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NASA NTRS2013

Application of Adjoint Methodology to Supersonic Aircraft Design Using Reversed Equivalent Areas

This paper presents an approach to shape an aircraft to equivalent area based objectives using the discrete adjoint approach. Equivalent areas can be obtained either using reversed augmented Burgers equation or direct conversion of off-body pressures into equivalent area. Formal coupling with CFD allows computation of sensitivities of equivalent area objectives with respect to aircraft shape parameters. The exactness of the adjoint sensitivities is verified against derivatives obtained using the complex step approach. This methodology has the benefit of using designer-friendly equivalent areas in the shape design of low-boom aircraft. Shape optimization results with equivalent area cost functionals are discussed and further refined using ground loudness based objectives.

Rallabhandi, Sriram K.↗
NASA NTRS2014

A Walsh Function Module Users' Manual

The solution of partial differential equations (PDEs) with Walsh functions offers new opportunities to simulate many challenging problems in mathematical physics. The approach was developed to better simulate hypersonic flows with shocks on unstructured grids. It is unique in that integrals and derivatives are computed using simple matrix multiplication of series representations of functions without the need for divided differences. The product of any two Walsh functions is another Walsh function - a feature that radically changes an algorithm for solving PDEs. A FORTRAN module for supporting Walsh function simulations is documented. A FORTRAN code is also documented with options for solving time-dependent problems: an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the usage of the Walsh function module including such features as operator overloading, Fast Walsh Transforms in multi-dimensions, and a Fast Walsh reciprocal.

Gnoffo, Peter A.↗
NASA NTRS2015

Advanced Concept Studies for Supersonic Commercial Transports Entering Service in the 2018-2020 Period Phase 2

Lockheed Martin Aeronautics Company (LM), working in conjunction with General Electric Global Research (GE GR) and Stanford University, executed a 19 month program responsive to the NASA sponsored "N+2 Supersonic Validation: Advanced Concept Studies for Supersonic Commercial Transports Entering Service in the 2018-2020 Period" contract. The key technical objective of this effort was to validate integrated airframe and propulsion technologies and design methodologies necessary to realize a supersonic vehicle capable of meeting the N+2 environmental and performance goals. The N+2 program is aligned with NASA's Supersonic Project and is focused on providing system level solutions capable of overcoming the efficiency, environmental, and performance barriers to practical supersonic flight. The N+2 environmental and performance goals are outlined in the technical paper, AIAA-2014-2138 (Ref. 1) along with the validated N+2 Phase 2 results. Our Phase 2 efforts built upon our Phase 1 studies (Ref. 2) and successfully demonstrated the ability to design and test realistic configurations capable of shaped sonic booms over the width of the sonic boom carpet. Developing a shaped boom configuration capable of meeting the N+2 shaped boom targets is a key goal for the N+2 program. During the LM Phase 1 effort, LM successfully designed and tested a shaped boom trijet configuration (1021) capable of achieving 85 PLdB under track (forward and aft shock) and up to 28 deg off-track at Mach 1.6. In Phase 2 we developed a refined configuration (1044-2) that extended the under 85 PLdB sonic boom level over the entire carpet of 52 deg off-track at a cruise Mach number of 1.7. Further, the loudness level of the configuration throughout operational conditions calculates to an average of 79 PLdB. These calculations rely on propagation employing Burger's (sBOOM) rounding methodology, and there are indications that the configuration average loudness would actually be 75 PLdB. We also added