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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
NASA NTRS1984

A Kosloff/Basal method, 3D migration program implemented on the CYBER 205 supercomputer

Conventional finite difference migration has relied on approximations to the acoustic wave equation which allow energy to propagate only downwards. Although generally reliable, such approaches usually do not yield an accurate migration for geological structures with strong lateral velocity variations or with steeply dipping reflectors. An earlier study by D. Kosloff and E. Baysal (Migration with the Full Acoustic Wave Equation) examined an alternative approach based on the full acoustic wave equation. The 2D, Fourier type algorithm which was developed was tested by Kosloff and Baysal against synthetic data and against physical model data. The results indicated that such a scheme gives accurate migration for complicated structures. This paper describes the development and testing of a vectorized, 3D migration program for the CYBER 205 using the Kosloff/Baysal method. The program can accept as many as 65,536 zero offset (stacked) traces.

Pyle, L. D.↗
DOE OSTI2020

Moving beyond the Solvent-Tip Approximation to Determine Site-Specific Variations of Interfacial Water Structure through 3D Force Microscopy

Although interfacial solution structure impacts environmental, biological, and technological phenomena, including colloidal stability, protein assembly, heterogeneous nucleation, and water desalination, its molecular details remain poorly understood. Here, we visualize the three-dimensional (3D) hydration structure at the boehmite(010)-water interface using fast force mapping (FFM). Using a self-consistent scheme to decouple long-range tip-sample interactions from short-range solvation forces, we obtain the solution structure with lattice resolution. The results are benchmarked against molecular dynamics simulations that explicitly include the effects of the tip with different levels of approximation and systematically account for tip size, chemistry, and confinement effects. We find four laterally structured water layers within 1 nm of the surface, with the highest water densities at sites adjacent to hydroxyl groups. The key features beyond the first two layers can only be predicted using a full-scale simulation of the boehmite-water-silica system. Our findings further reveal a complex relationship between site-specific chemistry, water density, and long-range particle interactions; and present important advances toward quantitative data interpretation in 3D FFM.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Machine Learning-Based Model Predictive Control of Two-Time-Scale Systems

In this study, we present a general form of nonlinear two-time-scale systems, where singular perturbation analysis is used to separate the dynamics of the slow and fast subsystems. Machine learning techniques are utilized to approximate the dynamics of both subsystems. Specifically, a recurrent neural network (RNN) and a feedforward neural network (FNN) are used to predict the slow and fast state vectors, respectively. Moreover, we investigate the generalization error bounds for these machine learning models approximating the dynamics of two-time-scale systems. Next, under the assumption that the fast states are asymptotically stable, our focus shifts toward designing a Lyapunov-based model predictive control (LMPC) scheme that exclusively employs the RNN to predict the dynamics of the slow states. Additionally, we derive sufficient conditions to guarantee the closed-loop stability of the system under the sample-and-hold implementation of the controller. A nonlinear chemical process example is used to demonstrate the theory. In particular, two RNN models are constructed: one to model the full two-time-scale system and the other to predict solely the slow state vector. Both models are integrated within the LMPC scheme, and we compare their closed-loop performance while assessing the computational time required to execute the LMPC optimization problem.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2004

Adjoint Formulation for an Embedded-Boundary Cartesian Method

Many problems in aerodynamic design can be characterized by smooth and convex objective functions. This motivates the use of gradient-based algorithms, particularly for problems with a large number of design variables, to efficiently determine optimal shapes and configurations that maximize aerodynamic performance. Accurate and efficient computation of the gradient, however, remains a challenging task. In optimization problems where the number of design variables dominates the number of objectives and flow- dependent constraints, the cost of gradient computations can be significantly reduced by the use of the adjoint method. The problem of aerodynamic optimization using the adjoint method has been analyzed and validated for both structured and unstructured grids. The method has been applied to design problems governed by the potential, Euler, and Navier-Stokes equations and can be subdivided into the continuous and discrete formulations. Giles and Pierce provide a detailed review of both approaches. Most implementations rely on grid-perturbation or mapping procedures during the gradient computation that explicitly couple changes in the surface shape to the volume grid. The solution of the adjoint equation is usually accomplished using the same scheme that solves the governing flow equations. Examples of such code reuse include multistage Runge-Kutta schemes coupled with multigrid, approximate-factorization, line-implicit Gauss-Seidel, and also preconditioned GMRES. The development of the adjoint method for aerodynamic optimization problems on Cartesian grids has been limited. In contrast to implementations on structured and unstructured grids, Cartesian grid methods decouple the surface discretization from the volume grid. This feature makes Cartesian methods well suited for the automated analysis of complex geometry problems, and consequently a promising approach to aerodynamic optimization. Melvin e t al. developed an adjoint formulation for the TRANAIR code, which is based on the full-potential equation with viscous corrections. More recently, Dadone and Grossman presented an adjoint formulation for the Euler equations. In both approaches, a boundary condition is introduced to approximate the effects of the evolving surface shape that results in accurate gradient computation.

