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Maurizio Quadrio - One of the best experts on this subject based on the ideXlab platform.
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reynolds dependence of turbulent skin friction Drag Reduction induced by spanwise forcing
arXiv: Fluid Dynamics, 2015Co-Authors: Davide Gatti, Maurizio QuadrioAbstract: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 {\em et al. J. Fluid Mech.}, vol. 627, 2009, pp. 161--178). The study builds upon an extensive Drag-Reduction database created with Direct Numerical Simulation of a turbulent channel flow for two, 5-fold separated values of $Re$, namely $Re_\tau=200$ and $Re_\tau=1000$. 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 for example riblets, 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.
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what happens to turbulent skin friction Drag Reduction at high re
arXiv: Fluid Dynamics, 2012Co-Authors: Davide Gatti, Maurizio QuadrioAbstract:We address one of the capital problems in the field of turbulent skin-friction Drag Reduction, i.e. the performance of the known techniques at high values of the Reynolds number $Re$. We limit ourselves to considering open-loop techniques based on spanwise forcing (oscillating wall, streamwise-travelling waves), and explore via Direct Numerical Simulations (DNS) how quickly the Drag Reduction and net energy savings decrease when the friction Reynolds number is increased. We suggest an unexpected and interesting scenario where the performance of the Drag-Reduction technique degrade with $Re$ with a markedly different rate depending on the parameters. In particular, the known optimal region turns out to be such at low-$Re$ only, since there Drag Reduction degrades quite fast with $Re$, in line with available results. However, other regions are much less sensitive to $Re$, or insensitive at all. If one considers that the energy required to create the forcing presents a slightly favorable trend with $Re$, the possibility exists of net energy saving at very high Reynolds numbers. This interesting scenario remains speculative in nature, owing to the spatial truncation implied by the limited domain size. However, a few full-scale DNS for the traveling waves at $Re_\tau=400$ have been carried out, and their results fully support the suggested scenario, which, though appealing, would force us to rethink our current understanding of how these Drag Reduction techniques work and behave at high $Re$.
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Drag Reduction in turbulent boundary layers by in plane wall motion
Philosophical Transactions of the Royal Society A, 2011Co-Authors: Maurizio QuadrioAbstract:Drag-Reduction techniques capable of reducing the level of turbulent friction through wall-parallel movement of the wall are described, with special emphasis placed on spanwise movement. The discussion is confined to active open-loop control strategies, although feedback control is briefly mentioned with regard to peculiarities of spanwise sensing and/or actuation. Theoretical considerations are first given to explain why spanwise motion is expected to be particularly effective in skin-friction Drag Reduction. A review of the spanwise oscillating-wall technique is given next, with discussion of recent results and prospects. Last, waves of spanwise velocity are addressed, either spanwise- or streamwise-travelling. The latter include the oscillating wall as a special case. The generalized Stokes layer--i.e. the laminar, transverse oscillating boundary layer that develops under the action of the streamwise-travelling waves--is described, and its importance in determining turbulent Drag Reduction discussed. Finally, open issues like energetic efficiency and its dependence on Reynolds number are addressed.
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wall oscillation conditions for Drag Reduction in turbulent channel flow
International Journal of Heat and Fluid Flow, 2008Co-Authors: Pierre Ricco, Maurizio QuadrioAbstract:Abstract The Drag Reduction properties of a turbulent channel flow modified by spanwise sinusoidal oscillations of the walls are investigated by direct numerical simulations. The work is based on the linear relation between the Drag Reduction and the parameter S, function of the maximum wall velocity and the period of the oscillation. This quantity, first determined by Choi et al. [Choi, J.-I, Xu, C.-X., Sung, H. J., 2002. Drag Reduction by spanwise wall oscillation in wall-bounded turbulent flows. AIAA J. 40 (5), 842–850] and later studied by Quadrio and Ricco [Quadrio, M., Ricco, P., 2004. Critical assessment of turbulent Drag Reduction through spanwise wall oscillations. J. Fluid Mech. 521, 251–271], has been found through physical arguments pertaining to the action of the oscillating Stokes layer on the near-wall turbulence dynamics. The predictive potential of the scaling parameter is exploited to gain insight into the Drag-reducing effects of the oscillating-wall technique. The period of oscillation which guarantees the maximum Drag Reduction for a given maximum wall displacement is studied for the first time. The issue of the minimum intensity of wall forcing required to produce a non-zero Drag Reduction effect and the dependence of the Drag Reduction on the Reynolds number are also addressed. The Drag Reduction data available in the literature are compared with the prediction given by the scaling parameter, thus attaining a comprehensive view of the state of the art.
