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1 Metadata of the chapter that will be visualized in SpringerLink Book Title Series Title Chapter Title Copyright Year 2016 Copyright HolderName Progress in Wall Turbulence On the Extension of Polymer Molecules in Turbulent Viscoelastic Flows: Statistical and Tens Investigation Springer International Publishing Switzerland Cresponding Auth Family Name Pereira Particle Given Name Prefix Suffix Anselmo Soeiro Division Labatoire de Mécanique de Lille (LML), CNRS, UMR 8107 Organization Address Auth Family Name Martins Particle Given Name Prefix Suffix École Polytechnique Universitaire de Lille, Université Lille Nd de France F59655, Villeneuve D ascq, France anselmo.pereira@polytech-lille.fr Ramon Silva Division Labatoire de Mécanique de Lille (LML), CNRS, UMR 8107 Organization Address Auth Family Name Mompean Particle Given Name Prefix Suffix École Polytechnique Universitaire de Lille, Université Lille Nd de France F59655, Villeneuve D ascq, France ramon.martins@polytech-lille.fr Gilmar Division Labatoire de Mécanique de Lille (LML), CNRS, UMR 8107 Organization Address Auth Family Name Thais Particle Given Name Prefix Suffix École Polytechnique Universitaire de Lille, Université Lille Nd de France F59655, Villeneuve D ascq, France gilmar.mompean@polytech-lille.fr Laurent Division Labatoire de Mécanique de Lille (LML), CNRS, UMR 8107 Organization Address École Polytechnique Universitaire de Lille, Université Lille Nd de France F59655, Villeneuve D ascq, France

2 Auth Family Name Thompson Particle Given Name Prefix Suffix Division Organization Address Roney Leon Labatório de Mecânica Teórica Aplicada (LMTA), Department of Mechanical Engineering Universidade Federal Fluminense Rua Passo da Pátria 156, Niterói, RJ, , Brazil Abstract In the present wk, direct numerical simulations of turbulent channel flow of a viscoelastic FENE-P fluid, at zero-shear friction Reynolds number equal to 180, are used to analyze the polymer extension mechanism. As a primary focus, the relative polymer stretch and the probability distribution function of the alignment between the confmation tens and other relevant entities are investigated. In near-wall regions, polymers present a strong tendency to ient along the streamwise direction of the flow. Furtherme, the polymer extension seems to be strongly crelated with alignment between both confmation tens and the velocity fluctuations product tens, (defined as ). Joint probability density functions show that large positive polymer wk fluctuations,, are closely related with positive growth rate of the product of streamwise velocity fluctuations,. In contrast, small negative fluctuations of polymer wk are observed in the regions of negative rate of. However, in both cases, polymers are predominantly iented along the principal direction of, which indicates the relevance of this tens f the polymer-turbulence interaction mechanism.

3 Auth Proof On the Extension of Polymer Molecules in Turbulent Viscoelastic Flows: Statistical and Tens Investigation Anselmo Soeiro Pereira, Ramon Silva Martins, Gilmar Mompean, Laurent Thais and Roney Leon Thompson Abstract In the present wk, direct numerical simulations of turbulent channel flow of a viscoelastic FENE-P fluid, at zero-shear friction Reynolds number equal to 180, are used to analyze the polymer extension mechanism. As a primary focus, the relative polymer stretch and the probability distribution function of the alignment between the confmation tens and other relevant entities are investigated. In near-wall regions, polymers present a strong tendency to ient along the streamwise direction of the flow. Furtherme, the polymer extension seems to be strongly crelated with alignment between both confmation tens and the velocity fluctuations product tens, τ (defined as u iu j). Joint probability density functions show that large positive polymer wk fluctuations, E x, are closely related with positive growth rate of the product of streamwise velocity fluctuations, t u 2 x. In contrast, small negative fluctuations of polymer wk are observed in the regions of negative rate of u 2 x. However, in both cases, polymers are predominantly iented along the principal direction of τ, which indicates the relevance of this tens f the polymer-turbulence interaction mechanism. A.S. Pereira (B) R.S. Martins G. Mompean L. Thais Labatoire de Mécanique de Lille (LML), CNRS, UMR 8107, École Polytechnique Universitaire de Lille, Université Lille Nd de France, F59655 Villeneuve D ascq, France anselmo.pereira@polytech-lille.fr R.S. Martins ramon.martins@polytech-lille.fr G. Mompean gilmar.mompean@polytech-lille.fr L. Thais laurent.thais@polytech-lille.fr R.L. Thompson Labatório de Mecânica Teórica Aplicada (LMTA), Department of Mechanical Engineering, Universidade Federal Fluminense, Rua Passo da Pátria 156, Niterói, RJ , Brazil rthompson@id.uff.br Springer International Publishing Switzerland 2016 M. Stanislas et al. (eds.), Progress in Wall Turbulence, ERCOFTAC Series 23, DOI / _15 1

