Implicit Time Accurate Method for Unsteady Computations

Abstract

A dual time-stepping algorithm has been developed for the efficient computation of unsteady fluid dynamics. The algorithm is effective over a wide range of low Mach numbers and physical time scales, when working together with a preconditioning technique which offers a way to formulate the Euler and Navier-Stokes equations such that convergence can be made independent of Mach number. This enhancement is shown to provide more accurate solutions for unsteady subsonic and low-speed flows through several test cases. I. Introduction ime-accurate computations at all speeds are essential since many practical flows of engineering interest are inherently transient and may range from the incompressible limit to supersonic speeds. There are two popular methods for fluid flow simulations: pressure-based methods (1) for incompressible flows and density-based methods (2) for compressible flows. So far, flow problems at all speeds can be addressed by a well assessed flow solver employing a single solution algorithm. Two approaches are probably the most successful for the single solution algorithms. One is the extension of the pressure-based methods for high speed compressible flows using pressure- velocity-density correction algorithms. The other is based on low Mach number preconditioning technique for the density-based approach to equilibrate the eigenvalues, which are critical factors in the coupled system so that the stiffness is alleviated, which occurs when the flow speed is very small in comparison to acoustic speed. Both explicit and implicit algorithms are commonly used for unsteady computations. When applied as non- iterative, time-marching methods, these algorithms frequently lose temporal accuracy unless small physical time- step sizes are used. This is particularly true for complex flow fields involving strong non-linear behavior such as shock waves and chemical reaction. Furthermore, stability criteria impose further limitations on the physical time- step size, especially for explicit algorithms. Similar time-step restrictions also exist for implicit algorithms in multi- dimensions because of errors associated with the approximate-inversion methods that are typically used. In the presence of strong local grid-stretching or in low Mach numbers flows, such time-step restrictions may severely impair the usefulness of the algorithm. For these reasons, unsteady algorithms usually adopt some sort of an iterative procedure at each physical time-level that ensures temporal accuracy and, in the case of implicit schemes, also serves to eliminate the linearization and approximate-factorization errors introduced by the scheme. To achieve