Temporal stability analysis of jets of lobed geometry

Abstract

A 2D temporal incompressible stability analysis is carried out for lobed jets. The jet base flow is assumed to be parallel and of a vortex-sheet type. The eigenfunctions of this simplified stability problem are expanded using the eigenfunctions of a round jet.The original problem is then formulated as an innovative matrix eigenvalue problem, which can be solved in a very robust and efficient manner. The results show that the lobed geometry changes both the convection velocity and temporal growth rate of the instability waves. However, different modes are affected differently. In particular, mode 0 is not sensitive to the geometry changes, while modes of higher-orders can be changed significantly. The changes become more pronounced as the number of lobes N and the penetration ratio ǫ increase. Moreover, the lobed geometry can cause a previously degenerate eigenvalue (λ n = λ −n ) to become non-degenerate (λ n = λ −n ) and lead to opposite changes to the stability characteristics of the corresponding symmetric (n) and antisymmetric (−n) modes. It is also shown that each eigen-mode changes its shape in response to the lobes of the vortex sheet, and the degeneracy of an eigenvalue occurs when the vortex sheet has more symmetric planes than the corresponding mode shape (including both symmetric and antisymmetric planes). The new approach developed in this paper can be used to study the stability characteristics of jets of other arbitrary geometries in a robust and efficient manner.

Near-field coherent structures in circular and fractal orifice jets

“…From the discussion of the results presented up to this point, the fractal orifice leads to the following modifications of the near-field jet structure when compared to the circular orifice: (1) an increase of the streamwise extent of the potential core, (2) a reduction of the decay rate of the streamwise velocity, and (3) a strong attenuation of the transversal velocity rms and a mild attenuation of the streamwise velocity rms. These experimental findings are consistent with recent results from linear stability analysis presented in Lyu and Dowling [46] and in Lajús Jr et al [47] on jets issuing from noncircular (lobed) nozzles. These two studies concluded that the temporal growth of near-field instabilities decreases for increasing number of lobes.…”

“…The marginal importance of the radial POD modes in the fractal jet is a consequence of the reduced growth rate of the Kelvin-Helmholtz instabilities. As discussed in the recent stability analyses by Lyu and Dowling [46] and Lajús Jr et al [47], the stronger is the deviation from the axisymmetry of the jet orifice, the lower is the growth rate of the near-field instabilities. For this reason, at the downstream position of z 0 /D e = 2, 044612-13 the energy of the radial velocity fluctuations is much larger in the round jet than in the fractal jet, as the rms in Figs.…”

“…These two studies concluded that the temporal growth of near-field instabilities decreases for increasing number of lobes. Additionally, Lyu and Dowling [46] observed that also a larger penetration ratio of the lobes of the jet nozzle produces a decrease in the temporal growth rate.…”

An experimental study of the effects of lobed nozzles on installed jet noise

“…As mentioned in Sect. 1, the earlier work of the authors (Lyu and Dowling 2018b) shows that the stability characteristics of base flows of a lobed vortex-sheet type jet are different from those of an axisymmetric jet. The differences consist of changes to both the convection velocity and the temporal growth rate of instability waves.…”

“…Another alternative method seeking to control installed jet noise by modifying jet instability waves is to use lobed nozzles. The recent work by Lyu and Dowling (2018b) shows that the lobed structure of a jet can indeed change the characteristics of instability waves, including both the temporal growth rate and convection velocity. These changes, however, depend on the azimuthal orders of the instability waves, the number of lobes and their penetration ratio.…”