Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-02T23:29:49.328Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic study of linear instability in a viscoelastic pipe flow

Published online by Cambridge University Press:  31 January 2022

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]
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 addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

It is recently found that viscoelastic pipe flows can be linearly unstable, leading to the possibility of a supercritical transition route, in contrast to Newtonian pipe flows. Such an instability is referred to as the centre mode, which was studied numerically by Chaudhary et al. (J. Fluid Mech., vol. 908, 2021, p. A11) based on an Oldroyd-B model. In this paper, we are interested in expanding the parameter space investigated and exploring the asymptotic scalings related to this centre instability in the Oldroyd-B viscoelastic pipe flow. It is found from the asymptotic analysis that the centre mode exhibits a three-layered asymptotic structure in the radial direction, a wall layer, a main layer and a central layer, which are driven by pure viscosity, axial and/or radial pressure gradient, and a combined effect of viscosity and elasticity, respectively. Depending on the relations of the control parameters, two regimes, the long-wavelength and short-wavelength centre instabilities, emerge, for which the central-layer thicknesses are of different orders of magnitude. Our large-Reynolds-number asymptotic predictions are compared to the numerical solutions of the original eigenvalue system, and favourable agreement is achieved, especially when the parameters approach their individual limits. In addition to revealing the dominant factors and their balances, the asymptotic analysis describes the instability system by reducing the number of control parameters, and furthermore explaining the collapse of the numerical results for different re-scalings.

JFM classification

Non-Newtonian Flows: Viscoelasticity Non-Newtonian Flows: Polymers Instability: Shear-flow instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 935 , 25 March 2022 , A28
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

