No 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.

This work is concerned with the numerical solution of the K-BKZ integral constitutive equation for two-dimensional time-dependent free surface flows. The numerical method proposed herein is a finite difference technique for simulating flows possessing moving surfaces that can interact with solid walls. The main characteristics of the methodology employed are: the momentum and mass conservation equations are solved by an implicit method; the pressure boundary condition on the free surface is implicitly coupled with the Poisson equation for obtaining the pressure field from mass conservation; a novel scheme for defining the past times t is employed; the Finger tensor is calculated by the deformation fields method and is advanced in time by a second-order Runge-Kutta method. This new technique is verified by solving shear and uniaxial elongational flows. Furthermore, an analytic solution for fully developed channel flow is obtained that is employed in the verification and assessment of convergence with mesh refinement of the numerical solution. For free surface flows, the assessment of convergence with mesh refinement relies on a jet impinging on a rigid surface and a comparison of the simulation of a extrudate swell problem studied by Mitsoulis (2010) [44] was performed. Finally, the new code is used to investigate in detail the jet buckling phenomenon of K-BKZ fluids.

The aim of this work is to develop a numerical technique for simulating incompressible, isothermal, free surface (also confined) viscoelastic flows of fluids governed by the integral models of Maxwell and K-BKZ (Kaye-Bernstein, Kearsley and Zapas). The numerical technique described herein is an extension of the GENSMAC method (Tome and McKee, J. Comput. Phys., 110, pp. 171-186, 1994) to the solution of the momentuum and mass conservation equations together with the integral constitutive Maxwell and K-BKZ equations. The governing equations are solved by the finite difference method on a staggered grid using a Marker-and-Cell approach. The fluid is represented by marker particles on the fluid surface only. This provides the visualization and location of the fluid free surface so that the free surface stress conditions can be applied. The Finger tensor B t (t) is computed using the ideias of the deformation fields method (Peters et al. J. Non-Newtonian Fluid Mech., 89, pp. 209-228, 2001) so that it is not necessary to track a fluid particle in order to calculate its deformation history. However, in this work modifications to the deformation fields method are introduced: the past time is discretized using a different formula, the Finger tensor B t (x, t) is obtained by a second order method and the stress tensor τ (x, t) is computed by a second order quadrature formula. The numerical method presented in this work is validated by simulating the flow of a Maxwell fluid in a two-dimensional channel and the numerical solutions of the velocity and the stress components are compared with the respective analytic solutions providing a good agreement. Further, the flow through a 4:1 planar contraction of a specific fluid studied experimentally by Quinzani et al. (J. Non-Newtonian Fluid Mech., 52, pp. 1-36, 1994) was simulated and the numerical results were compared qualitatively and quantitatively with the experimental results and very good agreement was obtained. The Maxwell and the K-BKZ models were applied to simulate the 4:1 planar contraction problem using various Weissenberg numbers and the numerical results were in agreement with those published in the literature. Finally, numerical results of free surface flows using the Maxwell and K-BKZ integral constitutive equations are presented. In particular, the numerical simulation of jet buckling using several Weissenberg numbers and various Reynolds numbers are presented.

Resumo: Esse trabalho tem por objetivo apresentar um método numérico para resolver as equações governantes para escoamentos viscoelásticos, com superfícies livres em movimento, governados pela equação constitutiva integral K-BKZ. Considera-se escoamentos bidimensionais e um algoritmo para resolver as equações governantes, utilizando o método de diferenças finitas,é apresentado. A metodologia propostaé verificada comparando-se a solução numérica do escoamento totalmente desenvolvido em um canal. Resultados provenientes da simulação do problema conhecido como 'jet-buckling' são apresentados.Palavras-chave: Modelo integral K-BKZ, diferenças finitas, escoamentos viscoelásticos, superfície livre.