The radiation of fan generated noise to the far field from a nacelle of realistic geometry is investigated using the finite element method. Several innovations have been introduced to minimize the computational requirements and create a highly efficient numerical scheme. The innovations include: (1) formulation of the problem in terms of velocity potential and density in such a way that no inlet mean flow velocity derivatives are required in the field equations, (2) the use of "wave envelope" elements in an outer region permitting a grid much coarser than would be used for conventional finite elements, (3) the use of a mesh which deforms with an increase of forward flight speed so that mesh lines are always lines of constant phase and rays for a point source, permitting the use of wave envelope elements and simple boundary conditions for any case of forward velocity, (4) an efficient scheme for introducing the noise source via modal amplitude coefficients, and (5) the use of a frontal solution technique which for physically realistic problems drastically reduces the active storage requirements. The finite element scheme is outlined, as are the specific details of the innovations. Results are given for cases where comparable experimental data are available.

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The problem of acoustic radiation from turbofan engine inlets in flow has not lent itself fully to analysis by numerical means because of the large domains and high frequencies involved. The current work has extended the use of finite elements and wave envelope elements, elements that simulate decay and wavelike behavior in their interpolation functions, from the no-flow case in which they have proven, to cases incorporating mean flow. By employing an irrotational mean flow assumption, the acoustics problem has been posed in an axisymmetric formulation in terms of acoustic velocity potential, thus minimizing computer solution storage requirements. The results obtained from the numerical procedures agree well with known analytical solutions, static experimental jet engine inflow data, and flight test results. Nomenclaturea£aj~ = incident and reflected modal amplitudes, respectively c = nondimensional speed of sound in flow e£ef = positive and negative uniform duct eigenfunctions, respectively m = spinning mode number TV,-= interpolation function p = pressure t = time u = axial component of local Mach number U = Mach number v = radial component of local Mach number v = acoustic velocity V -velocity x = axial coordinate 7= ratio of specific heats e = penalty parameter TJ, £ = local finite element coordinates K,A = local and axial wave numbers, respectively IJL = uniform duct eigenvalue p = density = velocity potential, trial function $(Q) = trial function space $ = full nondimensional velocity potential \l/ =test function ¥ (fi) = test function space co = harmonic frequency Superscripts ( )' = nondimensional quantity, derivative (see text) e = quantity defined inside a finite element

This paper presents a study of acoustic tone radiation patterns from a small turbofan engine in flight and compares results with similar static test stand data and a recently developed radiation theory. An interaction tone was induced for test and evaluation purposes by placing a circumferential array of inlet rods just upstream of the fan blades. Overhead and sideline flight directivity patterns showed cut-on of a dominant single-mode tone occurred at the predicted fan speed, and there was an absence of any other significant circumferential or radial modes. In general, good agreement was found between measured flight and static data, with small differences being attributed to inlet geometry and/or forward speed effects. Good agreement was also obtained between flight data and theory for directivity pattern shape, however, the theory consistently predicted higher values for peak radiation angle over a wide range of frequency. a B c f Km,n k M th M tip Nl (m, R r x Nomenclature = radius of inlet at throat, = 9.95 in. = number of fan blades = speed of sound at inlet throat, in./s = frequency, Hz = mode eigenvalue = free-space wave number, = 2irf/c, in. ~1 = Mach number at inlet throat = Mach number of fan tip = Mach number of aircraft or freestream Mach number = fan speed, rpm = (circumferential, radial) duct mode numbers = radial distance to far field in spherical coordinates, in. = cylindrical radial coordinate, in. = cylindrical axial coordinate, in. = directivity angle measured from inlet axis in spherical coordinates, deg _________ = cutoff ratio, =BM i[p /K m>fJL^J l -M? h= sideline or polar angle, deg (see Fig. 5)

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