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The centre-mode instability of viscoelastic plane Poiseuille flow

Published online by Cambridge University Press:  12 March 2021

Mohammad Khalid ,
Indresh Chaudhary Open the ORCID record for Indresh Chaudhary [Opens in a new window] ,
Piyush Garg Open the ORCID record for Piyush Garg [Opens in a new window] ,
V. Shankar Open the ORCID record for V. Shankar [Opens in a new window]  and
Ganesh Subramanian Open the ORCID record for Ganesh Subramanian [Opens in a new window]
Mohammad Khalid
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kanpur208016, India
Indresh Chaudhary
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kanpur208016, India
Piyush Garg
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore560064, India
V. Shankar*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kanpur208016, India
Ganesh Subramanian*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore560064, India
*
†Email addresses for correspondence: vshankar@iitk.ac.in, sganesh@jncasr.ac.in
†Email addresses for correspondence: vshankar@iitk.ac.in, sganesh@jncasr.ac.in
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Abstract

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 (Garg et al., Phys. Rev. Lett., vol. 121, 2018, 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.

JFM classification

Instability: Transition to turbulence Non-Newtonian Flows: Viscoelasticity
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A43
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© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Avila, K., Moxey, D., Lozar, A.D., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333, 192–196.CrossRefGoogle ScholarPubMed
Bertola, V., Meulenbroek, B., Wagner, C., Storm, C., Morozov, A., van Saarloos, W. & Bonn, D. 2003 Experimental evidence for an intrinsic route to polymer melt fracture phenomena: a nonlinear instability of viscoelastic Poiseuille flow. Phys. Rev. Lett. 90, 114502.CrossRefGoogle ScholarPubMed
Bistagnino, A., Boffetta, G., Celani, A., Mazzino, A., Puliafito, A. & Vergassola, M. 2007 Nonlinear dynamics of the viscoelastic Kolmogorov flow. J. luid Mech. 590, 61–80.Google Scholar
Bodiguel, H., Beaumont, H., Machado, A., Martinie, L., Kellay, H. & Colin, A. 2015 Flow enhancement due to elastic turbulence in channel flows of shear thinning fluids. Phys. Rev. Lett. 114, 028302(5).CrossRefGoogle ScholarPubMed
Boyd, J.P. 1999 Chebyshev and Fourier Spectral Methods, 2nd edn. Springer.Google Scholar
Brandi, A.C., Mendonça, M.T. & Souza, L.F. 2019 DNS and LST stability analysis of Oldroyd-B fluid in a flow between two parallel plates. J. Non-Newtonian Fluid Mech. 267, 14–27.CrossRefGoogle Scholar
Budanur, N.B., Short, K.Y., Farazmand, M., Willis, A.P. & Cvitanović, P. 2017 Relative periodic orbits form the backbone of turbulent pipe flow. J. luid Mech. 833, 274–301.Google Scholar
Chandra, B., Mangal, R., Das, D. & Shankar, V. 2019 Instability driven by shear thinning and elasticity in the flow of concentrated polymer solutions through microtubes. Phys. Rev. Fluids 4, 083301.CrossRefGoogle Scholar
Chandra, B., Shankar, V. & Das, D. 2018 Onset of transition in the flow of polymer solutions through microtubes. J. luid Mech. 844, 1052–1083.Google Scholar
Chandra, B., Shankar, V. & Das, D. 2020 Early transition, relaminarization and drag reduction in the flow of polymer solutions through microtubes. J. luid Mech. 885, A47.Google Scholar
Chaudhary, I., Garg, P., Shankar, V. & Subramanian, G. 2019 Elasto-inertial wall mode instabilities in viscoelastic plane Poiseuille flow. J. luid Mech. 881, 119–163.Google Scholar
