highly asymmetric profiles of mean velocity and turbulent Reynolds stresses. In accordance with the experimental observations of Johnston, Halleen & Lezius (1972), the mean velocity profile exhibited an appreciable region with slope 2Ω. At Ro = 0.50 the Reynolds stresses vanished in the vicinity of the stabilized side, and the nearly complete suppression of the turbulent agitation was confirmed by marker particle trackings and two-point velocity correlations. Rotational-induced Taylor-Görtler-like counter-rotating streamwise vortices have been identified, and the simulations suggest that the vortices are shifted slightly towards the pressure side with increasing rotation rates, and the number of vortex pairs therefore tend to increase with Ro.

Measurements of air and solid velocities were made in an air-solid two-phase flow in a horizontal pipe by the use of a laser-Doppler velocimeter (LDV). The pipe was 30 mm inner diameter, and two kinds of plastic particles, 0.2 and 3.4 mm in diameter, were conveyed in addition to fine particles (ammonium chloride) for air-flow detection. The air velocities averaged over the pipe cross section ranged from 6 to 20m/s and the solid-to-air mass-flow ratio was up to 6. Simultaneous measurements of both air and 0.2 mm particle velocities were found possible by setting threshold values against the pedestal and Doppler components of the photomultiplier signal.

As the loading ratio increased and the air velocity decreased, mean-velocity distributions of both phases increased asymmetrical tendency. I n the presence of 0.2mm particles, a flattening of the velocity profile was remarkable. The effects of the solid particles on air-flow turbulence varied greatly with particle size. That is, 3.4 mm particles increased the turbulence markedly, while 0-2 mm ones reduced it. The probability-density function of the air flow deviated from the normal distribution (Gaussian) in the presence of particles. Finally, the frequency spectra of air-flow turbulence were obtained in the presence of 0.2 mm particles by using a fast Fourier transform (FFT). As a result, it was found that t,he higher-frequency components increased with increasing loading ratio.

Time-averaged LDA measurements and time-resolved numerical flow predictions were performed to investigate the laminar flow induced by the harmonic in-line oscillation of a circular cylinder in water at rest. The key parameters, Reynolds number Re and Keulegan–Carpenter number KC, were varied to study three parameter combinations in detail. Good agreement was observed for Re=100 and KC=5 between measurements and predictions comparing phase-averaged velocity vectors. For Re=200 and KC=10 weakly stable and non-periodic flow patterns occurred, which made repeatable time-averaged measurements impossible. Nevertheless, the experimentally visualized vortex dynamics was reproduced by the two-dimensional computations. For the third combination, Re=210 and KC=6, which refers to a totally different flow regime, the computations again resulted in the correct fluid behaviour. Applying the widely used model of Morison et al. (1950) to the computed in-line force history, the drag and the added-mass coefficients were calculated and compared for different grid levels and time steps. Using these to reproduce the force functions revealed deviations from those originally computed as already noted in previous studies. They were found to be much higher than the deviations for the coarsest computational grid or the largest time step. The comparison of several in-line force coefficients with results obtained experimentally by Kühtz (1996) for β=35 confirmed that force predictions could also be reliably obtained by the computations.

A study is made of the propagation of sound in both a constant gradient shear flow and a turbulent shear flow above a flat surface. Curves are presented showing how, in the case of downstream propagation, the flow gradient tends to channel the sound energy into a narrow layer next to the wall. These results are used in estimating the effect of a flow on the attenuation of sound in a duct with absorbing side walls.

In immiscible two-phase flows, the contact line denotes the intersection of the fluid–fluid interface with the solid wall. When one fluid displaces the other, the contact line moves along the wall. A classical problem in continuum hydrodynamics is the incompatibility between the moving contact line and the no-slip boundary condition, as the latter leads to a non-integrable singularity. The recently discovered generalized Navier boundary condition (GNBC) offers an alternative to the no-slip boundary condition which can resolve the moving contact line conundrum. We present a variational derivation of the GNBC through the principle of minimum energy dissipation (entropy production), as formulated by Onsager for small perturbations away from equilibrium. Through numerical implementation of a continuum hydrodynamic model, it is demonstrated that the GNBC can quantitatively reproduce the moving contact line slip velocity profiles obtained from molecular dynamics simulations. In particular, the transition from complete slip at the moving contact line to near-zero slip far away is shown to be governed by a power-law partial-slip regime, extending to mesoscopic length scales. The sharp (fluid–fluid) interface limit of the hydrodynamic model, together with some general implications of slip versus no slip, are discussed.

