Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T12:39:05.389Z Has data issue: false hasContentIssue false
  • English
  • Français

On the large-Weissenberg-number scaling laws in viscoelastic pipe flows

Published online by Cambridge University Press:  28 June 2022

Dongdong Wan ,
Ming Dong Open the ORCID record for Ming Dong [Opens in a new window]  and
Mengqi Zhang Open the ORCID record for Mengqi Zhang [Opens in a new window]
Dongdong Wan*
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117575, Singapore
Ming Dong
Affiliation:
State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China
Mengqi Zhang
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117575, Singapore
*
†Email address for correspondence: mpezmq@nus.edu.sg
Get access Rights & Permissions [Opens in a new window]

Abstract

This work explains a scaling law of the first Landau coefficient of the derived Ginzburg–Landau equation in the weakly nonlinear analysis of axisymmetric viscoelastic pipe flows in the large-Weissenberg-number ($Wi$) limit, recently reported in Wan et al. (J. Fluid Mech., vol. 929, 2021, A16). Using an asymptotic method, we derive a reduced system, which captures the characteristics of the linear centre-mode instability near the critical condition in the large-$Wi$ limit. Based on the reduced system we then conduct a weakly nonlinear analysis using a multiple-scale expansion method, which readily explains the aforementioned scaling law of the Landau coefficient and some other scaling laws. Particularly, the equilibrium amplitude of disturbance near linear critical conditions is found to scale as $Wi^{-1/2}$, which may be of interest to experimentalists. The current analysis reduces the numbers of parameters and unknowns and exemplifies an approach to studying the viscoelastic flow at large $Wi$, which could shed new light on the understanding of its nonlinear dynamics.