significant fidelity to the design of the configuration in this phase by performing a low speed wind tunnel test at our LTWT facility in Palmdale, by more complete modelling of propulsion effects in our sonic boom analysis, and by refining our configuration packaging and performance assessments. Working with General Electric, LM performed an assessment of the impact of inlet and nozzle effects on the sonic boom signature of the LM N+2 configurations. Our results indicate that inlet/exhaust streamtube boundary conditions are adequate for conceptual design studies, but realistic propulsion modeling at similar stream-tube conditions does have a small but measurable impact on the sonic boom signature. Previous supersonic transport studies have identified aeroelastic effects as one of the major challenges associated with the long, slender vehicles particularly common with shaped boom aircraft (Ref. 3). Under the Phase 2 effort, we have developed a detailed structural analysis model to evaluate the impact of flexibility and structural considerations on the feasibility of future quiet supersonic transports. We looked in particular at dynamic structural modes and flutter as a failure that must be avoided. We found that for our N+2 design in particular, adequate flutter margin existed. Our flutter margin is large enough to cover uncertainties like large increases in engine weight and the margin is relatively easy to increase with additional stiffening mass. The lack of major aeroelastic problems probably derives somewhat from an early design bias. While shaped boom aircraft require long length, they are not required to be thin. We intentionally developed our structural depths to avoid major flexibility problems. So at the end of Phase 2, we have validated that aeroelastic problems are not necessarily endemic to shaped boom designs. Experimental validation of sonic boom design and analysis techniques was the primary objective of the N+2 Supersonic Validations contract; and in this Phase, LM participated in four high speed wind tunnel tests. The first so-called Parametric Test in the Ames 9x7 tunnel did an exhaustive look at variation effects of the parameters: humidity, total pressure, sample time, spatial averaging distance and number of measurement locations, and more. From the results we learned to obtain data faster and more accurately, and made test condition tolerances easy to meet (eliminating earlier 60 percent wasted time when condition tolerances could not be held). The next two tests used different tunnels. The Ames 11 ft tunnel was used to test lower Mach numbers of 1.2 and 1.4. There were several difficulties using this tunnel for the first time for sonic boom including having to shift the measurement Mach numbers to 1.15 and 1.3 to avoid flow problems. It is believed that the 11 ft could be used successfully to measure sonic boom but there are likely to be a number of test condition restrictions. The Glenn 8x6 ft tunnel was used next and the tunnel has a number of desirable features for sonic boom measurement. While the Ames 9x7 can only test Mach 1.55 to 2.55 and the 11 ft can only test Mach 1.3 and lower, the Glenn 8x6 can test continuously from Mach 0.3 to 2.0. Unfortunately test measurement accuracy was compromised by a reference pressure drift. Post-test analysis revealed that the drift occurred when Mach number drifted slightly. Test measurements indicated that if Mach number drift is eliminated, results from the 8x6 would be more accurate, especially at longer distances, than results from the 9x7. The fourth test in the 9x7, called LM4, used everything we learned to comprehensively and accurately measure our new 1044-02 configuration with a full-carpet shaped signature design. Productivity was 8 times greater than our Phase 1 LM3 test. Measurement accuracy and repeatability was excellent out to 42 in. However, measurements at greater distances require the rail in the aft position and become substantially less accurate. Further signature processing or measurement improvements are needed for beyond near-field signature validation.

