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NASA NTRS2010

Roles of Engineering Correlations in Hypersonic Entry Boundary Layer Transition Prediction

Efforts to design and operate hypersonic entry vehicles are constrained by many considerations that involve all aspects of an entry vehicle system. One of the more significant physical phenomenon that affect entry trajectory and thermal protection system design is the occurrence of boundary layer transition from a laminar to turbulent state. During the Space Shuttle Return To Flight activity following the loss of Columbia and her crew of seven, NASA's entry aerothermodynamics community implemented an engineering correlation based framework for the prediction of boundary layer transition on the Orbiter. The methodology for this implementation relies upon the framework of correlation techniques that have been in use for several decades. What makes the Orbiter boundary layer transition correlation implementation unique is that a statistically significant data set was acquired in multiple ground test facilities, flight data exists to assist in establishing a better correlation and the framework was founded upon state of the art chemical nonequilibrium Navier Stokes flow field simulations. The basic tenets that guided the formulation and implementation of the Orbiter Return To Flight boundary layer transition prediction capability will be reviewed as a recommended format for future empirical correlation efforts. The validity of this approach has since been demonstrated by very favorable comparison of recent entry flight testing performed with the Orbiter Discovery, which will be graphically summarized. These flight data can provide a means to validate discrete protuberance engineering correlation approaches as well as high fidelity prediction methods to higher confidence. The results of these Orbiter engineering and flight test activities only serve to reinforce the essential role that engineering correlations currently exercise in the design and operation of entry vehicles. The framework of information-related to the Orbiter empirical boundary layer transition prediction capability will be utilized to establish a fresh perspective on this role, to illustrate how quantitative statistical evaluations of empirical correlations can and should be used to assess accuracy and to discuss what the authors' perceive as a recent heightened interest in the application of high fidelity numerical modeling of boundary layer transition. Concrete results will also be developed related to empirical boundary layer transition onset correlations. This will include assessment of the discrete protuberance boundary layer transition onset data assembled for the Orbiter configuration during post-Columbia Return To Flight. Assessment of these data will conclude that momentum thickness Reynolds number based correlations have superior coefficients and uncertainty in comparison to roughness height based Reynolds numbers, aka Re(sub k) or Re(sub kk). In addition, linear regression results from roughness height Reynolds number based correlations will be evaluated, leading to a hypothesis that non-continuum effects play a role in the processes associated with incipient boundary layer transition on discrete protuberances.

Campbell, Charles H.↗

NASA NTRSDate not supplied

Aeroheating Environments of Aerocapture Systems for Uranus Orbiters

Aeroheating environments for an aerocapture system enabling flagship-class science at Uranus are presented. Applicability of a low lift-to-drag entry vehicle aeroshell with flight heritage for Martian entries is considered in the context of an end-to-end Uranus orbiter and probe mission design using aerocapture for orbit insertion. A feasible trajectory space for aerocapture orbit insertion based on various launch opportunities and interplanetary trajectories yield atmosphere-relative entry velocities between 22 km/s and 31 km/s with maximum freestream densities on the order of 10−5 kg/m3. A preliminary study of the aerothermal environments for the entire trajectory space is discussed followed by a more detailed study of the nominal design trajectories. Design conditions are based on laminar, thermochemical nonequilibrium Navier-Stokes flowfield simulations with coupled radiation transport for full lift-up (deep) and lift-down (shallow) design trajectories. Convective heating is found to be the dominant heating mode, with radiation accounting for only 1 % to 5 % of the total heat flux for the design trajectories. The peak design conditions are found to be 422 W/cm2 for heat flux, 10.5 kPa for pressure, 149 Pa for shear stress, and 88.1 kJ/cm2 for the total heat load.