4 2 A.S. Pereira et al. Auth Proof AQ Introduction The addition of a small amount of high molecular weight polymers can lead to a pressure drop decrease in turbulent flows. Since this first observation [1 3] numerous experimental studies have been conducted in attempts to make practical use of polymer-induced drag reduction including long-distance transptation of liquids, oil well operations, firefighting, transpt of suspensions and slurries, and biomedical applications [4]. In a remarkable and pioneering wk, Virk et al. [5], who perfmed careful analyses with an experimental turbulent pipe flow apparatus, showed that whether the friction drag f pipe flows is plotted in Prandtl-Kármán codinates, polymer-induced drag reduction (DR) departs from Prandtl-Kármán law (onset of DR) to its bound, so-called maximum drag reduction (MDR) Virk asymptote, as a result of Reynolds number, polymer concentration polymer molecular weight increases. Over the years, researchers have successfully analyzed relevant aspects of this phenomenon [6]. However, up to now, there has been no definitive consensus concerning the interactions between the turbulent energy and polymer defmations. Phenomenological polymer drag reduction explanations gravitate around two maj theies. Accding to the viscous they independently proposed by Lumley [7] and Seyer and Metzner [8], and suppted by Ryskin [9], polymer stretching in a turbulent flow produces an increase in the effective viscosity in the region outside of the viscous sublayer and in the buffer layer which suppress turbulent fluctuations, increasing the buffer layer thickness and reducing the wall friction. The elastic they postulated by Tab and de Gennes [10] assumes that the elastic energy sted by the polymer becomes comparable to the kinetic energy in the buffer layer. Since the cresponding viscoelastic length scale is larger than the Kolmogov scale, the usual energy cascade is inhibited, which thickens the buffer layer and reduces the drag. In an attempt to quantify the viscous scenario, L vov et al. [11] used conservation principles to show that an additional effective viscosity growing linearly with the distance from the wall in the buffer layer has similar effects to those observed by the addition of flexible polymers in turbulent flows. This theetical prediction was later confirmed by De Angelis et al. [12], who perfmed DNS of Newtonian turbulent flows with an added viscosity profile obtaining results previously observed in viscoelastic FENE-P simulations. Additionally, the auths showed that, using this simple linear viscosity model, they were capable to predict the maximum drag reduction asymptote, a point discussed in detail by Benzi et al. [13]. However, it is imptant to note that the elastic they has been actively expled. Min et al. [14] conducted DNS of turbulent drag reducing channel flows in which the dilute polymer solution is simulated with an Oldroyd-B model. Their results showed good agreement with previous theetical and experimental predictions of the onset of DR at specific friction Weissenberg numbers, which is interpreted based on elastic they. Min et al. [14], as well as Dallas et al. [15], describe an elastic scenario in which the elastic energy sted in the near-wall region due to the uncoiling of polymer chains is transpted to and, in some ption, released in the buffer and log layers. This