Morgenstern, John↗
NASA NTRS2004

Atomic Force Microscope Observation of Growth and Defects on As-Grown (111) 3C-SiC Mesa Surfaces

This paper presents experimental atomic force microscope (AFM) observations of the surface morphology of as-grown (111) silicon-face 3C-SiC mesa heterofilms. Wide variations in 3C surface step structure are observed as a function of film growth conditions and film defect content. The vast majority of as-grown 3C-SiC surfaces consisted of trains of single bilayer height (0.25 nm) steps. Macrostep formation (i.e., step-bunching) was rarely observed, and then only on mesa heterofilms with extended crystal defects. As supersaturation is lowered by decreasing precursor concentration, terrace nucleation on the top (111) surface becomes suppressed, sometimes enabling the formation of thin 3C-SiC film surfaces completely free of steps. For thicker films, propagation of steps inward from mesa edges is sometimes observed, suggesting that enlarging 3C mesa sidewall facets begin to play an increasingly important role in film growth. The AFM observation of stacking faults (SF's) and 0.25 nm Burgers vector screw component growth spirals on the as-grown surface of defective 3C films is reported.

Neudeck, Philip G.↗
NASA NTRS2015

Unsteady Solution of Non-Linear Differential Equations Using Walsh Function Series

Walsh functions form an orthonormal basis set consisting of square waves. The discontinuous nature of square waves make the system well suited for representing functions with discontinuities. The product of any two Walsh functions is another Walsh function - a feature that can radically change an algorithm for solving non-linear partial differential equations (PDEs). The solution algorithm of non-linear differential equations using Walsh function series is unique in that integrals and derivatives may be computed using simple matrix multiplication of series representations of functions. Solutions to PDEs are derived as functions of wave component amplitude. Three sample problems are presented to illustrate the Walsh function series approach to solving unsteady PDEs. These include an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the use of the Walsh function solution algorithms, exploiting Fast Walsh Transforms in multi-dimensions (O(Nlog(N))). Details of a Fast Walsh Reciprocal, defined here for the first time, enable inversion of aWalsh Symmetric Matrix in O(Nlog(N)) operations. Walsh functions have been derived using a fractal recursion algorithm and these fractal patterns are observed in the progression of pairs of wave number amplitudes in the solutions. These patterns are most easily observed in a remapping defined as a fractal fingerprint (FFP). A prolongation of existing solutions to the next highest order exploits these patterns. The algorithms presented here are considered a work in progress that provide new alternatives and new insights into the solution of non-linear PDEs.

Gnoffo, Peter A.↗
NASA NTRS2017

Feature Detection and Curve Fitting Using Fast Walsh Transforms for Shock Tracking: Applications

Walsh functions form an orthonormal basis set consisting of square waves. Square waves make the system well suited for detecting and representing functions with discontinuities. Given a uniform distribution of 2p cells on a one-dimensional element, it has been proven that the inner product of the Walsh Root function for group p with every polynomial of degree < or = (p - 1) across the element is identically zero. It has also been proven that the magnitude and location of a discontinuous jump, as represented by a Heaviside function, are explicitly identified by its Fast Walsh Transform (FWT) coefficients. These two proofs enable an algorithm that quickly provides a Weighted Least Squares fit to distributions across the element that include a discontinuity. The detection of a discontinuity enables analytic relations to locally describe its evolution and provide increased accuracy. Time accurate examples are provided for advection, Burgers equation, and Riemann problems (diaphragm burst) in closed tubes and de Laval nozzles. New algorithms to detect up to two C0 and/or C1 discontinuities within a single element are developed for application to the Riemann problem, in which a contact discontinuity and shock wave form after the diaphragm bursts.

Gnoffo, Peter A.↗
NASA NTRS2017

Cart3D Simulations for the Second AIAA Sonic Boom Prediction Workshop

Simulation results are presented for all test cases prescribed in the Second AIAA Sonic Boom Prediction Workshop. For each of the four nearfield test cases, we compute pressure signatures at specified distances and off-track angles, using an inviscid, embedded-boundary Cartesian-mesh flow solver with output-based mesh adaptation. The cases range in complexity from an axisymmetric body to a full low-boom aircraft configuration with a powered nacelle. For efficiency, boom carpets are decomposed into sets of independent meshes and computed in parallel. This also facilitates the use of more effective meshing strategies - each off-track angle is computed on a mesh with good azimuthal alignment, higher aspect ratio cells, and more tailored adaptation. The nearfield signatures generally exhibit good convergence with mesh refinement. We introduce a local error estimation procedure to highlight regions of the signatures most sensitive to mesh refinement. Results are also presented for the two propagation test cases, which investigate the effects of atmospheric profiles on ground noise. Propagation is handled with an augmented Burgers' equation method (NASA's sBOOM), and ground noise metrics are computed with LCASB.

Prediction↗
NASA NTRS2018

Sphene Emotional: How Titanite Was Shocked When the Dinosaurs Died

Accessory mineral geochronometers such as zircon, monazite, baddeleyite, and xenotime are increasingly being recognized for their ability to preserve diagnostic microstructural evidence of hypervelocity processes. However, little is known about the response of titanite to shock metamorphism, even though it is a widespread accessory phase and U-Pb geochronometer. Here we report two new mechanical twin modes in titanite within shocked granitoids from the Chicxulub impact structure, Mexico. Titanite grains in the newly acquired International Ocean Discovery Program Site expedition 364 M0077A core preserve multiple sets of polysynthetic twins, most commonly with composition planes (K1), = ~{1̅11}, and shear direction (η1) = <110>, and less commonly with the mode K1 = {130}, η1 = ~<522>. In some grains, {130} deformation bands have formed concurrently with shock twins, indicating dislocation glide with Burgers vector b = [341] can be active at shock conditions. Twinning of titanite in these modes, the presence of planar deformation features in shocked quartz, and lack of diagnostic shock microstructures in zircon in the same samples highlights the utility of titanite as a shock indicator for a shock pressure range between ~12 and ~17 GPa. Given the challenges of identifying ancient impact evidence on Earth and other bodies, microstructural analysis of titanite is here demonstrated to be a new avenue for recognizing impact deformation in materials where other impact evidence may be erased, altered, or did not manifest due to low shock pressure.

