No abstract

A number of numerical implementations of the Helmholtz integral equation exist today that can predict routinely the field scattered by a volume-holding body, such as the ellipsoidal “core” of a typical airborne or submerged vehicle stripped of its thin appendages, i.e., stripped of control surfaces, etc. The reason for these exclusions has often been an inherent limitation of the cited modeling tools, rather than a rational dismissal of the potential effect of the neglected protrusions on the complete body's expected scattering cross section. The limitation of existing techniques is this: The standard form of Gauss' theorem on which they are based, which leads to the Helmholtz integral, becomes meaningless when the volume of the shape addressed tapers down to zero even over only part of the structure. This paper explains, analytically, the origin of this thin-shape breakdown (TSB), and develops an alternate boundary element formulation for its cure.

A three-dimensional aeroacoustic model is developed to predict the sound pulse radiated by the passage of a helicopter blade over a potential vortex. The linearized analysis assumes that either the blade-vortex separation is small or that the blade-tip Mach number is close to 1, or both, so that an acoustically noncompact situation exists. The three-dimensional blade loading due to blade-vortex interaction is constructed through a span wise superposition of two-dimensional solutions with strength linearly increasing from hub to tip. Such a loading overestimates somewhat the strength of tip region dipoles in the acoustic calculation that follows. The final expression for the predicted far-field signature is obtained in closed form, and thus permits a relatively inexpensive calculation of the directivity of peak acoustic pressures in three dimensions. Nomenclature a =V-/x 2 foi> 2 <0 arg = argument, or phase, of a complex number b = blade semichord, reference length c 0 = dimensional sound speed C = complex constant, Eqs. (28) and (29) h = nondimensional vertical blade-vortex separation Im = imaginary part of a complex number K =k x M/(\-M 2 ) k x = reduced frequency or nondimensional chprdwise wave number k y = A^tanA, the spanwise wave number L = nondimensional blade length M =freestream Mach number and blade-tip Mach number in three-dimensional theory M eff = effective Mach number, M/sinA P,P* = perturbation pressures, Eqs. (6) and (15), respectively P = Fourier transform of P*, Eq. (19b) p = predicted pulse r = nondimensional distance to observer f = nondimensional vector position of observer Re = real part of a complex number t = dimensional time U -dimensional freestream velocity, or blade-tip velocity W = Fourier transform of vortex-induced upwash w 0 = FfVcosA w = dimensional vortex upwash x,y,z =chordwise, spanwise, and normal to blade nondimensional sp>atial_ coordinates A = blade-vortex interaction angle ju, = frequency parameter, Eq. (8) £ =.x a cosA+ > ysinA P 0 = freestream density o> = acoustic frequency fi = rotor angular velocity 0,7 = angles in spherical coordinate system fixed in the still fluid, Fig. 3 F = dimensional vortex strength

This paper develops theoretical models to predict the radiation of low-frequency and impulsive sound from horizontal-axis wind turbines due to three sources: (i) steady blade loads, (ii) unsteady blade loads due to operation in a ground shear, (iii) unsteady loads felt by the blades as they cross the tower wake. These models are then used to predict the acoustic output of MOD-I, the large wind turbine operated near Boone, N. C. Predicted acoustic time signals are compared to those actually measured near MOD-I; good agreement is obtained.

The problem of diffraction by a rigid, open-ended pipe of negligible thickness and finite length is formulated and solved using the methodology of lifting-surface theory. The analysis assumes a still acoustic medium and considers incident fields corresponding to specified dipole sources. Among the possible applications is the preliminary but rigorous estimate of the shielding effect through pure diffraction of a cylindrical shroud on unsteady propeller forces prescribed within it . The present study confines itself to axisymmetric situations, including the simple case of a single harmonic force applied to the fluid along the cylinder's axis. The assumed zero thickness for the pipe wall results in a virtual source representation of unsteady doublets whose integrated component of radial velocity must cancel that of the incident field over the diffracting surface. For the single on-axis insonifying dipole the diffraction loading solution, or pressure difference between points just outside and inside the pipe, is checked by computing the resistive component of the acoustic admittance at the spatial origin of incident sound, and by verifying numerically that for each sample frequency this value of source input power matches the system's total radiated power.Nomenclature cylinder radius (nondimensional) virtual load coefficient sectional blade load at hub sectional blade load at effective tip position Bessel, Hankel, and modified functions of order v\ superscript (1) on H v is omitted throughout for simplicity of notation acoustic wavenumber (nondimensional) cylinder half length and normalizing constant incident pressure field diffraction loading, p° -p l , the difference between pressures on r = a + , a ~ for lzl