On the noise generated by an imperfectly expanded supersonic jet

Abstract

This paper examines in detail a theoretical model describing the generation of sound by an imperfectly expanded supersonic jet. Such a jet consists of a steady system of ‘shock cells’, and sound is produced by their interaction with turbulence convected in the mixing region of the jet. The dominant convection velocity of the large-scale eddies is shown to be a principal factor determining the total radiated sound power, essentially independently of the details of the interaction mechanism. The ideal laminar jet constitutes a wave guide in which are trapped the steady waves of a super-critical flow. Imperfections of the wave guide cause it to leak. Turbulence provides those imperfections and thus acts as a catalyst enabling the energy of the nominally steady wave system to leak out as sound. When appropriately viewed, the sound power is equal to the power of the wave system in the steady jet. The computation of that power is consequently insensitive to details of the jet turbulence. The spectrum of the sound is characterized by a sequence of peaks produced by coherent scattering from the ends of successive cells. In addition, the theory reveals the existence of a more broadband component which is related to the effects of multiple scattering. The directivity of the radiated sound predicted from the model exhibits encouraging agreement with experiment. The total sound power associated with the presence of the system of shock cells is estimated and shown to be between 0.2 and 1.2 % of the jet exhaust power for a jet operating under conditions appropriate to those of a supersonic transport at take-off. This is entirely consistent with available experimental evidence.