New Acoustic Analogies
As noted in Section 3.1 little success has been achieved with the prediction of the noise spectra in the peak radiation direction based on acoustic analogies. Two recent approaches appear to have overcome some these earlier problems. Goldstein and Leib (2008) and Karabasov et al. (2010) have extended earlier approaches and achieved some success. However, it should be noted that both studies were limited to convectively subsonic, unheated jets. The work by Goldstein and Leib (2008) builds on the generalized acoustic analogy developed by Goldstein (2003). Goldstein and Leib (2008) rearranged the equations of continuity, momentum, and energy for an ideal gas into a system of five formally linear equations. The equivalent sources on the right hand sides of the system of equations have zero time average. The problem is split into two parts: first, find the vector Green’s function for the system of equations, and then model the statistical properties of the noise sources. Goldstein and Leib (2008) stress the importance of accounting for the slow divergence of the jet flow in the Green’s function. They base their statistical model for the sources on their expected symmetry properties as well as experimental measurements. Predictions are presented for unheated jets with Mach numbers 0.5, 0.9 and 1.4 and the comparisons with measurements are very encouraging. Karabasov et al. (2010) take a slightly different approach. They also used the generalized acoustic analogy of Goldstein (2003), but they calculate the vector Green’s function numerically using an adjoint approach (see Tam (1998a)). In addition, they use a companion Large Eddy Simulation (LES) to provide source statistics, including the general shape of the two-point space-time cross correlations as well as the relative amplitude of the different components. However, they choose to use very simple Gaussian functions to model the correlations. They present results for an unheated jet with Mach number 0.75 and compare with noise measurements. Again, the agreement is quite good. The authors argue that the important effects to be included are; the divergence of the mean flow, the use of non-isotropic source models, the inclusion of the radial variation of the Green’s function, and the use of the same observer location as in the experiments. A careful examination of these two works indicates that different assumptions are made in order to achieve similar agreement with measurements. The two studies do agree on the importance of including the mean flow divergence in the propagation calculation and the need for an anisotropic source model. The other differences have yet to be reconciled.