Sound generated by instability waves of supersonic flows. Part 2. Axisymmetric jets

Abstract

A solution describing the spatial evolution of small-amplitude instability waves and their associated sound field of axisymmetric supersonic jets is found using the method of matched asymptotic expansions (see Part 1, Tam & Burton 1984). The inherent axisymmetry of the problem allows the instability waves to be decomposed into azimuthal wave modes. In addition, it is found that because of the cylindrical geometry of the problem the gauge functions of the inner expansion, unlike the case of two-dimensional mixing layers, are no longer just powers of ε. Instead they contain logarithmic terms. To test the validity of the theory, numerical results of the solution are compared with the experimental measurements of Troutt (1978) and Troutt & McLaughlin (1982). Two series of comparisons at Strouhal numbers 0.2 and 0.4 for a Mach-number 2.1 cold supersonic jet are made. The data compared include hot-wire measurements of the axial distribution of root-mean-squared jet centreline mass-velocity fluctuations and radial and axial distributions of near-field pressure-level contours measured by microphones. The former is used to test the accuracy of the inner (or instability-wave) solution. The latter is used to verify the correctness of the outer solution. Very favourable overall agreements between the calculated results and the experimental measurements are found. These very favourable agreements strongly suggest that the method of solution developed in Part 1 paper is indeed valid. Furthermore, they also offer concrete support to the proposition made previously by a number of investigators that instability waves are important noise sources in supersonic jets.