Uranus↗

NASA NTRSDate not supplied

Aeroheating Environments of Aerocapture Systems for Uranus Orbiters

Aeroheating environments for an aerocapture system enabling flagship-class science at Uranus are presented. Applicability of a low lift-to-drag entry vehicle aeroshell with flight heritage for Martian entries is considered in the context of an end-to-end Uranus orbiter and probe mission design using aerocapture for orbit insertion. A feasible trajectory space for aerocapture orbit insertion based on various launch opportunities and interplanetary trajectories yield atmosphere-relative entry velocities between 22 km/s and 31 km/s with maximum freestream densities on the order of 10 −5 kg/m 3 . A preliminary study of the aerothermal environments for the entire trajectory space is discussed followed by a more detailed study of the nominal design trajectories. Design conditions are based on laminar, thermochemical nonequilibrium Navier-Stokes flowfield simulations with coupled radiation transport for full lift-up (deep) and lift-down (shallow) design trajectories. Convective heating is found to be the dominant heating mode, with radiation accounting for only 1 % to 5 % of the total heat flux for the design trajectories. The peak design conditions are found to be 422 W/cm 2 for heat flux, 10.5 kPa for pressure, 149 Pa for shear stress, and 88.1 kJ/cm 2 for the total heat load.

Aerocapture↗

NASA NTRS1990

Plume impingement study for reaction control system of the Orbital Maneuvering Vehicle

A study has been performed to assess plume impingement effects from the reaction control system of NASA's OMV. A key aspect of this modeling effort has been the use of the direct Monte Carlo method as an integral part of the nozzle/plume flowfield modeling and to compute the impingement effects for configurations in which vehicle structure or multiple plumes resulted in highly three-dimensional rarefied plume flowfields. The calculation of the flowfield for a pair of interacting thruster plumes has shown that backflow in the region between the thrusters is greatly enhanced by the interaction of the two plumes. Although the present analyses have focussed on impingement force and heating effects, this methodology would also be valuable for contamination assessments since it properly accounts for the species separation effects that are inherent in the nonequilibrium nature of rarefied gas flows.

Mcgregor, R. D.↗

NASA NTRS1971

Research and investigation of gas dynamic lasers

Chemical mixing laser investigations were conducted (1) to investigate the properties of a bimolecular exchange laser system pumped by the H + Cl2 yields HCl(v) + Cl reaction, initiated by arc-dissociated H2, with lasing occurring between wavelengths of 3.4 and 4.0 microns, and (2) to establish the feasibility of an atom recombination-transfer laser employing recombination of arc-dissociated nitrogen with subsequent transfer of vibrational energy to CO2 for lasing at 10.6 microns. One-dimensional analytical results indicate higher results should be obtained with up to v = 3 to 2 transitions participating. Diagnostic and analytical results show that the reaction mechanism during mixing, a back reaction of HCl(v) with H atoms, reaction of Cl with H2(v), moderately fast V-V, V-T processes, and possible HCl(o) initial contaminant level may explain the low performance. N2-CO2 thermal mixing laser studies were extended to measure the efficiency of transfer of recombination energy in such a nonequilibrium N2 source to 10.6 microns optical energy. The low level of efficiency suggests that V-T decay processes may prevent vibrational energy freezing until much lower temperatures are achieved and that trapping of energy in long-lived electronic excitation of N2 may be a factor.

Boedeker, L. R.↗

NASA NTRS1994

Theoretical studies of a molecular beam generator

We consider the problem of modeling a high temperature diatomic gas N2 flowing through a converging-diverging high expansion nozzle. The problem is modeled in two ways. The first model uses a single temperature with variable specific heats as functions of this temperature. For the second model we assume that the various degrees of freedom all have a Boltzmann population distribution which means that each degree of freedom has its own temperature and consequently each system state can be characterized by these temperatures. This suggests the formulation of a second model with a vibrational degree of freedom as having its own temperature along with a rotational-translational degree of freedom with its own temperature. Initially the vibrational degree of freedom is excited by heating the gas to a high temperature. As the high temperature gas expands through the nozzle throat there is a sudden drop in the rotational-translational temperature along with a finite relaxation time for the vibrational degree of freedom to achieve equilibrium with the rotational-translational degree of freedom. That is, the temperature change that occurs when the N2 gas passes through the nozzle throat is so great that the changes in the vibrational degree of freedom lags behind the rotational-translational energy changes. The resulting relaxation time is finite. It is in this context that the term nonequilibrium is used. That is, the term nonequilibrium denotes the fact that the energy content of the various degrees of freedom are characterized by two temperatures. We neglect any chemical reactions resulting from the high temperatures which could also add nonequilibrium effects. We develop the basic equations for the two models in various forms in order to check the derivations with other sources. The final form which is solved numerically consists of the scaled equations in a conservative dimensionless form.

Heinbockel, John H.↗

NASA NTRS2022

VULCAN-CFD User Manual: Ver. 7.2.0

VULCAN-CFD offers a comprehensive set of capabilities to enable the simulation of continuum flowfields from subsonic to hypersonic conditions. The governing equations that are employed include allowances for both chemical and thermal nonequilibrium processes, coupled with a wide variety of turbulence models for both Reynolds-averaged and large eddy simulations. The software package can simulate two-dimensional, axisymmetric, or three-dimensional problems on structured multiblock meshes or fully unstructured meshes. A parabolic (i.e., space-marching) treatment can also be used for any subset of a structured mesh that can accommodate this solution strategy. The flow solver provides a significant level of geometric flexibility for structured grid simulations by allowing for arbitrary face-to-face C(0) continuous and non-C(0) continuous block interface connectivities. The unstructured grid paradigm allows for mixed element unstructured meshes that contain any combination of tetrahedral, prismatic, pyramidal, and hexahedral cell elements. The flow solver is also fully parallelized using MPI (Message Passing Interface) libraries in a data-parallel fashion, allowing for efficient simulations on modern High Performance Computing (HPC) systems. This document provides information related to the installation and execution of the VULCAN-CFD software package. A detailed description of the physical and numerical models available in the software are provided in the VULCAN-CFD Theory Manual.

VULCAN-CFD User Manual↗

NASA NTRSDate not supplied

Aerothermal Analysis of the Dragonfly Titan Entry

The Dragonfly mission will send a rotorcraft lander to the surface of Saturn’s moon Titan as part of the New-Frontiers program. This will be the first spacecraft to land on Titan since the Huygens probe’s descent in January of 2005 and only the second spacecraft to enter Titan’s atmosphere. The Dragonfly entry capsule is significantly larger than the Huygens probe and will experience higher aerothermal environments. This poster will provide an overview of the aerothermodynamic models used in the Dragonfly aeroshell design process. Titan provides a unique entry environment that has several fundamental differences from the environments at the more common entry destinations of Earth and Mars. The lack of atmospheric oxygen significantly reduces heatshield recession (expected to be negligible), which simplifies material response analysis but also reduces the efficiency of the thermal protection system. The atmos-phere is composed of mostly nitrogen and trace amounts of methane. During hypersonic entry this methane dissociates and leads to the formation of the molecule CN, which is known to radiate strongly in the shock layer. Current Dragonfly heating predictions estimate that radiation contributes about 50% of the total heat flux along the forebody and up to 90% on the aftbody, making radiation a key element of aerothermal analysis. To properly address the importance of radiation, a unique radiative-heating correlation was developed that incorporates normal-shock equilibrium chemistry predictions based on work from the Mars 2020 program. These new heating correlations agree well with Computational Fluid Dynamics (CFD) predictions and enable large-scale trajectory analyses by providing rapid heating predictions along candidate trajectories. Dragonfly aerothermal environments are generated with the Data Parallel Line Relaxation (DPLR) finite-volume Navier-Stokes solver alongside the Nonequilibrium Radiative Transport and Spectra (NEQAIR) radiation transport code. DPLR produces shock-aligned flowfields that incorporate the effects of both chemical and thermal nonequilibrium while NEQAIR solves the radiation transport equations along lines-of-sight through the flowfield solutions to predict the radiative heating on the aeroshell. Titan’s atmosphere produces complex thermal and chemical nonequilibria and so classical radiation approximations that extrapolate from a single line-of-sight, such as the Tangent-Slab method, were found to overpredict peak radiative heat fluxes by 10-20%. Therefore, all radiative heating is directly calculated using 3-D transport for both the forebody and aftbody to reduce unnecessary conservatism. A 21-species finite-rate chemistry model, including electrons and ionized species, is used in the CFD to ensure accurate vehicle heating predictions (the level of ionization is particularly for accurate radiative flux predictions). This poster will cover: 1) The models and assumptions used to simulate entry into Titan’s atmosphere, 2) Dragonfly aerothermal engineering correlations alongside statistical heating values, and 3) Driving aerothermal considerations for a Titan entry such as CN formation and radiation-flowfield coupling.