Chaudhary, I., Garg, P., Subramanian, G. & Shankar, V. 2021 Linear instability of viscoelastic pipe flow. J. luid Mech. 908, A11.Google Scholar
Chilcott, M.D. & Rallison, J.M. 1988 Creeping flow of dilute polymer solutions past cylinders and spheres. J. Non-Newtonian Fluid Mech. 29 (C), 381–432.CrossRefGoogle Scholar
Chokshi, P. & Kumaran, V. 2009 Stability of the plane shear flow of dilute polymeric solutions. Phys. Fluids 21, 014109.CrossRefGoogle Scholar
Choueiri, G.H., Lopez, J.M. & Hof, B. 2018 Exceeding the asymptotic limit of polymer drag reduction. Phys. Rev. Lett. 120, 124501.CrossRefGoogle ScholarPubMed
Darbyshire, A.G. & Mullin, T. 1995 Transition to turbulence in constant-mass-flux pipe flow. J. luid Mech. 289, 83–114.Google Scholar
De Angelis, E., Casciola, C.M. & Piva, R. 2002 DNS of wall turbulence: dilute polymers and self-sustaining mechanisms. Comput. Fluids 31, 495–507.CrossRefGoogle Scholar
Doi, M. & Edwards, S.F. 1986 The Theory of Polymer Dynamics. Clarendon.Google Scholar
Drazin, P.G. & Reid, W.H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Dubief, Y., Page, J., Kerswell, R.R., Terrapon, V.E. & Steinberg, V. 2020 A first coherent structure in elasto-inertial turbulence. arXiv:2006.06770.Google Scholar
Dubief, Y., Terrapon, V.E. & Soria, J. 2013 On the mechanism of elasto-inertial turbulence. Phys. Fluids 25 (11), 110817.CrossRefGoogle ScholarPubMed
Dubief, Y., White, C.M., Terrapon, V.E., Shaqfeh, E.S.G., Moin, P. & Lele, S.K. 2004 On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows. J. luid Mech. 514, 271–280.Google Scholar
Eckhardt, B., Schneider, T.M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.CrossRefGoogle Scholar
Garg, P., Chaudhary, I., Khalid, M., Shankar, V. & Subramanian, G. 2018 Viscoelastic pipe flow is linearly unstable. Phys. Rev. Lett. 121, 024502.CrossRefGoogle ScholarPubMed
Garg, P., Shankar, V. & Subramanian, G. 2020 Weakly nonlinear analysis of the center-mode instability in viscoelastic plane Poiseuille flow. In Preparation.Google Scholar
Gorodtsov, V.A. & Leonov, A.I. 1967 On a linear instability of a plane parallel Couette flow of viscoelastic fluid. Z. Angew. Math. Mech. 31 (2), 310–319.CrossRefGoogle Scholar
Graham, M.D. 1998 Effect of axial flow on viscoelastic Taylor–Couette instability. J. luid Mech. 360, 341–74.Google Scholar
Graham, M.D. 2014 Drag reduction and the dynamics of turbulence in simple and complex fluids. Phys. Fluids 26 (10), 101301.CrossRefGoogle Scholar
Ho, T.C. & Denn, M.M. 1977 Stability of plane Poiseuille flow of a highly elastic liquid. J. Non-Newtonian Fluid Mech. 3 (2), 179–195.CrossRefGoogle Scholar
Hoda, N., Jovanovic, M.R. & Kumar, S. 2008 Energy amplification in channel flows of viscoelastic fluids. J. luid Mech. 601, 407–424.Google Scholar
Hoda, N., Jovanovic, M.R. & Kumar, S. 2009 Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids. J. luid Mech. 625, 411–434.Google Scholar
Hof, B., Juel, A. & Mullin, T. 2003 Scaling of the turbulence transition threshold in a pipe. Phys. Rev. Lett. 91, 244502.CrossRefGoogle Scholar
James, D.F. 2009 Boger fluids. Annu. Rev. Fluid Mech. 41 (1), 129–142.CrossRefGoogle Scholar
Jovanovic, M.R. & Kumar, S. 2010 Transient growth without inertia. Phys. Fluids 22, 023101.CrossRefGoogle Scholar
Jovanovic, M.R. & Kumar, S. 2011 Nonmodal amplification of stochastic disturbances in strongly elastic channel flows. J. Non-Newtonian Fluid Mech. 166, 755–778.CrossRefGoogle Scholar
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.Google Scholar
Kulicke, W.M., Kniewske, R. & Klein, J. 1982 Preaparation, characterization, solution properties and rheological behaviour of polyacrylamide. Prog. Polym. Sci. 8, 373–468.CrossRefGoogle Scholar