5 On the Extension of Polymer Molecules in Turbulent Viscoelastic Flows 3 Auth Proof stage of energy around near-wall vtices was confirmed by Dubief et al. [16], who perfmed DNS of turbulent polymer solutions in a channel using the FENE-P model, although, in contrast with Min et al. [14] and Dallas et al. [15], they proposed an autonomous regeneration cycle of polymer wall turbulence in which coherent release of energy occurs in the very near-wall region, just above the viscous layer. Despite the discrepancies between the two most prominent theies, what seems to be in accdance with both scenarios is the relevance of the coil stretch polymer process, which further imposes a transient behavi on drag reduction as well as subsequent polymer degradation as a consequence of high polymer elongation. In the present wk, we investigate the process of polymer coil stretch with the aid of direct numerical simulations of turbulent channel flow of a viscoelastic FENE-P fluid, at zero-shear friction Reynolds number equal to 180. Tens and statistical analyses are developed. The relative polymer stretch and the alignment between the confmation tens and other relevant entities are studied. Additionally, joint probability density functions are used in der to crelate the polymer turbulence exchanges of energy and polymer ientations. 2 Methodology A turbulent channel flow of an incompressible dilute polymer solution is considered. The channel streamwise direction is x 1 =x, the wall-nmal direction is x 2 = y, and the spanwise direction is x 3 = z. The instantaneous velocity field in the respective directions is (u 1, u 2, u 3 ) = (u, v, w). The governing equations are scaled with the channel half-gap h, the bulk velocity U b, and the fluid density ρ. The scaled momentum equations get the fm u i u i + u j = p + β 0 2 u i + (1 β 0) Ξ ij + e i δ i1. (1) t x j x i Re b Re b x j x 2 j In Eq. 1, β 0 is the ratio of the Newtonian solvent viscosity (ν N ) to the total zero-shear viscosity (ν 0 = ν N + ν p0 ), and the bulk Reynolds number is Re b = ρu b h/ν 0.The extra-stress tens is denoted by Ξ ij and the quantity e i δ i1 represents the non-periodic pressure gradient driving the flow in the streamwise direction. The fmalism of 1 includes the assumption of a unifm polymer concentration which is governed by the viscosity ratio β 0 where β 0 = 1 yields the limiting behavi of the Newtonian case. The extra-stress tens (Ξ ij )ineq.1holds the polymer contribution to the solution tension. Such contribution is accounted f with a single spring-dumbbell model. We consider here the kinetic they Finitely Extensible Non-linear Elastic in the Peterlin approximation (FENE-P) model. The FENE-P model is mostly preferred due to its physically realistic finite extensibility of the polymer molecules and to its relatively simple second-der closure. This model considers the phase-averaged

6 4 A.S. Pereira et al. Auth Proof confmation tens c ij = q i q j, where q i are the components of the end-to-end vect of each individual polymer molecule. The extra-stress tens is then Ξ ij = { f (tr (C)) c ij δ ij }/We b in which We b = λu b /h is the bulk Weissenberg number (λ being the relaxation time scale), δ ij is the Kronecker delta operat and f (tr (C)) is given by the Peterlin approximation f (tr (C)) = ( L 2 3 ) / ( L 2 tr (C) ), where L is the maximum polymer molecule extensibility and tr(.) represents the trace operat. The governing equation f the confmation tens is C ij C ij + u k u i C kj u j C ki + f (tr (C)) c ( ) ij δ ij 1 2 C ij = t x k x k x k We b Pr c Re b xk 2. (2) in which Pr c = ν 0 /ρκ c is a stress Prandtl number defined as the ratio of the total kinematic zero-shear rate viscosity (ν 0 /ρ) to an artificial stress diffusivity κ c.this explicit elliptic diffusion term included in 2 is necessary to remove non-physical high wave-number instabilities typically induced by the chaotic nature of viscoelastic turbulent flows. This dissipative term was first introduced in this context by Sureshkumar and Beris [17], and the methodology subsequently validated under a variety of flow and material parameter values [18]. The numerical scheme f DNS used here was carefully detailed by Thais et al. [19]. 3 Results and Discussion A viscoelastic fluid can have significantly different mean and turbulent statistical behavi than a Newtonian fluid. F a given turbulence level as parameterized by the zero-shear friction Reynolds number, Re τ, this effect can vary with the friction Weissenberg number, We τ, and maximum polymer extension length, L. Inthis wk, one Newtonian flow and four viscoelastic FENE-P flows are examined keeping Re τ = 180 fixed and using two different values of We τ and L (We τ = 50; We τ = 115; L = 30; L = 100). Our main results are separated into two parts. In Sect. 3.1, we analyze the evolution of polymer stretch along the wall distance, y +, of which the effects on near-wall vtices and dependence on L and We τ are investigated as well. Tens and statistical investigations, and polymer turbulence energy exchange analysis are conducted in Sect. 3.2 in an attempt to verify the alignment between the confmation tens and other relevant entities and its effects on turbulent energy transfers. 3.1 Polymer Stretch The three-dimensional structures showed in Fig. 1 represent isosurfaces of vtical regions defined as the positive second invariant of velocity gradient tens, u,

