An efficient iterative solver and high-resolution computations of the Oldroyd-B fluid flow past a confined cylinder

An efficient iterative solution method is developed to obtain high-resolution solutions of viscoelastic flow problems, and the accuracy as well as the performance is investigated for highly refined finite elements. The formulation is based on DEVSS-G/SUPG, and the problem size is expanded up to the extremely large-scale problem composed of 1,362,480 unknowns with 84,531 finite elements, much more than the previous results [F.P.T. Baaijens, An iterative solver for the DEVSS/DG method with application to smooth and non-smooth flows of the upper convected Maxwell fluid, J. Non-Newtonian Fluid Mech. 75 (1998) 119; A.E. Caola, Y.L. Joo, R.C. Armstrong, R.A. Brown, Highly parallel time integration of viscoelastic flows, J. Non-Newtonian Fluid Mech. 100 (2001) 191]. An iterative solver [Y.S. Nam, H.G. Choi, J.Y. Yoo, AILU preconditioning for the finite element formulation of the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng. 191 (2002) 4323] with a mixed formulation originally developed for Newtonian flow simulation and BiCGSTAB method is implemented as the solution method. The preconditioner of adaptive incomplete LU [Y.S. Nam, H.G. Choi, J.Y. Yoo, AILU preconditioning for the finite element formulation of the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng. 191 (2002) 4323] is modified for DEVSS-G/SUPG. The modified adaptive incomplete LU(1)/BiCGSTAB with the variable reordering scheme is excellently applicable for a wide range of Weissenberg numbers (We) with the finest mesh (We ∼ 0.875). The solution method is applied to the well-known two-dimensional Oldroyd-B fluid flow past a confined cylinder problem. The obtained dimensionless drag with different meshes varies within 0.013% at all converged We and shows a mesh convergent behavior of extra stresses in the wake region up to We = 0.7. At We larger than 0.7, complete mesh convergence is not achieved, but the prediction of dimensionless drag agrees well with some of the previous reports based on other methods. As there exists a discrepancy in the prediction of drag at high We among the research groups, it is meaningful that some of the results agree well with each other, even though they are computed from different methods. In addition, we could observe a singular behavior of extra stresses at high elasticity with highly refined meshes, and this singular behavior is suspected to be the origin of the failure of convergence at high We. This may be related with the unboundedness of the extensional viscosity in the Oldroyd-B model or with other numerical origin in the very refined meshes. To figure out whether this singular behavior or the failure originates from the constitutive equation or from other numerical origin in the very refined meshes, it will be necessary to solve the problems with other constitutive equations or with much more refined meshes.