A nonconservative nonlinear flowfield splitting method for 3-D unsteady fluid dynamics

This paper describes a new way to write and solve the governing equations of fluid dynamics that is very well suited to aeroacoustics problems. The flow field variables are split into a mean and a fluctuating part, and then inserted into the governing equations. If the mean flowfield satisfies the steady-state governing equations, then the resulting equations can be written in a very convenient form. These equations are then solved using a method that is second or fourth order accurate in space and time, on a Cartesian grid. Several aeroacoustic benchmark problems are solved, some duct propagation problems are presented, and a longrange scattering problem is solved. The code is written in Fortran-90 using the Message Passing Interface (MPI), and performance results are presented for a cluster of workstations. INTRODUCTION There have been dramatic advances in computational aeroacoustics (CAA) over the last 10 years. Non-reflecting boundary conditions [1, 2], higher order accurate schemes [3-9], and parallel computers [10] have all combined to produce solutions for problems beyond reach several years ago. In particular, problems that are amenable to using the linear Euler equations are fairly easy to solve now. However, there are other problems where nonlinear effects, and where the mean flow field must be computed, which are still very difficult to solve. In addition, the linearized Euler equations are not always well behaved, e.g. at stagnation points. Another problem is that CAA codes are designed to solve wave propagation problems, and cannot efficiently compute a mean flow. They are designed to have low dispersion and dissipation, which means they cannot rapidly converge to a mean flow. On the other hand, traditional computational fluid dynamics (CFD), especially multi-grid or implicit, codes are designed to rapidly converge to a steady state, but they have large dispersion and dissipation errors, and so they are not suitable for CAA. This paper describes a method that is designed to take advantage of the best features of existing CFD and _______________________________________ Copyright © 2000 by L. N. Long. Published by AIAA with Permission. CAA algorithms and codes. We split the flow field into a mean and a fluctuating part, and each can be solved using a different algorithm/code. The mean flow field can be obtained from a traditional CFD code, while the unsteady effects can be solved using the code described herein. While we have been using the Nonlinear Disturbance Equations (NLDE) for several years [6, 11, 12], the approach described herein is different from NLDE. The main difference is that the NLDE approach uses the conservative form of the governing equations, while the approach described here uses the nonconservative equations. This results in a very different algorithm and code, and has some advantages. In particular, it results in a much simpler set of equations. In addition to using a novel set of equations, we also use simple Cartesian grids, although this is not necessary for the equations described below. No grid generation is required. We simply need to know if the grid point is inside or outside of the body. The goal here was to have a code analogous to the FDTD code used in electromagnetics [13, 14] for fluid dynamics or aeroacoustics. GOVERNING EQUATIONS In the recent past, we have presented several computational results from a method called the Nonlinear Disturbance Equation (NLDE) approach [6]. This has included unsteady flow around naval ships [12], engine liners [11], and supersonic turbulent jets [15], and more recently [16-18]. This method starts with the conservative form of the governing equations: 0 H G F q = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ z y x t Where: