Improved both sides diffusion (iBSD): A new and straightforward stabilization approach for viscoelastic fluid flows

Abstract

This paper reports the developments made to improve the numerical stability of the open-source finite-volume computational library OpenFOAM ® developed for the numerical computation of viscoelastic fluid flows described by differential constitutive models. The improvements are based on the modification of the both-sides diffusion technique, named improved both-sides diffusion (iBSD), which promotes the coupling between velocity and stress fields. Calculations for two benchmark 2D case studies of an upper-convected Maxwell (UCM) fluid are presented and compared with literature results, namely the 4:1 planar contraction flow and the flow around a confined cylinder. The results obtained for the first case are computed in five meshes with different refinement levels and are compared with literature results. In this case study it was possible to achieve steady-state converged solutions in the range of Deborah numbers tested, = De {0, 1, 2, 3, 4, 5}, for all meshes. The corner vortex size predictions agree well with the literature and a relative error below 0.6% is obtained for De ≤ 5. In the flow around a confined cylinder, steady-state converged solutions were obtained in the range of Deborah numbers tested, = De {0, 0.3, 0.6, 0.8}, in four consecutively refined meshes. The predictions of the drag coefficient on the cylinder are similar to reference data with a relative error below 0.08%. For both test cases the developed numerical method was shown to have a convergence order between 1 and 2, in general very close to the latter. Moreover, the results presented for both case studies clearly extend the previous ones available in the literature in terms of accuracy. This was a direct consequence of the capability of performing the calculation with more refined meshes, than the ones employed before.

A finite volume method for the simulation of elastoviscoplastic flows and its application to the lid-driven cavity case

“…[49,50]), similarly to the original momentum interpolation [45], derived the coefficient from the SIMPLE matrix of the linearised discrete constitutive equation, arriving at complicated expressions. The present simpler approach is essentially equivalent to that adopted by [51,52] who, for viscoelastic flows, set the coefficient η a equal to the polymeric viscosity. The aim is that D τ + f and D τ − f are of the same order of magnitude as the EVP stress acting on the cell face, and this can be achieved using a characteristic viscosity η a defined as the ratio of a typical stress to a typical rate of strain for the given problem.…”

A coupled finite volume flow solver for the solution of incompressible viscoelastic flows

“…where ⋆ is a viscosity coefficient proportional to the polymer viscosity P , here defined as ⋆ = . The new stress-velocity coupling formulation developed by Fernandes et al [50] is used to couple the velocity and stress fields. In practice, the stress-velocity coupling is achieved by replacing the explicit diffusive term of the BSD technique with a special second-order derivative of the velocity field, as described in Section 3.1 , a methodology known as iBSD [50] , improved both-sides diffusion.…”

“…Finally, the explicit diffusion term present in Eq. 9is given by a special second-order derivative, which contributes to the coupling between the velocity and the extra-stress tensor fields, as described in detail in Fernandes et al [50] . The discretization of this term is given by…”

“…These works showed that when dealing with inelastic fluids the computational time can be reduced using a fully coupled solver. Therefore, it is worth to expand the method to viscoelastic flows for which the segregated algorithm, employed until now, presents several stability issues [50] , that requires the use of large relaxation factors and limits the maximum achievable Deborah/Weissenberg numbers if the log-conformation approach is not used [43] .…”

“…The performance of the coupled viscoelastic solver is assessed with three case studies, namely the Oldroyd-B Poiseuille, the UCM lid-driven cavity and the LPTT 4:1 sudden contraction planar flows. The main target is to assess the performance, in terms of the number of iterations and CPU time, of the coupled algorithm with increasing mesh density and Deborah number, when compared with the previously developed segregated solver [50] based on the SIMPLE algorithm. Additionally, for assessment purposes, the results obtained with the developed code are compared with analytical solutions, results obtained with the segregated viscoelastic solver [50] , and with some additional solutions presented in the scientific literature.…”