Turbulence,dynamic similarity and scale effects in high-velocity free-surface flows above a stepped chute

In high-velocity free-surface flows, air entrainment is common through the interface, and intense interactions take place between turbulent structures and entrained bubbles. Two-phase flow properties were measured herein in high-velocity open channel flows above a stepped chute. Detailed turbulence measurements were conducted in a large-size facility, and a comparative analysis was applied to test the validity of the Froude and Reynolds similarities. The results showed consistently that the Froude similitude was not satisfied using a 2:1 geometric scaling ratio. Lesser number of entrained bubbles and comparatively greater bubble sizes were observed at the smaller Reynolds numbers, as well as lower turbulence levels and larger turbulent length and time scales. The results implied that small-size models did underestimate the rate of energy dissipation and the aeration efficiency of prototype stepped spillways for similar flow conditions. Similarly a Reynolds similitude was tested. The results showed also some significant scale effects. However a number of self-similar relationships remained invariant under changes of scale and confirmed the analysis of Chanson and Carosi (Exp Fluids 42:385-401, 2007). The finding is significant because self-similarity may provide a picture general enough to be used to characterise the air–water flow field in large prototype channels.     

Testing of Model Equations for the Mean Dissipation using Kolmogorov Flows

Direct Numerical Simulations (DNS) of Kolmogorov flows are performed at three different Reynolds numbers Re _λ between 110 and 190 by imposing a mean velocity profile in y-direction of the form U(y) = F sin(y) in a periodic box of volume (2π)³. After a few integral times the turbulent flow turns out to be statistically steady. Profiles of mean quantities are then obtained by averaging over planes at constant y. Based on these profiles two different model equations for the mean dissipation ε in the context of two-equation RANS (Reynolds Averaged Navier–Stokes) modelling of turbulence are compared to each other. The high Reynolds number version of the k-ε-model (Jones and Launder, Int J Heat Mass Transfer 15:301–314, 1972), to be called the standard model and a new model by Menter et al. (2006), to be called the Menter–Egorov model, are tested against the DNS results. Both models are solved numerically and it is found that the standard model does not provide a steady solution for the present case, while the Menter–Egorov model does. In addition a fairly good quantitative agreement of the model solution and the DNS data is found for the averaged profiles of the kinetic energy k and the dissipation ε. Furthermore, an analysis based on flow-inherent geometries, called dissipation elements (Wang and Peters, J Fluid Mech 608:113–138, 2008), is used to examine the Menter–Egorov ε model equation. An expression for the evolution of ε is derived by taking appropriate moments of the equation for the evolution of the probability density function (pdf) of the length of dissipation elements. A term-by-term comparison with the model equation allows a prediction of the constants, which with increasing Reynolds number approach the empirical values.     

The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 2: single-crystal plasticity

Variational formulations are constructed for rate-independent problems in small-deformation single-crystal strain-gradient plasticity. The framework, based on that of Gurtin (J Mech Phys Solids 50: 5–32, 2002), makes use of the flow rule expressed in terms of the dissipation function. Provision is made for energetic and dissipative microstresses. Both recoverable and non-recoverable defect energies are incorporated into the variational framework. The recoverable energies include those that depend smoothly on the slip gradients, the Burgers tensor, or on the dislocation densities (Gurtin et al. J Mech Phys Solids 55:1853–1878, 2007), as well as an energy proposed by Ohno and Okumura (J Mech Phys Solids 55:1879–1898, 2007), which leads to excellent agreement with experimental results, and which is positively homogeneous and therefore not differentiable at zero slip gradient. Furthermore, the variational formulation accommodates a non-recoverable energy due to Ohno et al. (Int J Mod Phys B 22:5937–5942, 2008), which is also positively homogeneous, and a function of the accumulated dislocation density. Conditions for the existence and uniqueness of solutions are established for the various examples of defect energy, with or without the presence of hardening or slip resistance.     

Extended self-similar scaling law of multi-scale eddy structure in wall turbulence

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On the correlation between enstrophy and energy dissipation rate in a turbulent wake

All three components of the vorticity fluctuation have been measured simultaneously in a turbulent wake using a new eight-sensor vorticity probe. The vorticity fluctuation spectra agree reasonably well with those from a direct numerical simulation of a turbulent channel flow at high wavenumbers. The similarity between the instantaneous energy dissipation rate ε and the instantaneous enstrophy ω² is examined using spectra and probability density functions. The correlation between ω² and ε is evaluated in some detail. The homogeneous value of ε is strongly correlated with ω². The full value of ε and, more especially its isotropic value, are less well correlated with the enstrophy. Conditional averaging indicates that high enstrophy regions are associated with high energy dissipation rate regions.     

