Aerodynamic Sound Emission as a Singular Perturbation Problem

Sound emission from an eddy region involves three length scales: the eddy size I, wavelength λ of the sound, and a dimension L ofthe region. They are related by the Mach number M = l/λ, small for nearly incompressible eddies, and a parameter Λ = L/λ which plays no apparent role in current theories of aerodynamic sound. The theories of Lighthill and Ribner are examined in the case M ≪ 1, Λ ˜ 1. Ribner's result is found to contain an unacceptable improper integral. The utility of Lighthill's solution is found to depend on properties of the quadrupole moment Tij that can be established only by studying the flow in more detail than Lighthill's theory allows. The general problem is posed in the form: given the body force f and vorticity ω find the density ρ and potential φ of the velocity u = ∇ × ψ{ω} + ∇φ The problem is solved for M ≪ 1, Λ ˜ 1 by matching a compressible eddy core scaled on I to a surrounding acoustic field scaled on λ. Lighthill's solution for ρ is shown to be adequate in both regions if Tij is approximated by ρ0υiυj, with v = ∇ × ψ. The situation M ≪ 1, Λ ≫ 1 is studied, and the conclusion is reached that sound emission from large bodies of turbulence is an open problem, Lighthill's theory notwithstanding.