Hydrodynamic Stability of Periodically Unsteady Axisymmetric and Swirling Jets
Axisymmetric and swirling jets are generic flows that characterize many natural and man-made flows. These include cylindrical shear layer/mixing layer flows, aircraft jets and wakes, shedding of leading edge and wing tip vortices, tornadoes, astrophysical plasma flows and flows in mechanical devices such as supersonic combustion chambers and cyclone separators. These and other applications have resulted in a high level of interest in the stability of axisymmetric and swirling jets. To date, the majority of studies on stability of axisymmetric and swirling jets have been completed under the assumption of steady flow in both axial and azimuthal (swirl) directions. Yet, flows such as the ones mentioned above can have an inherent unsteadiness. Moreover, such unsteadiness can be used to control stability and thus flow characteristics in axisymmetric and swirling jets. In this work effects of periodic variations on the temporal stability of axisymmetric and swirling jets is examined. The unsteadiness is introduced in the former as a periodic variation of the axial velocity component of the flow, and in the latter as a periodic variation of the azimuthal (swirl) velocity component of the flow.
The temporal linear hydrodynamic stability of both steady inviscid axisymmetric and swirling jets is reviewed. An analytical dispersion relation is obtained in both cases and solved numerically. In the case of the steady axisymmetric jet, growth rate and celerity of unstable axisymmetric and helicalmodes are determined as functions of axial wavenumber. Results show that the inviscid axisymmetric jet is unstable to all values of axisymmetric and helical modes. In the case of the steady swirling jet, growth rate and celerity of axisymmetric modes are determined as functions of the axial wavenumber and swirl number. Results show that the inviscid swirling jet is unstable to all values of axial and azimuthal wavenumber, however, it is shown that increasing the swirl decreases the growth rate and increases the celerity of axisymmetric disturbances. The effects of periodic variations on the stability of a mixing layer is also reviewed. Results show that when the instability time scale is much smaller than the mean time scale a transformation of the time variable may be taken that, when the quasi-steady approach works, will reduce the unsteady field to that of the corresponding steady field in the new time scale. The price paid for this transformation, however, is a modulation of the amplitude and phase of the unsteady modes.
Extending the results from the unsteady mixing layer, the stability of a periodically unsteady inviscid axisymmetric jet is considered. An analytical dispersion relation is obtained and results show that for the unsteady inviscid axisymmetric jet, the quasi-steady approach works. Following this, the stability of a periodically unsteady swirling jet is considered and an analytical dispersion relation is obtained. It is shown that for the unsteady inviscid swirling jet, the quasi-steady approach does not work. Resulting modulations of unsteady modes are shown via a numerical solution to the unsteady dispersion relation. In both cases, using established results for unsteady mixing layers, these results are substantiated analytically by showing that the unsteady axisymmetric jet can be reduced the the exact equational form of the steady axisymmetric jet in a new time scale, whereas the unsteady swirling jet cannot.