The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 1: polycrystalline plasticity

A general set of flow laws and associated variational formulations are constructed for small-deformation rate-independent problems in strain-gradient plasticity. The framework is based on the thermodynamically consistent theory due to Gurtin and Anand (J Mech Phys Solids 53:1624–1649, 2005), and includes as variables a set of microstresses which have both energetic and dissipative components. The flow law is of associative type. It is expressed as a normality law with respect to a convex but otherwise arbitrary yield function, or equivalently in terms of the corresponding dissipation function. Two cases studied are, first, an extension of the classical Hill-Mises or J ₂ flow law and second, a form written as a linear sum of the magnitudes of the plastic strain and strain gradient. This latter form is motivated by work of Evans and Hutchinson (Acta Mater 57:1675–1688, 2009) and Nix and Gao (J Mech Phys Solids 46:411–425, 1998), who show that it leads to superior correspondence with experimental results, at least for particular classes of problems. The corresponding yield function is obtained by a duality argument. The variational problem is based on the flow rule expressed in terms of the dissipation function, and the problem is formulated as a variational inequality in the displacement, plastic strain, and hardening parameter. Dissipative components of the microstresses, which are indeterminate, are absent from the formulation. Existence and uniqueness of solutions are investigated for the generalized Hill-Mises and linear-sum dissipation functions, and for various combinations of defect energy. The conditions for well-posedness of the problem depend critically on the choice of dissipation function, and on the presence or otherwise of a defect energy in the plastic strain or plastic strain gradient, and of internal-variable hardening.     

Reynolds Number Effects on the Coherent Dynamics of the Turbulent Horseshoe Vortex System

The adverse pressure gradient induced by a surface-mounted obstacle in a turbulent boundary layer causes the approaching flow to separate and form a dynamically rich horseshoe vortex system (HSV) in the junction of the obstacle with the wall. The Reynolds number of the flow (Re) is one of the important parameters that control the rich coherent dynamics of the vortex, which are known to give rise to low-frequency, bimodal fluctuations of the velocity field (Devenport and Simpson, J Fluid Mech 210:23–55, 1990; Paik et al., Phys Fluids 19:045107, 2007). We carry out detached eddy simulations (DES) of the flow past a circular cylinder mounted on a rectangular channel for Re = 2.0 × 10⁴ and 3.9 × 10⁴ (Dargahi, Exp Fluids 8:1–12, 1989) in order to systematically investigate the effect of the Reynolds number on the HSV dynamics. The computed results are compared with each other and with previous experimental and computational results for a related junction flow at a much higher Reynolds number (Re = 1.15 × 10⁵) (Devenport and Simpson, J Fluid Mech 210:23–55, 1990; Paik et al., Phys Fluids 19:045107, 2007). The computed results reveal significant variations with Re in terms of the mean-flow quantities, turbulence statistics, and the coherent dynamics of the turbulent HSV. For Re = 2.0 × 10⁴ the HSV system consists of a large number of necklace-type vortices that are shed periodically at higher frequencies than those observed in the Re = 3.9 × 10⁴ case. For this latter case the number of large-scale vortical structures that comprise the instantaneous HSV system is reduced significantly and the flow dynamics becomes quasi-periodic. For both cases, we show that the instantaneous flowfields are dominated by eruptions of wall-generated vorticity associated with the growth of hairpin vortices that wrap around and disorganize the primary HSV system. The intensity and frequency of these eruptions, however, appears to diminish rapidly with decreasing Re. In the high Re case the HSV system consists of a single, highly energetic, large-scale necklace vortex that is aperiodically disorganized by the growth of the hairpin mode. Regardless of the Re, we find pockets in the junction region within which the histograms of velocity fluctuations are bimodal as has also been observed in several previous experimental studies.     

