Vortex behavior of the Oldroyd-B fluid in the 4-1 planar contraction simulated with the streamfunction-log-conformation formulation

Abstract

In this paper, we present numerical solutions of the Oldroyd-B fluid flowing through a 4:1 planar contraction, for Weissenberg numbers (Wi) up to 20. The incompressible viscoelastic flows are simulated with the streamfunction-log-conformation methodology. The log-conformation representation guarantees by construction the positive-definiteness of the conformation tensor, which circumvents the appearance of the high Weissenberg number problem. In addition, the streamfunction flow formulation removes the pressure variable from the governing equations and automatically satisfies the mass conservation. Thus, the streamfunction-log-conformation reformulation is beneficial for the accuracy and stability of the numerical algorithm. The resulting governing equations are solved with a high-resolution finite-volume method. Our numerical results for the reattachment length and the intensity of the recirculation vortices produced at the contraction plane are in excellent agreement with the benchmark solutions, available in the literature for Weissenberg numbers up to 3. For highly elastic flows, our results agree qualitatively well with the data of Afonso et al. (2011) [53]. Our simulations predict the reduction of the vortex size with increasing Wi, up to Wi approximate to 5. Moreover, we observe a periodic third vortex growth and annihilation regime for Wi >= 15. The periodic vortex growth and annihilation is correlated with the accumulation of elastic strain in the cavity upstream of the contraction. This elastic instability is viewed as a mechanism that releases the elastic energy accumulated in the Oldroyd-B fluid at the fringe of the recirculation vortices. The dimensionless period of the third vortex annihilation appears to be independent on the Weissenberg number.