Nemec, Marian↗
NASA NTRS2005

Solving Nonlinear Euler Equations with Arbitrary Accuracy

A computer program that efficiently solves the time-dependent, nonlinear Euler equations in two dimensions to an arbitrarily high order of accuracy has been developed. The program implements a modified form of a prior arbitrary- accuracy simulation algorithm that is a member of the class of algorithms known in the art as modified expansion solution approximation (MESA) schemes. Whereas millions of lines of code were needed to implement the prior MESA algorithm, it is possible to implement the present MESA algorithm by use of one or a few pages of Fortran code, the exact amount depending on the specific application. The ability to solve the Euler equations to arbitrarily high accuracy is especially beneficial in simulations of aeroacoustic effects in settings in which fully nonlinear behavior is expected - for example, at stagnation points of fan blades, where linearizing assumptions break down. At these locations, it is necessary to solve the full nonlinear Euler equations, and inasmuch as the acoustical energy is of the order of 4 to 5 orders of magnitude below that of the mean flow, it is necessary to achieve an overall fractional error of less than 10-6 in order to faithfully simulate entropy, vortical, and acoustical waves.

Dyson, Rodger W.↗
DOE OSTI2021

Effects of overlapping sources on cosmic shear estimation: Statistical sensitivity and pixel-noise bias

The next generation of dark-energy imaging surveys — so called “Stage-IV” surveys, such as that of the Rubin Observatory Legacy Survey of Space and Time (LSST) — will cross a threshold in the number density of detected sources on the sky that requires qualitatively different image analysis and measurement techniques compared to the current generation of Stage-III surveys. In Stage-IV surveys, a significant amount of the cosmologically useful information is due to sources whose images overlap with those of other sources on the sky. Here, we focus on the weak gravitational lensing probe, for which we expect the largest impact since the cosmic shear signal is primarily encoded in the estimated shapes of observed galaxies and thus directly impacted by overlaps. We introduce a framework based on the Fisher formalism to analyze the effect of the overlapping sources (“blending”) on the estimation of cosmic shear. This method gives concrete predictions for the minimum loss of information due to noise and blending for any choice of “deblending” scheme and shape-measurement algorithm. Our studies account for undetected sources but do not address their full effects and biases they may introduce. We use simulated images and predict this impact of blending for three surveys: the Dark Energy Survey (DES), the Hyper-Suprime Cam Subaru Strategic Program (HSC-SSP), and the Rubin LSST. Our methodology successfully estimates the statistical sensitivity to weak lensing for DES and HSC early results. For LSST, we present the expected loss in statistical sensitivity for the ten-year survey due to blending. We find that for approximately 62% of galaxies that are likely to be detected in full-depth LSST images, at least 1% of the flux in their pixels is from overlapping sources. We also find that the statistical correlations between measures of overlapping galaxies and, to a much lesser extent (0.2%) the higher shot noise level due to their presence, decrease the effective number density of galaxies, N eff , by ~ 18%. We calculate an upper limit on N eff of 39.4 galaxies per arcmin 2 in r band. We study the impact of stars on as a function of stellar density and illustrate the diminishing returns of extending the survey into lower Galactic latitudes. We extend the simulation-based Fisher formalism to predict the expected increase in pixel-noise bias due to blending for maximum-likelihood (ML) shape estimators. We find that noise bias depends sensitively on the particular shape estimator and measure of ensemble-average shape that is used, and properties of the galaxy that include redshift-dependent quantities such as size and luminosity. The source code for these studies is available online.[The documented software developed for the catalog-level studies are available in the open-source LSST DESC github repository https://github.com/LSSTDESC/WeakLensingDeblending. The software for analyzing one or two galaxies with user-defined parameters is in the open-source github repository https://github.com/ismael-mendoza/ShapeMeasurementFisherFormalism.]