EDL↗

NASA NTRS2019

Boundary Layer Stability and Transition in a Chemically Reacting Martian Atmosphere Using LASTRAC

The Thermal Protection System (TPS) is a significant portion of the mass of reentry vehicles, planetary probes, and Martian entry vehicles. Reducing this mass has benefits in terms of decreased fuel requirements and increased payload; however, due to the high risk and uncertainty, the TPS or heat shields are designed conservatively by assuming fully turbulent flow. Laminar flow results in reduced heat flux, and improved transition prediction has the potential to reduce TPS mass and uncertainty in aerothermodynamic predictions. Limited previous research exists examining the problem of transition prediction for Martian atmospheric entry, with studies available on transition on the Mars Science Laboratory TPS. Although transition was demonstrated in wind-tunnel tests, uncertainty in the transition location resulted in a TPS designed for fully turbulent flow, and therefore greater mass than required for a partially laminar condition. In this work, we extend the boundary layer stability code LASTRAC, recently modified to include chemical and thermal nonequilibrium capabilities, to a model of the Martian atmosphere. LASTRAC provides Parabolized Stability Equations (PSE) as well as Linear Stability Theory (LST) to predict the stability of a boundary layer and transition with semi-empirical e(sup N) methods. Results included in this work compare disturbance growth characteristics between air and Martian atmosphere at similar non-dimensional freestream conditions on a simple flat plate geometry. Both chemical nonequilibrium and thermochemical nonequilibrium, as well as both PSE and LST, are used.

Kline, H. L.↗

NASA NTRSDate not supplied

Boundary Layer Stability and Transition in a Chemically Reacting Martian Atmosphere using LASTRAC

The Thermal Protection System (TPS) is a significant portion of the mass of reentry vehicles, planetary probes, and Martian entry vehicles. Reducing this mass has benefits in terms of decreased fuel requirements and increased payload; however, due to the high risk and un-certainty, the TPS or heat shields are designed conservatively by assuming fully turbulent flow. Laminar flow results in reduced heat flux, and improved transition prediction has the potential to reduce TPS mass and uncertainty in aerothermodynamic predictions. Limited previous research exists examining the problem of transition prediction for Martian atmospheric entry, with studies available on transition on the Mars Science Laboratory TPS. Although transition was demonstrated in wind-tunnel tests, uncertainty in the transition location resulted in a TPS designed for fully turbulent flow, and therefore greater mass than required for a partially-laminar condition. In this work, we extend the boundary layer stability code LASTRAC, recently modified to include chemical and thermal nonequilibrium capabilities, to a model of the Martian atmosphere. LASTRAC provides Parabolized Stability Equations (PSE) as well as Linear Stability Theory (LST) to predict the stability of a boundary layer and transition with semi-empirical eN methods. Results included in this work compare disturbance growth characteristics between air and Martian atmosphere at similar nondimensional freestream conditions on a simple flat plate geometry. Both chemical nonequilibrium and thermochemical nonequilibrium, as well as both PSE and LST, are used.

Heather L Kline↗

NASA NTRSDate not supplied

Thermomechanical Relaxation of Vibrationally Excited NO in a Hypersonic Turbulent Boundary Layer

The effects of thermal nonequilibrium are studied using a DC glow discharge plasma to selectively modify the vibrational temperature of an otherwise cold flow. These experiments were performed at 𝑴=5.7and 𝑹𝒆=6×10 6 /m in the Actively Controlled Expansion tunnel at Texas A&M University on a canonical 2.75° half-angle wedge test article. A∼47 W plasma was generated across spanwise electrodes, and removable trips produced either a laminar or turbulent boundary layer. NO planar laser-induced fluorescence (PLIF) quantified the extent of the NO 𝑿 2 𝚷 (𝒗=1) vibrational perturbation due to the plasma and its subsequent interplay with the boundary layer, the first such data of which the authors are aware. NOPLIF measurements were taken at three locations along the test article, and quasi-temperature measurements are reported at each. Despite not quantifiably perturbing the boundary layer, the plasma was shown to alter the NO vibrational temperature profile therein. Independent analysis of the thermodynamics and fluid mechanics of the system suggests each set of physics could explain the results observed near and far from the wall, respectively.