On the fluid–structure interaction of a splitter plate: vibration modes and Reynolds number effects

Previous work (Eloranta et al. in Exp Fluids 39:841–855, 2005) has shown that flow separation from the trailing edge of a splitter plate in a convergent channel involves a fluid–structure interaction (FSI), which modifies the fundamental instability related to vortex shedding. Under certain conditions, the FSI induces cellular vortex shedding from the trailing edge. This paper reports detailed measurements of the plate vibration mode and studies the effect of the Reynolds number on the FSI. Experimental techniques including laser vibrometer and digital imaging are used to measure the response of the plate and particle image velocimetry is used to measure the flow field in the near wake. Combining data from these techniques, the development of the vibration frequency and mode can be addressed together with the imprint of the vibration mode in the flow. The results show that over most of the Reynolds numbers measured, the plate vibrates in a distinct mode characterized by a spanwise standing wave along the plate trailing edge. The vibration frequency and the spacing between the nodes of the standing wave depend on the Reynolds number. As the Reynolds number is increased, the frequency of the dominant vibration mode does not increase linearly. The plot of the vibration frequency as a function of the Reynolds number shows that the vibration tends to lock to a rather constant frequency over of range of Reynolds numbers. After certain Reynolds number if threshold is exceeded, the frequency jumps to a new level, which also involves a new vibration mode.     

A comparison of PIV measurements of canopy turbulence performed in the field and in a wind tunnel model

Particle image velocimetry (PIV) has been used to compare between turbulence characteristics just within and above a mature corn canopy and those of a model canopy setup in a wind tunnel (WT). The laboratory normalized mean velocity profile is adjusted using variable mesh screens to match the normalized mean shear of the corn field (CF) data. The smallest resolved scale in the field is about 15 times the Kolmogorov length scale (η_CF ≈ 0.4 mm), whereas in the WT it is 5 times η_WT (η_WT ≈ 0.15 mm). In both cases, the mean velocity and turbulence statistics are consistent with those measured using single point sensors. However, the profiles of normalized Reynolds shear stress in the field and the laboratory differ. Turbulent spectral densities calculated from PIV spatial and time series in the field display an inertial range spanning three decades. In the laboratory due to lower Reynolds numbers, the inertial range shrinks to two decades. Quadrant-Hole analysis is applied to Reynolds shear stress, vorticity magnitude and dissipation rates. In quadrants 1–3, the WT and field conditionally sampled stresses show similar trends. However, a conflicting trend is found in the sweep quadrant. The analysis confirms that sweep and ejections dominate the momentum flux and dissipation rate.The content of this paper, entitled “Applying PIV for Measuring Turbulence just within and above a Corn Canopy,” was presented at the 6th International Symposium on Particle Image Velocimetry at Pasadena, CA, USA, September 21–23, 2005.     

The Structure Functions of the Velocity and Temperature Fields from the Perspective of Dimensional Scaling

In this paper, dimensional scaling is used to describe the turbulence structure of the velocity and temperature fields in the inertial range and the far dissipation range as well as the intermediate transition range under locally isotropic conditions at sufficiently large Reynolds numbers. This kind of scaling is expressed in a strictly mathematical manner employing dimensional π -invariants analysis. It is shown that in the case of the asymptotic solutions for either the inertial range or the far dissipation range only one π number occurs that has to be considered as a non-dimensional universal constant. This π number may be determined theoretically or/and empirically. In the case of the transition range two π numbers occur. Consequently, a universal function is established that has to be derived theoretically or/and empirically, too. Here, Batchelor's [7] classical interpolation formula for the turbulence structure of the velocity field and the empirical one of Stolovitzky et al. [59], both may serve as universal functions, are compared with the results provided by numerical solutions of Kolmogorov′s [32] structure equation for the velocity field. It is shown that these interpolation formulae match not only the asymptotic solutions of the inertial range and the far dissipation range, respectively, but also these numerical results in an excellent manner. The former may be considered as necessary condition and the latter as sufficient condition. In the case of the temperature field results of the corresponding universal function are predicted using Yaglom's [63] structure equation. These results also match the corresponding asymptotic solutions of both the inertial range and the far dissipation range. However, in contrast to the case of the velocity field, the predicted universal function for the temperature field may notably overshoot its asymptotic solution for the inertial range. This overshooting occurs in the transition range and may be considered as an analogue to the so-called Hill ‘bump’ that usually occurs in the high-wave number portion of the temperature spectrum.     

Evaluation of drag correction factor for spheres settling in associative polymers

Drag correction factors are calculated for the creeping motion of spheres descending in various associative polymers of different concentration with various sphere-container ratios and Weissenberg numbers. The simple-shear rheology and linear viscoelasticity of these polymeric fluids have been previously presented and modeled with the BMP (Bautista–Manero–Puig) equation of state (Mendoza-Fuentes et al., Phys Fluids 21:033104, 2009). The drag on the sphere is initially kept nearly constant for small Weissenberg numbers, We < 0.1. As the Weissenberg number increases, We < 0.1, a reduction in drag is found. Experimental results show the presence of a critical Weissenberg number at which a drag reduction occurs. The reduction in the drag correction factor is associated to the onset of extension-thinning, which coincides with the formation of a negative wake. No increase in the drag correction factor was observed, due to the simultaneous opposing effects of extension-thickening and shear-thinning viscosity. The shape of the drag correction factor curve may be predicted considering the extensional properties of the solutions, as suggested elsewhere (Chen and Rothstein, J Non-Newton Fluid Mech 116:205–215, 2004).     

On dislocation models of grain boundaries

Dislocation models of grain boundaries was suggested by Bragg (Proc Phys Soc 52:54–55, 1940) and Burgers (Proc Phys Soc 52:23–33, 1940). The first quantitative study of these models was given by Read and Shockley (Phys Rev 78(3):275–289, 1950). They obtained a formula for the dependence of the grain boundary energy on the misorientation of the neighboring grains, which became a cornerstone of the grain boundary theory. The Read–Shockley formula was based on a proposition that the grain boundary energy is the sum of energies of the two sets of dislocations that come from the two neighboring grains. This proposition was proved under an assumption on a quite special geometry of the slip planes. This paper aims to show that the assumption is not necessary and the proposition holds for arbitrary geometry of slip planes. Another goal of this paper is to provide all basic formulas of the theory: though the dislocation model of grain boundaries is considered in all treatises on dislocation theory, a complete analysis, including the relations for lattice rotations and displacements, has not been given. This analysis shows, in particular, that continuum theory does not yield the proper relations for the lattice misorientations, and these relations must be introduced by an independent ansatz.     

Finite-Amplitude Equilibrium Solutions for Plane Poiseuille–Couette Flow

Two-dimensional nonlinear equilibrium solutions for the plane Poiseuille–Couette flow are computed by directly solving the full Navier–Stokes equations as a nonlinear eigenvalue problem. The equations are solved using the two-point fourth-order compact scheme and the Newton–Raphson iteration technique. The linear eigenvalue computations show that the combined Poiseuille–Couette flow is stable at all Reynolds numbers when the Couette velocity component σ₂ exceeds 0.34552. Starting with the neutral solution for the plane Poiseuille flow, the nonlinear neutral surfaces for the combined Poiseuille–Couette flow were mapped out by gradually increasing the velocity component σ₂. It is found that, for small σ₂, the neutral surfaces stay in the same family as that for the plane Poiseuille flow, and the nonlinear critical Reynolds number gradually increases with increasing σ₂. When the Couette velocity component is increased further, the neutral curve deviates from that for the Poiseuille flow with an appearance of a new loop at low wave numbers and at very low energy. By gradually increasing the σ₂ values at a constant Reynolds number, the nonlinear critical Reynolds numbers were determined as a function of σ₂. The results show that the nonlinear neutral curve is similar in shape to a linear case. The critical Reynolds number increases slowly up to σ₂∼ 0.2 and remains constant until σ₂∼ 0.58. Beyond σ₂ > 0.59, the critical Reynolds number increases sharply. From the computed results it is concluded that two-dimensional nonlinear equilibrium solutions do not exist beyond a critical σ₂ value of about 0.59. Received: 26 November 1996 and accepted 12 May 1997     

Heat convection from a cylinder performing steady rotation or rotary oscillation – Part I: Steady rotation

In this paper, the problem of laminar, two dimensional heat convection from a circular cylinder performing steady rotation is investigated. The cylinder is␣placed with its axis horizontal in a quiescent fluid of infinite extent. Because of viscous dissipation, the flow process is confined to the region adjacent to the cylinder and is mainly driven by shear and buoyancy forces. The study is based on the solution of the full conservation equations of mass, momentum and energy for Rayleigh numbers up to 10⁴ and Reynolds numbers (based on surface velocity) up to 400 while Prandtl number ranges between 0.7 and 7.0. For the range of parameters considered, the study revealed that the rate of heat transfer increases with the increase of Rayleigh number and decreases with the increase of speed of rotation. The increase of Prandtl number resulted in an appreciable increase in the average Nusselt number only at low Reynolds numbers. The effect of Prandtl number at high Reynolds number is negligibly small. The resulting flow field in all cases is steady with no vortex shedding. The streamlines and isotherms are plotted for a number of cases to show the details of the velocity and thermal fields. Received on 15 December 1997     