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Photon (Non)Conservation in the Reduced Speed of Light Approximation and How to (Almost) Fix It

The "Reduced Speed of Light" (RSL) approximation is commonly used to speed up radiative transfer calculations in cosmological simulations. However, it has been shown previously that the RSL approximation leads to photon non-conservation when the radiation field is rapidly evolving in time. I show that these missing photons can be counted exactly for some numerical schemes. Adding them back into a simulation, however, is a much harder task. I show one example of such a scheme, which achieves sub-percent accuracy on simple tests. Unfortunately, the scheme performs much worse on semi-realistic simulations of cosmic reionization, leading to a faster overlap and significant errors in the point-wise comparison of the RSL radiation field with the reference simulation that maintains the full speed of light for the radiative transfer.

Gnedin, Nickolay Y. [Fermilab; Chicago U., KICP; C↗
NASA NTRS1981

Full potential solution of a transonic quasi-3-D flow through a cascade using artificial compressibility

A reliable method is presented for calculating the flowfield about a cascade of arbitrary 2-D airfoils. The method approximates the three-dimensional flow in a turbomachinery blade row by correcting for streamtube convergence and radius change in the throughflow direction. The method is a fully conservative solution of the full potential equation incorporating the finite volume technique on a body-fitted periodic mesh, with an artificial density imposed in the transonic region to ensure stability and the capture of shock waves. Comparison of results for several supercritical blades shows good agreement with their hodograph solutions. Other calculations for these profiles as well as standard NACA blade sections indicate that this is a useful scheme for analyzing both the design and off-design performance of turbomachinery blading.

Farrell, C.↗
DOE OSTI2020

Improved opacity expansion for medium-induced parton splitting

We present a new expansion scheme to compute the rate for parton splittings in dense and finite QCD media. In contrast to the standard opacity expansion, our expansion is performed around the harmonic oscillator whose characteristic frequency depends on the typical transverse momentum scale generated in the splitting. The first two orders account for the high frequency regime that is dominated by single hard scatterings together with the regime of multiple soft scatterings at low frequency. This work generalizes the findings of Ref. [1] beyond the leading logarithmic approximation allowing to account also for the Bethe-Heitler regime and compare to the full numerical results from [2]. We investigate the sensitivity of our results to varying the separation scale that defines the leading order. Finally, the application to Monte Carlo event generators is discussed.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS2014

High-Order Implicit-Explicit Multi-Block Time-stepping Method for Hyperbolic PDEs

This work seeks to explore and improve the current time-stepping schemes used in computational fluid dynamics (CFD) in order to reduce overall computational time. A high-order scheme has been developed using a combination of implicit and explicit (IMEX) time-stepping Runge-Kutta (RK) schemes which increases numerical stability with respect to the time step size, resulting in decreased computational time. The IMEX scheme alone does not yield the desired increase in numerical stability, but when used in conjunction with an overlapping partitioned (multi-block) domain significant increase in stability is observed. To show this, the Overlapping-Partition IMEX (OP IMEX) scheme is applied to both one-dimensional (1D) and two-dimensional (2D) problems, the nonlinear viscous Burger's equation and 2D advection equation, respectively. The method uses two different summation by parts (SBP) derivative approximations, second-order and fourth-order accurate. The Dirichlet boundary conditions are imposed using the Simultaneous Approximation Term (SAT) penalty method. The 6-stage additive Runge-Kutta IMEX time integration schemes are fourth-order accurate in time. An increase in numerical stability 65 times greater than the fully explicit scheme is demonstrated to be achievable with the OP IMEX method applied to 1D Burger's equation. Results from the 2D, purely convective, advection equation show stability increases on the order of 10 times the explicit scheme using the OP IMEX method. Also, the domain partitioning method in this work shows potential for breaking the computational domain into manageable sizes such that implicit solutions for full three-dimensional CFD simulations can be computed using direct solving methods rather than the standard iterative methods currently used.

