SPH simulations of a viscoelastic flow around a periodic array of cylinders confined in a channel A. Vázquez-Quesada, M. Ellero ⇑ Lehrstuhl für Aerodynamik und Strömungsmechanik, Technische Universität München, 85747 Garching, Germany article info Article history: Received 8 August 2011 Received in revised form 12 September 2011 Accepted 13 September 2011 Available online 5 October 2011 Keywords: Smoothed Particle Hydrodynamics Oldroyd-B model Elastic instabilities abstract In this paper a numerical study on the performance of the Smoothed Particle Hydrodynamics method (SPH) in the case of a flow of a viscoelastic liquid around a linear array of cylinders confined in a channel is presented. Numerical convergence in the case of a low Reynolds number Newtonian flow was demon- strated by Ellero et al. in [Int. J. Numer. Methods Eng. 86 (2011) 1027]. Here viscoelastic effects are incor- porated in the SPH scheme according to the Oldroyd-B model presented in [Phys. Rev. E 79 (2009) 056707]. Good agreement of the dimensionless drag force acting on the cylinder with literature data is observed for a wide range of Weissenberg numbers We. The case of closely spaced cylinders is also inves- tigated and the impact of We on the solution discussed. It turns out that in this case the Newtonian solu- tion exhibits a stable secondary flow represented by two counter-rotating vortices. When elasticity is considered, above a critical Weissenberg number We c these vortices become unstable, breaking the plane symmetry and producing a quasi-periodic flow of mass in and out of the gap region between cylinders. Correspondingly, the flow becomes increasingly unsteady and a dramatic increase in the cylinder’s drag is observed. The results are in qualitative agreement with experimental observations made on Boger liquids under similar flow conditions. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction The study of viscoelastic flows in complex geometries is of cru- cial importance for the proper modelling and accurate prediction of complex liquid behaviours. Flow of polymeric suspensions through porous media, contraction/expansion geometries are em- ployed in a variety of engineering applications including composite manufacturing processes, paper coating and recently also in micro- fluidics devices [1,2]. Generally, the rheological behaviour of com- plex fluids can be very well characterised under simple viscometric conditions, that is, under a pure shear or extensional flow. How- ever, modelling the flow in geometries where the two components are simultaneously present represents a big challenge both, from the mathematical and numerical point of view. Recently, several complex flows have been studied which include, for example, the flow through channels with variable cross-section, flow around cylinder arrays both unbounded and confined by plane walls. In particular, the latter test case has been investigated experimentally [3–6] in relation to the occurrence of purely elastic instabilities, namely flow instabilities arising in absence of inertia. The study of elastic instabilities in complex fluids has been reviewed in [7,8] and is receiving increasing attention also due to the strict connection with the phenomenon of elastic turbulence [9]. An empirical explanation has been attempted by McKinley et al. in terms of the possible destabilizing mechanism linking curved flow streamlines to the presence of normal elastic stresses [10], how- ever a detailed numerical analysis is still lacking. Elastic instabili- ties show up generally together with a modification of the global flow behaviour: as a critical Weissenberg number is achieved, flow quantities (e.g. mean flow velocity, pressure drop, etc.) start to ex- hibit fluctuations which increase in magnitude as the effect of li- quid elasticity becomes more pronounced. Moreover, this phenomenon is accompanied by an abrupt increase in the flow resistance which is now believed to be related to a non-linear tran- sition from a steady-state towards a more dissipative time-depen- dent flow structure [5,6]. Although much experimental evidence of this kind of process can be found in the literature, few numerical computations have been able to reproduce these results. For exam- ple, the elastic transition of an Upper-Convected-Maxwell fluid (UCM) towards a steady secondary asymmetric flow field has been simulated in a cross-channel geometry by Alves et al. [11] and found in agreement with previously reported experimental obser- vations [12]. Very recently the instability of a two-dimensional vis- coelastic liquid flowing through a wavy-walled channel has been also investigated [13]. In the context of a cylinder array structure, however, no numer- ical simulations have been able to reproduce, at least qualitatively, 0377-0257/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2011.09.002 ⇑ Corresponding author. E-mail address: marco.ellero@aer.mw.tum.de (M. Ellero). Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 1–8 Contents lists available at SciVerse ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: http://www.elsevier.com/locate/jnnfm