7 On the Extension of Polymer Molecules in Turbulent Viscoelastic Flows 5 Auth Proof Fig. 1 The three-dimensional structures represent isosurfaces of vtical regions defined as the positive second invariant of velocity gradient tens, u. The cols in figures indicate the polymer stretch, tr ( ) C L 2 in Newtonian (a) and viscoelastic (b) flows. F incompressible flows, the second invariant of u, so-called Q-criterion [20], is simplified Q = 1/2 ( W 2 D 2), which indicates spatial regions where the Euclidean nm of the rate of rotation tens, ( W, dominates that of the rate of strain, D.BothDand W are defined as 1 2 u + u T ) and 2 1 ( u u T ), respectively. Comparing Fig. 1a, b it is observed that the number of vtices with a value of Q criterion equal to 0.7 decreases with increasing of viscoelasticity (We τ and L). In viscoelastic flows the vtical structures are significantly weaker than in the Newtonian flow, which is considered a fundamental evidence of the polymer turbulence interactions and the consequently drag reduction [6, 21, 22]. As the viscoelasticity increases, some vtices characteristic change. Their thicknesses and streamwise lengths increase, while their strengths weaken. Furtherme, vtices become me parallels to the wall. It has been experimentally and numerically shown that, in drag reducing flows, the streamwise component of the Reynolds nmal stresses increase relative to the Newtonian case, while the other components of the Reynolds stress tenss decrease [23, 24]. These variations seem to be closely connected with the coil stretch polymer transition and the following vtex structural changes [25]. The cols in Fig. 2b indicate the relative polymer stretch, tr ( ) C L 2. The yz-planes

8 6 A.S. Pereira et al. Auth Proof Fig. 2 Evolution of xz-plane average ( ) polymer stretch, tr C, along the wall L 2 xz distance, y +, nmalized with the local friction velocity, u τ show that polymers are me stretched close to the wall (yellow and red regions). In contrast, polymer extensions are less pronounced in the middle of channel (blue regions). The isosurface cols and those of the intersections between vtical structures and yz-planes show that polymers present a me significant extension around the near-wall vtices. The polymer stretch can be seen me clearly in Fig. 2, where the evolution of ( xz-plane average nmalized trace of the instantaneous confmation tens, tr C, along the wall distance, y +, is displayed (solid lines) together with the xz L 2 ) nmalized streamwise nmal component of the confmation tens, ( Cxx ( ) C L 2 L 2 ) xz (open symbols). The percentage polymer extension, tr, is relatively high xz at the wall, achieving a peak in the near-wall region, of which the exact location vary with L and We τ. This peak is commonly ( ) associated with the streamwise vtices [15, 16, 25]. After this point, tr C starts to decrease until reaching a L 2 xz minimum value at the channel center. In ( comparing both gray and red solid lines in Fig. 2, it is clearly observed that tr C decreases with increasing L, keeping fixed Re τ and We τ, which suggests xz L 2 ) that large polymer molecules could be less susceptible to chain scission degradation [26, 27]. A further comparison of red and green solid lines reveals that the relative polymer extension become greater as the friction Weissenberg number increases, since higher values of polymer time scale are influenced from a wider spectrum of flow time scales [15]. Figure 2 also shows that ( the) dominant contribution ) in confmation tens trace comes from C xx, i.e., tr C ( Cxx (especially near L 2 xz L 2 xz the wall). This distribution suggests a significant stretching of the polymeric chain

9 On the Extension of Polymer Molecules in Turbulent Viscoelastic Flows 7 Auth Proof in the streamwise direction. The analysis of the confmation ( tens ) trace reveals two locations of interest: y + = 8.2, approximately where tr C is maximum; L 2 xz y + = 180, where the confmation tens trace reaches its minimum value. 3.2 Tens and Statistical Analysis of Polymer Orientation and Polymer Turbulence Exchanges of Energy Figure 3 shows the evolution of xz-plane average cosine of the angle Φ between the principal direction of relevant entities, e 1 (the eigenvect related to the largest eigenvalue) and the streamwise direction, e x, along y +. These relevant entities crespond to the confmation tens, C, the velocity fluctuations product tens, τ,the rate of strain tens, D, and the vticity vect, ω, which is defined as the non-null components of W. F all viscoelastic fluids investigated here, the confmation tens (open gray balls) exhibits an imptant alignment along the streamwise direction, which is me pronounced at the wall, achieving a minimum value in the middle of channel (a). This minimum value grows with increasing of both We τ and L. Consequently, in the most viscoelastic case (b), cos ( e1 C, e x) xz 1 f any wall distance. The rate of strain tens (red diamonds) presents a similar behavi of that described f C. The average angle between D and e 1 is maximum and equal to 45 o within the viscous sublayer (0 < y + < 5) since the main component of the strain rate in this flow region comes from the wall-nmal shear u. As the wall dis- (a) (b) Fig. 3 Xz-plane average cosines of the angles between the principal direction of a given tens and the unit vect e x along the nmalized wall distance, y + y