Casey J. Broslawski↗

NASA NTRSDate not supplied

Thermomechanical Relaxation of Vibrationally Excited NO in a Hypersonic Turbulent Boundary Layer

Casey J. Broslawski↗

NASA NTRS2002

Aeroheating Environments for a Mars Smart Lander

A proposed Mars Smart Lander is designed to reach the surface via lifting-body atmospheric entry (alpha = 16 deg) to within 10 km of the target site. CFD (computational fluid dynamics) predictions of the forebody aeroheating environments are given for a direct entry from a 2005 launch. The solutions were obtained using an 8-species gas in thermal and chemical nonequilibrium with a radiative-equilibrium wall temperature boundary condition. Select wind tunnel data are presented from tests at NASA Langley Research Center. Turbulence effects are included to account for both smooth body transition and turbulence due to heatshield penetrations. Natural transition is based on a momentum-thickness Reynolds number value of 200. The effects of heatshield penetrations on turbulence are estimated from wind tunnel tests of various cavity sizes and locations. Both natural transition and heatshield penetrations are predicted to cause turbulence prior to the nominal trajectory peak heating time. Laminar and turbulent CFD predictions along the trajectory are used to estimate heat rates and loads. The predicted peak turbulent heat rate of 63 W/sq cm on the heatshield leeward flank is 70% higher than the laminar peak. The maximum integrated heat load for a fully turbulent heat pulse is 38% higher than the laminar load on the heatshield nose. The predicted aeroheating environments with uncertainty factors will be used to design a thermal protection system.

Edquist, Karl T.↗

NASA NTRS1978

Analysis of turbulent free jet hydrogen-air diffusion flames with finite chemical reaction rates

The nonequilibrium flow field resulting from the turbulent mixing and combustion of a supersonic axisymmetric hydrogen jet in a supersonic parallel coflowing air stream is analyzed. Effective turbulent transport properties are determined using the (K-epsilon) model. The finite-rate chemistry model considers eight reactions between six chemical species, H, O, H2O, OH, O2, and H2. The governing set of nonlinear partial differential equations is solved by an implicit finite-difference procedure. Radial distributions are obtained at two downstream locations of variables such as turbulent kinetic energy, turbulent dissipation rate, turbulent scale length, and viscosity. The results show that these variables attain peak values at the axis of symmetry. Computed distributions of velocity, temperature, and mass fraction are also given. A direct analytical approach to account for the effect of species concentration fluctuations on the mean production rate of species (the phenomenon of unmixedness) is also presented. However, the use of the method does not seem justified in view of the excessive computer time required to solve the resulting system of equations.

Sislian, J. P.↗

NASA NTRS2000

Hierarchical Statistical 3D ' Atomistic' Simulation of Decanano MOSFETs: Drift-Diffusion, Hydrodynamic and Quantum Mechanical Approaches

