Quadrupole correlations governing the pattern of jet noise

Abstract

The effects of convection and refraction dominate the heart-shaped pattern of jet noise. These can be corrected out to yield the small ‘basic directivity’ of the eddy noise generators. The observed quasi-ellipsoidal pattern was predicted by Ribner (1963, 1964) in a variant of the Lighthill theory postulating isotropic turbulence superposed on a mean shear flow. This had the feature of dealing with the joint effects of the quadrupoles without displaying them individually. The present paper reformulates the theory so as to calculate the relative contributions of the different quadrupole self and cross-correlations to the sound emitted in a given direction. Some minor errors are corrected.Of the thirty-six possible quadrupole correlations only nine yield distinct non-vanishing contributions to the axisymmetric noise pattern of a round jet. The correlations contribute either cos4θ, cos2θ sin2θ or sin4θ directional patterns, where θ is the angle with the jet axis. A separation into parts called ‘self noise’ (from turbulence alone) and ‘shear noise’ (jointly from turbulence and mean flow) may be made.The nine self-noise patterns combine as $A\; cos^4\theta(1)+A\; cos^2sin^2\theta(\frac{7}{8}+\frac{7}{8}+\frac{1}{8}+\frac{1}{8})+A sin^2 \theta (\frac {12}{32} & + & \frac{12}{32}+\frac{7}{32}+\frac{1}{32})\\ & = & A(cos^2\theta+sin^\theta)^2 = A;$ this is uniform in all directions as it must be, arising from isotropic turbulence. The two non-vanishing shear-noise correlation patterns combine as $B\;cos^4\theta (1) + B\;cos^2\theta sin^2\theta(\frac{1}{2})=B(cos^2\theta+sin^2\theta)^2 = A;$The overall ‘basic’ pattern (self noise plus shear noise) thus has the form A + B(cos2θ + cos4θ)/2; this is a slight change from the previous result. The dimensional constants A and B are of comparable magnitude; the pattern in any plane through the jet axis thus resembles an ellipse of modest eccentricity.Frequency spectra are also discussed, following the earlier work. Since the self noise depends quadratically on turbulent velocity components and the shear noise only linearly, there is a relative shift of the self noise to higher frequencies. This in conjunction with refraction figures in the explanation of the deeper pitch of jet noise radiated at small angles to the axis.Finally, the predictions are shown to be compatible with recent experimental results.