An accurate and efficient technique for unsteady viscous flows in two dimensions

In the study of the physics of vortex dominated flows, vorticity plays a fundamental role, its dynamics being the primary tool which enables one to understand the time evolution of vortical structures; therefore, in the numerical solution of the time-dependent incompressible Navier-Stokes equations, it is convenient to select vorticity as one of the dependent variables. In order to be able to solve two-dimensional as well as three-dimensional flows, the velocity vector is used as the second dependent variable, as originally proposed by Fasel [1]. For such a formulation of the incompressible Navier-Stokes equations the authors have recently developed a numerical method for solving steady flows in two-dimensional general boundary-fitted curvilinear coordinates [2]. The governing equations are written as a system of a scalar vorticity transport equation and a second-order equation for the velocity vector, obtained by combining the vorticity definition and the incompressibility condition. The equations are discretized in space by a staggered-grid second-order-accurate finite-volume scheme, the main merit of the method being its remarkable accuracy, achieved by satisfying the discrete counterparts of the vorticity definition and of the continuity equation, exactly, i.e., within machine accuracy. The method has been then extended to the case of unsteady flows, by solving the vorticity transport equation by means of an alternating direction implicit technique and the elliptic equations for the velocity components by means of a multigrid line-Gauss-Seidel relaxation procedure [3]. Such an approach requires an inner iteration to evaluate the vorticity boundary values at each new time level in order to achieve second-order accuracy in space and time. Here a new method is proposed, which achieves the same accuracy at a markedly lower computational cost. The vorticity transport equation is discretized in time by means of a fully implicit three-level scheme coupled with the velocity vector equations and the resulting large linear system is solved by a dual-time stepping technique: at each fictitious time step the vorticity and velocity fields are iterated to convergence by means of an efficient multigrid line-Gauss-Seidel relaxation scheme. In the following, the numerical method is described in detail; then, it is used to cumpute a test problem with an exact solution, the vortex shedding phenomenon around a circular cylinder, for Reynolds number equal to 100, and the impulsively started flow past a circular cylinder for Reynolds number equal to 3000.