Timms, Nicholas E.↗
NASA NTRS2019

Recent Enhancements to NASA’s PCBoom Sonic Boom Propagation Code

This paper presents the recent updates the author has made to PCBoom sonic boom propagation code on its computational efficiency, predictive capability, and file operations. The updated version has recently been released and has a Burgers’ equation solver, which is computationally more efficient than that in the older version by 2 to 3 orders of magnitude. The updated version also enhances the accuracy of the sonic boom propagation prediction by accounting for the full wind effects on the shock wave propagation. Predictions from the updated PCBoom are compared with predictions from the older version and with data from flight tests. Other upgrades and changes that have been made in the latest version include support for Unix-like operating systems, bug fixes, and generation of new types of output files that allow for better manipulation of PCBoom-generated data.

Lonzaga, Joel B.↗
NASA NTRSDate not supplied

Modelling Sonic Boom Propagation through Planetary Boundary Layer Turbulence near the Lateral Extent of the Carpet

Nonlinear acoustic propagation of sonic booms in the atmospheric boundary layer is considered in the context of a one-way solution to a third order wave equation. A split-step approach is utilized to compute different physical effects efficiently. Unlike previous approaches, the diffraction effects are computed exactly in the forward direction with no restriction on the angle of propagation. This results in more accurate modelling of sonic boom near the carpet edge. Heterogenous flow effects are incorporated with a wide-angle parabolic approximation. Nonlinear propagation is computed with a Burgers-Hayes method. Turbulence in the medium is constructed by the method of random Fourier modes. The turbulence spectra are constructed using an altitude dependent 3-dimensional von Karman spectrum. The turbulence field is considered frozen, as the eddy turnover time in the atmospheric boundary layer is generally much larger than the propagation time of the acoustic wave. Comparisons with benchmark predictions from the PCBoom software are conducted. The benchmark cases provide insight into the accuracy of the current prediction code for non-turbulent atmospheres. Future work is discussed regarding the prediction of shaped booms, boundary conditions, and predictions near the lateral extent of the boom carpet where previous approaches fail to capture the diffraction effects.

sonic boom↗
NASA NTRSDate not supplied

Cartesian Mesh Simulations for the Third AIAA Sonic Boom Prediction Workshop

Simulation results are presented for all cases from the Third AIAA Sonic Boom Prediction Workshop. An inviscid, embedded-boundary Cartesian-mesh flow solver is used in conjunction with adjoint-based mesh adaptation to compute nearfield pressure signatures. Specialized techniques are applied to maximize accuracy and minimize cost on Cartesian meshes. The Richardson-based error estimate highlights regions of the signatures most sensitive to mesh refinement. Timing results and coarse, medium, and fine mesh sizes for nearfield cases demonstrate that the parallel decomposition approach is efficient in both computational time and wall-clock. Pressure signals are propagated to the ground using an augmented Burgers’ equation solver to predict boom carpets. Ground signatures and loudness metrics are presented for a standard atmosphere as well as more realistic atmospheric profiles, which affect overall noise levels and can significantly widen the boom carpet. Mesh convergence studies show that high sampling frequencies, around 500 kHz, are required for propagation, and the sampling frequency increases at large off-track angles with longer acoustic ray paths and propagation times. The numerical methods yield accurate results for predicting low sonic boom signatures while being among the least computationally expensive of the workshop.

