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The centre-mode instability of viscoelastic plane Poiseuille flow. (English) Zbl 1461.76025
Summary: A modal stability analysis shows that plane Poiseuille flow of an Oldroyd-B fluid becomes unstable to a ‘centre mode’ with phase speed close to the maximum base-flow velocity, \(U_{\max}\). The governing dimensionless groups are the Reynolds number \(Re = \rho U_{\max} H/\eta\), the elasticity number \(E = \lambda \eta /(H^2 \rho)\) and the ratio of solvent to solution viscosity \(\beta = \eta_s/\eta\); here, \(\lambda\) is the polymer relaxation time, \(H\) is the channel half-width and \(\rho\) is the fluid density. For experimentally relevant values (e.g. \(E \sim 0.1\) and \(\beta \sim 0.9)\), the critical Reynolds number, \(Re_c\), is around \(200\), with the associated eigenmodes being spread out across the channel. For \(E(1-\beta ) \ll 1\), with \(E\) fixed, corresponding to strongly elastic dilute polymer solutions, \(Re_c \propto (E(1-\beta ))^{-3/2}\) and the critical wavenumber \(k_c \propto (E(1-\beta ))^{-1/2}\). The unstable eigenmode in this limit is confined in a thin layer near the channel centreline. These features are largely analogous to the centre-mode instability in viscoelastic pipe flow [P. Garg et al., “Viscoelastic pipe flow is linearly unstable”, Phys. Rev. Lett. 121, No. 2, Article ID 024502, 6 p. (2018; doi:10.1103/PhysRevLett.121.024502)], and suggest a universal linear mechanism underlying the onset of turbulence in both channel and pipe flows of sufficiently elastic dilute polymer solutions. Although the centre-mode instability continues down to \(\beta \sim 10^{-2}\) for pipe flow, it ceases to exist for \(\beta < 0.5\) in channels. Whereas inertia, elasticity and solvent viscous effects are simultaneously required for this instability, a higher viscous threshold is required for channel flow. Further, in the opposite limit of \(\beta \rightarrow 1\), the centre-mode instability in channel flow continues to exist at \(Re \approx 5\), again in contrast to pipe flow where the instability ceases to exist below \(Re \approx 63\), regardless of \(E\) or \(\beta \). Our predictions are in reasonable agreement with experimental observations for the onset of turbulence in the flow of polymer solutions through microchannels.
76A10 Viscoelastic fluids
76F06 Transition to turbulence
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[1] Avila, K., Moxey, D., Lozar, A.D., Barkley, D. & Hof, B.2011The onset of turbulence in pipe flow. Science333, 192-196. · Zbl 1411.76035
[2] Bertola, V., Meulenbroek, B., Wagner, C., Storm, C., Morozov, A., Van Saarloos, W. & Bonn, D.2003Experimental evidence for an intrinsic route to polymer melt fracture phenomena: a nonlinear instability of viscoelastic Poiseuille flow. Phys. Rev. Lett.90, 114502.
[3] Bistagnino, A., Boffetta, G., Celani, A., Mazzino, A., Puliafito, A. & Vergassola, M.2007Nonlinear dynamics of the viscoelastic Kolmogorov flow. J. luid Mech.590, 61-80. · Zbl 1141.76335
[4] Bodiguel, H., Beaumont, H., Machado, A., Martinie, L., Kellay, H. & Colin, A.2015Flow enhancement due to elastic turbulence in channel flows of shear thinning fluids. Phys. Rev. Lett.114, 028302(5).
[5] Boyd, J.P.1999Chebyshev and Fourier Spectral Methods, 2nd edn. Springer.
[6] Brandi, A.C., Mendonça, M.T. & Souza, L.F.2019DNS and LST stability analysis of Oldroyd-B fluid in a flow between two parallel plates. J. Non-Newtonian Fluid Mech.267, 14-27.
[7] Budanur, N.B., Short, K.Y., Farazmand, M., Willis, A.P. & Cvitanović, P.2017Relative periodic orbits form the backbone of turbulent pipe flow. J. luid Mech.833, 274-301. · Zbl 1419.76261
[8] Chandra, B., Mangal, R., Das, D. & Shankar, V.2019Instability driven by shear thinning and elasticity in the flow of concentrated polymer solutions through microtubes. Phys. Rev. Fluids4, 083301.
