No abstract

Noise measurements of heated axisymmetric jets at fixed supersonic acoustic Mach number indicate that the acoustic spectrum reduces when the temperature ratio increases. The 'spectral quietening' effect has been observed both experimentally and computationally using Large Eddy Simulations (LES). It was explained by Afsar et al. (M. Z. Afsar and M. E. Goldstein & A. M. Fagan AIAAJ., Vol. 49, p. 2522, 2011) through the cancellation introduced by enthalpy flux/momentum flux coupling term using the generalized acoustic analogy formulation. But the parallel flow assumption is known to give inaccurate predictions at high jet speeds. In this paper we therefore extend the non-parallel flow asymptotic theory of Goldstein et al.

This paper is concerned with the small-amplitude unsteady motion of an inviscid non-heat-conducting compressible fluid on a transversely sheared mean flow. It extends previous analyses (Goldstein, show that the hydrodynamic component of the motion is determined by two arbitrary convected quantities in the absence of solid surfaces and hydrodynamic instabilities. These results can be used to specify appropriate upstream boundary conditions for unsteady surface interaction problems on transversely sheared mean flows in the same way that the vortical component of the Kovasznay (J. Aero. Sci., vol. 20, 1953, pp. 657-674) decomposition is used to specify these conditions for surface interaction problems on uniform mean flows. But unlike Kovasznay's result, the arbitrary convected quantities no longer bear a simple relation to the physical variables. A major purpose of this paper is to complete the formalism developed in Goldstein's earlier two papers by obtaining the necessary relations between these quantities and the measurable flow variables. The results are important because they enable the complete extension of non-homogeneous rapid distortion theory to transversely sheared mean flows. Another purpose of the paper is to derive a generalization of the famous Ffowcs Williams and Hall (J. Fluid Mech., vol. 40, 1970, pp. 657-670) formula for the sound produced by the interaction of turbulence with an edge, which is frequently used as a starting point for predicting sound generation by turbulence-solid surface interactions. We illustrate the utility of this result by using it to calculate the sound radiation produced by the interaction of a two-dimensional jet with the downstream edge of a flat plate.

It is now well-known that there is an exact formula relating the far-field jet noise spectrum to the convolution product of a propagator (that accounts for the mean flow interactions) and a generalized Reynolds stress autocovariance tensor (that accounts for the turbulence fluctuations). The propagator depends only on the mean flow and an adjoint vector Green’s function for a particular form of the linearized Euler equations. Recent numerical calculations of Karabasov, Bogey & Hynes (AIAA Paper 2011-2929) for a Mach 0.9 jet show use of the true non-parallel flow Green’s function rather than the more conventional locally parallel flow result leads to a significant increase in the predicted low-frequency sound radiation at observation angles close to the downstream jet axis. But the non-parallel flow appears to have little effect on the sound radiated at $9{0}^{\ensuremath{\circ} } $ to the downstream axis. The present paper is concerned with the effects of non-parallel mean flows on the adjoint vector Green’s function. We obtain a low-frequency asymptotic solution for that function by solving a very simple second-order hyperbolic equation for a composite dependent variable (which is directly proportional to a pressure-like component of this Green’s function and roughly corresponds to the strength of a monopole source within the jet). Our numerical calculations show that this quantity remains fairly close to the corresponding parallel flow result at low Mach numbers and that, as expected, it converges to that result when an appropriately scaled frequency parameter is increased. But the convergence occurs at progressively higher frequencies as the Mach number increases and the supersonic solution never actually converges to the parallel flow result in the vicinity of a critical- layer singularity that occurs in that solution. The dominant contribution to the propagator comes from the radial derivative of a certain component of the adjoint vector Green’s function. The non-parallel flow has a large effect on this quantity, causing it (and, therefore, the radiated sound) to increase at subsonic speeds and decrease at supersonic speeds. The effects of acoustic source location can be visualized by plotting the magnitude of this quantity, as function of position. These ‘altitude plots’ (which represent the intensity of the radiated sound as a function of source location) show that while the parallel flow solutions exhibit a single peak at subsonic speeds (when the source point is centred on the initial shear layer), the non-parallel solutions exhibit a double peak structure, with the second peak occurring about two potential core lengths downstream of the nozzle. These results are qualitatively consistent with the numerical calculations reported in Karabasov et al. (2011).