Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc. A98-30893 AIAA-98-2361 High-Order Accurate Dual Time-Stepping Algorithm for Viscous Aeroacoustic Simulations Chingwei M. Shieh* and Philip J. Morris t Department of Aerospace Engineering The Pennsylvania State University University Park, PA 16802, U.S.A. The field of computational aeroacoustics (CAA) has shown great promise in the so- lution of primarily inviscid noise generation and wave propagation problems that are governed by the Euler equations. The present research extends these ideas to the solu- tion of the Navier-Stokes equations in multi-dimensions where the acoustic and flow fields may be influenced by viscous effects. In this paper, efficient acceleration techniques typ- ical of explicit steady-state solvers are extended to time-accurate calculations. Stability restrictions based on the grid spacing are relaxed greatly with the implementation of a fully implicit time discretization. Some simplified problems that address the important issues of viscous calculations are investigated. A two-dimensional, Navier-Stokes code, written in Fortran 90 with the Message Passing Library (MPI) as a parallel implemen- tation, is used to perform these calculations. As an example calculation, scattering of an acoustic source by a flat plate in the presence of a mean flow including a viscous boundary layer is presented. Introduction I N aeroacoustic simulations, the phenomena of in- terest are inherently unsteady and consist of a wide range of frequencies and amplitudes. Hence, to repro- duce the physics of these phenomena faithfully and to capture all the scales in the flow field, new algorithms have been proposed for very accurate calculations of wave propagation. Typical CAA algorithms consist of high-order finite difference spatial discretizations (fourth-order 1 , sixth-order 2 ) and high-order explicit time-stepping techniques (high-order explicit linear multistep methods 3 , Runge-Kutta 4 ) to minimize both dispersion and dissipation errors. For simple acous- tic model problems and inviscid calculations, such algorithms are efficient and accurate as long as the numerical stability criterion is met. However, these algorithms are no longer efficient when calculations of the acoustic and flow fields for more realistic problems are performed. In a recent study, Lockard 5 performed parallel com- putations of acoustic scattering by a three-dimensional wing. The unsteady part of the calculations were carried out with the use of a seven-point Dispersion- Relation-Preserving (DRP) finite difference stencil and a low dispersion and dissipation Runge Kutta scheme. However, after 91 equivalent single CPU hours, the acoustic pulse had not propagated out of the compu- tational domain. This is because a very small tune step was taken for the unsteady calculation due to the numerical stability criterion based on the tiny grid "Graduate Research Assistant, Student Member, AIAA t Boeing/A. D. Welliver Professor, Associate Fellow, AIAA Copyright ©1998 by C. M. Shieh and P. J. Morris. Pub- lished by the Confederation of European Aerospace Societies, with permission. spacing at the tip of the wing. Although the algorithm is still very accurate for this inviscid problem, it is no longer efficient. Therefore, in order to extend CAA methodologies to the calculation of the Navier-Stokes equations in multi-dimensions where the acoustic and flow fields may be influenced by viscous effects, a dif- ferent approach needs to be taken. Many popular algorithms for unsteady simulations in CFD use an implicit method. Unlike explicit meth- ods, one of the advantages in the use of an implicit method is its ability to advance solutions with a very large time step. This is especially important in Navier- Stokes calculations since grid cells need to be highly clustered in the vicinity of physical boundaries, and a very small time step has to be employed to avoid the violation of the stability criterion if an explicit method is adopted. However, a major drawback with low- order implicit methods is that numerical dispersion and dissipation are high. The Euler implicit method is only first-order accurate in time. When compared with traditional CAA approaches, which are fourth- order accurate in time if a classical Runge-Kutta time integration technique is used, the wave propagation properties of implicit methods are indeed inferior. In the current analysis, a high-order accurate dual time-stepping algorithm is proposed as a viable al- ternative for viscous aeroacoustic calculations. This algorithm is formally second-order accurate in time and sixth-order accurate in space. In order to perform acoustic calculations, all the transients in the mean flows have to be eliminated from the computational domain. A time-independent mean flow solution is obtained from an efficient steady-state solver in the current approach. This mean flow so- lution is then used as the initial condition, and an 912