[9] Chandra, B., Shankar, V. & Das, D.2018Onset of transition in the flow of polymer solutions through microtubes. J. luid Mech.844, 1052-1083. · Zbl 1429.76058
[10] Chandra, B., Shankar, V. & Das, D.2020Early transition, relaminarization and drag reduction in the flow of polymer solutions through microtubes. J. luid Mech.885, A47. · Zbl 07154409
[11] Chaudhary, I., Garg, P., Shankar, V. & Subramanian, G.2019Elasto-inertial wall mode instabilities in viscoelastic plane Poiseuille flow. J. luid Mech.881, 119-163. · Zbl 1430.76037
[12] Chaudhary, I., Garg, P., Subramanian, G. & Shankar, V.2021Linear instability of viscoelastic pipe flow. J. luid Mech.908, A11. · Zbl 1461.76018
[13] Chilcott, M.D. & Rallison, J.M.1988Creeping flow of dilute polymer solutions past cylinders and spheres. J. Non-Newtonian Fluid Mech.29 (C), 381-432. · Zbl 0669.76016
[14] Chokshi, P. & Kumaran, V.2009Stability of the plane shear flow of dilute polymeric solutions. Phys. Fluids21, 014109. · Zbl 1183.76150
[15] Choueiri, G.H., Lopez, J.M. & Hof, B.2018Exceeding the asymptotic limit of polymer drag reduction. Phys. Rev. Lett.120, 124501.
[16] Darbyshire, A.G. & Mullin, T.1995Transition to turbulence in constant-mass-flux pipe flow. J. luid Mech.289, 83-114.
[17] De Angelis, E., Casciola, C.M. & Piva, R.2002DNS of wall turbulence: dilute polymers and self-sustaining mechanisms. Comput. Fluids31, 495-507. · Zbl 1075.76556
[18] Doi, M. & Edwards, S.F.1986The Theory of Polymer Dynamics. Clarendon.
[19] Drazin, P.G. & Reid, W.H.1981Hydrodynamic Stability. Cambridge University Press. · Zbl 0449.76027
[20] Dubief, Y., Page, J., Kerswell, R.R., Terrapon, V.E. & Steinberg, V.2020 A first coherent structure in elasto-inertial turbulence. arXiv:2006.06770.
[21] Dubief, Y., Terrapon, V.E. & Soria, J.2013On the mechanism of elasto-inertial turbulence. Phys. Fluids25 (11), 110817.
[22] Dubief, Y., White, C.M., Terrapon, V.E., Shaqfeh, E.S.G., Moin, P. & Lele, S.K.2004On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows. J. luid Mech.514, 271-280. · Zbl 1067.76052
[23] Eckhardt, B., Schneider, T.M., Hof, B. & Westerweel, J.2007Turbulence transition in pipe flow. Annu. Rev. Fluid Mech.39, 447-468.
[24] Garg, P., Chaudhary, I., Khalid, M., Shankar, V. & Subramanian, G.2018Viscoelastic pipe flow is linearly unstable. Phys. Rev. Lett.121, 024502.
[25] Garg, P., Shankar, V. & Subramanian, G.2020 Weakly nonlinear analysis of the center-mode instability in viscoelastic plane Poiseuille flow. In Preparation.
[26] Gorodtsov, V.A. & Leonov, A.I.1967On a linear instability of a plane parallel Couette flow of viscoelastic fluid. Z. Angew. Math. Mech.31 (2), 310-319. · Zbl 0166.45101
[27] Graham, M.D.1998Effect of axial flow on viscoelastic Taylor-Couette instability. J. luid Mech.360, 341-74. · Zbl 0947.76023
[28] Graham, M.D.2014Drag reduction and the dynamics of turbulence in simple and complex fluids. Phys. Fluids26 (10), 101301.
[29] Ho, T.C. & Denn, M.M.1977Stability of plane Poiseuille flow of a highly elastic liquid. J. Non-Newtonian Fluid Mech.3 (2), 179-195. · Zbl 0414.76008
[30] Hoda, N., Jovanovic, M.R. & Kumar, S.2008Energy amplification in channel flows of viscoelastic fluids. J. luid Mech.601, 407-424. · Zbl 1151.76372
[31] Hoda, N., Jovanovic, M.R. & Kumar, S.2009Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids. J. luid Mech.625, 411-434. · Zbl 1171.76364
[32] Hof, B., Juel, A. & Mullin, T.2003Scaling of the turbulence transition threshold in a pipe. Phys. Rev. Lett.91, 244502. · Zbl 1049.76511
[33] James, D.F.2009Boger fluids. Annu. Rev. Fluid Mech.41 (1), 129-142.
[34] Jovanovic, M.R. & Kumar, S.2010Transient growth without inertia. Phys. Fluids22, 023101.
[35] Jovanovic, M.R. & Kumar, S.2011Nonmodal amplification of stochastic disturbances in strongly elastic channel flows. J. Non-Newtonian Fluid Mech.166, 755-778. · Zbl 1282.76052
[36] Khalid, M., Chaudhary, I., Shankar, V. & Subramanian, G.2020 Role of solvent viscous effects and finite extensibility on elasto-inertial wall modes in viscoelastic channel flow. In Preparation.
