A solver using Newton's method for unsteady viscous flows

Abstract

A solver is developed for time-accurate computations of viscous flows based on the conception of Newton's method. A set of pseudo-time derivatives are added into governing equations and the discretized system is solved using GMRES algorithm. Due to some special properties of GMRES algorithm, the solution procedure for unsteady flows could be regarded as a kind of Newton iteration. The physical-time derivatives of governing equations are discretized using two different approaches, i.e., 3-point Euler backward, and Crank-Nicolson formulas, both with 2nd-order accuracy in time but with different truncation errors. The turbulent eddy viscosity is calculated by using a version of Spalart-Allmaras one-equation model modified by authors for turbulent flows. Two cases of unsteady viscous flow are investigated to validate and assess the solver, i.e., low Reynolds number flow around a row of cylinders and transonic bi-circular-arc airfoil flow featuring the vortex shedding and shock buffeting problems, respectively. Meanwhile, comparisons between the two schemes of time-derivative discretizations are carefully made. It is illustrated that the developed unsteady flow solver shows a considerable efficiency and the Crank-Nicolson scheme gives better results compared with Euler method.