International audienceThis study is devoted to the three-dimensional numerical simulation of developing secondary flows of Newtonian and viscoelastic fluids through a curved duct of square cross-section. The Phan-Thien-Tanner (PTT) model is used to represent viscoelastic effects. The numerical method uses a finite volume discretization with a staggered grid, and the equations are written in general orthogonal coordinates. The numerical simulations produced for 3 different Dean numbers (125, 137 and 150) show clearly the presence of two steady Dean cells and the upstream development of a four-cell pattern when the centrifugal forces become significant. The comparison between Newtonian and PTT flows shows that the transition from twin-cells to four-cells is anticipated for the viscoelastic fluid

Gilmar Mompean

Laurent Thais

Lionel Helin

Mohammed Boutabaa

Comptes Rendus Mécanique (CRM)

C. R. Mecanique 337 (2009) 40–47http://france.elsevier.com/direct/CRAS2B/Numerical study of Dean vortices in developing Newtonian andviscoelastic flows through a curved duct of square cross-sectionMohammed Boutabaa a, Lionel Helin b, Gilmar Mompean b,∗, Laurent Thais ba Département de mécanique, faculté des sciences de l’ingénieur, Université de Chlef, Algérieb Laboratoire de mécanique de Lille, UMR-CNRS 8107, Polytech-Lille, cité scientifique, 59655 Villeneuve d’Ascq cedex, FranceReceived 21 March 2007; accepted after revision 14 November 2008Available online 13 December 2008Presented by Patrick HuerreAbstractThis study is devoted to the three-dimensional numerical simulation of developing secondary flows of Newtonian and viscoelasticfluids through a curved duct of square cross-section. The Phan-Thien–Tanner (PTT) model is used to represent viscoelastic effects.The numerical method uses a finite volume discretization with a staggered grid, and the equations are written in general orthogonalcoordinates. The numerical simulations produced for 3 different Dean numbers (125, 137 and 150) show clearly the presenceof two steady Dean cells and the upstream development of a four-cell pattern when the centrifugal forces become significant.The comparison between Newtonian and PTT flows shows that the transition from twin-cells to four-cells is anticipated for theviscoelastic fluid. To cite this article: M. Boutabaa et al., C. R. Mecanique 337 (2009).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméÉtude numérique des vortex de Dean de fluides Newtonien et viscoélastique en écoulement non établi dans une conduitecourbe à section carrée. Cette étude est consacrée à la simulation numérique tridimensionnelle des écoulements secondaires deDean au sein d’un fluide Newtonien et d’un fluide viscoélastique de Phan-Thien–Tanner s’écoulant dans une conduite courbe desection carrée. La méthode des volumes finis avec un maillage décalé est utilisée pour résoudre les équations de conservation demasse, de quantité de mouvement et l’équation constitutive de PTT écrites en coordonnées orthogonales généralisées. Les calculsfaits pour des nombres de Dean de 125, 137 et 150 montrent clairement la présence de deux cellules de Dean et le développementde quatre cellules en aval de la conduite lorsque les forces centrifuges deviennent importantes. Les résultats montrent égalementque le passage du mode « deux cellules » au mode « quatre cellules » est anticipé pour le fluide viscoélastique. Pour citer cetarticle : M. Boutabaa et al., C. R. Mecanique 337 (2009).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Keywords: Fluid mechanics; Viscoelastic fluid; Dean vortices; Curved duct; Finite volume methodMots-clés : Mécanique des fluides ; Fluide viscoélastique ; Vortex de Dean ; Conduite courbe ; Méthode des volumes finis* Corresponding author.E-mail address: gilmar.mompean@polytech-lille.fr (G. Mompean).1631-0721/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crme.2008.11.001M. Boutabaa et al. / C. R. Mecanique 337 (2009) 40–47 411. IntroductionDean vortices are the result of centrifugal forces. They are present in a variety of practical applications includingtechnological and physical problems: internal turbine blades, cooling passages, biological systems, ducting in internalcombustion engines and heat exchangers.The first mathematical analysis made by Dean [1] shows the onset of a pair of counter-rotating vortex cells in aNewtonian fluid flow in a curved channel resulting from the interaction between centrifugal and viscous forces. Inthis paper, Dean introduces a dimensionless number Dn = Re√a/R1 (subsequently named the Dean number), whereRe is the Reynolds number (Re = ρaU/μ,ρ is the fluid density, μ is the dynamic viscosity, a is the channel gapand U is the bulk velocity), and R1 the inner curvature radius of the channel. The Dean number represents a ratioof the centrifugal forces to the viscous forces, and measures the intensity of the secondary flows. Henceforth, manydefinitions of the Dean number have been used in the analysis of flows in curved channels and a review is given byBerger et al. [2].Several works, both theoretical and experimental, have been performed on Dean instabilities, treating the hy-drodynamic aspect, thermo-hydrodynamic interactions, and impact on mixture phenomena. For Newtonian fluids,a theoretical analysis performed by Ito [3] shows the existence of secondary flows in a curved channel of rectangu-lar cross section. Targett et al. [4] obtained numerical solutions for the flow in an annulus between two concentriccylinders, of large and infinite aspect ratio, and very small gap. They found a critical Dean number, 37.3, close to thetheoretically predicted value by Dean [1]. Cheng and Akiyama [5] used a finite-difference formulation to calculatethe secondary flows in curved rectangular ducts. They calculated the familiar two-vortex cell and only mentioned theexistence of a new four-vortex cell beyond a critical value of the Dean number. The four-vortex pattern was laterpresented by Cheng et al. [6]. The numerical investigation by Joseph et al. [7] also showed the switch from the twincounter-rotating vortices to the four-vortex pattern above a critical Dean number. They confirmed the presence of thefour-vortex pattern by flow visualization. The appearance of more than one pair of cells was demonstrated for squaresections by Soh [8] and Bara et al. [9].Many researchers extended the study to non-Newtonian fluids, as the analytical study of second order fluids flow incurved pipes made by Jitchote and Robertson [10]. Chen et al. [11] studied viscoelastic flow in rotating curved pipes.Phan-Thien and Zheng [12] present a similarity solution for the Oldroyd-B fluid flow in a narrow curved channel.Joo and Shaqfeh [13] showed the existence of a purely elastic instability in an Oldroyd-B fluid flow. Xue et al. [14]used an implicit finite volume method to investigate numerically the pattern and strength of the secondary flow inrectangular pipes for a general class of viscoelastic model. The present work is focused on exploring Dean vortices inNewtonian and viscoelastic Phan-Thien–Tanner flows in a curved duct of square section, with an average curvatureratio a/Rc = 0.066, where Rc is the medium radius of curvature of the channel.2. Governing equationsThe problem is governed by the mass and momentum conservation equations, and the constitutive equation of thePhan-Thien–Tanner viscoelastic fluid which connects the extra-stress components to the velocity field.2.1. Mass conservation equationFor an incompressible flow:∇ · u = 0 (1)where u is the velocity vector and ∇ the gradient operator.2.2. Momentum conservation equationρ(Du/Dt) = ∇ · (−pI + 2ηsS + τ ) (2)where D/Dt is the material derivative, ρ the fluid density, p the Lagrange multiplier arising from incompressibility,I the matrix identity, ηs the solvent viscosity, τ the polymeric contribution to the extra-stress tensor and S = 1/2(∇u+∇uT ) the (symmetric) rate-of-deformation tensor.42 M. Boutabaa et al. / C. R. Mecanique 337 (2009) 40–472.3. Constitutive equation of Phan-Thien–Tanner modelThe constitutive equation of the Phan-Thien–Tanner fluid can be expressed in the following general form:f({τ })τ + λ(Dτ/Dt − τ∇u − ∇utτ ) = 2ηpS (3)We used the exponential form presented in Phan-Thien [15] where the function f is defined asf({τ }) = exp((ελ/ηp){τ }) (4)where ηp represents the polymeric viscosity, {τ } the trace of tensor τ , λ the relaxation time and ε a parameter char-acterizing the elongational behaviour of the model. For ε = 0, the model reduces to the Oldroyd-B fluid.2.4. Governing equations in general orthogonal coordinatesThe use of general orthogonal coordinates carries out a particular interest to simulate flows in ducts presentingcurves or acute borders and flows around bluff bodies. This method was applied by Pope [16] to solve a 2D turbulentrecirculating flow in a diffuser. Thais et al. [17] used a similar technique to simulate a laminar unsteady flow past acircular cylinder. By adopting generalized orthogonal coordinates, the problem’s equations in tensorial form become(see Pope [16] and Aris [18] for the rules of transformation).2.4.1. Mass conservation∑i∇ ·(i) (Vi) = 0 (5)where Vi represents the physical curvilinear velocity components obtained from normalization of the contravariantvelocity components.2.4.2. Momentum conservation∂(ρVj )∂t+∑i∇ ·(i) (ρViVj − Tij ) = − ∂p∂ξj−∑iH