[37] Kulicke, W.M., Kniewske, R. & Klein, J.1982Preaparation, characterization, solution properties and rheological behaviour of polyacrylamide. Prog. Polym. Sci.8, 373-468.
[38] Kumar, A.S. & Shankar, V.2005Instability of high-frequency modes in viscoelastic plane Couette flow past a deformable wall at low and finite Reynolds number. J. Non-Newtonian Fluid Mech.125 (2), 121-141. · Zbl 1187.76656
[39] Larson, R.G.1988Constitutive Equations for Polymer Melts and Solutions. Butterworths.
[40] Lee, K.C. & Finlayson, B.A.1986aStability of plane Poiseuille and Couette flow of a Maxwell fluid. J. Non-Newtonian Fluid Mech.21, 65-78. · Zbl 0587.76059
[41] Lee, K.C. & Finlayson, B.A.1986bStability of plane Poiseuille and Couette flow of a Maxwell fluid. J. Non-Newtonian Fluid Mech.21 (1), 65-78. · Zbl 0587.76059
[42] Li, W. & Graham, M.D.2007Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow. Phys. Fluids19 (8), 083101. · Zbl 1182.76452
[43] Li, W., Xi, L. & Graham, M.D.2006Nonlinear travelling waves as a framework for understanding turbulent drag reduction. J. luid Mech.565, 353-362. · Zbl 1177.76169
[44] Lopez, J.M., Choueiri, G.H. & Hof, B.2019Dynamics of viscoelastic pipe flow at low Reynolds numbers in the maximum drag reduction limit. J. luid Mech.874, 699-719. · Zbl 1419.76040
[45] Meseguer, A. & Trefethen, L.N.2003Linearized pipe flow to Reynolds number \(10^7\). J. Comput. Phys.186 (1), 178-197. · Zbl 1047.76565
[46] Meulenbroek, B., Storm, C., Morozov, A.N. & Van Saarloos, W.2004Weakly nonlinear subcritical instability of viscoelastic Poiseuille flow. J. Non-Newtonian Fluid Mech.116, 235-268. · Zbl 1106.76367
[47] Morozov, A.N. & Van Saarloos, W.2005Subcritical finite-amplitude solutions for plane Couette flow of viscoelastic fluids. Phys. Rev. Lett.95, 024501.
[48] Morozov, A.N. & Saarloos, W.2007An introductory essay on subcritical instabilities and the transition to turbulence in visco-elastic parallel shear flows. Phys. Rep.447 (3), 112-143.
[49] Page, J., Dubief, Y. & Kerswell, R.R.2020Exact travelling wave solutions in viscoelastic channel flow. Phys. Rev. Lett.125, 154501.
[50] Pan, L., Morozov, A., Wagner, C. & Arratia, P.E.2013Nonlinear elastic instability in channel flows at low Reynolds numbers. Phys. Rev. Lett.110, 174502.
[51] Patel, V.C. & Head, M.R.1969Some observations on skin friction and velocity profiles in fully developed pipe and channel flows. J. luid Mech.38, 181-201.
[52] Pfenniger, W.1961 Transition in the inlet length of tubes at high Reynolds numbers. In Boundary Layer and Flow Control (ed. G.V. Lachman), pp. 970-980. Pergamon.
[53] Picaut, L., Ronsin, O., Caroli, C. & Baumberger, T.2017Experimental evidence of a helical, supercritical instability in pipe flow of shear thinning fluids. Phys. Rev. Fluids2, 083303.
[54] Poole, R.J.2016Elastic instabilities in parallel shear flows of a viscoelastic shear-thinning liquid. Phys. Rev. Fluids1, 041301.
[55] Poole, R.J., Alves, M.A. & Oliveira, P.J.2007Purely elastic flow asymmetries. Phys. Rev. Lett.99, 164503. · Zbl 1274.76124
[56] Porteous, K.C. & Denn, M.M.1972Linear stability of plane Poiseuille flow of viscoelastic liquids. Trans. Soc. Rheol.16 (2), 295-308. · Zbl 0362.76079
[57] Sadanandan, B. & Sureshkumar, R.2002Viscoelastic effects on the stability of wall-bounded shear flows. Phys. Fluids14, 41-48. · Zbl 1184.76468
[58] Samanta, D., Dubief, Y., Holzner, M., Schäfer, C., Morozov, A.N., Wagner, C. & Hof, B.2013Elasto-inertial turbulence. Proc. Natl Acad. Sci. USA110 (26), 10557-10562.
