ALBERT

Description: 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 90° 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 lowfrequency 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). © Cambridge University Press 2012.

Description: 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,J. Fluid Mech., vol. 84, 1978b, pp. 305–329; Goldstein,J. Fluid Mech., vol. 91, 1979a, pp. 601–632), which 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.

Description: This paper is a continuation of the work begun in Goldstein et al. (J. Fluid Mech., vol. 644, 2010, p. 123), who constructed an asymptotic high-Reynolds-number solution for the flow over a spanwise periodic array of relatively small roughness elements with (spanwise) separation and plan form dimensions of the order of the local boundary-layer thickness. While that paper concentrated on the linear problem, here the focus is on the case where the flow is nonlinear in the immediate vicinity of the roughness with emphasis on the intermediate wake region corresponding to streamwise distances that are large in comparison with the roughness dimension, but small in comparison with the distance between the roughness array and the leading edge. An analytical O(h2) asymptotic solution is obtained for the limiting case of a small roughness height parameter h. These weakly nonlinear results show that the spanwise variable component of the wall-pressure perturbation decays as x -5/3 ln x when x →∞ (where x denotes the streamwise distance scaled on the roughness dimension), but the corresponding component of the streamwise velocity perturbation (i.e. the wake velocity) exhibits an O(x1/3 ln x) algebraic/transcendental growth in the main boundary layer. Numerical solutions for h = O(1) demonstrate that the wake velocity perturbation for the fully nonlinear case grows in the same manner as the weakly nonlinear prediction-which is considerably different from the strictly linear result obtained in Goldstein et al. (2010). © 2011 Cambridge University Press.

Description: It has come to our attention that in the printed version of Goldstein et al. (2011) figures 8, 9, 10 and 11 were printed in black and white when they should have appeared in colour. The reader is referred to the online version of this article where the colour is correct for all figures.

Description: We consider a periodic array of relatively small roughness elements whose spanwise separation is of the order of the local boundary-layer thickness and construct a local asymptotic high-Reynolds-number solution that is valid in the vicinity of the roughness. The resulting flow decays on the very short streamwise length scale of the roughness, but the solution eventually becomes invalid at large downstream distances and a new solution has to be constructed in the downstream region. This latter result shows that the roughness-generated wakes can persist over very long streamwise distances, which are much longer than the distance between the roughness elements and the leading edge. Detailed numerical results are given for the far wake structure. Copyright © 2010 Cambridge University Press.