Input-output analysis of high-speed turbulent jet noise

Abstract

We use input-output analysis to predict and understand the aeroacoustics of high-speed turbulent jets. We consider linear perturbations about Reynolds-averaged Navier-Stokes (RANS) solutions of ideally expanded, axisymmetric, compressible turbulent jets under various operating conditions. For jet noise, a key aspect of our method is the ability to spatially separate near-field input forcing (driven by nonlinear turbulence) from far-field acoustic output. Precisely the same idea, namely the separation of sources and outputs, forms the basis of traditional acoustic analogies. Different from the usual statistical descriptions of the acoustic source terms, input-output analysis provides a dynamical description based on modes correlated over significant distances within the flow. Specifically, we compute optimal and sub-optimal harmonic forcing functions and their corresponding linear responses governed either by the linearized Euler equations (LEE) or by the linearized Navier-Stokes (LNS) equations, using singular value decomposition of the resolvent operator. For supersonic jets, the optimal response closely resembles a wavepacket in both the near-field and the far-field such as those obtained by the parabolized stability equations (PSE), and this mode dominates the response. For subsonic jets, however, the singular values indicate that the contributions of sub-optimal modes to noise generation are nearly equal to that of the optimal mode, explaining why the PSE do not fully capture the far-field sound in this case. Furthermore, we utilize a high-fidelity large-eddy simulation (LES) data to assess the prevalence of sub-optimal modes in the unsteady data. By projecting the LES source term data onto input modes and the LES acoustic far-field onto output modes, respectively, we demonstrate that sub-optimal modes of both types are physically relevant. Far-field acoustics generated from turbulent jets are further modeled, using a Ffowcs Williams-Hawkings (FW-H) solver implemented directly within linear input-output analysis framework. Our hybrid input-output/FW-H method efficiently connects input fluctuations embedded in the jet turbulence to pressure outputs in the far-field, and recovers a significant portion of the LES acoustic energy. By repeating input-output analysis over a wide range of frequencies, we find that the far-field acoustic spectra broaden with increasing the radiation angles, as observed in experiments. To distill acoustically relevant sources, input forcings are further restricted by introducing a new weighting matrix, which selects forcing functions only in the region that contains high turbulent kinetic energy (TKE). We then find that input modes correspond exactly to wavepackets with asymmetric pseudo-Gaussian envelope functions. Furthermore, wavepackets obtained by input-output analysis collapse to a single shape when scaled by $St^{-0.5}$, where $St$ is the jet Strouhal number. This explains the success of recent theoretical models based on stochastic similarity wavepackets.