J. Non-Newtonian Fluid Mech. 138 (2006) 111–133Time-dependent simulations of non-axisymmetric patterns inTaylor–Couette flow of dilute polymer solutionsD.G. Thomas 1 , U.A. Al-Mubaiyedh 2 , R. Sureshkumar ∗ , B. KhomamiDepartment of Chemical Engineering and the Center for Materials Innovation, Washington University, St. Louis, MO 63130, USAReceived 25 July 2005; received in revised form 24 March 2006; accepted 10 April 2006AbstractNonlinear dynamics that ensue after the inception of viscoelastic flow instabilities in homogeneous, curvilinear shear flows remain largelyunexplored. In this work, we have developed an efficient, operator splitting influence matrix spectral (OSIMS) algorithm for the simulation ofthree-dimensional and transient viscoelastic flows. The OSIMS algorithm is applied to explore, for the first time, the post-critical dynamics ofviscoelastic Taylor–Couette flow of dilute polymeric solutions utilizing the Oldroyd-B constitutive equation. Linear stability theory predicts thatthe flow is unstable to non-axisymmetric and time-dependent disturbances with critical conditions depending on the flow elasticity, E, defined asthe ratio of the characteristic time scales of fluid relaxation to viscous diffusion. Two types of secondary flow patterns emerge near the bifurcationpoint,namely,ribbonsandspirals.Wehavedemonstratedviatime-dependentsimulationsfornarrowandmoderategapwidths,ribbon-likepatternsare generally stable at and above the linear stability threshold for 0.05≤E≤0.15. For an inner to outer cylinder radius ratio of 0.8, the bifurcationto ribbons at E=0.1 and 0.125 occurs through a subcritical transition while the transition is supercritical at smaller E values.© 2006 Elsevier B.V. All rights reserved.Keywords: Stability; Non-axisymmetric; Viscoelastic; Taylor–Couette; Influence matrix; Spectral; Bifurcation; Artificial diffusivity; Elastic turbulence1. IntroductionThe addition of high molecular weight polymers to a Newto-nian flow can qualitatively alter the sequence of flow transitionsas well as the critical points. Depending on the flow elasticity,characterized by the ratio of the time scale of fluid relaxationto that of viscous diffusion, and inertia, such flow transitionscanleadtoturbulentstateswhosetime-averagedproperties(e.g.velocity profile, drag) and (energy) spectral characteristics aredramatically different from those in the Newtonian case. In fact,it has been shown experimentally that viscoelastic flow transi-tions in curvilinear shear flows could result in the establishmentof turbulent flow states even when the Reynolds number, Re(ratio of inertial to viscous forces) is vanishingly small [1]. Thisphenomenon, referred to as “elastic turbulence”, has eludedexplanation based on first principle modeling/simulations. To∗Corresponding author.E-mail addresses: suresh@che.wustl.edu (R. Sureshkumar),bam@che.wustl.edu (B. Khomami).1Present address: Department of Biomedical Engineering Washington Uni-versity, St. Louis, MO 63130, USA.2Present address: Department of Chemical Engineering, King Fahd Univer-sity of Petroleum and Minerals, Dahran, Saudi Arabia.date, the literature on the direct numerical simulation (DNS) ofthree-dimensional (3D) and time-dependent viscoelastic flowsislimitedtohomogeneousshear[2]andpressure-drivenchannel[3,4] flows focusing primarily on the study of polymer-inducedturbulent drag reduction. In this paper, we report the first suc-cessful simulation of 3D, time-dependent flow patterns in aviscoelastic, curvilinear shear flow.For several decades, unidirectional shear flows with (primar-ily)curvedstreamlinesofsimple(Newtonian)andcomplex(e.g.polymeric liquids) fluids have served as classical paradigmsfor the investigation of hydrodynamic instabilities and pat-tern formation [5]. Prominent among the paradigms used forcurvilinear flows is the Taylor–Couette flow in which a fluidconfined between two long concentric cylinders is sheared bytheir relative rotation. Taylor [6], in a landmark paper in 1923,showed both theoretically as well as experimentally that theprimary (base) azimuthal shear flow of a Newtonian fluid ina Taylor–Couette cell with the inner cylinder rotating and theouterstationary,becomesunstabletoaxisymmetricdisturbanceswhen Re √ d/R 1 ≈ 41 where d is the radial gap width and R 1 istheinnercylinderradius.Theinstabilitymanifestsintheformofsteadytoroidalvortexcells(Taylorvortices)intheaxialdirectionwhich replace the circular, base Couette flow. Theoretically, thestabilitythresholdisdeterminedbasedonanormalmodepertur-0377-0257/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.jnnfm.2006.04.013
