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Asymptotic Solution
An asymptotic solution of the present problem may be found by a straightforward application of Fourier transform. Details of the analysis are given in the Proceedings of the Third CAA Workshop on Benchmark Problems (NASA CP-2000-209790, August 2000; Dahl, M. D. ed.). The directivity of sound radiation with respect to a spherical polar coordinate system (R, в, ф) with the polar axis coinciding with the axis of the rotor is
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Lim R2p2
12.4.1.5 Numerical Results
Eqs. (12.34) are discretized according to the multi-size-mesh mult-time-step DRP scheme and marched in time to a time periodic state. To start the computation, a zero initial condition is used. Figure 12.13 shows a comparison of the directivity at
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Figure 12.13. Directivity of sound field at R = 50; D = 0.85;– , numerical;….. , asymptotic. |
R = 50 obtained computationally against the asymptotic solution for D = 0.85, the subsonic tip speed case. As expected, most of the acoustic radiation is concentrated in the plane of rotation. There is good agreement between the numerical results and the asymptotic solution. Figure 12.14 shows the directivity at supersonic tip speed with D = 1.15. At higher frequency, the acoustic wavelength is shorter. This case, therefore, offers a more stringent test of the accuracy of the entire computational algorithm. The agreement with asymptotic solution is reasonably good.
