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A novel analysis technique is developed for the time-accurate analysis of noise sources in compressible jet flows. By adopting a Lagrangian point of view, the finite-time Lyapunov exponent (FTLE) method provides a means of linking discrete events in the vicinity of a high-subsonic jet shear layer to aft- and sideline-radiated noise. The FTLE is first validated in the compressible flow regime, as it was originally developed for incompressible flows, where it has shown substantial success in analyzing the dynamics of Lagrangian coherent structures. It is demonstrated by theoretical and numerical experiments that judicious choice of temporal intergration parameters highlights distinct features in the flow field. Due to the non-solenoidal nature of compressible flows, Lagrangian FTLE coefficients computed in forward and backward time can extract wave dynamics from the velocity field in addition to the flow-organizing convective features. Integration time, thus, acts as a pseudo-filter, smoothly separating coherent (convective) structures from propagating waves.Results are confirmed by first examining forward and backward FTLE coefficients for several simple, well-known acoustic fields and in the propagation of a two-dimensional acoustic pulse. Each coefficient is shown to capture a distinct portion of the traveling acoustic waves, with the forward FTLE (sf) focusing on the positive stroke and the negative portion extracted by the backward- integrated (sb) coefficient. Analysis of the mono-chromatic acoustic flow fields shows the peak FTLE magnitudes directly scale with the acoustic frequency and a complete reconstruction of the underlying dilatational field can be achieved by conducting the Lagrangian analysis over a single time step for each time instant. An increase in integration time subsequently leads to both damping and phase-shifting of the resolved acoustic waves as a consequence of the contribution from multiple temporal snapshots.Having established the theoretical foundation of the Lagrangian dynamics in compressible flows, the technique is then applied to identify events associated with intermittency in jet noise pressure probe data. Although intermittent events are known to be dominant causes of jet noise, their direct source in turbulent jet flows has remained unexplained. To this end, Large-Eddy Simulation (LES) data of several Mach 0.9 jets are subjected to FTLE to simultaneously examine, and thus expose, the causal relationship between coherent structures and the corresponding acoustic waves. The analysis illustrates that intermittent events are associated with entrainment in the initial roll up region and emissive events downstream of the potential-core collapse. Instantaneous acoustic disturbances are observed to be primarily induced near the collapse of the potential core and continue propagating towards the far-field at the experimentally-observed, approximately thirty-degree angle relative to the jet axis.Analysis of a fully-turbulent jet LES, where inflow turbulence was accounted for with a Reynolds- averaged-Navier-Stokes-initiated digital filter, further demonstrates that repelling Lagrangian structures given by the forward-integrated FTLE play a dominant role in establishing the near-acoustic field. By examining the evolution of the FTLE coefficients and the Lagrangian reconstruction of dilatation along the direction of maximum noise directivity, it is shown that sf contributes the bulk of the acoustic energy in the LES 'far-field'; and is also better correlated with the Eulerian dilatation at these locations than the backward-integrated field, despite the fact that the latter field is associated with the large-scale coherent structures. A two- point correlation analysis between the farfield dilatation and the Lagrangian coefficients along the time-mean potential core and shear layer boundaries further demonstrates that the farfield signals are intimately linked to the dynamics of coherent structures in these regions. Vortex merging events in the turbulent jet are also shown to be significant contributors to the genesis of farfield-radiating disturbances. In particular, they play a key role in establishing and modulating wavepackets within the potential core, which are known to be an important component of aft-radiated noise. The modulation of the wavepackets and the entrainment and emmission of acoustic energy to and from the ambient is highly influenced by the presence of strong repelling structures in the turbulent shear layer. |
