In this thesis two distinct features of coherent structures are investigated. In Part I a model for the pinch-off process in vortex ring formation is developed. The predicted nondimensional stroke length L/D (referred to as "formation number") satisfactorily matches experimental observations. The model introduces two nondimensional parameters that govern the limiting value of the formation number: a nondimensional energy and circulation, E_(nd) and Γ_(nd), respectively. The predicted value of E_(nd) also matches well with the experimental data. The limiting value for the new nondimensional circulation is predicted to be in the range 1.77 ≲ Γ_(nd) ≲ 2.07. We perform detailed computations of vortex ring formation by nonconservative forcing. The validity of the assumptions in our model is verified in these computations. Some techniques for generating fat rings are successfully investigated, resulting in generation of vortex rings with Hill's like properties. We consider thermodynamics of the vorticity density field (w/r), and we develop a statistical equilibrium theory for axisymmetric flows. It is shown that the statistical equilibrium of an axisymmetric flow is the state that maximizes an entropy functional constrained to the invariants of motion. Furthermore, it is shown that the final equilibrium state satisfies a variational principle similar to Kelvin's variational principle. In Part II Mach wave radiation mechanisms in a fully expanded supersonic jet is studied. We compare a direct numerical simulation (DNS) of a 1.92 Mach number jet with a linearized Navier-Stokes (LNS) simulation. The numerical integration technique, inflow boundary conditions, and grid distributions are the same in both simulations. We found that the generated noise in the DNS calculation is dominated by the first two azimuthal modes, and contributions from all other azimuthal modes were limited to less than 1.5 dB in the acoustic field. The total directivity of the sound field in the LNS matches reasonably well with the sound field of the DNS data. At the peak Strouhal frequency, particularly for the azimuthal mode n = 1, the amplification of flow variables in the LNS closely matches that of the DNS data. However, for frequencies away from the peak Strouhal number the DNS data shows amplification rates comparable to those of the peak Strouhal number, while in the LNS data any disturbances away from the peak Strouhal number are highly damped. These extra noise sources in the DNS data have the characteristics of a nonlinear interaction among various modes. The noise generated by the first two modes in the linearized computation is substantially weaker than in the DNS. For example, in the near acoustic field, at a distance of 6 jet diameters from the jet centerline, the sound pressure level in the linearized computation is as much as 8 db smaller than the DNS results. We observed that the maximum amplification rate for the DNS data occurs at an axial location further downstream than for the LNS data, which corresponds to regions around and beyond the end of the potential core. Our results indicate that the missing sound generation mechanisms in the LNS computation at the frequencies considered in this study can be attributed to the non-linear sound generation mechanisms, that are not captured in linear theories.
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