Semi-implicit stress formulation for viscoelastic models: Application to three-dimensional contraction flows

In this work we propose a new formulation for the divergence of the viscoelastic stress for the collocated (cell-centered) finite-volume method. The reformulation allows for a semi-implicit handling of the constitutive equation, which promotes the numerical stability. Simulations of a three-dimensional planar and a square-square contraction show the robustness of this technique. The new formulation is completely devoid of unphysical checkerboard patterns of the velocity, which are present when using standard approximations for the divergence in conjunction with non-staggered grid methods. The consistency is ensured by giving results, which are independent of the time-step δ. t for steady-state problems. The results for the planar contraction are generally in good agreement with experimental data for velocity, shear stress and first-normal stress difference. Stable simulations for the square-square contraction could be performed over a wide range of Deborah numbers. The vortex length is in agreement with the experimental results in the Newtonian-like and vortex-enhancement flow regime, however, the results deviate from the experiments in the diverging streamline regime.