A physics-preserving pure streamfunction formulation and high-order compact solver with high-resolution for three-dimensional steady incompressible flows

Abstract

In this paper, a pure streamfunction high-order compact (HOC) difference solver is proposed for three-dimensional (3D) steady incompressible flows. A physics-preserving pure streamfunction formulation is first introduced for the steady 3D incompressible Navier–Stokes (NS) equations without in-flow and out-flow boundary conditions, where the divergence of streamfunction ∇ · ψ is maintained in the convective and the vortex-stretching terms together in the nonlinear term of equations to reduce the physics-informed loss. Moreover, taking the streamfunction-vector components and their first-order partial derivatives as unknown variables, some fourth-order compact schemes are suggested for the partial derivatives that appear in the streamfunction formulation, and a high-resolution HOC scheme is introduced for approximating the pure third-order partial derivatives in the convective term. Meanwhile, a new HOC scheme is proposed for the first-type boundary conditions of the streamfunction. Finally, a fourth-order compact difference scheme and its algorithm are established for the 3D steady incompressible NS equations in the streamfunction form, subject to no in-flow and out-flow boundary conditions. Several numerical examples are carried out to validate and prove the accuracy, convergence, and efficiency of the proposed new method. Numerical results reveal that the proposed method not only can achieve fourth-order accuracy but also has excellent convergence, high-resolution, and low computational cost at higher Reynolds number.