ij (ρViVj − Tij ) +∑iHji (ρViVi − Tii) (6)where Tij = τij + 2ηsSij is the sum of the physical curvilinear components of the polymeric extra-stress tensor τijand of the Newtonian stress components and Hji represent the stretching factors of the coordinate transform.2.4.3. Phan-Thien–Tanner constitutive equationf({τij })τij + λ{∂τij∂t+∑k∇ ·(k) (Vkτij ) −∑kH ikVkτkj +∑kHki Viτkj−∑kHjk Vkτik +∑kHkj Vj τik −∑kLikτkj −∑kLjkτki}= 2ηpSij (7)with Lij the components of the generalized velocity gradient operator.3. Numerical method3.1. Spatial discretizationThe finite volume method is employed to discretize the equations in space with a staggered grid, as proposed byPatankar [19]. The pressure and the normal stress components of the viscoelastic tensor are located at the centre ofthe control volumes. The velocities are stored and evaluated at the center of the faces of the control volumes. Theoff-diagonal terms of the viscoelastic tensor are located at the mid-edges of the control volume faces.M. Boutabaa et al. / C. R. Mecanique 337 (2009) 40–47 43Fig. 1. Geometry of the channel and boundary conditions.The diffusion terms of the momentum equations are calculated with the second-order accurate centred differencescheme. The non-linear terms (convective fluxes) of the momentum equations and the advection terms of the consti-tutive equation for the PTT model are evaluated with an upwind quadratic scheme. To improve stability, the EVSS(elastic viscous split stress) algorithm developed by Rajagopalan et al. [20] is used.3.2. Time discretizationThe decoupling procedure used for the pressure is derived from the Marker and Cell algorithm of Harlow andWelch [21]. Details of this procedure are well documented in Mompean and Deville [22]. The symmetric linearsystem obtained for the pressure is solved by direct Choleski factorization.3.3. Geometry, mesh and boundary conditionsThe channel, as shown in Fig. 1, is divided in three parts: (i) a square straight channel of width a = 1 and lengthLe = 10a at the entrance; (ii) a connected 180◦ curved channel with internal radius R1 = 14.6a and external radiusR2 = 15.6a; and (iii) another square straight channel at the exit having the same dimensions as the inlet channel. Theorigin of the Cartesian coordinates is taken at the center O of the curved channel. A non-uniform spaced mesh hasbeen used: 95 nodes in the streamwise direction (11 nodes located on both straight parts of the channel and the restuniformly spaced in the curved part), 31 nodes in the normal direction and 16 nodes in the spanwise direction. Theanalytical solution of a fully developed Newtonian square duct flow is imposed at the inlet and at the outlet of thechannel. This solution consists of a rapidly converging power series expansion with a maximum velocity located atthe center of the duct 2.096 times the bulk velocity. As mentioned in Fig. 1, the no-slip condition is employed on thewalls, and a symmetry condition is employed along the median plane of the duct.4. ResultsIn this section we present the evolution of Dean vortices along the curved duct (average curvature ratio a/Rc =0.066) for Newtonian and PTT fluids at Dean numbers Dn = 125, 137 and 150. In all figures we show contour linesof the non-dimensional stream function Ψ ; to facilitate inter-comparison the extreme values of Ψ are the same on eachframe.4.1. Newtonian flowAt Dn = 125, the flow keeps a two-vortex structure until the exit of the curved part of the duct. In Fig. 2 we representvortex structures at positions θ = 40◦, 90◦ and 180◦. For this Dean number, the centrifugal forces are not large enough44 M. Boutabaa et al. / C. R. Mecanique 337 (2009) 40–47Fig. 2. Stream function at Dean number Dn = 125 for Newtonian fluid. (a) θ = 40◦ , (b) θ = 90◦ , (c) θ = 180◦ .Fig. 3. Stream function at Dean number Dn = 137 for Newtonian fluid. (a) θ = 90◦ , (b) θ = 100◦ , (c) θ = 180◦ .to cause the formation of the additional vortices so the two-vortex flow structure remains intact. However, at Dn = 137(Fig. 3) viscous effects can no longer retain the two-vortex structure, so an additional pair of vortices starts to appear atθ = 90◦. This result is in agreement with Bara et al. [9]. At Dn = 150 (Fig. 4) the centrifugal forces are even stronger,and the additional pair of vortices starts to form at θ = 80◦ (Fig. 4(b)), and is clearly seen at θ = 90◦ (Fig. 4(c)). Alsothe growth rate of the additional vortices is faster than at Dn = 137. Bara et al. [9] also observed the appearance ofadditional vortices at θ = 80◦ for Dn = 150. Fig. 4(d)–(f) shows the development of the four-cell pattern until the exitof the curved duct.4.2. Non-Newtonian flowFor the non-Newtonian flow of the