An Acoustic-Analogy Model for Jet-Noise Prediction

Abstract

The solution of the steady Reynolds-averaged Navier Stokes equations (RANS) oers an easy-access model for the time-averaged statistics of turbulent jets. The spatial distribution of mean velocity, turbulent kinetic energy, turbulence dissipation rate, etc. can be modelled by means of commercially-available RANS solvers. This paper introduces an acoustic-analogy model which uses as input these spatial distributions of turbulent-jet statistical variables to generate a prediction of the far-eld jet-noise spectrum. Asymptotic low-frequency and highfrequency Green functions of the Lilley acoustic analogy are used to relate the far eld to the forcing terms of the Lilley analogy. The source expression introduced by Goldstein for a perfect gas is adopted. Both applied-stress and applied-force equivalent sources are retained; thus the model includes both the Reynolds-stress source and the density-inhomogeneity source. The far-eld acoustic spectra associated with a given turbulence-statistics eld are predicted by dividing the eld into small uncorrelated volumes. The o w-variable statistics are assumed not to vary inside each small volume. In this case the power spectral density (PSD) of far-eld pressure which is associated with each small volume can be evaluated in three steps. Firstly the two-point cross power spectral density (CPSD) of the Reynoldsstress (quadrupole) and the density-inhomogeneity (dipole) sources is modelled. As second step the CPSD model is appropriately phase shifted and spatially integrated across the local region of coherence to obtain a local measure of the source strength per unit volume, in each direction and at each frequency. Finally the Green functions (low- and high-frequency approximations) appropriate to the particular source and observer locations are applied. The resulting far-eld PSD contributions from each volume element are integrated across the turbulent jet to generate the PSD for the whole jet. For the rst step of the procedure above, a model for the CPSD of both equivalent sources is required. A modied-distance model for the Reynolds-stress 2-point correlation is used. This model enables tting of published Reynolds-stress measurements. The Fourier transform of the 2-point correlation model gives the CPSD. By scaling, the CPSD model is extended to all the Reynolds-stress components and to the density-uctuation vector. The implemented model is suitable for the prediction of noise from axisymmetric turbulent jets, both single stream and coaxial. A commercial RANS code with Reynolds stress model (RSM) closure is used to provide turbulence statistics, including components of the mean Reynolds stress. Predicted jet-noise 1/3-octave spectra are compared to corresponding experimental data at the 90-degree polar angle. Isothermal single-stream and coaxial jets are tted well by the model. Preliminary results are also shown for hot single-stream jets.