A Simple Method for Computing Far-FieldSound in Aeroacoustic Computations

When computing aerodynamic noise from moderate Mach number flows, compressible flow equations are often solved numerically in a finite region of space containing the noise sources and the near acoustic field [1]. However, whether this is done by direct numerical simulation or with some modeling approximations (e.g., large-eddy simulation or unsteady Reynolds averaging), such simulations typically involve a range of length and time scales and are therefore expensive. To cope with this, the computational domain is often truncated in an acoustic region that is not too far beyond the unsteady flow. The sound at greater distances is then computed by solving relatively simple acoustic equations. Several numerical methods for extending the solution in this fashion are reviewed by Shih et al. [2]. One approach is to use wave equation solutions formulated as surface integrals, so-called Kirchhoff or Ffowcs Williams–Hawkings methods [3–5]. The integral evaluation operations scale as O(N 2) per “measurement.” So, if the acoustic field is needed at a series of N times in an N 3 volume, the total scaling is O(N 6). Providing this data is at times impractical due to computer memory limitations; and, though binning techniques are available, application is complicated by the fact that data are required at retarded times. When extensive sound field data are needed it is at times better to solve a wave equation (or similar set of acoustic equations) into the far field using direct methods. After an initial transient, this approach scales like O(N 4) (three spatial dimensions plus time) for O(N 4) measurements.1 The implementation of the direct solver is simpler because the data are now only needed sequentially. Existing direct methods are formulated either as an