Comparison of turbulent channel and pipe flows with varying Reynolds number

Single normal hot-wire measurements of the streamwise component of velocity were taken in fully developed turbulent channel and pipe flows for matched friction Reynolds numbers ranging from 1,000 ≤ Re _τ ≤ 3,000. A total of 27 velocity profile measurements were taken with a systematic variation in the inner-scaled hot-wire sensor length l ⁺ and the hot-wire length-to-diameter ratio (l/d). It was observed that for constant l ⁺ = 22 and l/d >~200l/d \gtrsim 200, the near-wall peak in turbulence intensity rises with Reynolds number in both channels and pipes. This is in contrast to Hultmark et al. in J Fluid Mech 649:103–113, (2010), who report no growth in the near-wall peak turbulence intensity for pipe flow with l ⁺ = 20. Further, it was found that channel and pipe flows have very similar streamwise velocity statistics and energy spectra over this range of Reynolds numbers, with the only difference observed in the outer region of the mean velocity profile. Measurements where l ⁺ and l/d were systematically varied reveal that l ⁺ effects are akin to spatial filtering and that increasing sensor size will lead to attenuation of an increasingly large range of small scales. In contrast, when l/d was insufficient, the measured energy is attenuated over a very broad range of scales. These findings are in agreement with similar studies in boundary layer flows and highlight the need to carefully consider sensor and anemometry parameters when comparing flows across different geometries and when drawing conclusions regarding the Reynolds number dependency of measured turbulence statistics. With an emphasis on accuracy, measurement resolution and wall proximity, these measurements are taken at comparable Reynolds numbers to currently available DNS data sets of turbulent channel/pipe flows and are intended to serve as a database for comparison between physical and numerical experiments.     

Very high Reynolds number boundary layers over 3D sparse roughness and obstacles: the mean flow

Hot-wire and oil-film interferometry measurements are taken for 3D rough wall boundary layers at very high Reynolds numbers (61,000 < Re θ < 120,000) with low blockage ratios, 10 < δ/H < 135, and high roughness, 100 < H ⁺ < 4,900. The results cover flows over both rough walls and over obstacles and are compared with and provide extension to recent lower Reynolds number results. The validity of the Townsend ‘wall similarity hypothesis’ in relation to consistently increasing 3D roughness is interrogated. In agreement with recent work, Schultz and Flack (J Fluid Mech 580:381–405, 2007) and Castro (J Fluid Mech 585:469–485, 2007) found that, for relatively low roughness, Townsend’s hypothesis holds for the mean velocity field. With increasing roughness, the equilibrium layer diminishes and gradually vanishes. The viscous component of the wall shear stress decreases, while the turbulent component increases as the roughness effects extend across the boundary layer.     

Experimental assessment of scale effects affecting two-phase flow properties in hydraulic jumps

A hydraulic jump is the rapid transition from a supercritical to subcritical free-surface flow. It is characterised by strong turbulence and air bubble entrainment. New air–water flow properties were measured in hydraulic jumps with partially developed inflow conditions. The data set together with the earlier data of Chanson (Air bubble entrainment in hydraulic jumps. Similitude and scale effects, 119 p, 2006) yielded similar experiments conducted with identical inflow Froude numbers Fr ₁ = 5 and 8.5, but Reynolds numbers between 24,000 and 98,000. The comparative results showed some drastic scale effects in the smaller hydraulic jumps in terms of void fraction, bubble count rate and bubble chord time distributions. The present comparative analysis demonstrated quantitatively that dynamic similarity of two-phase flows in hydraulic jumps cannot be achieved with a Froude similitude. In experimental facilities with Reynolds numbers up to 10⁵, some viscous scale effects were observed in terms of the rate of entrained air and air–water interfacial area.     