[59] Schmid, P.J. & Henningson, D.S.2001Stability and Transition in Shear Flows. Springer.
[60] Shaqfeh, E.S.G.1996Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech.28 (1), 129-185.
[61] Shekar, A., Mcmullen, R.M., Mckeon, B.J. & Graham, M.D.2020Self-sustained elastoinertial Tollmien-Schlichting waves.J. Fluid Mech.897, A3. · Zbl 1460.76401
[62] Shekar, A., Mcmullen, R.M., Wang, S., Mckeon, B.J. & Graham, M.D.2019Critical-layer structures and mechanisms in elastointurbulence. Phys. Rev. Lett.122, 124503.
[63] Sibilla, S. & Baron, A.2002Polymer stress statistics in the near-wall turbulent flow of a drag-reducing solution. Phys. Fluids14, 1123-1136.
[64] Sid, S., Terrapon, V.E. & Dubief, Y.2018Two-dimensional dynamics of elasto-inertial turbulence and its role in polymer drag reduction. Phys. Rev. Fluids3, 011301.
[65] Srinivas, S.S. & Kumaran, V.2017Effect of viscoelasticity on the soft-wall transition and turbulence in a microchannel. J. luid Mech.812, 1076-1118. · Zbl 1383.76193
[66] Stone, P.A. & Graham, M.D.2003Polymer dynamics in a model of the turbulent buffer layer. Phys. Fluids15, 1247-1256. · Zbl 1186.76502
[67] Stone, P.A., Roy, A., Larson, R.G., Waleffe, F. & Graham, M.D.2004Polymer drag reduction in exact coherent structures of plane shear flow. Phys. Fluids16, 3470-3482. · Zbl 1187.76502
[68] Stone, P.A., Waleffe, F. & Graham, M.D.2002Toward a structural understanding of turbulent drag reduction: nonlinear coherent states in viscoelastic shear flows. Phys. Rev. Lett.89, 208301.
[69] Sureshkumar, R. & Beris, A.N.1995aEffect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows. J. Non-Newtonian Fluid Mech.60, 53-80.
[70] Sureshkumar, R. & Beris, A.N.1995bLinear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm. J. Non-Newtonian Fluid Mech.56 (2), 151-182.
[71] Sureshkumar, R., Beris, A.N. & Handler, R.A.1997Direct numerical simulation of the turbulent channel flow of a polymer solution. Phys. Fluids9, 743-755.
[72] Toms, B.A.1977On the early experiments on drag reduction by polymers. Phys. Fluids20 (10), S3-S5.
[73] Varshney, A. & Steinberg, V.2017Elastic wake instabilities in a creeping flow between two obstacles. Phys. Rev. Fluids2, 051301(R).
[74] Varshney, A. & Steinberg, V.2018aDrag enhancement and drag reduction in viscoelastic flow. Phys. Rev. Fluids3, 103302.
[75] Varshney, A. & Steinberg, V.2018bMixing layer instability and vorticity amplification in a creeping viscoelastic flow. Phys. Rev. Fluids3, 103303.
[76] Virk, P.S.1975Drag reduction fundamentals. AIChE J.21 (4), 625-656.
[77] Waleffe, F.1998Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett.81, 4140-4143.
[78] Waleffe, F.2001Exact coherent structures in channel flow. J. luid Mech.435, 93-102. · Zbl 0987.76034
[79] Wedin, H. & Kerswell, R.R.2004Exact coherent structures in pipe flow: travelling wave solutions. J. luid Mech.508, 333-371. · Zbl 1065.76072
[80] Weideman, J.A. & Reddy, S.C.2000A MATLAB differentiation matrix suite. ACM Trans. Math. Softw.26 (4), 465-519.
[81] White, C.M. & Mungal, M.G.2008Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech.40, 235-256. · Zbl 1229.76043
[82] Wilson, H.J., Renardy, M. & Renardy, Y.1999Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids. J. Non-Newtonian Fluid Mech.80, 251-268. · Zbl 0956.76025
[83] Xi, L.2019Turbulent drag reduction by polymer additives: fundamentals and recent advances. Phys. Fluids31 (12), 121302.
[84] Xi, L. & Graham, M.D.2010Active and hibernating turbulence in minimal channel flow of Newtonian and polymeric fluids. Phys. Rev. Lett.104, 218301.
[85] Xi, L. & Graham, M.D.2012Dynamics on the laminar-turbulent boundary and the origin of the maximum drag reduction asymptote. Phys. Rev. Lett.108, 028301.
[86] Zhang, M., Lashgari, I., Zaki, T.A. & Brandt, L.2013Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids. J. luid Mech.737, 249-279. · Zbl 1294.76119
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