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wall oscillation conditions for Drag Reduction in turbulent channel flow
arXiv: Fluid Dynamics, 2008Co-Authors: Pierre Ricco, Maurizio QuadrioAbstract:The Drag Reduction properties of a turbulent channel flow modified by spanwise sinusoidal oscillations of the walls are investigated by direct numerical simulations. The work is based on the linear relation between the Drag Reduction and the parameter $S$, function of the maximum wall velocity and the period of the oscillation. This quantity has been found through physical arguments pertaining to the action of the oscillating Stokes layer on the near-wall turbulence dynamics. The predictive potential of the scaling parameter is exploited to gain insight into the Drag-reducing effects of the oscillating wall technique. The period of oscillation which guarantees the maximum Drag Reduction for a given maximum wall displacement is studied for the first time. The issue of the minimum intensity of wall forcing required to produce a non-zero Drag Reduction effect and the dependence of the Drag Reduction on the Reynolds number are also addressed. The Drag Reduction data available in the literature are compared with the prediction given by the scaling parameter, thus attaining a comprehensive view of the state of the art.
Detlef Lohse - One of the best experts on this subject based on the ideXlab platform.
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Drag Reduction in boiling taylor couette turbulence
Journal of Fluid Mechanics, 2019Co-Authors: Rodrigo Ezeta, Chao Sun, Dennis Bakhuis, Sander G Huisman, Detlef LohseAbstract:We create a highly controlled laboratory environment – accessible to both global and local monitoring – to analyse turbulent boiling flows and in particular their shear stress in a statistically stationary state. By precisely monitoring the Drag of strongly turbulent Taylor–Couette flow (the flow in between two coaxially rotating cylinders, Reynolds number ) during its transition from non-boiling to boiling, we show that the intuitive expectation, namely that a few volume per cent of vapour bubbles would correspondingly change the global Drag by a few per cent, is wrong. Rather, we find that for these conditions a dramatic global Drag Reduction of up to 45 % occurs. We connect this global result to our local observations, showing that for major Drag Reduction the vapour bubble deformability is crucial, corresponding to Weber numbers larger than one. We compare our findings with those for turbulent flows with gas bubbles, which obey very different physics from those of vapour bubbles. Nonetheless, we find remarkable similarities and explain these.
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the influence of wall roughness on bubble Drag Reduction in taylor couette turbulence
arXiv: Fluid Dynamics, 2018Co-Authors: Ruben A Verschoof, Chao Sun, Dennis Bakhuis, Sander G Huisman, Pim A Bullee, Detlef LohseAbstract:We experimentally study the influence of wall roughness on bubble Drag Reduction in turbulent Taylor-Couette flow, i.e.\ the flow between two concentric, independently rotating cylinders. We measure the Drag in the system for the cases with and without air, and add roughness by installing transverse ribs on either one or both of the cylinders. For the smooth wall case (no ribs) and the case of ribs on the inner cylinder only, we observe strong Drag Reduction up to $DR=33\%$ and $DR=23\%$, respectively, for a void fraction of $\alpha=6\%$. However, with ribs mounted on both cylinders or on the outer cylinder only, the Drag Reduction is weak, less than $DR=11\%$, and thus quite close to the trivial effect of reduced effective density. Flow visualizations show that stable turbulent Taylor vortices --- large scale vortical structures --- are induced in these two cases, i.e. the cases with ribs on the outer cylinder. These strong secondary flows move the bubbles away from the boundary layer, making the bubbles less effective than what had previously been observed for the smooth-wall case. Measurements with counter-rotating smooth cylinders, a regime in which pronounced Taylor rolls are also induced, confirm that it is really the Taylor vortices that weaken the bubble Drag Reduction mechanism. Our findings show that, although bubble Drag Reduction can indeed be effective for smooth walls, its effect can be spoiled by e.g.\ biofouling and omnipresent wall roughness, as the roughness can induce strong secondary flows.
