Aerodynamic Sound Emission as a Singular Perturbation Problem

Source abstract

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 T ij 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 T ij is approximated by ρ 0 υ i υ j , 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.