Sound generated by instability waves of supersonic flows. Part 1. Two-dimensional mixing layers

Abstract

The problem of acoustic radiation generated by a spatially growing instability wave of a supersonic two-dimensional mixing layer is studied. It is shown that at high supersonic Mach numbers the classical locally parallel-flow hydrodynamic stability theory as well as the more recent theories based on the method of multiple scales (e.g. Saric & Nayfeh 1975; Crighton & Gaster 1976; Plaschko 1979; Tam & Morris 1980) would fail to give even a first-order instability wave solution. Physically, at these high flow speeds the radiated sound field is no longer an insignificant part of the total phenomenon. The disturbances associated with the flow-instability process now extend from the mixing layer all the way to the far field. The problem is therefore global in nature. Methods of solution which are predicated on local approximations such as the classical locally parallel-flow hydrodynamic-stability theory or the method of multiple scales are hence inappropriate and inapplicable. A global solution based on the method of matched asymptotic expansions is constructed. The outer solution is valid outside the mixing layer. It provides a mathematical description of the radiated acoustic field and the pressure near field. The near field in this case consists of both the acoustic and the hydrodynamic (non-propagating) fluctuation components. The inner solution is valid inside and in the immediate vicinity of the mixing layer. Physically it represents the instability wave of the flow. Matching is carried out according to the intermediate matching principle of Van Dyke (1975) and Cole (1968). Matching terms to order unity gives the basic instability-wave solution. Matching terms to the next order gives the instability- and acoustic-wave amplitude equation. For low-Mach-number flows it is found that the present results agree with the multiple-scales solution of Tam & Morris (1980).