Speaker
Description
Non-constant viscosity plays a crucial role in numerous incompressible flow problems of interest in science and engineering, with prime examples being polymer or blood flow. In such applications, the fluid’s viscosity depends on the local shear rate, where generalized Newtonian fluid models are considered an acceptable middle-ground between complexity and model capability. Standard discretization based on coupled velocity–pressure formulations in an implicit finite element scheme lead to saddle-point systems posing a challenge to state-of-the-art solvers and preconditioners.
Projection schemes, on the contrary, decouple the balance equations governing velocity and pressure, giving rise to multiple, simpler, and sequentially solved problems incorporating standard linear systems. Following such an approach requires caution when deriving suitable boundary conditions for the intermediate steps to preserve accuracy. For Newtonian incompressible fluids, several projection and splitting methods of high-order accuracy are available (see, e.g. [1, 2]). The extension to generalised Newtonian fluids, however, is often found to be non-trivial, as such schemes often build on the assumption of constant viscosity.
In this contribution we address this shortcoming and present an extension of the work by Karniadakis et al. [1] towards generalized Newtonian fluids. The presented method is thus based on an explicit-implicit treatment of convective, viscous, and pressure terms and includes the typical pressure Poisson and projection steps. Numerical results obtained through a matrix-free implementation in ExaDG [3] showcase temporal stability, accuracy, and efficiency of the higher-order splitting scheme in challenging numerical examples of practical interest.
REFERENCES
[1] Karniadakis, G. E., Israeli, M. and Orszag, S. A. High-order splitting methods for theincompressible Navier-Stokes equations. J. Comput. Phys., 1991.
[2] Timmermans, L.J.P., Minev, P.D. and Van de Vosse, F. N. An approximate projectionscheme for incompressible flow using spectral elements. Int. J. Numer. Methods Fluids,Vol. 22(7), pp. 673–688, 1996.
[3] Arndt, D., Fehn, N., Kanschat, G. and others. ExaDG: High-order discontinuous Galerkin for the exa-scale. In Bungartz, H.-J., Reiz, S., Uekermann and others (eds.), Software for exascale computing-SPPEXA 2016–2019, pp 189–224, 2020, Cham, Springer.
