Higher approximations in the asymptotic theory of propeller noise

Abstract

This paper extends earlier work on the asymptotic theory of the acoustic radiation from a single-rotatio n propeller (in the many-blade limit B —> oc) in two ways. First, a correction is found to the dominant Mach radius field in supersonic operating conditions; the correction is associated with the blade tip and its inclusion brings the asymptotic prediction into fine agreement with numerical evaluation of the radiation integrals, in addition to giving physical insight and the possibility of control, at negligible computing expense. Second, an expression is derived that provides a transition across the Mach condition (from subsonic to supersonic radiation characteristics), and from this, composite expressions are given, valid across the entire range of operating conditions. Nomenclature Ai = Airy function B = number of blades JmB = Bessel function of first kind and order mB Mrt = tip relative Mach number Mt = tip rotational Mach number Mx = flight (axial) Mach number m = harmonic index Pm = normalized complex amplitude of far-field pressure in rath harmonic S(z) = normalized radial source function z — normalized radial (spanwise) coordinate z0 = normalized hub radius z* = Mach radius, defined by Eq. (2) 0, = sech-'Cl/z*), seeEq. (16) A = coordinate defined by Eq. (18) 0 = angular location of observer at emission time A, - sec-!(l/z*), see Eq. (14) JJL = tip-source exponent