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bubble Drag Reduction requires large bubbles
Bulletin of the American Physical Society, 2016Co-Authors: Ruben A Verschoof, Roeland C A Van Der Veen, Chao Sun, Detlef LohseAbstract:In the maritime industry, the injection of air bubbles into the turbulent boundary layer under the ship hull is seen as one of the most promising techniques to reduce the overall fuel consumption. However, the exact mechanism behind bubble Drag Reduction is unknown. Here we show that bubble Drag Reduction in turbulent flow dramatically depends on the bubble size. By adding minute concentrations (6 ppm) of the surfactant Triton X-100 into otherwise completely unchanged strongly turbulent Taylor-Couette flow containing bubbles, we dramatically reduce the Drag Reduction from more than 40% to about 4%, corresponding to the trivial effect of the bubbles on the density and viscosity of the liquid. The reason for this striking behavior is that the addition of surfactants prevents bubble coalescence, leading to much smaller bubbles. Our result demonstrates that bubble deformability is crucial for bubble Drag Reduction in turbulent flow and opens the door for an optimization of the process.
Itamar Procaccia - One of the best experts on this subject based on the ideXlab platform.
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colloquium theory of Drag Reduction by polymers in wall bounded turbulence
Reviews of Modern Physics, 2008Co-Authors: Itamar Procaccia, Victor S Lvov, R BenziAbstract:The flow of fluids in channels, pipes, or ducts, as in any other wall-bounded flow (like water along the hulls of ships or air on airplanes) is hindered by a Drag, which increases manyfold when the fluid flow turns from laminar to turbulent. A major technological problem is how to reduce this Drag in order to minimize the expense of transporting fluids like oil in pipelines, or to move ships in the ocean. It was discovered that minute concentrations of polymers can reduce the Drag in turbulent flows by up to 80%. While experimental knowledge had accumulated over the years, the fundamental theory of Drag Reduction by polymers remained elusive for a long time, with arguments raging whether this is a "skin" or a "bulk" effect. In this Colloquium the phenomenology of Drag Reduction by polymers is summarized, stressing both its universal and nonuniversal aspects, and a recent theory is reviewed that provides a quantitative explanation of all the known phenomenology. Both flexible and rodlike polymers are treated, explaining the existence of universal properties like the maximum Drag Reduction asymptote, as well as nonuniversal crossover phenomena that depend on the Reynolds number, on the nature of the polymer and on its concentration. Finally other agents for Drag Reduction are discussed with a stress on the important example of bubbles.
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Drag Reduction by polymers in wall bounded turbulence
Physical Review Letters, 2004Co-Authors: Victor S Lvov, Itamar Procaccia, Anna Pomyalov, Vasil TiberkevichAbstract:We elucidate the mechanism of Drag Reduction by polymers in turbulent wall-bounded flows: while momentum is produced at a fixed rate by the forcing, polymer stretching results in the suppression of momentum flux to the wall. On the basis of the equations of fluid mechanics we develop the phenomenology of the ``maximum Drag Reduction asymptote'' which is the maximum Drag Reduction attained by polymers. Based on Newtonian information only we demonstrate the existence of Drag Reduction, and with one experimental parameter we reach agreement with the experimental measurements.
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theory of concentration dependence in Drag Reduction by polymers and of the maximum Drag Reduction asymptote
Physical Review Letters, 2004Co-Authors: R Benzi, Emily S C Ching, Nizan Horesh, Itamar ProcacciaAbstract:A simple model of the effect of polymer concentration on the amount of Drag Reduction in turbulence is presented, simulated, and analyzed. The qualitative phase diagram of Drag coefficient versus Reynolds number (Re) is recaptured in this model, including the theoretically elusive onset of Drag Reduction and the maximum Drag Reduction (MDR) asymptote. The Re-dependent Drag and the MDR are analytically explained, and the dependence of the amount of Drag on material parameters is rationalized.