10 8 A.S. Pereira et al. Auth Proof tance increases, u y becomes less imptant and cos ( e1 D, e x) xz decreases smoothly from the buffer layer (5 < y + < 30) to the outer layer (y + > 50). However, as We τ and L rise, this 45 degrees angle is maintained, even at y + far from the wall (see Fig. 3b), a fact linked whit the extension of the buffer layer region into the channel caused by the polymers [24]. The ientation of τ along the streamwise direction is relatively accentuated at the wall, achieving a peak in the ( near-wall ) region, of which the( exact location ) vary with L and We τ as well as tr C. After this point, cos e τ L 2 xz 1, e x stars to xz decrease until reaching a minimum ( value) at the channel center. Comparing Fig. 1a, b, it is clearly observed that cos e τ 1, e x increases when the viscoelasticity is xz incremented. Looking at the relative ientation between the vticity vect and the streamwise direction (green squares), it can be concluded that the peak magnitude of cos ( e1 ω, e x) xz is smaller than that f the other curves in Fig. 1, presenting a zero wall value and growing with increasing wall distance. This peak magnitude falls with increasing viscoelasticity since ω becomes me aligned along the spanwise direction with increasing We τ and L (not shown here). It is wth noting that the preferential ientation of both C and τ along the x direction reveals a strong coaxiality between these two tenss. This alignment seems to play an imptant role in both coil stretch process and polymer turbulence exchanges of energy, which is clarify in Fig. 4. F this figure, let us consider the Reynods stress equation, ( ) 1 u x 2 u x u j = u x u p x + β 0 u u x 2 x } 2 {{ t } x j x }{{}}{{} Re b 2 + (1 β 0) u Ξ xj x, (3) x j Re b x j }{{}}{{} T x A x P x 213 where the instantaneous amount of energy which is sted (E x < 0) released 214 (E x > 0) by polymers from the streamwise velocity fluctuation, u x, is represented 215 by E x. The complementary wk terms denote de advection A x, the pressure redis- 216 tribution P x, and the viscous stress V x, all of them in the streamwise direction. The 217 sum A x + P x + V x is referred here as to Newtonian wk, N x [16]. 218 Figure 4 shows three different joint probability density functions (JPF) considering the xz-plane located close to the wall, at y = 8.2 (approximately where the relative 220 polymer extension is maximum), f the less viscoelastic case analyzed here (We τ = 50, L = 30). The black solid line refers to the JPF of E x versus T x, whereas the dashed line indicates the JPF linking E x and N x. Lastly, the dot-dash line represents 223 the JPF which considers the polymer wk fluctuation and the cosine of the angle between the principal directions C and τ 224. The wk terms were nmalized by their 225 respective spatial root mean square (rms, which considers the whole channel). 226 First, with regard to Fig.4, it is imptant to observe that polymers are allowed 227 to coil within this region due to the reduction of turbulent kinetic energy (T x )by V x E x

11 On the Extension of Polymer Molecules in Turbulent Viscoelastic Flows 9 Auth Proof Fig. 4 Joint probability density functions of polymer wk versus other relevant quantities at y + = 8.2. F each wk term, fluctuations are nmalized by the respective standard global deviation viscous dissipation (V x ), as exposed by Dubief et al. [16]. In comparing Newtonian 229 wk to polymer wk, it is apparent that large positive polymer wk fluctuations 230 occur in regions where the Newtonian turbulent wk is negative. In contrast, E x 231 is closely related with positive growth rate of the product of streamwise velocity fluctuations, t u x, indicating an imptant injection of energy into the flow. Quite 233 surprisingly, in these both opposite scenarios, ( ) polymers are predominantly iented 234 along the principal direction of τ (cos e1 C, eτ 1 1), which reveals the relevance of 235 this tens f the polymer turbulence interactions. These observations are equally 236 valid f the other viscoelastic cases (not shown here) Final Remarks We investigated the process of polymer coil stretch with the aid of direct numerical simulations of turbulent channel flows. One Newtonian flow and four viscoelastic FENE-P flows were examined keeping Re τ = 180 fixed and using two different values of We τ and L (We τ = 50; We τ = 115; L = 30; L = 100). Polymers present a strong tendency to ient along the streamwise direction of the flow in near-wall regions (where their extension are accentuated), as well as the velocity fluctuations product tens, τ. Joint probability density functions show that large positive polymer wk fluctuations, E x, are closely related with positive growth rate of the product of streamwise velocity fluctuations, t u 2 x. In contrast, small negative fluctuations of polymer wk are observed in regions of negative rate of u 2 x. However, in both cases, polymers are predominantly iented along the principal direction of τ.