ARMD↗
NASA NTRSDate not supplied

Uncertainty Estimates for Sonic-Boom Pressure Signatures and Loudness Carpets

A non-intrusive uncertainty quantification method is applied to computational analysis of supersonic, low-boom aircraft. The mean and standard deviation statistics of the pressure waveforms and loudness metrics are evaluated through use of numerical quadrature. The probability density function (p.d.f.) of these outputs is evaluated via kernel density estimation. The simulations use an inviscid, embedded-boundary Cartesian-mesh flow solver in the nearfield combined with an augmented Burgers’ equation solver for propagation in the farfield. The results show that the p.d.f. of the waveform is bimodal at shocks, which makes the mean and standard deviation statistics inappropriate. Despite this limitation, we show that the moment statistics can provide effective assessment of discrepancies when comparing with experimental data. This is demonstrated by presenting uncertainty analysis of a wind-tunnel test and showing that we significantly improve the predictions when we include the test uncertainties in the simulation. Normal distributions are obtained for the ground signature and loudness metrics, which is primarily due to the careful shaping of the low-boom waveform. Separation of variables and error control are used to reduce computational cost. We demonstrate that this is an efficient approach in the sense of balancing numerical errors in the statistics quadrature with discretization errors in the solvers.

ARMD↗
NASA NTRSDate not supplied

Numerical Fluxes and Shocks

Accurately describing shocks within numerical partial differential equation solvers has been a significant computational hurdle for the field of applied mathematics. In our work we present and discuss the background and history of how shocks have been dealt with. We have analyzed numerical schemes and fluxes for the purpose of education and improved computational speed and accuracy. We demonstrate the need for specific methods to deal with shocks by showcasing some methods in a simple one dimensional case for burger’s equation. We aim to extend our knowledge to the euler equations.

Partial Differential Equations↗
NASA NTRSDate not supplied

Uncertainty Quantification of CFD Model Assumptions Against Sonic Boom Noise Prediction of a Commercial Supersonic Transport

This paper presents the results of uncertainty modeling of sonic boom noise generation from commercial supersonic transport considering the Spalart-Allmaras (SA) turbulence modeling parameters as well as Mach number, angle of attack and altitude. Sample generation and analysis for this uncertainty model was performed by UQPCE, which is a software package developed at the NASA Langley Research Center. To build the uncertainty model, 42 cases of sonic boom noise calculation were performed. Computation of the ground noise can be briefly summarized in two steps. First, the near field pressure waveforms are sampled from CFD calculation using the NASA Langley’s FUN3D solver. Second, this information is passed to an atmospheric propagation code, sBOOM, which solves an augmented Burger’s equation and simulates how the near field waveforms will change while passing through the atmosphere. The ground signature is further processed to obtain the perceived loudness, PLdB. Having a high spatial resolution near the shockwave in the CFD calculation is critical in sonic boom noise prediction. Because the variation in the input parameters for the current uncertainty quantification (UQ) study is likely to lead to change in shock location, angle and strength, the grid adaptation for shock capturing is independently applied for each condition. The final mesh used in the CFD calculation consists of approximately 420 million cells. The pressure signatures are sampled at three, four and five body lengths away from the aircraft to make sure the three dimensional effects around the aircraft are resolved. The results of the UQ analysis shows that within the three aleatory variables, the angle of attack had the most impact against ground noise, followed by the altitude and the Mach number. Between the two SA model parameters, the Kármán constant (𝜅) was significantly more important than the turbulent Prandtl number (𝜎), but these two parameters were only marginally significant in the overall prediction variance in ground noise. The UQ procedure explained in this paper can be widely applied to other model parameters.

Uncertainty Quantification↗
NASA NTRSDate not supplied

Uncertainty Quantification of CFD Model Assumptions Against Sonic Boom Noise Prediction of a Commercial Supersonic Transport

This paper presents the results of uncertainty modeling of sonic boom noise generation from commercial supersonic transport considering the Spalart-Allmaras (SA) turbulence modeling parameters as well as Mach number, angle of attack and altitude. Sample generation and analysis for this uncertainty model was performed by UQPCE, which is a software package developed at the NASA Langley Research Center. To build the uncertainty model, 42 cases of sonic boom noise calculation were performed. Computation of the ground noise can be briefly summarized in two steps. First, the near field pressure waveforms are sampled from CFD calculation using the NASA Langley’s FUN3D solver. Second, this information is passed to an atmospheric propagation code, sBOOM, which solves an augmented Burger’s equation and simulates how the near field waveforms will change while passing through the atmosphere. The ground signature is further processed to obtain the perceived loudness, PLdB. Having a high spatial resolution near the shock wave in the CFD calculation is critical in sonic boom noise prediction. Because the variation in the input parameters for the current uncertainty quantification (UQ) study is likely to lead to change in shock location, angle and strength, the grid adaptation for shock capturing is independently applied for each condition. The final mesh used in the CFD calculation consists of approximately 420 million cells. The pressure signatures are sampled at three, four and five body lengths away from the aircraft to make sure the three dimensional effects around the aircraft are resolved. The results of the UQ analysis shows that within the three aleatory variables, the angle of attack had the most impact against ground noise, followed by the altitude and the Mach number. Between the two SA model parameters, the Kármán constant (𝜅) was significantly more important than the turbulent Prandtl number (𝜎), but these two parameters were only marginally significant in the overall prediction variance in ground noise. The UQ procedure explained in this paper can be widely applied to other model parameters.

