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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 235 records · Page 13
All sourcesNASA NTRSDOE OSTI
NASA NTRS1999

Computational Investigation of Fluidic Counterflow Thrust Vectoring

A computational study of fluidic counterflow thrust vectoring has been conducted. Two-dimensional numerical simulations were run using the computational fluid dynamics code PAB3D with two-equation turbulence closure and linear Reynolds stress modeling. For validation, computational results were compared to experimental data obtained at the NASA Langley Jet Exit Test Facility. In general, computational results were in good agreement with experimental performance data, indicating that efficient thrust vectoring can be obtained with low secondary flow requirements (less than 1% of the primary flow). An examination of the computational flowfield has revealed new details about the generation of a countercurrent shear layer, its relation to secondary suction, and its role in thrust vectoring. In addition to providing new information about the physics of counterflow thrust vectoring, this work appears to be the first documented attempt to simulate the counterflow thrust vectoring problem using computational fluid dynamics.

Hunter, Craig A.↗
NASA NTRS1999

New approaches to virtual environment surgery

This research focused on two main problems: 1) low cost, high fidelity stereoscopic imaging of complex tissues and organs; and 2) virtual cutting of tissue. A further objective was to develop these images and virtual tissue cutting methods for use in a telemedicine project that would connect remote sites using the Next Generation Internet. For goal one we used a CT scan of a human heart, a desktop PC with an OpenGL graphics accelerator card, and LCD stereoscopic glasses. Use of multiresolution meshes ranging from approximately 1,000,000 to 20,000 polygons speeded interactive rendering rates enormously while retaining general topography of the dataset. For goal two, we used a CT scan of an infant skull with premature closure of the right coronal suture, a Silicon Graphics Onyx workstation, a Fakespace Immersive WorkBench and CrystalEyes LCD glasses. The high fidelity mesh of the skull was reduced from one million to 50,000 polygons. The cut path was automatically calculated as the shortest distance along the mesh between a small number of hand selected vertices. The region outlined by the cut path was then separated from the skull and translated/rotated to assume a new position. The results indicate that widespread high fidelity imaging in virtual environment is possible using ordinary PC capabilities if appropriate mesh reduction methods are employed. The software cutting tool is applicable to heart and other organs for surgery planning, for training surgeons in a virtual environment, and for telemedicine purposes.

NASA Discipline Neuroscience↗
NASA NTRS2000

Software Fault Tolerance: A Tutorial

Because of our present inability to produce error-free software, software fault tolerance is and will continue to be an important consideration in software systems. The root cause of software design errors is the complexity of the systems. Compounding the problems in building correct software is the difficulty in assessing the correctness of software for highly complex systems. After a brief overview of the software development processes, we note how hard-to-detect design faults are likely to be introduced during development and how software faults tend to be state-dependent and activated by particular input sequences. Although component reliability is an important quality measure for system level analysis, software reliability is hard to characterize and the use of post-verification reliability estimates remains a controversial issue. For some applications software safety is more important than reliability, and fault tolerance techniques used in those applications are aimed at preventing catastrophes. Single version software fault tolerance techniques discussed include system structuring and closure, atomic actions, inline fault detection, exception handling, and others. Multiversion techniques are based on the assumption that software built differently should fail differently and thus, if one of the redundant versions fails, it is expected that at least one of the other versions will provide an acceptable output. Recovery blocks, N-version programming, and other multiversion techniques are reviewed.

Torres-Pomales, Wilfredo↗
NASA NTRS2020

Search and Rescue under the Forest Canopy using Multiple UAS

We consider the problem of multi-robot search and rescue under the forest canopy. Forest is a particularly challenging environment for collaborative mapping and exploration, mainly due to the existence of severe perceptual aliasing, which hinders reliable mutual localization and map fusion. Our proposed system features unmanned aerial vehicles (UAVs) with onboard sensing and autonomy. Each UAV runs a lightweight filtering algorithm for local state estimation, and a dynamic-aware frontier selection algorithm for fast exploration. The essential and computationally intensive task of collaborative simultaneous localization and mapping (CSLAM) is performed at a central ground station. To handle perceptual aliasing, we make use of stable landmarks extracted from trees, which significantly improve precision and recall during place recognition. Furthermore, to recover from incorrect pairwise data associations during loop closure, we propose a novel procedure for global data association based on recently developed techniques on cycle consistent multiway matching. Our algorithm returns a global data association that is guaranteed to be cycle consistent, and is shown to significantly improve precision compared to the input pairwise associations. The overall multi-UAV system is extensively validated during real-world collaborative exploration missions in a forest at NASA Langley Research Center.

