A novel Lagrangian–Eulerian weighted-least squares scheme coupled with other stable techniques for multi-physical fluid flow around complex obstacle

Abstract

In this paper, a stable novel meshless coupled method is proposed to simulate the non-isothermal magnetohydrodynamics (MHD) flow problems (multi-physics quantities) inside a lid-driven cavity around complex obstacle. The proposed method is mainly motivated by a Lagrangian–Eulerian (L–E) weighted-least squares (WLS) scheme combined with a stream function-vorticity (SFV) and other stable techniques, and it is further to investigate the non-isothermal MHD flow around an airfoil obstacle at large Hartmann (Ha) or Reynolds (Re) number, for the first time. In the present meshless coupled approach (named L–E WLS–SFV), the traditional MHD equations are derived as another form with an SFV method under divergence-free constraint, which can avoid the tedious treatment of pressure on complex irregular obstacle. Then, a stable L–E WLS coupled algorithm is proposed to approximate the space derivatives of multi-physical quantities (velocity, magnetic, temperature, etc.), in which a corrected particle shifting technique is employed to improve the tensile instability among Lagrangian particles moving inside the domain and a second-order upwind scheme is adopted to stabilize large Re number problem in Eulerian fixed nodes near the boundary. Several benchmarks are simulated to show the numerical accuracy and convergence rates of the proposed WLS scheme for MHD flow at different parameters. Subsequently, the case of the non-isothermal MHD flow around a square obstacle under large parameters is simulated by the proposed L–E WLS–SFV method and compared with other numerical results to demonstrate the validity and capacity of the proposed method for multi-physical flow and the necessity of imposing the above two stable techniques. Finally, the case of non-isothermal MHD flow around the circular or airfoil obstacle is numerically investigated, and the important effects of the Hartmann, Rayleigh, and Reynolds numbers on the multi-physical quantities (stream function, vorticity, temperature, and magnetic field) are discussed. The advantages of the proposed method for the muti-physical flow around irregular obstacles are also exemplified. All the numerical results show that the proposed L–E WLS–SVF method is robust and accurate to simulate the multi-physical fluid flow around complex obstacles.