A study of the interactions between large-scale coherent structures and fine-grained turbulence in a round jet

Our recent studies of the interactions between large-scale structures and fine-grained turbulence in plane mixing layers have shown that many physical features of the problem may easily be obtained from an approximate energy integral description (Liu & Merkine 1976; Alper & Liu 1978), and this is confirmed by computational models (Gatski & Liu 1980). In this work, therefore, the approximate description is used to study at length the development of large-scale coherent structures in the technologically important problem of the round turbulent jet. The analysis begins from the radially integrated form of the kinetic energy equations of the mean flow, the large scale structure and the fine-grained turbulence, which are obtained through the use of the usual Reynolds time average and a conditional average with reference to the frequency of the idealized monochromatic component of the large-scale wavelike structure. This forms the basis for obtaining the ‘amplitude equations’ for the three components of the flow in terms of the mean flow momentum thickness, the large-scale structure kinetic energy and the fine-grained turbulence kinetic energy across the jet. These are obtained via the accompanying shape assumptions which also implicitly address the closure problems. The large-scale structure is also characterized by the Strouhal number St = fd/U e , where f is the frequency, d is the jet diameter and U e is the jet exit velocity, and by the azimuthal wave number n . The calculations are compared with the well controlled forced-jet observations of Binder & Favre-Marinet (1973), Favre-Marinet (1975), Favre-Marinet & Binder (1979) and Moore (1977). Although the present approximate considerations are not directed at structural details, the comparisons with observations on this aspect are most encouraging. Further theoretical work is presented that addresses the fundamental understanding of the mechanisms leading to the development of large-scale structures in turbulent jets. In general, large-scale structures in the range 0.02 ^ St St is increased, the streamwise location of the peak signal moves upstream, and the streamwise lifespan shortens. Consequently, high-frequency components of the large-scale structure dominate upstream while low-frequency components prevail further downstream. The Strouhal number that gives rise to maximum amplification is about 0.70 for weak initial levels of the large-scale structure and decreases to about 0.35 for very strong initial levels. As the initial energy level of the large-scale structure increases, its maximum relative amplification decreases until a level is reached beyond which the large-scale structure decays immediately downstream. This is explained in terms of the modification of the mean flow by the increasingly high energy levels of the large-scale structure in such a manner that it chokes off its own energy supply from the mean flow. At low Strouhal numbers the n = 1 helical component amplifies initially more than the n = 0 axisymmetric component for the same initial energy level. However, the n = 1 mode decays subsequently much faster than the n = 0 mode for all Strouhal numbers. This is attributable to the azimuthally-related wave-induced turbulent shear stresses in the n = 1 mode which give rise to additional mechanisms for energy transfer to the fine-grained turbulence. The possible control of the large-scale structure is fully explored through considering adjustments of the energy levels of the fine-grained turbulence and changes in initial mean velocity profile through changes in the momentum thickness at the nozzle exit. Increasing the initial turbulence levels and smoothing the mean nozzle exit velocity profile places restraints upon the downstream amplification of the large-scale structure. The non-equilibrium development of the large-scale structure is sensitive to its own initial conditions and spectral content, the initial condition of the fine-grained turbulence and the mean flow. Physical and quantitative studies of the large-scale structure in turbulent shear flows thus necessitate that the nature of such an initial environment be established and understood.