Cartesian grid method with a consistent direct discretization approach for numerical simulation of non-Newtonian fluid flow

Abstract

This paper proposes an extended version of the discretization scheme for a consistent direct discretization approach in a Cartesian grid method for flow simulations of incompressible non-Newtonian fluids. Based on the concept of the original method, which was developed for Newtonian fluid flow, the Navier–Stokes and the pressure Poisson equations are discretized directly in the vicinity of a solid boundary while maintaining the consistency between the velocity and pressure fields. The main difficulty in the discretization arises from the evaluation of the velocity gradients on the fluid–solid interface, which is necessary for handling the viscous term for spatially distributed shear-dependent viscosity. In the extended method, a new procedure for the calculation of the velocity gradient using the relation that holds on a rigidly moving solid surface is proposed. The validity of the method is ascertained in fundamental test problems of non-Newtonian fluids: laminar Poiseuille/Couette flows in a concentric annulus and flows around a circular cylinder. It is found that a second-order spatial accuracy with the desired numerical stability is achieved in both the shear-thinning and shear-thickening viscous properties, and the numerical error is considerably decreased compared with some conventional methods. As a realistic application of the extended method, a polymer mixing simulation inside a co-rotating twin-screw extruder is demonstrated by applying the method to the moving boundary problem. In a screw arrangement with a sequential combination of a kneading disk (KD), a cylindrical ring (CR), and a backward mixing screw (BMS), large- and small-scale structures of bifurcating and converging flows are respectively formed in the KD and BMS zones, whereas elongational flow structures and velocity fluctuations are still maintained in the CR zone due to flow interaction with the upstream and downstream screw elements, which presumably contributes to moderate mixing at low shear stress. • A Cartesian grid method for non-Newtonian fluid flow is developed. • The present method extends a consistent direct discretization approach. • A procedure for evaluating velocity gradients on an immersed boundary is proposed. • Second order spatial accuracies for velocity and its derivatives are ascertained. • A polymer mixing simulation in a co-rotating twin-screw extruder is demonstrated.