An improved splitting algorithm for unsteady generalized Newtonian fluid flow problems with natural boundary conditions

Generalized Newtonian fluids are challenging to solve using the standard projection or fractional-step methods which split the diffusion term from the incompressibility constraint during the time integration process. Most of this class numerical methods already suffer from some inconsistencies, even in the Newtonian case, due to unphysical pressure boundary conditions which deteriorate the quality of approximations especially when open boundary conditions are prescribed in the problem under study. The present study proposes an improved viscosity-splitting approach for solving the generalized Newtonian fluids in which the viscosity follows a nonlinear generic rheological law. This method consists of decoupling the convective effects from the incompressibility while keeping a diffusion term in the last step allowing to enforce consistent boundary conditions. We provide a full algorithmic description of the method accounting for both Dirichlet and Neumann boundary conditions. To evaluate the computational performance of the proposed viscosity-splitting algorithm, we present numerical results for an example with manufactured exact solution and for the benchmark problems of lid-driven cavity flow and flow past a circular cylinder. We also assess the accuracy of the method for an unsteady flow around an arrangement of two cylinders in tandem and comparisons with results obtained using a monolithic approach reveal good general agreement.