JFM classification

Non-Newtonian Flows: Viscoelasticity Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 944 , 10 August 2022 , A21
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333 (6039), 192–196.10.1126/science.1203223CrossRefGoogle ScholarPubMed
Bird, R.B., Curtiss, C.F., Armstrong, R.C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, Volume 2: Kinetic Theory. Wiley.Google Scholar
Burger, E.D., Chorn, L.G. & Perkins, T.K. 1980 Studies of drag reduction conducted over a broad range of pipeline conditions when flowing prudhoe bay crude oil. J. Rheol. 24 (5), 603–626.CrossRefGoogle Scholar
Buza, G., Beneitez, M., Page, J. & Kerswell, R.R. 2022 a Finite-amplitude elastic waves in viscoelastic channel flow from large to zero Reynolds number. Preprint. arXiv:2202.08047.Google Scholar
Buza, G., Page, J. & Kerswell, R.R. 2022 b Weakly nonlinear analysis of the viscoelastic instability in channel flow for finite and vanishing Reynolds numbers. J. Fluid Mech. 940, A11.CrossRefGoogle Scholar
Chandra, B., Shankar, V. & Das, D. 2020 Early transition, relaminarization and drag reduction in the flow of polymer solutions through microtubes. J. Fluid Mech. 885, A47.CrossRefGoogle Scholar
Chapman, S.J. 2002 Subcritical transition in channel flows. J. Fluid Mech. 451, 35–97.CrossRefGoogle Scholar
Chaudhary, I., Garg, P., Subramanian, G. & Shankar, V. 2021 Linear instability of viscoelastic pipe flow. J. Fluid Mech. 908, A11.CrossRefGoogle Scholar
Choueiri, G.H., Lopez, J.M., Varshney, A., Sankar, S. & Hof, B. 2021 Experimental observation of the origin and structure of elastoinertial turbulence. Proc. Natl Acad. Sci. USA 118 (45), 1–5.10.1073/pnas.2102350118CrossRefGoogle ScholarPubMed
Datta, S.S., et al. 2021 Perspectives on viscoelastic flow instabilities and elastic turbulence. arXiv:2108.09841.Google Scholar
Davey, A. & Drazin, P.G. 1969 The stability of Poiseuille flow in a pipe. J. Fluid Mech. 36 (2), 209–218.CrossRefGoogle Scholar
Dong, M. & Zhang, M. 2022 Asymptotic study of linear instability in a viscoelastic pipe flow. J. Fluid Mech. 935, A28.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T.M., Hof, B. & Westerweel, J. 2006 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39 (1), 447–468.CrossRefGoogle Scholar
Fujimura, K. 1989 The equivalence between two perturbation methods in weakly nonlinear stability theory for parallel shear flows. Proc. R. Soc. Lond. A. Math. Phys. Sci. 424 (1867), 373–392.Google Scholar
Garg, P., Chaudhary, I., Khalid, M., Shankar, V. & Subramanian, G. 2018 Viscoelastic pipe flow is linearly unstable. Phys. Rev. Lett. 121 (2), 024502.CrossRefGoogle ScholarPubMed
Graham, M.D. 2014 Drag reduction and the dynamics of turbulence in simple and complex fluids. Phys. Fluids 26 (10), 625–656.CrossRefGoogle Scholar
Graham, M.D. & Floryan, D. 2021 Exact coherent states and the nonlinear dynamics of wall-bounded turbulent flows. Annu. Rev. Fluid Mech. 53, 227–253.10.1146/annurev-fluid-051820-020223CrossRefGoogle Scholar
Hameduddin, I., Gayme, D.F. & Zaki, T.A. 2019 Perturbative expansions of the conformation tensor in viscoelastic flows. J. Fluid Mech. 858, 377–406.10.1017/jfm.2018.777CrossRefGoogle Scholar
Hameduddin, I., Meneveau, C., Zaki, T.A. & Gayme, D.F. 2018 Geometric decomposition of the conformation tensor in viscoelastic turbulence. J. Fluid Mech. 842, 395–427.CrossRefGoogle Scholar
Hameduddin, I. & Zaki, T.A. 2019 The mean conformation tensor in viscoelastic turbulence. J. Fluid Mech. 865, 363–380.10.1017/jfm.2019.46CrossRefGoogle Scholar
Herbert, T. 1980 Nonlinear stability of parallel flows by high-ordered amplitude expansions. AIAA J. 18 (3), 243–248.10.2514/3.50755CrossRefGoogle Scholar
Herbert, T. 1983 On perturbation methods in nonlinear stability theory. J. Fluid Mech. 126, 167–186.10.1017/S0022112083000099CrossRefGoogle Scholar
Hof, B., Juel, A. & Mullin, T. 2003 Scaling of the turbulence transition threshold in a pipe. Phys. Rev. Lett. 91 (24), 244502.CrossRefGoogle Scholar
Jha, N.K. & Steinberg, V. 2020 Universal coherent structures of elastic turbulence in straight channel with viscoelastic fluid flow. Preprint. arXiv:2009.12258.Google Scholar
Jovanović, M.R. & Kumar, S. 2010 Transient growth without inertia. Phys. Fluids 22 (2), 023101.CrossRefGoogle Scholar
Kerswell, R.R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18 (6), R17–R44.CrossRefGoogle Scholar
Khalid, M., Chaudhary, I., Garg, P., Shankar, V. & Subramanian, G. 2021 a The centre-mode instability of viscoelastic plane Poiseuille flow. J. Fluid Mech. 915, A43.CrossRefGoogle Scholar
