A Galerkin finite element algorithm based on third‐order Runge‐Kutta temporal discretization along the uniform streamline for unsteady incompressible flows. (18th March 2019)

• Record Type: Journal Article
• Authors: Liao, Shaokai; Zhang, Yan; Chen, Da
• Is Part Of: International journal for numerical methods in fluids. Volume 90:Number 7(2019)
• Page Start: 323
• Page End: 339
• Publication Date: 2019-03-18
• Subjects: accuracy -- cost -- Navier‐Stokes equations -- Runge‐Kutta method -- streamline; Fluid dynamics -- Mathematics -- Periodicals; 532
• DOI: 10.1002/fld.4722
• Languages: English
• ISSNs: 0271-2091

Abstract

Summary: In this paper, for two‐dimensional unsteady incompressible flow, the Navier‐Stokes equations without convection term are derived by the coordinate transformation along the streamline characteristic. The third‐order Runge‐Kutta method along the streamline is introduced to discrete the alternative Navier‐Stokes equations in time, and spacial discretization is carried out by the Galerkin method, and then, the third‐order accuracy finite element method is obtained. Meanwhile, the streamline velocity is uniformly approximated by initial velocity in each time step in order to reduce update frequency of total element matrix and improve calculation efficiency. Finally, some classic unsteady flow examples are calculated and analyzed by different calculation methods, which further demonstrate that the present method has more advantages in stability, permissible time step, dissipation, computational cost, and accuracy. The code can be downloaded athttps://doi.org/10.13140/RG.2.2.27706.44484 . Abstract : Based on the direction derivation along the streamline, we directly derive the Navier‐Stokes equations without convection term under the moving coordinate system. We use the third‐order Runge‐Kutta method along the streamline for temporal discretization and the Galerkin method for spacial discretization. We introduce the Taylor expansion along the uniform streamline to obtain the corresponding express under the static coordinate system. Finally, we discuss the stability, permissible time step, dissipation, computational cost, and accuracy of the present method.