Multi-robot systems↗
NASA NTRS2004

An Interferometry Imaging Beauty Contest

We present a formal comparison of the performance of algorithms used for synthesis imaging with optical/infrared long-baseline interferometers. Six different algorithms are evaluated based on their performance with simulated test data. Each set of test data is formated in the interferometry Data Exchange Standard and is designed to simulate a specific problem relevant to long-baseline imaging. The data are calibrated power spectra and bispectra measured with a ctitious array, intended to be typical of existing imaging interferometers. The strengths and limitations of each algorithm are discussed.

aperture synthesis↗
NASA NTRSDate not supplied

Demonstration of Capability to Simulate Particle Irregular Shape and Poly-Disperse Mixtures Within Lunar Lander Plume-Surface Interaction

Plume-Surface Interaction (PSI) between lander engine plumes and regolith soil creates hazards in obscuration and contamination by particle clouds, high-energy ejecta streams, and landing area cratering damage. The MSFC Fluid Dynamics Branch is developing simulation tools to offer a predictive PSI capability to NASA customers such as the Human Lander System (HLS) and Commercial Lunar Payload Services (CLPS). The Gas-Granular Flow Solver (GGFS) is the main application tool for coupled gas-particle two-phase flow simulations to predict the range of PSI effects from onset of surface erosion to deep crater formation. GGFS features an Eulerian-Eulerian modeling approach, treating both gas and granular material as interacting continuum phases. Modeling the lunar regolith granular material fluidic characteristics poses special challenges due to complex particle shapes and mixture composition. The lunar regolith is poorly sorted with broad particle size distributions and large fines content. It has significant cohesion, due to interlocking jagged particle shapes. Eulerian granular material flow modeling requires closure formulations for the granular material constitutive models (stress, friction, collisional and kinetic energy dissipation, drag, etc.). While closure models for spherical particles are available from particle kinetic theory, closure models for realistic non-spherical particles must be extracted from unit physics Discrete Element Model (DEM) particle interaction simulations and provided in the form of tabular datasets. The effects of particle irregular shape (non-spherical shape factors, angular particle surface roughness, and interlocking features) are simulated by approximating the particle features in the form of grouped elemental spheres to form composite particles in the DEM simulations. The effects of the wide range of regolith mixture particle sizes and the strong effects of the presence of the small particle sizes results in high cohesion and low porosity of the regolith mixture. The range of particle sizes is simulated by binning the particle sizes into an appropriate finite number of particle-size species and solving the problem as a species mixture. Combining these two modeling approaches enables simulations to capture both, the contributions of the irregular particle shape and the particle size distribution. The integration and maturation of the DEM-based constitutive model database generation process and poly-disperse mixture binning approach into the GGFS simulation framework are proceeding under funding by the NASA Game Changing Development program. The status of current capabilities will be presented in comparisons of crater characteristics resulting for spherical and irregular shape particles, and for mono-, bi-, and tri-disperse mixture simulations of Apollo LM plume-surface interaction. The computational results confirm the significance of including the particle shape and mixture effects. Going forward plans for the full implementation of the general poly-disperse regolith modeling capability and maturation towards NASA project application readiness under the GCD program will be presented.

Peter A Liever↗
NASA NTRSDate not supplied

Demonstration of Capability to Simulate Particle Irregular Shape and Poly-Disperse Mixtures Within Lunar Lander Plume-Surface Interaction

Plume-Surface Interaction (PSI) between lander engine plumes and regolith soil creates hazards in obscuration and contamination by particle clouds, high-energy ejecta streams, and landing area cratering damage. The MSFC Fluid Dynamics Branch is developing simulation tools to offer a predictive PSI capability to NASA customers such as the Human Lander System (HLS) and Commercial Lunar Payload Services (CLPS). The Gas-Granular Flow Solver (GGFS) is the main application tool for coupled gas-particle two-phase flow simulations to predict the range of PSI effects from onset of surface erosion to deep crater formation. GGFS features an Eulerian-Eulerian modeling approach, treating both gas and granular material as interacting continuum phases. Modeling the lunar regolith granular material fluidic characteristics poses special challenges due to complex particle shapes and mixture composition. The lunar regolith is poorly sorted with broad particle size distributions and large fines content. It has significant cohesion, due to interlocking jagged particle shapes. Eulerian granular material flow modeling requires closure formulations for the granular material constitutive models (stress, friction, collisional and kinetic energy dissipation, drag, etc.). While closure models for spherical particles are available from particle kinetic theory, closure models for realistic non-spherical particles must be extracted from unit physics Discrete Element Model (DEM) particle interaction simulations and provided in the form of tabular datasets. The effects of particle irregular shape (non-spherical shape factors, angular particle surface roughness, and interlocking features) are simulated by approximating the particle features in the form of grouped elemental spheres to form composite particles in the DEM simulations. The effects of the wide range of regolith mixture particle sizes and the strong effects of the presence of the small particle sizes results in high cohesion and low porosity of the regolith mixture. The range of particle sizes is simulated by binning the particle sizes into an appropriate finite number of particle-size species and solving the problem as a species mixture. Combining these two modeling approaches enables simulations to capture both, the contributions of the irregular particle shape and the particle size distribution. The integration and maturation of the DEM-based constitutive model database generation process and poly-disperse mixture binning approach into the GGFS simulation framework are proceeding under funding by the NASA Game Changing Development program. The status of current capabilities will be presented in comparisons of crater characteristics resulting for spherical and irregular shape particles, and for mono-, bi-, and tri-disperse mixture simulations of Apollo LM plume-surface interaction. The computational results confirm the significance of including the particle shape and mixture effects. Going forward plans for the full implementation of the general poly-disperse regolith modeling capability and maturation towards NASA project application readiness under the GCD program will be presented.

