A High-Order Method for Solving Unsteady Incompressible Navier-Stokes Equations with Implicit Time Stepping on Unstructured Grids

This paper reports development of an unstructured high-order compact method for solving two-dimensional incompressible flow. This method employs the gGA correction from Huynh, and falls under the class of methods now referred to as Flux Reconstruction/Correction Procedure via Reconstruction. The artificial compressibility method and a dual time stepping scheme are used to simulate unsteady incompressible flow. A lower-upper symmetric Gauss-Seidel scheme with backward Euler discretization is used to efficiently march the solution in pseudo time, while a second order backward Euler discretization is used to march in physical time. We demonstrate order of accuracy with steady Taylor-Couette flow. We further validate the solver with steady flow past a NACA-0012 airfoil at zero angle of attack and unsteady flow past a circle. The implicit time stepping scheme is proven efficient and effective for driving the pseudo time derivative term toward zero in the classical artificial compressibility formulation. As a result, this scheme is capable of quickly establishing the divergence-free velocity condition of the continuity equation when compared to an explicit scheme.