A new smoothing approach for accelerating the convergence of power-law preconditioning method in steady and unsteady flows simulation

A new explicit smoothing approach is presented to improve the convergence acceleration of laminar viscous steady-state and unsteady flows simulation over airfoils (with/without considering surface air injection). This approach is developed based on Jameson's residual smoothing technique, and it is called here as a Solution Smoothing Method (SSM). This scheme is accompanied by the progressive locally power-law preconditioning method. The steady/unsteady preconditioned governing equations are solved using Jameson's cell-centered finite volume method in which the stabilization is achieved by means of the second- and fourth-order artificial dissipation terms. An implicit dual-time algorithm is used for simulation of unsteady cases and an explicit four-stage Runge-Kutta time integration with local time step is applied to achieve the steady-state condition in pseudo-time in an inner loop. Stability of SSM is analyzed on a one-dimensional linear model equation, and the effects of different Courant numbers, artificial dissipation coefficients, and smoothing parameters are investigated. Numerical results in comparison with other classic preconditioning methods show a reduction of the iteration number required to compute the steady-state and unsteady solution. Results show about 55 ∼ 77 and 40 ∼ 72% reduction in iteration number for steady and unsteady flows simulation, respectively.