Peter A. Liever↗
NASA NTRS2005

Partially-Averaged Navier Stokes Model for Turbulence: Implementation and Validation

Partially-averaged Navier Stokes (PANS) is a suite of turbulence closure models of various modeled-to-resolved scale ratios ranging from Reynolds-averaged Navier Stokes (RANS) to Navier-Stokes (direct numerical simulations). The objective of PANS, like hybrid models, is to resolve large scale structures at reasonable computational expense. The modeled-to-resolved scale ratio or the level of physical resolution in PANS is quantified by two parameters: the unresolved-to-total ratios of kinetic energy (f(sub k)) and dissipation (f(sub epsilon)). The unresolved-scale stress is modeled with the Boussinesq approximation and modeled transport equations are solved for the unresolved kinetic energy and dissipation. In this paper, we first present a brief discussion of the PANS philosophy followed by a description of the implementation procedure and finally perform preliminary evaluation in benchmark problems.

Girimaji, Sharath S.↗
NASA NTRS2020

Juncture Flow Computations using kL-Based Turbulence Models

The development and implementation of kL-based Reynolds-Average Navier-Stokes (RANS) two-equation turbulence models are reported in this paper. The kL model is based on Abdol-Hamid's closure and Menter's modification to Rotta's two-equation model. Rotta showed that a reliable transport equation can be formed from the turbulence length scale L, and the turbulence kinetic energy k. Rotta's kL equation is well suited for a term-by-term modeling and displays useful features compared to other scale formulations. One of the important differences is the inclusion of higher-order velocity derivatives in the source terms of the scale equation. This can enhance the ability of RANS solvers to simulate unsteady flows in URANS mode. The k-kL scheme has been modified to include the Algebraic Reynolds Stress Model (ARSM) and the Quadratic Constitutive Relation (QCR) as nonlinear models. These models have gone through extensive validations using two-dimensional benchmark problems. The present study documents the application of the k-kL schemes to simulate flow around the Juncture Flow Model (JFM). The k-kL prediction results show generally good comparisons with measurements. The results from this formulation are similar to, or better than results using the SA-RC-QCR2000 two-equation turbulence model. The ARSM and the QCR formulations of k-kL show promise with a similar level of computational resources as basic two-equation turbulence models.

K S Abdol-Hamid↗
NASA NTRS2007

Understanding and Counteracting Fatigue in Flight Crews

The materials included in the collection of documents describe the research of the NASA Ames Fatigue Countermeasures Group (FCG), which examines the extent to which fatigue, sleep loss, and circadian disruption affect flight-crew performance. The group was formed in 1980 in response to a Congressional request to examine a possible safety problem of uncertain magnitude due to transmeridian flying and a potential problem due to fatigue in association with various factors found in air-transport operations and was originally called the Fatigue/Jet Lag Program. The goals of the FCG are: (1) the development and evaluation of strategies for mitigating the effects of sleepiness and circadian disruption on pilot performance levels; (2) the identification and evaluation of objective approaches for the prediction of alertness changes in flight crews; and (3) the transfer and application of research results to the operational field via classes, workshops, and safety briefings. Some of the countermeasure approaches that have been identified to be scientifically valid and operationally relevant are brief naps (less than 40 min) in the cockpit seat and 7-min activity breaks, which include postural changes and ambulation. Although a video-based alertness monitor based on slow eyelid closure shows promise in other operational environments, research by the FCG has demonstrated that in its current form at the time of this reporting, it is not feasible to implement it in the cockpit. Efforts also focus on documenting the impact of untreated fatigue on various types of flight operations. For example, the FCG recently completed a major investigation into the effects of ultra-long-range flights (20 continuous hours in duration) on the alertness and performance of pilots in order to establish a baseline set of parameters against which the effectiveness of new ultra-long-range fatigue remedies can be judged.