Spatial structures and scaling in the Convective Boundary Layer

We performed an investigation on spatial features of the Convective Boundary Layer (CBL) of the atmosphere, which was simulated in a laboratory model and analyzed by means of image analysis techniques. This flow is dominated by large, anisotropic vortical structures, whose spatial organization affects the scalar transport and therefore the fluxes across the boundary layer. With the aim of investigating the spatial structure and scaling in the Convective Boundary Layer, two-dimensional velocity fields were measured, on a vertical plane, by means of a pyramidal Lucas–Kanade algorithm. The coherent structures characterizing the turbulent convection were educed by analyzing the Finite-Time Lyapunov Exponent fields, which also revealed interesting phenomenological features linked to the mixing processes occurring in the Convective Boundary Layer. Both velocity and vorticity fields were analyzed in a scale-invariance framework. Data analysis showed that normalized probability distribution functions for velocity differences are dependent on the scale and tend to become Gaussian for large separations. Extended Self Similarity holds true for velocity structure functions computed within the mixing layer, and their scaling exponents are interpreted well in the phenomenological framework of the Hierarchical Structure Model. Specifically, β parameter, which is related to the similarity between weak and strong vortices, reveals a higher degree of intermittency for the vertical velocity component with respect to the horizontal one. On the other hand, the analysis of circulation structure functions shows that scaling exponents are fairly constant in the lowest part of the mixed layer, and their values are in agreement with those reported in Benzi et al. (Phys Rev E 55:3739–3742, 1997) for shear turbulence. Moreover, the relationship between circulation and velocity scaling exponents is analyzed, and it is found to be linear in the bottom part of the mixing layer. The investigation of the CBL spatial features, which has seldom been studied experimentally, has important implications for the comprehension of the mixing dynamics, as well as in turbulence closure models.     

Polynomial Filtration Laws for Low Reynolds Number Flows Through Porous Media

In this study, we use the method of homogenization to develop a filtration law in porous media that includes the effects of inertia at finite Reynolds numbers. The result is much different than the empirically observed quadratic Forchheimer equation. First, the correction to Darcy’s law is initially cubic (not quadratic) for isotropic media. This is consistent with several other authors (Mei and Auriault, J Fluid Mech 222:647–663, 1991; Wodié and Levy, CR Acad Sci Paris t.312:157–161, 1991; Couland et al. J Fluid Mech 190:393–407, 1988; Rojas and Koplik, Phys Rev 58:4776–4782, 1988) who have solved the Navier–Stokes equations analytically and numerically. Second, the resulting filtration model is an infinite series polynomial in velocity, instead of a single corrective term to Darcy’s law. Although the model is only valid up to the local Reynolds number, at the most, of order 1, the findings are important from a fundamental perspective because it shows that the often-used quadratic Forchheimer equation is not a universal law for laminar flow, but rather an empirical one that is useful in a limited range of velocities. Moreover, as stated by Mei and Auriault (J Fluid Mech 222:647–663, 1991) and Barree and Conway (SPE Annual technical conference and exhibition, 2004), even if the quadratic model were valid at moderate Reynolds numbers in the laminar flow regime, then the permeability extrapolated on a Forchheimer plot would not be the intrinsic Darcy permeability. A major contribution of this study is that the coefficients of the polynomial law can be derived a priori, by solving sequential Stokes problems. In each case, the solution to the Stokes problem is used to calculate a coefficient in the polynomial, and the velocity field is an input of the forcing function, F, to subsequent problems. While numerical solutions must be utilized to compute each coefficient in the polynomial, these problems are much simpler and robust than solving the full Navier–Stokes equations.     

Use of particle image velocimetry to study heterogeneous drag reduction

Large polymer filaments can form when drag reducing polymers are injected through wall slots. The presence of these structures enhances the performance of the drag reducing function by mechanisms which are not understood. This paper shows how particle image velocimetry (PIV) techniques can be used to study changes in the configuration of the injected polymer and in the structure of the velocity field with increasing drag reduction. The filaments are found to behave as solid bodies which break up in high shear regions close to a boundary. The breakup process provides an explanation of why the filaments are not observed close to a wall and offers the possibility of providing a heterogeneous distribution of small aggregates of polymers which could be more effective than uniformly distributed molecules as suggested by Hoyer and Gyr (J Non-Newton Fluid Mech 65:221–240, 1996; J Fluids Eng 120:818–823, 1998), Dunlop and Cox (Phys Fluids 20:203–213, 1977) and Vlachogiannis et al. (Phys Fluid 15:3786–3794, 2004). PIV measurements show dramatic qualitative changes in the velocity patterns at maximum drag reduction.