Marc Perlin - One of the best experts on this subject based on the ideXlab platform.
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skin friction Drag Reduction in the turbulent regime using random textured hydrophobic surfaces
Physics of Fluids, 2014Co-Authors: Rahul Anil Bidkar, Ambarish Jayant Kulkarni, Vaibhav Bahadur, Luc Stephane Leblanc, Steven L Ceccio, Marc PerlinAbstract:Technologies for reducing hydrodynamic skin-friction Drag have a huge potential for energy-savings in applications ranging from propulsion of marine vessels to transporting liquids through pipes. The majority of previous experimental studies using hydrophobic surfaces have successfully shown skin-friction Drag Reduction in the laminar and transitional flow regimes (typically Reynolds numbers less than ≃106 for external flows). However, this hydrophobicity induced Drag Reduction is known to diminish with increasing Reynolds numbers in experiments involving wall bounded turbulent flows. Using random-textured hydrophobic surfaces (fabricated using large-length scalable thermal spray processes) on a flat plate geometry, we present water-tunnel test data with Reynolds numbers ranging from 106 to 9 × 106 that show sustained skin-friction Drag Reduction of 20%–30% in such turbulent flow regimes. Furthermore, we provide evidence that apart from the formation of a Cassie state and hydrophobicity, we also need a low surface roughness and an enhanced ability of the textured surface to retain trapped air, for sustained Drag Reduction in turbulent flow regimes. Specifically, for the hydrophobic test surfaces of the present and previous studies, we show that Drag Reduction seen at lower Reynolds numbers diminishes with increasing Reynolds number when the surface roughness of the underlying texture becomes comparable to the viscous sublayer thickness. Conversely, test data show that textures with surface roughness significantly smaller than the viscous sublayer thickness and textures with high porosity show sustained Drag Reduction in the turbulent flow regime. The present experiments represent a significant technological advancement and one of the very few demonstrations of skin-friction Reduction in the turbulent regime using random-textured hydrophobic surfaces in an external flow configuration. The scalability of the fabrication method, the passive nature of this surface technology, and the obtained results in the turbulent regime make such hydrophobic surfaces a potentially attractive option for hydrodynamic skin-friction Drag Reduction.
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on the scaling of air layer Drag Reduction
Journal of Fluid Mechanics, 2013Co-Authors: Brian R Elbing, Marc Perlin, Simo A Makiharju, Andrew Wiggins, David R. Dowling, Steven L CeccioAbstract:Air-induced Drag Reduction was investigated on a 12.9 m long flat plate test model at a free stream speed of $6. 3~\mathrm{m} ~{\mathrm{s} }^{- 1} $ . Measurements of the local skin friction, phase velocity profiles (liquid and gas) and void fraction profiles were acquired at downstream distances to 11.5 m, which yielded downstream-distance-based Reynolds numbers above 80 million. Air was injected within the boundary layer behind a 13 mm backward facing step (BFS) while the incoming boundary layer was perturbed with vortex generators in various configurations immediately upstream of the BFS. Measurements confirmed that air layer Drag Reduction (ALDR) is sensitive to upstream disturbances, but a clean boundary layer separation line (i.e. the BFS) reduces such sensitivity. Empirical scaling of the experimental data was investigated for: (a) the critical air flux required to establish ALDR; (b) void fraction profiles; and (c) the interfacial velocity profiles. A scaling of the critical air flux for ALDR was developed from balancing shear-induced lift forces and buoyancy forces on a single bubble within a shear flow. The resulting scaling successfully collapses ALDR results from the current and past studies over a range of flow conditions and test model configurations. The interfacial velocity and void fraction profiles were acquired and scaled within the bubble Drag Reduction (BDR), ALDR and transitional ALDR regimes. The BDR interfacial velocity profile revealed that there was slip between phases. The ALDR results showed that the air layer thickness was nominally three-quarters of the total volumetric flux (per unit span) of air injected divided by the free stream speed. Furthermore, the air layer had an average void fraction of 0.75 and a velocity of approximately 0.2 times the free stream speed. Beyond the air layer was a bubbly mixture that scaled in a similar fashion to the BDR results. Transitional ALDR results indicate that this regime was comprised of intermittent generation and subsequent fragmentation of an air layer, with the resulting Drag Reduction determined by the fraction of time that an air layer was present.