12 10 A.S. Pereira et al. Auth Proof Acknowledgments The auths are grateful to Dr. Enrico Calzavarini and Dr. Stefano Berti from the Labatoire de Mécanique de Lille of Université Lille Nd de France f their useful comments and suggestions. This wk was granted access to the HPC resources of IDRIS under the allocation 2014-i20142b2277 made by GENCI. The auths would also like to express their acknowledgment and gratitude to the Brazilian Scholarship Program Science Without Bders, managed by CNPq (National Council f Scientific and Technological Development), f the partial financial suppt f this research. References 1. F. Frest, G.A. Grierson, Pap. Trade J. 92, 39 (1931) 2. B.A. Toms, Proceedings of the International Congress of Rheology, Section II (Holland, Nth- Holland, Amsterdam, 1948), pp K.J. Mysels, US Patent 2 492, 173, 27 December R.H.J. Sellin, J.W. Hoyt, J. Poliert, O. Scrivener, J. Hydraul. Res. 20, 235 (1982) 5. P.S. Virk, H.S. Mickley, K.A. Smith, J. Fluid Mech. 22, 22 (1967) 6. C.M. White, M.G. Mungal, Annu. Rev. Fluid Mech. 40, 235 (2008) 7. J.L. Lumley, Annu. Rev. Fluid Mech. 11, 367 (1969) 8. F.A. Seyer, A.B. Metzner, AIChE J. 492, 426 (1949) 9. G. Ryskin, Phys. Rev. Lett. 59, 2059 (1987) 10. M. Tab, P.G. de Gennes, Europhys. Lett. 7, 519 (1986) 11. V.S. L vov, A. Pomyalov, I. Procaccia, V. Tiberkevich, Phys. Rev. Lett. 92, (2004) 12. E. De Angelis, C. Casciola, V.S. L vov, A. Pomyalov, I. Procaccia, V. Tiberkevich, Phys. Rev. E 70, (2004) 13. R. Benzi, E.D. Angelis, V.S. L vov, I. Procaccia, Phys. Rev. Lett. 95, (2005) 14. T. Min, J.Y. Yoo, H. Choi, D.D. Joseph, J. Fluid Mech. 486, 213 (2003) 15. V. Dallas, J.C. Vassilicos, G.F. Hewitt, Phys. Rev. E 82, (2010) 16. Y. Dubief, C.M. White, V.E. Terrapon, E.S.G. Shaqfeh, P. Moin, S.K. Lele, J. Fluid Mech. 514, 271 (2004) 17. R. Sureshkumar, A.N. Beris, J. Non-Newton. Fluid Mech. 60, 53 (1995) 18. K.D. Housiadas, A.N. Beris, Phys. Fluids 15(8), 2369 (2003) 19. L. Thais, A. Tejada-Martinez, T.B. Gatski, G. Mompean, Comput. Fluids 43, 134 (2011) 20. J.C.R. Hunt, A.A. Wray, P. Moin, in Proceedings of Summer Program. Center f Turbulence Research, Rept CTR-S88 (1988) p K. Kim, C.F. Li, R. Sureshkumar, L. Balachandar, R.J. Adrian, J. Fluid Mech. 584, 281 (2007) 22. K. Kim, R.J. Adrian, L. Balachandar, R. Sureshkumar, Phys. Fluids 100, (2008) 23. M.D. Warholic, H. Massah, T.J. Hanratty, Exp. Fluids 27, 461 (1999) 24. L. Thais, T.B. Gatski, G. Mompean, J. Turbul. 13, 1 (2012) 25. C.D. Dimitropoulos, Y. Dubief, E.S.G. Shaqfeh, P. Moin, S.K. Lele, Phys. Fluids 17, 1 (2005) 26. A.S. Pereira, E.J. Soares, J. Non-Newton. Fluid Mech. 179, 9 (2012) 27. A.S. Pereira, R.M. Andrade, E.J. Soares, J. Non-Newton. Fluid Mech. 202, 72 (2013)

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