When MOSFETs are scaled to deep submicron dimensions the discreteness and randomness of the dopant charges in the channel region introduces significant fluctuations in the device characteristics. This effect, predicted 20 year ago, has been confirmed experimentally and in simulation studies. The impact of the fluctuations on the functionality, yield, and reliability of the corresponding systems shifts the paradigm of the numerical device simulation. It becomes insufficient to simulate only one device representing one macroscopical design in a continuous charge approximation. An ensemble of macroscopically identical but microscopically different devices has to be characterized by simulation of statistically significant samples. The aims of the numerical simulations shift from predicting the characteristics of a single device with continuous doping towards estimating the mean values and the standard deviations of basic design parameters such as threshold voltage, subthreshold slope, transconductance, drive current, etc. for the whole ensemble of 'atomistically' different devices in the system. It has to be pointed out that even the mean values obtained from 'atomistic' simulations are not identical to the values obtained from continuous doping simulations. In this paper we present a hierarchical approach to the 'atomistic' simulation of aggressively scaled decanano MOSFETs. A full scale 3D drift-diffusion'atomostic' simulation approach is first described and used for verification of the more economical, but also more restricted, options. To reduce the processor time and memory requirements at high drain voltage we have developed a self-consistent option based on a thin slab solution of the current continuity equation only in the channel region. This is coupled to the Poisson's equation solution in the whole simulation domain in the Gummel iteration cycles. The accuracy of this approach is investigated in comparison with the full self-consistent solution. At low drain voltage only single solution of the nonlinear Poisson equation is sufficient to extract the current with satisfactory accuracy. A pilot version of a hydrodynamic 'atomistic' simulator has been developed in order to study the effect of the nonequilibrium, non local transport in decanano MOSFETs on the random dopant induced current fluctuations. For the first time we have also applied the density gradient approach in 3D to investigate the effect of the quantum confinement on the threshold voltage fluctuations. The developed 'atomistic' simulation techniques have been applied to study various fluctuation resistant MOSFET architectures including epitaxial and delta doped devices.

Asenov, Asen↗

NASA NTRS2002

Assembly of Colloidal Materials Using Bioadhesive Interactions

We have pursued the use of biological crosslinking molecules of several types to make colloidal materials at relatively low volume fraction of colloidal particles. The objective is to make binary alloys of colloidal particles, made of two different colloidal particles coated with complementary biological lock-and-key binding molecules, which assemble due to the biological specificity. The long-term goal is to use low affinity lock-and-key biological interactions, so that the can anneal to form crystalline states. We have used a variety of different surface chemistries in order to make colloidal materials. Our first system involved using selectin-carbohydrate (sialyl-Lewis) interactions; this chemistry is derived from immune system. This chemical interaction is of relatively low affinity, with timescales for dissociation of several seconds. Furthermore, the adhesion mediated by these molecules can be reversed by the chelation of calcium atoms; thus assembled structures can be disassembled reversibly. Our second system employed avidin-biotin chemistry. This well-studied system is of high affinity, and is generally irreversible on a laboratory time-scale. Thus, we would expect selectin-carbohydrate interactions at high molecular density and avidin-biotin interactions to give kinetically-trapped structures; however, at low densities, we would expect significant differences in the structure and dynamics of the two materials, owing to their very different release rates. We have also begun to use a third chemistry - DNA hybridization. By attaching single stranded DNA oligonucleotide chains to beads, we can drive the assembly of colloidal materials by hybridization of complementary DNA chains. It is well known that DNA adenosine-thymine (A-T) and guanine-cytosine (G-C) bases hybridize pairwise with a Gibbs free energy change of 1.7 kcal/mol per base; thus, the energy of the assembly can be modulated by altering the number of complementary bases in the DNA chains. Using these different crosslinking molecules, we have assembled colloidal materials from different-sized colloidal particles, A and B. In the first sets of experiment, we used high densities of adhesion molecules, and 0.96 micron (A) and 5.5 micron (B) diameter particles. The high density of adhesion molecules means that the structures are kinetically trapped in nonequilibrium configurations. The structure of the suspension can be varied by changing the number ratio of the two types of colloidal particles, NA and NB, where A is the smaller particle. With carbohydrate-selectin or avidin-biotin interactions, large NA/NB leads to the formation of colloidal micelles, with the large center B particle surrounded by many smaller A particles. As the ratio NA/NB decreases, the structures become more extended, approaching the formation of macro-Rouse polymers - extended linear chains where A beads are connected with intervening small B linkers.