Mallis, Melissa↗
NASA NTRS2011

Waterhammer Transient Simulation and Model Anchoring for the Robotic Lunar Lander Propulsion System

Waterhammer transients have the potential to adversely impact propulsion system design if not properly addressed. Waterhammer can potentially lead to system plumbing, and component damage. Multi-thruster propulsion systems also develop constructive/destructive wave interference which becomes difficult to predict without detailed models. Therefore, it is important to sufficiently characterize propulsion system waterhammer in order to develop a robust design with minimal impact to other systems. A risk reduction activity was performed at Marshall Space Flight Center to develop a tool for estimating waterhammer through the use of anchored simulation for the Robotic Lunar Lander (RLL) propulsion system design. Testing was performed to simulate waterhammer surges due to rapid valve closure and consisted of twenty-two series of waterhammer tests, resulting in more than 300 valve actuations. These tests were performed using different valve actuation schemes and three system pressures. Data from the valve characterization tests were used to anchor the models that employed MSCSoftware.EASY5 v.2010 to model transient fluid phenomena by using transient forms of mass and energy conservation. The anchoring process was performed by comparing initial model results to experimental data and then iterating the model input to match the simulation results with the experimental data. The models provide good correlation with experimental results, supporting the use of EASY5 as a tool to model fluid transients and provide a baseline for future RLL system modeling. This paper addresses tasks performed during the waterhammer risk reduction activity for the RLL propulsion system. The problem of waterhammer simulation anchoring as applied to the RLL system is discussed with results from the corresponding experimental valve tests. Important factors for waterhammer mitigation are discussed along with potential design impacts to the RLL propulsion system.

Stein, William B.↗
NASA NTRS1999

Aftbody Closure Effects on the Reference H Configuration at Subsonic and Transonic Speeds

Experience with afterbody closure effects and accompanying test techniques issues on a High Speed Civil Transport (HSCT)-class configuration is described. An experimental data base has been developed which includes force, moment, and surface pressure data for the High Speed Research (HSR) Reference H configuration with a closed afterbody at subsonic and transonic speeds, and with a cylindrical afterbody at transonic and supersonic speeds. A supporting computational study has been performed using the USM3D unstructured Euler solver for the purposes of computational fluid dynamics (CFD) method assessment and model support system interference assessment with a focus on lower blade mount effects on longitudinal data at transonic speeds. Test technique issues related to a lower blade sting mount strategy are described based on experience in the National Transonic Facility (NTF). The assessment and application of the USM3D code to the afterbody/sting interference problem is discussed. Finally, status and plans to address critical test technique issues and for continuation of the computational study are presented.

Wahls, Richard A.↗
NASA NTRS1993

Development of a recursion RNG-based turbulence model

Reynolds stress closure models based on the recursion renormalization group theory are developed for the prediction of turbulent separated flows. The proposed model uses a finite wavenumber truncation scheme to account for the spectral distribution of energy. In particular, the model incorporates effects of both local and nonlocal interactions. The nonlocal interactions are shown to yield a contribution identical to that from the epsilon-renormalization group (RNG), while the local interactions introduce higher order dispersive effects. A formal analysis of the model is presented and its ability to accurately predict separated flows is analyzed from a combined theoretical and computational stand point. Turbulent flow past a backward facing step is chosen as a test case and the results obtained based on detailed computations demonstrate that the proposed recursion -RNG model with finite cut-off wavenumber can yield very good predictions for the backstep problem.

Zhou, YE↗
NASA NTRS2012

Large-Eddy / Reynolds-Averaged Navier-Stokes Simulations of a Dual-Mode Scramjet Combustor