Uncertainty Quantifications↗
NASA NTRSDate not supplied

Goal-Oriented Discretization Error Control in Coupled Nearfield-Farfield Low-Boom Simulations

The method of adjoint weighted residuals is used to determine the level of discretization error in loudness predictions of sonic booms on the ground. We analyze the standard nearfied-farfield domain decomposition approach. In the nearfield domain, the three-dimensional Euler equations are solved to obtain a pressure signature generated by the aircraft. In the farfield, this waveform is propagated through the atmosphere to the ground by solving the augmented Burgers’ equation. Loudness is characterized using weighted sound-exposure-level metrics. We formulate discretization error estimates for the ground signature and loudness metrics for this one-way coupled system. Although the nearfield solution is independent of the farfield, the adjoint formulation for the coupled system provides feedback from the farfield to identify high-error regions in the nearfield. The results demonstrate that the discrete adjoint implementation is asymptotically consistent and provides reliable error estimates. Furthermore, we show how the error can be controlled through adaptive refinement of the nearfield mesh. The approach is evaluated on two- and three-dimensional problems, including the X-59 flight demonstration aircraft.

CST↗
NASA NTRSDate not supplied

Nonlinear Acoustic Propagation in Low-Density Media

Nonlinear acoustic propagation in the atmosphere is usually modeled using an augmented Burgers equation accounting for atmospheric absorption and weak nonlinearity. However, the weak-nonlinearity assumption may not apply to acoustic signals propagating from the lower atmosphere that are subsequently refracted downward from the upper atmosphere (e.g., stratosphere and thermosphere) due to the decreasing air density with increasing altitude [Lonzaga, et al., Geophysical Journal International, 200(3), pp.1347-1361]. Consequently, this paper discusses the effects of a strong nonlinearity that lead to an amplitude-dependent increase in signal propagation speed. This propagation speed is obtained using a perturbation expansion where the leading-order term is simply the small-amplitude sound speed while the first-order term is the existing expression that gives rise to waveform steepening and stretching. Furthermore, the second-order term is proportional to the spectral density of the acoustic signal and is inversely proportional to the local density of the medium. Consequently, for an impulsive signal such as a sonic boom or an infrasound, the second-order effect causes a dispersion of the signal similar to the observed dispersion of acoustic signals from supersonic Concorde as well as from large explosions. This paper also discusses the extension of these results in general low-density propagation media.

sonic booms↗
NASA NTRS2024

Analytical Simulation of Effects of Local Mechanisms and Microstructure on Creep Response of Unidirectional Ceramic Matrix Composites

A micromechanics-based method has been developed to analyze the creep response of uncoated ceramic matrix minicomposites. Although the global stress level at which the creep response is analyzed is lower than the composite proportional limit, local stresses are assumed to be high enough that localized damage is present in the composite in the form of matrix microcracks. To model the composite creep response, including the effects of matrix cracking, a fiber shear lag-based methodology is employed. In this approach, stresses are assumed to vary in the fiber as a function of time and distance from the crack plane. The varying stresses are then used to compute the overall creep strain for the composite. Various assumptions regarding the level of matrix microcracking in the composite and the level of creep in the fiber and matrix in various portions of the composite unit cell are also examined. The creep response of the fiber is modeled using a linear Burgers model. The model is applied to a SiCf/SiC unidirectional minicomposite system. The computed creep results are compared to experimentally obtained values. The effects of the local fiber volume fraction on the overall creep response of the composite are also studied. This work will allow increased understanding of the key material damage mechanisms and load sharing that take place during creep conditions and can be expanded to provide improved analysis methods for full macrocomposites.

Micromechanics↗