A least-squares particle model with other techniques for 2D viscoelastic fluid/free surface flow

Abstract

In this work, a Lagrangian finite pointset model (FPM) is first developed to solve the 2D viscoelastic fluid governing equations, and then a corrected particle shifting technique (CPST) is introduced to eliminate the tensile instability in the long time simulations and added in the above FPM scheme (named as the FPMT). Subsequently, a coupled particle method (FPMT-SPH) is tentatively proposed to simulate the viscoelastic free surface flow, in which the FPMT method is used in the interior of fluid domain and the SPH method is adopted to treat the free surface near the boundary. The proposed FPMT method and FPMT-SPH method are motivated by: a) the spatial derivatives of the velocity and the viscoelastic stress are approximated and obtained by the weighted least squares method; b) the pressure is accurately solved by the application of projection method with a local iterative procedure; c) a corrected particle shifting technique is added to remedy the tensile instability (FPMT); d) an identification technique of free-surface particles is given in the FPMT-SPH method. The accuracy and the convergence of the proposed FPMT scheme for viscoelastic flow are first discussed by solving the planar flow based on the Oldroyd-B model, and compared with the analytical solutions. Secondly, the validity of the CPST is tested by several benchmarks, and compared with other numerical results. Thirdly, the viscoelastic lid-driven cavity flow at high Weissenberg number is simulated and compared with grid-based results, for further illustrating the robustness and the ability of the proposed FPMT method. Finally, the proposed coupled FPMT-SPH method is used to simulate the challenging free surface flow problem of a viscoelastic droplet impacting and spreading on rigid. All the numerical results show that the proposed particle method for the viscoelastic fluid or free surface flow is robust and reliable.