PTT fluid, calculations are made for a Deborah number De = λU/a = 0.3,a viscosity ratio ηs/(ηs + ηp) = 1/9 and a material parameter ε = 0.1. In Fig. 5, we plot the vortex structure atθ = 40◦, θ = 90◦ and θ = 180◦ at Dn = 125. A similar evolution to the Newtonian case is observed, a two-cellpattern is maintained along the curved duct. As shown in Fig. 6, at Dn = 137 the additional vortices appear at θ = 80◦(Fig. 6(b)) earlier than in the Newtonian flow for which the onset was observed at θ = 90◦. In Fig. 7 at Dn = 150,we show the switch from twin-cells to four-cells (Fig. 7(a)–(b)) and the development of the four-cell pattern until thecurved duct exit (Fig. 7(c)–(f)). At this Dean number, the same trend is observed, the additional vortices appear atθ = 70◦, earlier than in the Newtonian case for which they appear at θ = 80◦. We also remark that the strength andsize of the small vortex pair are slightly larger in the non-Newtonian flows.M. Boutabaa et al. / C. R. Mecanique 337 (2009) 40–47 45Fig. 4. Stream function at Dean number Dn = 150 for Newtonian fluid. (a) θ = 70◦ , (b) θ = 80◦ , (c) θ = 90◦ , (d) θ = 130◦ , (e) θ = 160◦ ,(f) θ = 180◦ .Fig. 5. Stream function at Dean number Dn = 125 and Deborah number De = 0.3 for PTT fluid. (a) θ = 40◦ , (b) θ = 90◦ , (c) θ = 180◦ .5. ConclusionsThis study shows that in both cases, Newtonian and non-Newtonian, the size and the number of Dean vorticesdepend on the Dean number. The development length (distance from the curved part of the duct to the position of46 M. Boutabaa et al. / C. R. Mecanique 337 (2009) 40–47Fig. 6. Stream function at Dean number Dn = 137 and Deborah number De = 0.3 for PTT fluid. (a) θ = 70◦ , (b) θ = 80◦ , (c) θ = 180◦ .Fig. 7. Stream function at Dean number Dn = 150 and Deborah number De = 0.3 for PTT fluid. (a) θ = 60◦ , (b) θ = 70◦ , (c) θ = 80◦ , (d) θ = 130◦ ,(e) θ = 160◦ , (f) θ = 180◦ .the onset of additional cells) decreases with increasing Dean number. The viscoelasticity affects secondary flows. Itis shown clearly that for the PTT fluid, the switch from twin-cells to four-cells appears earlier than in the Newtonianfluid. It is also observed that the growth rate of additional vortices is larger in the non-Newtonian case. This behaviouris similar to the perturbation analysis by Robertson and Muller [23] who also found that viscoelasticity tends toM. Boutabaa et al. / C. R. Mecanique 337 (2009) 40–47 47enhance secondary motions for an Oldroyd-B flow in curved pipes. For future work, it is suggested to undertake adetailed study of the persistence of the secondary flow in the straight outlet section (this would require a longer outletchannel).References[1] W.R. Dean, Fluid motion in a curved channel, Proc. R. Soc. London, Ser. A 121 (1928) 402–420.[2] S.A. Berger, L. Talbot, L.-S. Yao, Flow in curved pipes, Annu. Rev. Fluid Mech. 15 (1983) 461–512.[3] H. Ito, Theory on laminar flow through curved pipes of elliptic and rectangular cross sections, in: Rep. Inst. High Speed Mech., Tohoku Univ.,Sendai, Japan, vol. 1, 1951, pp. 1–16.[4] M.J. Targett, W.B. Retallick, S.W. Churchill, Flow through curved rectangular channels of large aspect ratio, AIChE J. 41 (1995) 1061–1070.[5] K.C. Cheng, M. Akiyama, Laminar forced convection heat transfer in curved rectangular channels, Int. J. Heat Mass Transfer 13 (1970)471–490.[6] K.C. Cheng, R.-C. Lin, J.-W. Ou, Fully developed laminar flow in curved rectangular channels, Trans. ASME C: J. Fluids Engrg. 98 (1976)41–48.[7] B. Joseph, E.P. Smith, R.J. Alder, Numerical treatment of laminar flow in helically coiled tubes of square cross section. Part 1. Stationaryhelically coiled tubes, AIChE J. 21 (1975) 965–974.[8] W.Y. Soh, Developing fluid flow in a curved duct of square cross-section and its fully developed dual solutions, J. Fluid Mech. 188 (1988)337–361.[9] B. Bara, K. Nandakumar, J.H. Masliyah, An experimental and numerical study of the Dean problem: flow development towards two-dimensional multiple solutions, J. Fluid Mech. 244 (1992) 339–376.[10] W. Jitchote, A.M. Robertson, Flow of second order fluids in curved pipes, J. Non-Newtonian Fluid Mech. 90 (2000) 91–116.[11] Y. Chen, H. Chen, J. Zhang, B. Zhang, Viscoelastic flow in rotating curved pipes, Phys. Fluids 18 (2006) 083103.[12] N. Phan-Thien, R. 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Numerical study of Dean vortices in developing Newtonian and viscoelastic flows through a curved duct of square cross-section

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Numerical study of Dean vortices in developing Newtonian and viscoelastic flows through a curved duct of square cross-section

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