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On the energy economics of air lubrication Drag Reduction
International Journal of Naval Architecture and Ocean Engineering, 2012Co-Authors: Simo A Makiharju, Marc Perlin, Steven L CeccioAbstract:Air lubrication techniques for frictional Drag Reduction on ships have been proposed by numerous researchers since the 19th century. However, these techniques have not been widely adopted as questions persist about their Drag Reduction performance beyond the laboratory, as well as energy and economic cost-benefit. This paper draws on data from the literature to consider the suitability of air lubrication for large ocean going and U.S. Great Lakes ships, by establishing the basic energy economic calculations and presenting results for a hypothetical air lubricated ship. All the assumptions made in the course of the analysis are clearly stated so that they can be refined when considering application of air lubrication to a specific ship. The analysis suggests that, if successfully implemented, both air layer and partial cavity Drag Reduction could lead to net energy savings of 10 to 20%, with corresponding Reductions in emissions.
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high reynolds number turbulent boundary layer friction Drag Reduction from wall injected polymer solutions
Journal of Fluid Mechanics, 2009Co-Authors: Eric S Winkel, Marc Perlin, David R. Dowling, Ghanem F Oweis, Siva A Vanapalli, Michael J Solomon, Steven L CeccioAbstract:A set of controlled high-Reynolds-number experiments has been conducted at the William B. Morgan Large Cavitation Channel (LCC) in Memphis, Tennessee to investigate the friction Drag Reduction achieved by injecting aqueous poly(ethylene oxide) (PEO) solutions at three different mean molecular weights into the near-zero-pressure-gradient turbulent boundary layer that forms on a smooth flat test surface having a length of nearly 11m. The test model spanned the 3.05m width of the LCC test section and had an overall length of 12.9m. Skin-friction Drag was measured with six floating-plate force balances at downstream-distance-based Reynolds numbers as high as 220 million and free stream speeds up to 20ms −1 . For a given polymer type, the level of Drag Reduction was measured for a range of free stream speeds, polymer injection rates and concentrations of the injected solution. Polymer concentration fields in the near-wall region (0 y + 3 ) were examined at three locations downstream of the injector using near-wall planar laser-induced-fluorescence imaging. The development and extent of Drag Reduction and polymer mixing are compared to previously reported results using the traditional K -factor scaling. Unlike smaller scale and lower speed experiments, speed dependence is observed in the K -scaled results for the higher molecular weight polymers and it is postulated that this dependence is caused by molecular aggregation and/or flow-induced polymer degradation (chain scission). The evolution of near-wall polymer concentration is divided into three regimes: (i) the development region near the injector where Drag Reduction increases with downstream distance and the polymer is highly inhomogeneous forming filaments near the wall, (ii) the transitional mixing region where Drag Reduction starts to decrease as the polymer mixes across the boundary layer and where filaments are less pronounced and (iii) the final region where the polymer mixing and dilution is set by the rate of boundary layer growth. Unlike pipe-flow friction-Drag Reduction, the asymptotic maximum Drag Reduction (MDR) either was not reached or did not persist in these experiments. Instead, the nearest approach to MDR was transitory and occurred between the development and transitional regions. The length of the development region was observed to increase monotonically with increasing polymer molecular weight, injection rate, concentration and decreasing free stream speed. And finally, the near-wall polymer concentration is correlated to the measured Drag Reduction for the three polymer molecular weights in the form of a proposed empirical Drag-Reduction curve.