Nielsen, Tanner B.↗
DOE OSTI2020

General Analysis of Data Collected from DOE Lidar Buoy Deployments Off Virginia and New Jersey

Pacific Northwest National Laboratory (PNNL) operates two AXYS WindSentinel lidar buoys for the U.S. Department of Energy’s Wind Energy Technologies. The purpose of these buoys is to collect hub-height winds and supporting meteorological and oceanographic information to facilitate the development of offshore wind energy in the U.S. In general, each buoy is deployed for a year or more at a given location in order to capture at least a full annual cycle of weather conditions. The initial deployment for one buoy was off the coast of Virginia beginning in 2015, and the other buoy was first deployed off the coast of New Jersey beginning in 2016. Over the last two years, PNNL has had an opportunity to analyze the data collected during these first two deployments. This report describes a substantial analysis of data collected by the two lidar buoys operated off the coasts of Virginia and New Jersey. The centerpiece instrument for each buoy as deployed off Virginia and New Jersey is a lidar system, which is designed to measure the horizontal wind vector from approximately 40 m to 200 m above the sea surface with vertical resolution of 40 m. Since the initial deployments, the original lidars have been replaced with more powerful Leosphere 866 v2 systems. The analyses in this report will apply to the original Vindicator systems. In addition to the wind profiles from the lidars, the buoys collect near-surface measurements of wind speed and direction, air temperature, relative humidity, barometric pressure, and solar irradiance. Oceanographic variables measured include the two-dimensional wave spectrum, water temperature and conductivity, and ocean current vectors to a depth of 90 m. An assessment of overall data recovery and a basic analysis of the data collected was provided in a previous report This report substantially extends that analysis. The various sections describe the development and application of an inertial measurement unit (IMU) data recovery scheme for the New Jersey deployment; a climatological analysis of winds at hub height and at the surface together with thermodynamic variables measured at the surface for the full deployment periods; an analysis of oceanographic observations describe sea state; the development of a refinement for NOAA’s WaveWatch III model to allow its application to near-shore areas; a basic climatology of ocean currents observed from the buoys; an analysis of observed winds in the framework of Monin-Obukhov Similarity Theory; and the development and evaluation of techniques to extract turbulence intensity and turbulence kinetic energy from the lidars. The analyses contained in this report provide a great deal of new information about offshore conditions on the U.S. East Coast. In addition, the experience gained will inform both configurations and analysis of data from future deployments of these lidar buoy systems.

17 WIND ENERGY↗
NASA NTRS2013

Inclusion of Linearized Moist Physics in Nasa's Goddard Earth Observing System Data Assimilation Tools

Inclusion of moist physics in the linearized version of a weather forecast model is beneficial in terms of variational data assimilation. Further, it improves the capability of important tools, such as adjoint-based observation impacts and sensitivity studies. A linearized version of the relaxed Arakawa-Schubert (RAS) convection scheme has been developed and tested in NASA's Goddard Earth Observing System data assimilation tools. A previous study of the RAS scheme showed it to exhibit reasonable linearity and stability. This motivates the development of a linearization of a near-exact version of the RAS scheme. Linearized large-scale condensation is included through simple conversion of supersaturation into precipitation. The linearization of moist physics is validated against the full nonlinear model for 6- and 24-h intervals, relevant to variational data assimilation and observation impacts, respectively. For a small number of profiles, sudden large growth in the perturbation trajectory is encountered. Efficient filtering of these profiles is achieved by diagnosis of steep gradients in a reduced version of the operator of the tangent linear model. With filtering turned on, the inclusion of linearized moist physics increases the correlation between the nonlinear perturbation trajectory and the linear approximation of the perturbation trajectory. A month-long observation impact experiment is performed and the effect of including moist physics on the impacts is discussed. Impacts from moist-sensitive instruments and channels are increased. The effect of including moist physics is examined for adjoint sensitivity studies. A case study examining an intensifying Northern Hemisphere Atlantic storm is presented. The results show a significant sensitivity with respect to moisture.

