Linear Instability Waves in Supersonic Turbulent Mixing Layers

The structures of supersonic turbulent mixing layers are assumed to be spatially growing instability waves superimposed on mean flows, which are also spatially diverging. The objective of the work presented here is to determine these waves and to evaluate the effect of certain parameters such as mean velocity and temperature profiles on the stability characteristics (growth rate, most amplified frequency, etc.). Linear inviscid stability theory of nonparallel mean flow has been used. Three-dimensional waves in the form of axisymmetric or helical modes interacting with the axisymmetric basic state are considered. The basic state is determined by solving the axisymmetric turbulent boundary-layer equations for the configuration of unconfined coaxial mixing. A mixing length model is used for turbulence closure. Increasing the Mach number of the outer stream or increasing the temperature of the inner stream (gas generator) significantly reduces the growth rate and gain along the mixing layer length. I. Introduction T HE scramjet engine cycle in which combustion of liquid fuels takes place at supersonic speeds has been recognized as an efficient propulsive cycle for missiles and airplanes cruising at hypersonic speeds.1"3 Since the early experimentation in the 1950's that demonstrated the feasibility of combustion of liquid fuels in a supersonic air stream, efforts have resulted in a hybrid engine cycle called the Dual-Combustion Ramjet (DCR).4 In this cycle, roughly 25% of the supersonic inlet flow is decelerated down to subsonic (or sonic) speed and directed into a "dump combust or** or gas generator. All of the fuel then is injected into the high-temperature subsonic stream resulting in a very fuel-rich mixture and, hence, ignition and combustion can be maintained. However, the combustion process of the unreacted fuel has to be completed in a supersonic combustor downstream of the gas generator after mixing with the supersonic airstream previously diverted by the inlet. The propulsive efficiency of the DCR cycle will be affected by the quality of mixing taking place in the shear layer that is formed by the subsonic (or sonic) stream issuing from the gas generator and the supersonic stream surrounding it. As pointed out by Waltrup, " Our current understanding of the mechanisms governing and/or controlling the mixing and combustion processes when two sonic or supersonic streams merge to form a free shear layer...is very limited."3 Thus, there is a need to investigate the structure, growth, and means of controlling supersonic turbulent mixing layers in nonreactive and reactive flows. The present work is a step toward the understanding of turbulent mixing layers. In this section we briefly discuss previous theoretical efforts that use hydrodynamic stability theory as a means of investigating the structures of mixing layers and their response to external excitations. Basic Features of Mixing Layers It has been experimentally established that the spreading rates (and, hence, entrainment and mixing) of supersonic turbulent mixing layers are substantially lower than those of incompressible layers (e.g., Refs. 5 and 6). Moreover, the basic