An Approximate Theory for Developing Turbulent Free Shear Layers

The development of a two-dimensional, free turbulent shear layer from an arbitrary initial velocity profile is analyzed theoretically. Included in the analysis are effects of both compressibility and heat transfer with unit turbulent Prandtl number. The mean flow is described by approximate velocity profiles containing an unknown position parameter which is dependent upon the development distance. Integral forms of the continuity and momentum equations are utilized to specify the flow characteristics along the streamline which separates the primary and secondary flow regions. By integrating a simplified form of the transverse motion equation for this dividing streamline, one is able to calculate the position parameter and thus complete the description of the developing flow field. For initial profiles of a power law type, the theory shows that the development distance required for any flow field variable to achieve a specified percentage of its asymptotic value is proportional to the free-stream Crocco number, to the power law exponent, and to the ratio of the ambient to jet stagnation temperatures. The theory is also utilized to estimate the effects of heat transfer and compressibility on the variation of growth rates for fully developed mixing zones.