Identifying an \u3cem\u3em\u3c/em\u3e-Ary Partition Identity through an \u3cem\u3em\u3c/em\u3e-Ary Tree

The Calkin-Wilf tree is well-known as one way to enumerate the rationals, but also may be used to count hyperbinary partitions of an integer, h2(n). We present an m-ary tree which is a generalization of the Calkin-Wilf tree and show how it may be used to count the hyper m-ary partitions of an integer, hm(n). We then use properties of the m-ary tree to prove an identity relating values of h2 to values of hm, showing that one sequence is a subsequence of the other. Finally, we give a bijection between the partitions to reprove our identity

Flight-Test Evaluation of Landing Gear Noise Reduction Technologies

Results from the third Acoustics Research Measurements flight test, conducted under the NASA Flight Demonstrations and Capabilities project, are presented and discussed. The test evaluated landing gear and gear cavity noise mitigation technologies installed on a NASA Gulfstream G-III. Aircraft configurations with and without main landing gear treatments were flown at several flap deflections to determine the acoustic performance of the technologies for aircraft equipped with conventional Fowler flaps. With the aircraft flying an approach path and engines at ground-idle, extensive acoustic measurements were acquired with a phased microphone array system. Computed beamform maps were used to examine the effectiveness of the tested technologies in reducing the strength of the noise sources generated by the main landing gear. Various integration regions were devised to extract the farfield noise spectra associated with the treated and untreated landing gear configurations. Analyses of the gathered acoustic data demonstrate that significant noise reduction was achieved. How- ever, the full noise reduction potential of the technologies could not be determined because of contamination from flap inboard edge noise and other secondary sources

Measured and Simulated Acoustic Signature of a Full-Scale Aircraft with Airframe Noise Reduction Technology Installed

Microphone phased-array and pole-mounted microphone data gathered during the NASA Acoustics Research Measurements flight tests were used to benchmark results from companion full-scale aeroacoustics simulations. Conducted with the lattice Boltzmann solver PowerFLOW, the simulations predicted the acoustic behavior of various tested aircraft configurations. Emphasis was placed on those flown during the third flight test - a Fowler flap-equipped Gulfstream G-III with and without noise abatement technology on the main landing gear. Direct comparisons between experimental and synthetic microphone phasedarray data were achieved by applying the same processing and deconvolution technique to both sets of data. To extend the validation of the computations to the metric used for noise certification, the Effective Perceived Noise Level, a high-fidelity digital model of the nose landing gear, which was excluded from earlier computations, was developed and integrated into the G-III aircraft geometry. The acoustic study presented here demonstrates that the simulated beamform maps and corresponding integrated farfield spectra accurately predict the locations and strengths of the prominent airframe noise sources present on the G-III aircraft

Effects of Geometric Details on Slat Noise Generation and Propagation

The relevance of geometric details to the generation and propagation of noise from leading-edge slats is considered. Typically, such details are omitted in computational simulations and model-scale experiments thereby creating ambiguities in comparisons with acoustic results from flight tests. The current study uses two-dimensional, computational simulations in conjunction with a Ffowcs Williams-Hawkings (FW-H) solver to investigate the effects of previously neglected slat "bulb" and "blade" seals on the local flow field and the associated acoustic radiation. The computations clearly show that the presence of the "blade" seal at the cusp significantly changes the slat cove flow dynamics, reduces the amplitudes of the radiated sound, and to a lesser extent, alters the directivity beneath the airfoil. Furthermore, it is demonstrated that a modest extension of the baseline "blade" seal further enhances the suppression of slat noise. As a side issue, the utility and equivalence of FW-H methodology for calculating far-field noise as opposed to a more direct approach is examined and demonstrated

Effects of Geometric Details on Slat Noise Generation and Propagation

The relevance of geometric details to the generation and propagation of noise from leading-edge slats is considered. Typically, such details are omitted in computational simulations and model-scale experiments thereby creating ambiguities in comparisons with acoustic results from flight tests. The current study uses two-dimensional, computational simulations in conjunction with a Ffowcs Williams-Hawkings (FW-H) solver to investigate the effects of previously neglected slat "bulb" and "blade" seals on the local flow field and the associated acoustic radiation. The computations show that the presence of the "blade" seal at the cusp in the simulated geometry significantly changes the slat cove flow dynamics, reduces the amplitudes of the radiated sound, and to a lesser extent, alters the directivity beneath the airfoil. Furthermore, the computations suggest that a modest extension of the baseline "blade" seal further enhances the suppression of slat noise. As a side issue, the utility and equivalence of FW-H methodology for calculating far-field noise as opposed to a more direct approach is examined and demonstrated