Hammer, Daniel A.↗

NASA NTRS1993

Towards a generalized computational fluid dynamics technique for all Mach numbers

Currently there exists no single unified approach for efficiently and accurately solving computational fluid dynamics (CFD) problems across the Mach number regime, from truly low speed incompressible flows to hypersonic speeds. There are several CFD codes that have evolved into sophisticated prediction tools with a wide variety of features including multiblock capabilities, generalized chemistry and thermodynamics models among other features. However, as these codes evolve, the demand placed on the end user also increases simply because of the myriad of features that are incorporated into these codes. In order for a user to be able to solve a wide range of problems, several codes may be needed requiring the user to be familiar with the intricacies of each code and their rather complicated input files. Moreover, the cost of training users and maintaining several codes becomes prohibitive. The objective of the current work is to extend the compressible, characteristic-based, thermochemical nonequilibrium Navier-Stokes code GASP to very low speed flows and simultaneously improve convergence at all speeds. Before this work began, the practical speed range of GASP was Mach numbers on the order of 0.1 and higher. In addition, a number of new techniques have been developed for more accurate physical and numerical modeling. The primary focus has been on the development of optimal preconditioning techniques for the Euler and the Navier-Stokes equations with general finite-rate chemistry models and both equilibrium and nonequilibrium thermodynamics models. We began with the work of Van Leer, Lee, and Roe for inviscid, one-dimensional perfect gases and extended their approach to include three-dimensional reacting flows. The basic steps required to accomplish this task were a transformation to stream-aligned coordinates, the formulation of the preconditioning matrix, incorporation into both explicit and implicit temporal integration schemes, and modification of the numerical flux formulae. In addition, we improved the convergence rate of the implicit time integration schemes in GASP through the use of inner iteration strategies and the use of the GMRES (General Minimized Resisual) which belongs to the class of algorithms referred to as Krylov subspace iteration. Finally, we significantly improved the practical utility of GASP through the addition of mesh sequencing, a technique in which computations begin on a coarse grid and get interpolated onto successively finer grids. The fluid dynamic problems of interest to the propulsion community involve complex flow physics spanning different velocity regimes and possibly involving chemical reactions. This class of problems results in widely disparate time scales causing numerical stiffness. Even in the absence of chemical reactions, eigenvalue stiffness manifests itself at transonic and very low speed flows which can be quantified by the large condition number of the system and evidenced by slow convergence rates. This results in the need for thorough numerical analysis and subsequent implementation of sophisticated numerical techniques for these difficult yet practical problems. As a result of this work, we have been able to extend the range of applicability of compressible codes to very low speed inviscid flows (M = .001) and reacting flows.

Walters, R. W.↗

NASA NTRS2013

Spurious Behavior of Shock-Capturing Methods: Problems Containing Stiff Source Terms and Discontinuities

The goal of this paper is to relate numerical dissipations that are inherited in high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities. For pointwise evaluation of the source term, previous studies indicated that the phenomenon of wrong propagation speed of discontinuities is connected with the smearing of the discontinuity caused by the discretization of the advection term. The smearing introduces a nonequilibrium state into the calculation. Thus as soon as a nonequilibrium value is introduced in this manner, the source term turns on and immediately restores equilibrium, while at the same time shifting the discontinuity to a cell boundary. The present study is to show that the degree of wrong propagation speed of discontinuities is highly dependent on the accuracy of the numerical method. The manner in which the smearing of discontinuities is contained by the numerical method and the overall amount of numerical dissipation being employed play major roles. Moreover, employing finite time steps and grid spacings that are below the standard Courant-Friedrich-Levy (CFL) limit on shockcapturing methods for compressible Euler and Navier-Stokes equations containing stiff reacting source terms and discontinuities reveals surprising counter-intuitive results. Unlike non-reacting flows, for stiff reactions with discontinuities, employing a time step and grid spacing that are below the CFL limit (based on the homogeneous part or non-reacting part of the governing equations) does not guarantee a correct solution of the chosen governing equations. Instead, depending on the numerical method, time step and grid spacing, the numerical simulation may lead to (a) the correct solution (within the truncation error of the scheme), (b) a divergent solution, (c) a wrong propagation speed of discontinuities solution or (d) other spurious solutions that are solutions of the discretized counterparts but are not solutions of the governing equations. The present investigation for three very different stiff system cases confirms some of the findings of Lafon & Yee (1996) and LeVeque & Yee (1990) for a model scalar PDE. The findings might shed some light on the reported difficulties in numerical combustion and problems with stiff nonlinear (homogeneous) source terms and discontinuities in general.

Yee, Helen M. C.↗