Kumar, A.S. & Shankar, V. 2005 Instability 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.CrossRefGoogle Scholar
Larson, R.G. 1988 Constitutive Equations for Polymer Melts and Solutions. Butterworths.Google Scholar
Lee, K.C. & Finlayson, B.A. 1986 a Stability of plane Poiseuille and Couette flow of a Maxwell fluid. J. Non-Newtonian Fluid Mech. 21, 65–78.CrossRefGoogle Scholar
Lee, K.C. & Finlayson, B.A. 1986 b Stability of plane Poiseuille and Couette flow of a Maxwell fluid. J. Non-Newtonian Fluid Mech. 21 (1), 65–78.CrossRefGoogle Scholar
Li, W. & Graham, M.D. 2007 Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow. Phys. Fluids 19 (8), 083101.CrossRefGoogle Scholar
Li, W., Xi, L. & Graham, M.D. 2006 Nonlinear travelling waves as a framework for understanding turbulent drag reduction. J. luid Mech. 565, 353–362.Google Scholar
Lopez, J.M., Choueiri, G.H. & Hof, B. 2019 Dynamics of viscoelastic pipe flow at low Reynolds numbers in the maximum drag reduction limit. J. luid Mech. 874, 699–719.Google Scholar
Meseguer, A. & Trefethen, L.N. 2003 Linearized pipe flow to Reynolds number $10^7$. J. Comput. Phys. 186 (1), 178–197.CrossRefGoogle Scholar
Meulenbroek, B., Storm, C., Morozov, A.N. & van Saarloos, W. 2004 Weakly nonlinear subcritical instability of viscoelastic Poiseuille flow. J. Non-Newtonian Fluid Mech. 116, 235–268.CrossRefGoogle Scholar
Morozov, A.N. & van Saarloos, W. 2005 Subcritical finite-amplitude solutions for plane Couette flow of viscoelastic fluids. Phys. Rev. Lett. 95, 024501.CrossRefGoogle ScholarPubMed
Morozov, A.N. & Saarloos, W. 2007 An introductory essay on subcritical instabilities and the transition to turbulence in visco-elastic parallel shear flows. Phys. Rep. 447 (3), 112–143.CrossRefGoogle Scholar
Page, J., Dubief, Y. & Kerswell, R.R. 2020 Exact travelling wave solutions in viscoelastic channel flow. Phys. Rev. Lett. 125, 154501.CrossRefGoogle Scholar
Pan, L., Morozov, A., Wagner, C. & Arratia, P.E. 2013 Nonlinear elastic instability in channel flows at low Reynolds numbers. Phys. Rev. Lett. 110, 174502.CrossRefGoogle ScholarPubMed
Patel, V.C. & Head, M.R. 1969 Some observations on skin friction and velocity profiles in fully developed pipe and channel flows. J. luid Mech. 38, 181–201.Google Scholar
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.Google Scholar
Picaut, L., Ronsin, O., Caroli, C. & Baumberger, T. 2017 Experimental evidence of a helical, supercritical instability in pipe flow of shear thinning fluids. Phys. Rev. Fluids 2, 083303.CrossRefGoogle Scholar
Poole, R.J. 2016 Elastic instabilities in parallel shear flows of a viscoelastic shear-thinning liquid. Phys. Rev. Fluids 1, 041301.CrossRefGoogle Scholar
Poole, R.J., Alves, M.A. & Oliveira, P.J. 2007 Purely elastic flow asymmetries. Phys. Rev. Lett. 99, 164503.CrossRefGoogle ScholarPubMed
Porteous, K.C. & Denn, M.M. 1972 Linear stability of plane Poiseuille flow of viscoelastic liquids. Trans. Soc. Rheol. 16 (2), 295–308.CrossRefGoogle Scholar
Sadanandan, B. & Sureshkumar, R. 2002 Viscoelastic effects on the stability of wall-bounded shear flows. Phys. Fluids 14, 41–48.CrossRefGoogle Scholar
Samanta, D., Dubief, Y., Holzner, M., Schäfer, C., Morozov, A.N., Wagner, C. & Hof, B. 2013 Elasto-inertial turbulence. Proc. Natl Acad. Sci. USA 110 (26), 10557–10562.CrossRefGoogle ScholarPubMed
Schmid, P.J. & Henningson, D.S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Shaqfeh, E.S.G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28 (1), 129–185.CrossRefGoogle Scholar
Shekar, A., McMullen, R.M., McKeon, B.J. & Graham, M.D. 2020 Self-sustained elastoinertial Tollmien–Schlichting waves. J. Fluid Mech. 897, A3.CrossRefGoogle Scholar