A new method is introduced for estimating the convection velocity of individual modes in turbulent shear flows that, in contrast to most previous ones, only requires spectral information in the temporal or spatial direction over which a modal decomposition is desired, while only using local derivatives in other directions. If no spectral information is desired, the method provides a natural definition for the average convection velocity, as well as a way to estimate the accuracy of the frozen-turbulence approximation. Existing data from numerical turbulent channels at friction Reynolds numbers Reτ ≤ 1900 are used to validate the new method against classical ones, and to characterize the dependence of the convection velocity on the eddy wavelength and wall distance. The results indicate that the small scales in turbulent channels travel at the local mean velocity, while large ‘global’ modes travel at a more uniform speed proportional to the bulk velocity. To estimate the systematic deviations introduced in experimental spectra by the use of Taylor's approximation with a wavelength-independent convection velocity, a semi-empirical fit to the computed convection velocities is provided. It represents well the data throughout the Reynolds number range of the simulations. It is shown that Taylor's approximation not only displaces the large scales near the wall to shorter apparent wavelengths but also modifies the shape of the spectrum, giving rise to spurious peaks similar to those observed in some experiments. To a lesser extent the opposite is true above the logarithmic layer. The effect increases with the Reynolds number, suggesting that some of the recent challenges to the kx−1 energy spectrum may have to be reconsidered.

The interaction between a turbulent gas flow and particle motion was investigated by numerical simulations of gas–particle turbulent downward flow in a vertical channel. In particular the effect of inter-particle collision on the two-phase flow field was investigated. The gas flow field was obtained by large-eddy simulation (LES). Particles were treated by a Lagrangian method, with inter-particle collisions calculated by a deterministic method. The spatial resolution for LES of gas–solid two-phase turbulent flow was examined and relations between grid resolution and Stokes number are presented. Profiles of particle mean velocity, particle wall-normal fluctuation velocity and number density are flattened as a result of inter-particle collisions and these results are in good agreement with experimental measurements. Calculated turbulence attenuation by particles agrees well with experimental measurements for small Stokes numbers, but not for large Stokes number particle. The shape and scale of particle concentrations calculated considering inter-particle collision are in good agreement with experimental observations.

Results are reported on direct numerical simulations of transition from two-dimensional to three-dimensional states due to secondary instability in the wake of a circular cylinder. These calculations quantify the nonlinear response of the system to three-dimensional perturbations near threshold for the two separate linear instabilities of the wake: mode A and mode B. The objectives are to classify the nonlinear form of the bifurcation to mode A and mode B and to identify the conditions under which the wake evolves to periodic, quasi-periodic, or chaotic states with respect to changes in spanwise dimension and Reynolds number. The onset of mode A is shown to occur through a subcritical bifurcation that causes a reduction in the primary oscillation frequency of the wake at saturation. In contrast, the onset of mode B occurs through a supercritical bifurcation with no frequency shift near threshold. Simulations of the three-dimensional wake for fixed Reynolds number and increasing spanwise dimension show that large systems evolve to a state of spatiotemporal chaos, and suggest that three-dimensionality in the wake leads to irregular states and fast transition to turbulence at Reynolds numbers just beyond the onset of the secondary instability. A key feature of these ‘turbulent’ states is the competition between self-excited, three-dimensional instability modes (global modes) in the mode A wavenumber band. These instability modes produce irregular spatiotemporal patterns and large-scale ‘spot-like’ disturbances in the wake during the breakdown of the regular mode A pattern. Simulations at higher Reynolds number show that long-wavelength interactions modulate fluctuating forces and cause variations in phase along the span of the cylinder that reduce the fluctuating amplitude of lift and drag. Results of both two-dimensional and three-dimensional simulations are presented for a range of Reynolds number from about 10 up to 1000.