Bird, R., Curtiss, C., Armstrong, R. & Hassager, O. 1987 Dynamics of Polymeric Liquids, Volume 2: Kinetic Theory. Wiley.Google Scholar
Buza, G., Page, J. & Kerswell, R.R. 2021 Weakly nonlinear analysis of the viscoelastic instability in channel flow for finite and vanishing Reynolds numbers. arXiv:2017.06191v1.Google Scholar
Chandra, B., Shankar, V. & Das, D. 2018 Onset of transition in the flow of polymer solutions through microtubes. J. Fluid Mech. 844, 1052–1083.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. & Hof, B. 2018 Exceeding the asymptotic limit of polymer drag reduction. Phys. Rev. Lett. 120, 124501.CrossRefGoogle ScholarPubMed
Choueiri, G.H., Lopez, J.M., Varshney, A., Sankar, S. & Hof, B. 2021 Experimental observation of the origin and structure of elasto-inertial turbulence. Proc. Natl Acad. Sci. USA 118, 1–5.CrossRefGoogle Scholar
Dong, M., Liu, Y. & Wu, X. 2020 Receptivity of inviscid modes in supersonic boundary layers due to scattering of freestream sound by wall roughness. J. Fluid Mech. 896, A23.CrossRefGoogle Scholar
Dong, M. & Wu, X. 2013 On continuous spectra of the Orr–Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances. J. Fluid Mech. 732, 616–659.CrossRefGoogle Scholar
Drazin, P.G. & Reid, W.H. 2004 Hydrodynamic Stability (Cambridge Mathematical Library). Cambridge University Press.CrossRefGoogle Scholar
Dubief, Y., Page, J., Kerswell, R.R., Terrapon, V.E. & Steinberg, V. 2021 A first coherent structure in elasto-inertial turbulence. arXiv:2006.06770v2.Google Scholar
Dubief, Y., Terrapon, V.E. & Soria, J. 2013 On the mechanism of elasto-inertial turbulence. Phys. Fluids 25 (11), 110817–110817.CrossRefGoogle ScholarPubMed
Garg, P., Chaudhary, I., Khalid, M., Shankar, V. & Subramanian, G. 2018 Viscoelastic pipe flow is linearly unstable. Phys. Rev. Lett. 121, 024502.CrossRefGoogle ScholarPubMed
Goldstein, M.E. 1985 Scattering of acoustic waves into Tollmien–Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech. 154, 509–530.CrossRefGoogle Scholar
Graham, M.D. 2014 Drag reduction and the dynamics of turbulence in simple and complex fluids. Phys. Fluids 26 (10), 101301.CrossRefGoogle Scholar
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405 (6782), 53–55.CrossRefGoogle 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.CrossRefGoogle 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
Kupferman, R. 2005 On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation. J. Non-Newtonian Fluid Mech. 127 (2), 169–190.CrossRefGoogle Scholar
Lin, C.C. 1946 On the stability of two-dimensional parallel flows. Part III. Stability in a viscous fluid. Q. Appl. Maths 3, 277–301.CrossRefGoogle Scholar
Liu, Y., Dong, M. & Wu, X. 2020 Generation of first Mack modes in supersonic boundary layers by slow acoustic waves interacting with streamwise isolated wall roughness. J. Fluid Mech. 888, A10.CrossRefGoogle 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. Fluid Mech. 874, 699–719.CrossRefGoogle Scholar
Mack, L.M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497–520.CrossRefGoogle Scholar
Meseguer, Á. & Trefethen, L.N. 2003 Linearized pipe flow to Reynolds number 107. J. Comput. Phys. 186 (1), 178–197.CrossRefGoogle Scholar
Mohseni, K. & Colonius, T. 2000 Numerical treatment of polar coordinate singularities. J. Comput. Phys. 157 (2), 787–795.CrossRefGoogle Scholar
Page, J., Dubief, Y. & Kerswell, R.R. 2020 Exact traveling wave solutions in viscoelastic channel flow. Phys. Rev. Lett. 125, 154501.CrossRefGoogle ScholarPubMed
Ram, A. & Tamir, A. 1964 Structural turbulence in polymer solutions. J. Appl. Polym. Sci. 8 (6), 2751–2762.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
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. 2021 Tollmien–Schlichting route to elastoinertial turbulence in channel flow. Phys. Rev. Fluids 6, 093301.CrossRefGoogle 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
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
Smith, S. 1979 On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366, 91–109.Google Scholar
Smith, F. 1981 The upper branch stability of the Blasius boundary layer, including non-parallel flow effects. Proc. R. Soc. Lond. A 375, 65–92.Google Scholar
Sureshkumar, R. & Beris, A.N. 1995 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 (1), 53–80.CrossRefGoogle Scholar
Terrapon, V.E., Dubief, Y. & Soria, J. 2014 On the role of pressure in elasto-inertial turbulence. J. Turbul. 16, 26–43.CrossRefGoogle Scholar
Trefethen, L. 2000 Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Virk, P.S. 1975 Drag reduction fundamentals. AIChE J. 21 (4), 625–656.CrossRefGoogle Scholar
Wan, D., Sun, G. & Zhang, M. 2021 Subcritical and supercritical bifurcations in axisymmetric viscoelastic pipe flows. J. Fluid Mech. 929, A16.CrossRefGoogle Scholar
White, C.M. & Mungal, M.G. 2008 Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40 (1), 235–256.CrossRefGoogle Scholar
Wu, X. & Dong, M. 2016 Entrainment of short-wavelength free-stream vortical disturbances in compressible and incompressible boundary layers. J. Fluid Mech. 797, 683–782.CrossRefGoogle Scholar
Xi, L. & Graham, M.D. 2010 a 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. 2010 b Turbulent drag reduction and multistage transitions in viscoelastic minimal flow units. J. Fluid Mech. 647, 421–452.CrossRefGoogle Scholar
Zhang, M. 2021 Energy growth in viscoelastic pipe flows. J. Non-Newtonian Fluid Mech. 294, 104581.CrossRefGoogle Scholar
Zhang, M., Lashgari, I., Zaki, T. & Brandt, L. 2013 Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids. J. Fluid Mech. 737, 279–279.CrossRefGoogle Scholar
Zhu, L. & Xi, L. 2021 Nonasymptotic elastoinertial turbulence for asymptotic drag reduction. Phys. Rev. Fluids 6, 014601.CrossRefGoogle Scholar