Shekar, A., McMullen, R.M., Wang, S., McKeon, B.J. & Graham, M.D. 2019 Critical-layer structures and mechanisms in elastointurbulence. Phys. Rev. Lett. 122, 124503.CrossRefGoogle Scholar
Sibilla, S. & Baron, A. 2002 Polymer stress statistics in the near-wall turbulent flow of a drag-reducing solution. Phys. Fluids 14, 1123–1136.CrossRefGoogle Scholar
Sid, S., Terrapon, V.E. & Dubief, Y. 2018 Two-dimensional dynamics of elasto-inertial turbulence and its role in polymer drag reduction. Phys. Rev. Fluids 3, 011301.CrossRefGoogle Scholar
Srinivas, S.S. & Kumaran, V. 2017 Effect of viscoelasticity on the soft-wall transition and turbulence in a microchannel. J. luid Mech. 812, 1076–1118.Google Scholar
Stone, P.A. & Graham, M.D. 2003 Polymer dynamics in a model of the turbulent buffer layer. Phys. Fluids 15, 1247–1256.CrossRefGoogle Scholar
Stone, P.A., Roy, A., Larson, R.G., Waleffe, F. & Graham, M.D. 2004 Polymer drag reduction in exact coherent structures of plane shear flow. Phys. Fluids 16, 3470–3482.CrossRefGoogle Scholar
Stone, P.A., Waleffe, F. & Graham, M.D. 2002 Toward a structural understanding of turbulent drag reduction: nonlinear coherent states in viscoelastic shear flows. Phys. Rev. Lett. 89, 208301.CrossRefGoogle Scholar
Sureshkumar, R. & Beris, A.N. 1995 a Effect 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.CrossRefGoogle Scholar
Sureshkumar, R. & Beris, A.N. 1995 b Linear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm. J. Non-Newtonian Fluid Mech. 56 (2), 151–182.CrossRefGoogle Scholar
Sureshkumar, R., Beris, A.N. & Handler, R.A. 1997 Direct numerical simulation of the turbulent channel flow of a polymer solution. Phys. Fluids 9, 743–755.CrossRefGoogle Scholar
Toms, B.A. 1977 On the early experiments on drag reduction by polymers. Phys. Fluids 20 (10), S3–S5.CrossRefGoogle Scholar
Varshney, A. & Steinberg, V. 2017 Elastic wake instabilities in a creeping flow between two obstacles. Phys. Rev. Fluids 2, 051301(R).CrossRefGoogle Scholar
Varshney, A. & Steinberg, V. 2018 a Drag enhancement and drag reduction in viscoelastic flow. Phys. Rev. Fluids 3, 103302.CrossRefGoogle Scholar
Varshney, A. & Steinberg, V. 2018 b Mixing layer instability and vorticity amplification in a creeping viscoelastic flow. Phys. Rev. Fluids 3, 103303.CrossRefGoogle Scholar
Virk, P.S. 1975 Drag reduction fundamentals. AIChE J. 21 (4), 625–656.CrossRefGoogle Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4143.CrossRefGoogle Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. luid Mech. 435, 93–102.Google Scholar
Wedin, H. & Kerswell, R.R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. luid Mech. 508, 333–371.Google Scholar
Weideman, J.A. & Reddy, S.C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26 (4), 465–519.CrossRefGoogle Scholar
White, C.M. & Mungal, M.G. 2008 Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40, 235–256.CrossRefGoogle Scholar
Wilson, H.J., Renardy, M. & Renardy, Y. 1999 Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids. J. Non-Newtonian Fluid Mech. 80, 251–268.CrossRefGoogle Scholar
Xi, L. 2019 Turbulent drag reduction by polymer additives: fundamentals and recent advances. Phys. Fluids 31 (12), 121302.Google Scholar
Xi, L. & Graham, M.D. 2010 Active and hibernating turbulence in minimal channel flow of Newtonian and polymeric fluids. Phys. Rev. Lett. 104, 218301.CrossRefGoogle ScholarPubMed
Xi, L. & Graham, M.D. 2012 Dynamics on the laminar-turbulent boundary and the origin of the maximum drag reduction asymptote. Phys. Rev. Lett. 108, 028301.CrossRefGoogle ScholarPubMed
Zhang, M., Lashgari, I., Zaki, T.A. & Brandt, L. 2013 Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids. J. luid Mech. 737, 249–279.Google Scholar