Instability wave models for the near-field fluctuations of turbulent jets

Abstract

Previous work has shown that aspects of the evolution of large-scale structures, particularly in forced and transitional mixing layers and jets, can be described by linear and nonlinear stability theories. However, questions persist as to the choice of the basic (steady) flow field to perturb, and the extent to which disturbances in natural (unforced), initially turbulent jets may be modelled with the theory. For unforced jets, identification is made difficult by the lack of a phase reference that would permit a portion of the signal associated with the instability wave to be isolated from other, uncorrelated fluctuations. In this paper, we investigate the extent to which pressure and velocity fluctuations in subsonic, turbulent round jets can be described as linear perturbations to the mean flow field. The disturbances are expanded about the experimentally measured jet mean flow field, and evolved using linear parabolized stability equations (PSE) that account, in an approximate way, for the weakly nonparallel jet mean flow field. We utilize data from an extensive microphone array that measures pressure fluctuations just outside the jet shear layer to show that, up to an unknown initial disturbance spectrum, the phase, wavelength, and amplitude envelope of convecting wavepackets agree well with PSE solutions at frequencies and azimuthal wavenumbers that can be accurately measured with the array. We next apply the proper orthogonal decomposition to near-field velocity fluctuations measured with particle image velocimetry, and show that the structure of the most energetic modes is also similar to eigenfunctions from the linear theory. Importantly, the amplitudes of the modes inferred from the velocity fluctuations are in reasonable agreement with those identified from the microphone array. The results therefore suggest that, to predict, with reasonable accuracy, the evolution of the largest-scale structures that comprise the most energetic portion of the turbulent spectrum of natural jets, nonlinear effects need only be indirectly accounted for by considering perturbations to the mean turbulent flow field, while neglecting any non-zero frequency disturbance interactions.

Wavepackets in the velocity field of turbulent jets

“…The linear operators A-E can be found in Gudmundsson & Colonius (2011). For brevity, the subscripts have been dropped from the shape function and wavenumber in the previous expression.…”

“…The same approach of Gudmundsson & Colonius (2011) was applied to obtain numerical solutions of the system of equations (4.3). The reader is referred to that paper for further details on the computational method.…”

“…The recent results of Suzuki & Colonius (2006) and Gudmundsson & Colonius (2011) show that the pressure field just outside the jet can be modelled quite accurately as a superposition of linear wavepackets with different frequencies and azimuthal modes, but it remains to be established if these models accurately predict the velocity field of these structures within the jet. The hydrodynamic waves have small amplitudes, no fixed phase reference, and their signature is therefore difficult to educe from the velocity field of unforced jets (a recent attempt has been made by Kerhervé et al 2012 using data from a large eddy simulation).…”

“…This is clearly not the case for natural jets with turbulent boundary layers at the nozzle exit. Gudmundsson & Colonius (2011) present a comparison between results of linear parabolized stability equations (hereafter PSE) and the velocity field of unforced jets, measured using PIV in cross-sections. In that work the PIV measurements were not time-resolved, and so comparison for individual frequencies was not possible; the wavepackets had to be superposed prior to comparison with experiment.…”

“…Linear PSE are used, as in the work of Gudmundsson & Colonius (2011), as this allows the evolution of wavepackets on a slowly diverging mean flow to be modelled. Linear PSE are used here as a reference model for hydrodynamic waves resulting from the Kelvin-Helmholtz instability in the initial region of the jet.…”