Convection↗
DOE OSTI2024

Investigation of Ring-heterogeneous Geometry Approximation for Efficient and Practical SFR Analysis

A novel geometry approximation scheme named ring-heterogeneous approximation for intermediate resolution multiphysics analysis of SFRs is presented. This approach particularly targets neutronics coupled with thermo-mechanics feedback, which dominate the passive safety of SFR designs. The ring-heterogeneous approximation casts an array of pins into volume-preserving bands of materials that constitute the pins. It allows to efficiently simulate SFRs with an unstructured mesh and also to explicitly consider the deformation of each material by retaining their heterogeneity. Verification results with single assembly and core problems of the ABTR benchmark demonstrated that the ring-heterogeneous approximation is a reasonable approximation for SFRs. For a 2D full core problem, the ring-heterogeneous approximation significantly reduced the number of elements by at least a factor of 70 from a fully-heterogeneous mesh while almost exactly preserving the reference fully-heterogeneous solutions with the RMS error of only 0.02% in the ring-wise fission reaction rate.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2025

A comparative study of calibration techniques for finite strain elastoplasticity: Numerically-exact sensitivities for FEMU and VFM

Accurate identification of material parameters is crucial for predictive modeling in computational mechanics. Here, the two primary approaches in the experimental mechanics community for calibration from full-field digital image correlation data are known as finite element model updating (FEMU) and the virtual fields method (VFM). In VFM, the objective function is a squared mismatch between internal and external virtual work or power. In FEMU, the objective function quantifies the weighted mismatch between model predictions and corresponding experimentally measured quantities of interest. It is minimized by iteratively updating the parameters of an FE model. While FEMU is seen as more flexible, VFM is commonly used instead of FEMU due to its considerably greater computational expense. However, comparisons between the two methods usually involve approximations of gradients or sensitivities with finite difference schemes, thereby making direct assessments difficult. Hence, in this study, we compare VFM and FEMU in the context of numerically-exact sensitivities obtained through local sensitivity analyses and the application of automatic differentiation software. To this end, we conduct a series of test cases to assess both methods under practical challenges using a finite strain elastoplasticity model.

Automatic differentiation↗
DOE OSTI2025

Assessing the difficulty of capturing the distribution function of neutrinos in neutron star merger simulations

The collision of two neutron stars is a rich source of information about nuclear physics. In particular, the kilonova signal following a merger can help us elucidate the role of neutron stars in nucleosynthesis, and informs us about the properties of matter above nuclear saturation. Approximate modeling of neutrinos remains an important limitation to our ability to make predictions for these observables. Part of the problem is the fermionic nature of neutrinos. By the exclusion principle, the expected value 𝑓 𝜈 for the number of neutrinos in a quantum state is at most 1. Any process producing neutrinos is suppressed by a blocking factor (1 −𝑓 𝜈 ). Recent simulations focused on neutrino physics mostly use a gray two-moment scheme to evolve neutrinos. This evolves integrals of 𝑓 𝜈 over momentum space, preventing direct calculations of blocking factors. Monte Carlo methods may be an attractive alternative, providing access to the full distribution of neutrinos. Their current implementation is, however, inadequate to estimate 𝑓 𝜈 : in our most recent simulations, a single Monte Carlo packet causes, in the worst cases, estimates of 𝑓 𝜈 to jump from 𝑓 𝜈 =0 to 𝑓 𝜈 ∼10 5 . While this is concerning, this brazen violation of the fermionic nature of neutrinos has been largely inconsequential, as the interactions used in simulations avoid direct calculations of 𝑓 𝜈 . We are, however, reaching a level of modeling at which this problem can no longer be ignored. Here, we discuss the relatively simple origin of this issue. We then show that very rough estimates of 𝑓 𝜈 can in theory be obtained in merger simulations, but that they will require a combination of unintuitive weighting schemes for Monte Carlo packets and smoothing of the neutrino distribution at coarser resolution than what the merger simulation uses.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1995