Khalid, M., Shankar, V. & Subramanian, G. 2021 b Continuous pathway between the elasto-inertial and elastic turbulent states in viscoelastic channel flow. Phys. Rev. Lett. 127, 134502.CrossRefGoogle ScholarPubMed
Landau, L.D. 1944 On the problem of turbulence. C. R. Acad. Sci. URSS 44 (31), 1–314.Google Scholar
Lemoult, G., Aider, J.-L. & Wesfreid, J.E. 2012 Experimental scaling law for the subcritical transition to turbulence in plane Poiseuille flow. Phys. Rev. E 85 (2), 025303.CrossRefGoogle ScholarPubMed
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46 (1), 493–517.CrossRefGoogle 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., Bertola, V., Wagner, C., Bonn, D. & van Saarloos, W. 2003 Intrinsic route to melt fracture in polymer extrusion: a weakly nonlinear subcritical instability of viscoelastic Poiseuille flow. Phys. Rev. Lett. 90 (2), 024502.CrossRefGoogle ScholarPubMed
Meulenbroek, B., Storm, C., Morozov, A.N. & van Saarloos, W. 2004 Weakly nonlinear subcritical instability of visco-elastic Poiseuille flow. J. Non-Newtonian Fluid Mech. 116 (2-3), 235–268.CrossRefGoogle Scholar
Mohseni, K. & Colonius, T. 2000 Numerical treatment of polar coordinate singularities. J. Comput. Phys. 157 (2), 787–795.CrossRefGoogle Scholar
Morozov, A. 2022 Coherent structures in plane channel flow of dilute polymer solutions. Preprint. arXiv:2201.01274.Google Scholar
Morozov, A.N. & van 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
Morozov, A. & van Saarloos, W. 2019 Subcritical instabilities in plane Poiseuille flow of an Oldroyd-B fluid. J. Stat. Phys. 175 (3), 554–577.10.1007/s10955-019-02268-6CrossRefGoogle Scholar
National Research Council 1997 Technology for the United States Navy and Marine Corps, 2000–2035: Becoming a 21st-Century Force, vol. 6. National Academies Press.Google Scholar
Page, J., Dubief, Y. & Kerswell, R.R. 2020 Exact traveling wave solutions in viscoelastic channel flow. Phys. Rev. Lett. 125, 154501.CrossRefGoogle ScholarPubMed
Pakdel, P. & McKinley, G.H. 1996 Elastic instability and curved streamlines. Phys. Rev. Lett. 77 (12), 2459.CrossRefGoogle ScholarPubMed
Philip, J., Svizher, A. & Cohen, J. 2007 Scaling law for a subcritical transition in plane Poiseuille flow. Phys. Rev. Lett. 98 (15), 154502.CrossRefGoogle ScholarPubMed
Reynolds, W. & Potter, M.C. 1967 Finite-amplitude instability of parallel shear flows. J. Fluid Mech. 27 (3), 465–492.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), 10557–10562.10.1073/pnas.1219666110CrossRefGoogle ScholarPubMed
Sánchez, H.A.C., Jovanović, M.R., Kumar, S., Morozov, A., Shankar, V., Subramanian, G. & Wilson, H.J. 2022 Understanding viscoelastic flow instabilities: Oldroyd-B and beyond. J. Non-Newtonian Fluid Mech. 302, 104742.CrossRefGoogle Scholar
Shekar, A., McMullen, R.M., McKeon, B.J. & Graham, M.D. 2021 Tollmien–Schlichting route to elastoinertial turbulence in channel flow. Phys. Rev. Fluids 6, 093301.10.1103/PhysRevFluids.6.093301CrossRefGoogle Scholar
Shekar, A., McMullen, R.M., Wang, S.-N., McKeon, B.J. & Graham, M.D. 2019 Critical-layer structures and mechanisms in elastoinertial turbulence. Phys. Rev. Lett. 122, 124503.CrossRefGoogle ScholarPubMed
Shnapp, R. & Steinberg, V. 2021 Non-modal elastic instability and elastic waves in weakly perturbed channel flow. arXiv:2106.01817v2.CrossRefGoogle Scholar
Steinberg, V. 2021 Elastic turbulence: an experimental view on inertialess random flow. Annu. Rev. Fluid Mech. 53, 27–58.CrossRefGoogle Scholar
Stuart, J.T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The basic behaviour in plane Poiseuille flow. J. Fluid Mech. 9 (3), 353–370.CrossRefGoogle Scholar
Toms, B.A. 1949 Some observations of the flow of linear polymer solution through straight tubes at large Reynolds numbers. In Proceedings of the First International Congress on Rheology (North-Holland, Amsterdam, 1949), pp. 135–141.Google Scholar
Trefethen, L.N. 2000 Spectral Methods in MATLAB. SIAM.10.1137/1.9780898719598CrossRefGoogle Scholar
Virk, P.S. 1975 Drag reduction fundamentals. AIChE J. 21 (4), 625–656.10.1002/aic.690210402CrossRefGoogle Scholar
Waleffe, F. & Wang, J. 2005 Transition threshold and the self-sustaining process. In IUTAM Symposium on Laminar-Turbulent Transition and Finite Amplitude Solutions, pp. 85–106. Springer.CrossRefGoogle Scholar
Wan, D., Sun, G. & Zhang, M. 2021 Subcritical and supercritical bifurcations in axisymmetric viscoelastic pipe flows. J. Fluid Mech. 929, A16.10.1017/jfm.2021.852CrossRefGoogle 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
Zhang, M. 2021 Energy growth in subcritical viscoelastic pipe flows. J. Non-Newtonian Fluid Mech. 294, 104581.CrossRefGoogle Scholar