Hierarchical radial basis functions method for solving the unsteady Navier–Stokes equations

Abstract

The Navier–Stokes equations (NSE) are essential equations in fluid dynamics that describe the motion of viscous fluids, accurately reflecting changes in fluid velocity and pressure. It is widely used in the fields of aerodynamics analysis in aerospace, fluid flow simulation and others. We use the hierarchical radial basis functions (H-RBFs) collocation method to simulate NSE in this paper, which is essentially a meshfree method that only requires knowledge of scattered data node information without the need to build the meshgrid. Then, the trial space of H-RBFs is constructed with the help of nested sets of points and scaling the support radii of compactly supported radial basis functions. Meanwhile, the numerical solution is found within the trial space to approximate the model’s solution. Numerical tests demonstrate that the method proposed in this paper achieves high accuracy in both regular and irregular domains. Compared to compactly supported radial basis functions collocation method, H-RBFs collocation method exhibits smaller errors, particularly when the collocation points become dense. Finally, a classical experiment above flow around the cylinder is presented, demonstrating that H-RBFs collocation method can effectively simulate the formation of vortex streets.