A Test Case: A Broadband Monopole Source

To provide a simple validation test for formula (14.102), reconsider the case of the monopole noise source problem discussed in section 14.5.2 (see Figure 14.12). Instead of a time periodic source, it is replaced by the broadband monopole source of Section 14.6. The half-apex angle of the conical surface 5 is again taken to be 10°. In this case, if Yr is a point on the conical surface, then

Y’ = (R’2 + x2 – 2R’xc cos 5)2.

The two-point space-time correlation function has the following form (see Eq. (14.91)): Ф(ІЇ, R", &,X) = F  The F-function as computed directly is shown in Figure 14.19. In this special case, Ф is independent of ©. For this reason, the integral over d& in Eq. (14.102) is zero except for n = 0. Thus the far-field noise spectrum obtained by continuation of the pressure field on the conical surface is as follows: 

A change of integration variable from X to n = (Y" – Yr – a0X)/D transforms the triple integrals in Eq. (14.103) into three separate integrals. This yields     s (Яв, ф,ю) D

Figure 14.21 shows a comparison of the far-field noise spectra at в = 150°, 90°, and 30° (exhaust angle) computed numerically according to Eq. (14.104) and the original noise spectrum of the broadband monopole source (the similarity spectrum of the noise of large turbulence structures of high-speed jets). The agreement is excellent. The excellent agreement provides strong support for the validity of the present continuation method. Figure 14.21. Comparisons between computed spectra (circles) and original spectra (full lines) at (a) в =150°, (b) 90°, and (c) 30°.

EXERCISE