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bubble induced skin friction Drag Reduction and the abrupt transition to air layer Drag Reduction
Journal of Fluid Mechanics, 2008Co-Authors: Brian R Elbing, Eric S Winkel, Steven L Ceccio, David R. Dowling, Marc PerlinAbstract:To investigate the phenomena of skin-friction Drag Reduction in a turbulent boundary layer (TBL) at large scales and high Reynolds numbers, a set of experiments has been conducted at the US Navy's William B. Morgan Large Cavitation Channel (LCC). Drag Reduction was achieved by injecting gas (air) from a line source through the wall of a nearly zero-pressure-gradient TBL that formed on a flat-plate test model that was either hydraulically smooth or fully rough. Two distinct Drag-Reduction phenomena were investigated; bubble Drag Reduction (BDR) and air-layer Drag Reduction (ALDR). The streamwise distribution of skin-friction Drag Reduction was monitored with six skin-friction balances at downstream-distance-based Reynolds numbers to 220 million and at test speeds to 20.0ms −1 . Near-wall bulk void fraction was measured at twelve streamwise locations with impedance probes, and near-wall (0 Y Results from the BDR experiments indicate that: significant Drag Reduction (>25%) is limited to the first few metres downstream of injection; marginal improvement was possible with a porous-plate versus an open-slot injector design; BDR has negligible sensitivity to surface tension; bubble size is independent of surface tension downstream of injection; BDR is insensitive to boundary-layer thickness at the injection location; and no synergetic effect is observed with compound injection. Using these data, previous BDR scaling methods are investigated, but data collapse is observed only with the ‘initial zone’ scaling, which provides little information on downstream persistence of BDR. ALDR was investigated with a series of experiments that included a slow increase in the volumetric flux of air injected at free-stream speeds to 15.3ms −1 . These results indicated that there are three distinct regions associated with Drag Reduction with air injection: Region I, BDR; Region II, transition between BDR and ALDR; and Region III, ALDR. In addition, once ALDR was established: friction Drag Reduction in excess of 80% was observed over the entire smooth model for speeds to 15.3ms −1 ; the critical volumetric flux of air required to achieve ALDR was observed to be approximately proportional to the square of the free-stream speed; slightly higher injection rates were required for ALDR if the surface tension was decreased; stable air layers were formed at free-stream speeds to 12.5ms −1 with the surface fully roughened (though approximately 50% greater volumetric air flux was required); and ALDR was sensitive to the inflow conditions. The sensitivity to the inflow conditions can be mitigated by employing a small faired step (10mm height in the experiment) that helps to create a fixed separation line.
R Benzi - One of the best experts on this subject based on the ideXlab platform.
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colloquium theory of Drag Reduction by polymers in wall bounded turbulence
Reviews of Modern Physics, 2008Co-Authors: Itamar Procaccia, Victor S Lvov, R BenziAbstract:The flow of fluids in channels, pipes, or ducts, as in any other wall-bounded flow (like water along the hulls of ships or air on airplanes) is hindered by a Drag, which increases manyfold when the fluid flow turns from laminar to turbulent. A major technological problem is how to reduce this Drag in order to minimize the expense of transporting fluids like oil in pipelines, or to move ships in the ocean. It was discovered that minute concentrations of polymers can reduce the Drag in turbulent flows by up to 80%. While experimental knowledge had accumulated over the years, the fundamental theory of Drag Reduction by polymers remained elusive for a long time, with arguments raging whether this is a "skin" or a "bulk" effect. In this Colloquium the phenomenology of Drag Reduction by polymers is summarized, stressing both its universal and nonuniversal aspects, and a recent theory is reviewed that provides a quantitative explanation of all the known phenomenology. Both flexible and rodlike polymers are treated, explaining the existence of universal properties like the maximum Drag Reduction asymptote, as well as nonuniversal crossover phenomena that depend on the Reynolds number, on the nature of the polymer and on its concentration. Finally other agents for Drag Reduction are discussed with a stress on the important example of bubbles.
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theory of concentration dependence in Drag Reduction by polymers and of the maximum Drag Reduction asymptote
Physical Review Letters, 2004Co-Authors: R Benzi, Emily S C Ching, Nizan Horesh, Itamar ProcacciaAbstract:A simple model of the effect of polymer concentration on the amount of Drag Reduction in turbulence is presented, simulated, and analyzed. The qualitative phase diagram of Drag coefficient versus Reynolds number (Re) is recaptured in this model, including the theoretically elusive onset of Drag Reduction and the maximum Drag Reduction (MDR) asymptote. The Re-dependent Drag and the MDR are analytically explained, and the dependence of the amount of Drag on material parameters is rationalized.