Aerodynamic sound of flow past an airfoil

The long term objective of this project is to develop a computational method for predicting the noise of turbulence-airfoil interactions, particularly at the trailing edge. We seek to obtain the energy-containing features of the turbulent boundary layers and the near-wake using Navier-Stokes Simulation (LES or DNS), and then to calculate the far-field acoustic characteristics by means of acoustic analogy theories, using the simulation data as acoustic source functions. Two distinct types of noise can be emitted from airfoil trailing edges. The first, a tonal or narrowband sound caused by vortex shedding, is normally associated with blunt trailing edges, high angles of attack, or laminar flow airfoils. The second source is of broadband nature arising from the aeroacoustic scattering of turbulent eddies by the trailing edge. Due to its importance to airframe noise, rotor and propeller noise, etc., trailing edge noise has been the subject of extensive theoretical (e.g. Crighton & Leppington 1971; Howe 1978) as well as experimental investigations (e.g. Brooks & Hodgson 1981; Blake & Gershfeld 1988). A number of challenges exist concerning acoustic analogy based noise computations. These include the elimination of spurious sound caused by vortices crossing permeable computational boundaries in the wake, the treatment of noncompact source regions, and the accurate description of wave reflection by the solid surface and scattering near the edge. In addition, accurate turbulence statistics in the flow field are required for the evaluation of acoustic source functions. Major efforts to date have been focused on the first two challenges. To this end, a paradigm problem of laminar vortex shedding, generated by a two dimensional, uniform stream past a NACA0012 airfoil, is used to address the relevant numerical issues. Under the low Mach number approximation, the near-field flow quantities are obtained by solving the incompressible Navier-Stokes equations numerically at chord Reynolds number of 104. The far-field noise is computed using Curle's extension to the Lighthill analogy (Curle 1955). An effective method for separating the physical noise source from spurious boundary contributions is developed. This allows an accurate evaluation of the Reynolds stress volume quadrupoles, in addition to the more readily computable surface dipoles due to the unsteady lift and drag. The effect of noncompact source distribution on the far-field sound is assessed using an efficient integration scheme for the Curle integral, with full account of retarded-time variations. The numerical results confirm in quantitative terms that the far-field sound is dominated by the surface pressure dipoles at low Mach number. The techniques developed are applicable to a wide range of flows, including jets and mixing layers, where the Reynolds stress quadrupoles play a prominent or even dominant role in the overall sound generation.

Wang, Meng↗
NASA NTRS1991

Development of a pressure based multigrid solution method for complex fluid flows

In order to reduce the computational difficulty associated with a single grid (SG) solution procedure, the multigrid (MG) technique was identified as a useful means for improving the convergence rate of iterative methods. A full MG full approximation storage (FMG/FAS) algorithm is used to solve the incompressible recirculating flow problems in complex geometries. The algorithm is implemented in conjunction with a pressure correction staggered grid type of technique using the curvilinear coordinates. In order to show the performance of the method, two flow configurations, one a square cavity and the other a channel, are used as test problems. Comparisons are made between the iterations, equivalent work units, and CPU time. Besides showing that the MG method can yield substantial speed-up with wide variations in Reynolds number, grid distributions, and geometry, issues such as the convergence characteristics of different grid levels, the choice of convection schemes, and the effectiveness of the basic iteration smoothers are studied. An adaptive grid scheme is also combined with the MG procedure to explore the effects of grid resolution on the MG convergence rate as well as the numerical accuracy.

Shyy, Wei↗
NASA NTRS1972

An implicit finite-difference solution to the viscous shock layer, including the effects of radiation and strong blowing

An implicit finite-difference scheme is developed for the fully coupled solution of the viscous, radiating stagnation-streamline equations, including strong blowing. Solutions are presented for both air injection and injection of carbon-phenolic ablation products into air at conditions near the peak radiative heating point in an earth entry trajectory from interplanetary return missions. A detailed radiative-transport code that accounts for the important radiative exchange processes for gaseous mixtures in local thermodynamic and chemical equilibrium is utilized in the study. With minimum number of assumptions for the initially unknown parameters and profile distributions, convergent solutions to the full stagnation-line equations are rapidly obtained by a method of successive approximations. Damping of selected profiles is required to aid convergence of the solutions for massive blowing. It is shown that certain finite-difference approximations to the governing differential equations stabilize and improve the solutions. Detailed comparisons are made with the numerical results of previous investigations. Results of the present study indicate lower radiative heat fluxes at the wall for carbonphenolic ablation than previously predicted.

Garrett, L. B.↗