Insight into the two-source structure of the jet noise spectrum using a generalized shell model of turbulence

There is a large body of experimental evidence which shows that the jet noise spectrum is composed of two sources. Our aim here is to prove, mathematically, that the two-source paradigm can be derived using a minimum number of self-consistent approximations based on our current knowledge of jet turbulence in cold flows. The starting point of the paper is Goldstein’s (2003) [20] exact re-arrangement of the Navier–Stokes equations, which shows that turbulence enters the acoustic spectrum formula through the Reynolds stress auto-covariance tensor. We extend the shell model of turbulence using a more general symmetry approximation that amounts to assuming that the Reynolds stress auto-covariance is, firstly, axi-symmetric, and secondly is equivalent to the same tensor only after it has been averaged (point-wise) over the azimuthal separation.As a consequence of these two assumptions, the space–time Fourier transform of the Reynolds stress auto-covariance (which we refer to as the spectral tensor) depends on the transverse wave vector only through the square of its magnitude and, moreover, is also an axi-symmetric tensor. This defines the generalized shell model (or GSM) and we apply it to the jet noise problem. The final result shows that the acoustic spectrum can be written as the sum of two groups of terms, one of which corresponds to the peak jet noise in the weakly non-parallel flow limit.