Application of FUN3D Solver for Aeroacoustics Simulation of a Nose Landing Gear Configuration

Numerical simulations have been performed for a nose landing gear configuration corresponding to the experimental tests conducted in the Basic Aerodynamic Research Tunnel at NASA Langley Research Center. A widely used unstructured grid code, FUN3D, is examined for solving the unsteady flow field associated with this configuration. A series of successively finer unstructured grids has been generated to assess the effect of grid refinement. Solutions have been obtained on purely tetrahedral grids as well as mixed element grids using hybrid RANS/LES turbulence models. The agreement of FUN3D solutions with experimental data on the same size mesh is better on mixed element grids compared to pure tetrahedral grids, and in general improves with grid refinement

Optical Geolocation for Small Unmanned Aerial Systems

This paper presents an airborne optical geolocation system using four optical targets to provide position and attitude estimation for a sUAS supporting the NASA Acoustic Research Mission (ARM), where the goal is to reduce nuisance airframe noise during approach and landing. A large precision positioned microphone array captures the airframe noise for multiple passes of a Gulfstream III aircraft. For health monitoring of the microphone array, the Acoustic Calibration Vehicle (ACV) sUAS completes daily flights with an onboard speaker emitting tones at frequencies optimized for determining microphone functionality. An accurate position estimate of the ACV relative to the array is needed for microphone health monitoring. To this end, an optical geolocation system using a downward facing camera mounted to the ACV was developed. The 3D positioning of the ACV is computed using the pinhole camera model. A novel optical geolocation algorithm first detects the targets, then a recursive algorithm tightens the localization of the targets. Finally, the position of the sUAS is computed using the image coordinates of the targets, the 3D world coordinates of the targets, and the camera matrix. A Real-Time Kinematic GPS system is used to compare the optical geolocation system

CFD-CAA Coupled Calculations of a Tandem Cylinder Configuration to Assess Facility Installation Effects

This paper presents a numerical assessment of acoustic installation effects in the tandem cylinder (TC) experiments conducted in the NASA Langley Quiet Flow Facility (QFF), an open-jet, anechoic wind tunnel. Calculations that couple the Computational Fluid Dynamics (CFD) and Computational Aeroacoustics (CAA) of the TC configuration within the QFF are conducted using the CFD simulation results previously obtained at NASA LaRC. The coupled simulations enable the assessment of installation effects associated with several specific features in the QFF facility that may have impacted the measured acoustic signature during the experiment. The CFD-CAA coupling is based on CFD data along a suitably chosen surface, and employs a technique that was recently improved to account for installed configurations involving acoustic backscatter into the CFD domain. First, a CFD-CAA calculation is conducted for an isolated TC configuration to assess the coupling approach, as well as to generate a reference solution for subsequent assessments of QFF installation effects. Direct comparisons between the CFD-CAA calculations associated with the various installed configurations allow the assessment of the effects of each component (nozzle, collector, etc.) or feature (confined vs. free jet flow, etc.) characterizing the NASA LaRC QFF facility

Aeroacoustic Simulations of a Nose Landing Gear with FUN3D: A Grid Refinement Study

A systematic grid refinement study is presented for numerical simulations of a partially-dressed, cavity-closed (PDCC) nose landing gear configuration that was tested in the University of Florida's open-jet acoustic facility known as the UFAFF. The unstructured-grid flow solver FUN3D is used to compute the unsteady flow field for this configuration. Mixed-element grids generated using the Pointwise (Registered Trademark) grid generation software are used for numerical simulations. Particular care is taken to ensure quality cells and proper resolution in critical areas of interest in an effort to minimize errors introduced by numerical artifacts. A set of grids was generated in this manner to create a family of uniformly refined grids. The finest grid was then modified to coarsen the wall-normal spacing to create a grid suitable for the wall-function implementation in FUN3D code. A hybrid Reynolds-averaged Navier-Stokes/large eddy simulation (RANS/LES) turbulence modeling approach is used for these simulations. Time-averaged and instantaneous solutions obtained on these grids are compared with the measured data. These CFD solutions are used as input to a FfowcsWilliams-Hawkings (FW-H) noise propagation code to compute the farfield noise levels. The agreement of the computed results with the experimental data improves as the grid is refined