A numerical study of steady and unsteady viscoelastic flow past bounded cylinders

Abstract

We consider two-dimensional, inertia-free, flow of a constant-viscosity viscoelastic fluid obeying the FENE-CR equation past a cylinder placed symmetrically in a channel, with a blockage ratio of 0.5. Through numerical simulations we show that the flow becomes unsteady when the Deborah number (using the usual definition) is greater than De ≈ 1.3, for an extensibility parameter of the model of L 2 = 144. The transition from steady to unsteady flow is characterised by a small pulsating recirculation zone of size approximately equal to 0.15 cylinder radius attached to the downstream face of the cylinder. There is also a rise in drag coefficient, which shows a sinusoidal variation with time. The results suggest a possible triggering mechanism leading to the steady three-dimensional Gortler-type vortical structures, which have been observed in experiments of the flow of a viscoelastic fluid around cylinders. The results reveal that the reason for failure of the search for steady numerical solutions at relatively high Deborah numbers is that the two-dimensional flow separates and eventually becomes unsteady. For a lower extensibility parameter, L 2 = 100, a similar recirculation is formed given rise to a small standing eddy behind the cylinder which becomes unsteady and pulsates in time for Deborah numbers larger than De ≈ 4.0-4.5.

A semi-staggered dilation-free finite volume method for the numerical solution of viscoelastic fluid flows on all-hexahedral elements

“…5 at W e = 0.9. Although Oliveira and Miranda [21] have recently showed a small time-periodic separation bubble behind the cylinder at a Deborah number of 1.3 for the FENE-CR model with L 2 = 144, we did not observe any similar flow structures for the Oldroyd-B fluid; our calculations beyond a Weissenberg number of 0.9 did not converge to any steady or time-periodic state on any of the meshes.…”

“…19 of [33] at x = 1.5R, just behind the cylinder. This is not seen in the two-dimensional numerical simulations of Oliveira and Miranda [21] with the FENE-CR model. The three dimensional effects may also be seen from the computed contour surfaces of stress component T xx and isobaric surfaces in Fig.…”

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“…Figure 4 also shows a comparison between our predictions of τ xx along the rear wake centreline with those of Dou and Phan-Thien (2003) on their mesh M4; a significant difference of the order of 50% is observed due to the high level of azimuthal refinement of M120 WR mesh. Previous works have shown mesh refinement not to be so crucial with constitutive models exhibiting shearthinning or bounded extensional viscosity (Oliveira et al 1998, Oliveira 2001, Oliveira and Miranda 2005and Alves et al 2000, 2001, such as FENE-CR, PTT or Giesekus models, and therefore, we decided not to use the finer mesh M120 WR and restrict the simulations with these models to meshes M60 and M60 WR . This decision is corroborated by the comparisons shown in Those authors also employed a "high resolution" FEM, and the matching against our data is excellent.…”

“…A slightly modified version of the FENE-CR model was proposed and used by Coates et al (1992) and later by a number of other authors in a variety of studies, such as in cylinder and sphere problems, both with uniform and Poiseuille approach flow conditions (McKinley et al 1993, Kim et al 2005a, Oliveira and Miranda 2005, Satrape and Crochet 1994, Harlen 2002and Dou and Phan-Thien 2004. In all these works, the constant viscosity FENE-CR model predicted the onset of negative wake in the velocity profiles for sufficiently low levels of the extensibility parameter.…”

“…The flow of a viscoelastic fluid past a cylinder has been extensively studied both experimentally (Manero and Mena (1981), Bush (1993), McKinley et al (1993) and Broadbent and Mena (1974)) and numerically (Bush (1993), Hu and Joseph (1990), Huang and Feng (1995), Liu et al (1998), Oliveira et al (1998), Fan et al (1999), Sun et al (1999), Alves et al (2001), Caola et al (2001), Owens et al (2002), Phan-Thien and Dou (1999), Dou and Phan-Thien (2003), Kim et al (2004Kim et al ( , 2005a, Oliveira and Miranda (2005) and Gerritsma (2006)). The large number of works in this geometry may be explained from two major motivations: (1) it is representative of the fundamental flow dynamics of viscoelastic fluids around solid bodies, and (2) it is intrinsically related to many processes in chemical engineering, namely, flows through porous media (McKinley et al 1993), enhanced oil recovery, composite and textile coating operations (Liu et al 1998) and food processes.…”