Investigation of a single-point nonlinearity indicator in the propagation of high-amplitude jet noise

There is evidence to suggest that nonlinearity is important in the propagation of high-amplitude jet noise [Gee et al., AIAA J. 43(6), 1398-1401 (2005)]. Typically, the power spectral density (PSD) is used to assess the impact of jet noise on the surrounding environment, but such an assessment requires multiple measurement locations to observe the nonlinear evolution of the PSD. The difference in the PSDs measured at different locations depends on a combination of source level, nozzle diameter, and propagation distance. As a result, full scale measurements have to be extended over large distances, and model scale measurements require high measurement bandwidths. These constraints complicate the measurement and make it difficult to observe nonlinear effects using the PSD. Here a different technique for determining the importance of nonlinearity is investigated. The imaginary part of the cross-spectral density of the pressure and the square of the pressure, also called the quadspectral density (QSD), is related to the rate of nonlinear change of the PSD. Thus, the extent to which the PSD is evolving nonlinearly can be determined at a single measurement location. In the absence of absorption, energy is conserved, and the integration of the product of the QSD with frequency over all frequencies must be zero; when nonlinearity is present, the value of the QSD can be nonzero for many frequencies. Because nonlinearity tends to transfer energy from low frequencies to high frequencies and the QSD is positive at frequency components that are losing energy, using the integral of the QSD over its positive values as a nonlinearity indicator eliminates the need for high bandwidth measurements. Experimental measurements were taken in a plane wave tube with a working length of 9.55 m in which boundary layer losses dominate over atmospheric absorption. Experimental and numerical results show that the ratio of the integral of the QSD over the frequencies for which it is positive (Q_pos) to the integral over the frequencies for which it is negative (Q_neg) is close to one. Also, because the QSD is third-order in pressure, normalizing its integral by the cube of the rms pressure yields a quantity that is easily compared across experimental conditions. Results for both periodic and broadband signals are presented and the practicality of using the QSD as a single-point indicator of nonlinearity addressed.