Numerical simulations of reacting and non-reacting flows within a scramjet combustor configuration experimentally mapped at the University of Virginia s Scramjet Combustion Facility (operating with Configuration A ) are described in this paper. Reynolds-Averaged Navier-Stokes (RANS) and hybrid Large Eddy Simulation / Reynolds-Averaged Navier-Stokes (LES / RANS) methods are utilized, with the intent of comparing essentially blind predictions with results from non-intrusive flow-field measurement methods including coherent anti-Stokes Raman spectroscopy (CARS), hydroxyl radical planar laser-induced fluorescence (OH-PLIF), stereoscopic particle image velocimetry (SPIV), wavelength modulation spectroscopy (WMS), and focusing Schlieren. NC State's REACTMB solver was used both for RANS and LES / RANS, along with a 9-species, 19- reaction H2-air kinetics mechanism by Jachimowski. Inviscid fluxes were evaluated using Edwards LDFSS flux-splitting scheme, and the Menter BSL turbulence model was utilized in both full-domain RANS simulations and as the unsteady RANS portion of the LES / RANS closure. Simulations were executed and compared with experiment at two equivalence ratios, PHI = 0.17 and PHI = 0.34. Results show that the PHI = 0.17 flame is hotter near the injector while the PHI = 0.34 flame is displaced further downstream in the combustor, though it is still anchored to the injector. Reactant mixing was predicted to be much better at the lower equivalence ratio. The LES / RANS model appears to predict lower overall heat release compared to RANS (at least for PHI = 0.17), and its capability to capture the direct effects of larger turbulent eddies leads to much better predictions of reactant mixing and combustion in the flame stabilization region downstream of the fuel injector. Numerical results from the LES/RANS model also show very good agreement with OH-PLIF and SPIV measurements. An un-damped long-wave oscillation of the pre-combustion shock train, which caused convergence problems in some RANS simulations, was also captured in LES / RANS simulations, which were able to accommodate its effects accurately.

Fulton, Jesse A.↗
NASA NTRS1993

The evolution equation for the flame surface density in turbulent premixed combustion

The mean reaction rate in flamelet models for turbulent premixed combustion depends on two basic quantities: a mean chemical rate, called the flamelet speed, and the flame surface density. Our previous work had been primarily focused on the problem of the structure and topology of turbulent premixed flames, and it was then determined that the flamelet speed, when space-averaged, is only weakly sensitive to the turbulent flow field. Consequently, the flame surface density is the key quantity that conveys most of the effects of the turbulence on the rate of energy release. In flamelet models, this quantity is obtained via a modeled transport equation called the Sigma-equation. Past theoretical work has produced a rigorous approach that leads to an exact but unclosed formulation for the turbulent Sigma-equation. In the exact Sigma-equation, it appears that the dynamical properties of the flame surface density are determined by a single parameter, namely the turbulent flame stretch. Unfortunately, the turbulent flame stretch as well as the flame surface density is not available from experiments, and, in the absence of experimental data, little is known on the validity of the closure assumptions used in current flamelet models. Direct Numerical Simulation (DNS) is the alternative approach to get basic information on these fundamental quantities. In the present work, three-dimensional DNS of premixed flames in isotropic turbulent flow is used to estimate the different terms appearing in the Sigma-equation. A new methodology is proposed to provide the source and sink terms for the flame surface density, resolved both temporally and spatially throughout the turbulent flame brush. Using this methodology, our objective is to extract the turbulent flame stretch from the DNS data base and then perform extensive comparisons with flamelet models. Thanks to the detailed information produced by the DNS-based analysis, it is expected that this type of comparison will not only underscore the shortcomings of current models, but also suggest ways to improve them.

Trouve, Arnaud↗
NASA NTRS1965

Postflight Evaluation of Atlas-Centaur AC-5 (Launched 2 March 1965)

The Atlas-Centaur AC-5 vehicle was launched from ETR Complex 36A on March 2., 1965 at 8:25.04 a.m. EST. Within about 1 second after launch the thrust of the Atlas booster engine decayed rapidly; the vehicle settled back on the launch pad and was quickly destroyed by fire and explosion. Considerable damage was sustained by the launch complex and its associated equipment. Loss of booster engine thrust was due to fuel depletion at the turbopump inlets, which is attributed to closure of the fuel prevalve or the staging valve. To preclude the recurrence of either of these fuel valving malfunctions, the following corrective action has been taken: The remote control actuator has been replaced by manual operation of the Atlas fuel prevalve; the internal passage dimensions in the staging valve have been increased to lessen the hydraulic load on the valve poppet. In addition to the Atlas fuel system malfunction, a failure in the power control circuitry of the Centaur guidance computer resulted in partial removal of power at umbilical ejection. To prevent such a guidance system failure on future flights some redundant circuitry has been eliminated and more rigorous checkout procedures have been adopted. No further anomalies were discovered in the telemetered data prior to the Atlas booster thrust decay. A prime objective of the AC-5 flight was to place a dynamic model of the Surveyor spacecraft in a simulated lunar transfer trajectory. An important facet of this problem is the demonstration of a launch-on-time capability in accordance with the proper Earth-moon relation. The window opening time was established at 8:25 a.m. EST; thus the actual launch occurred within 4 seconds of the planned time.

Source record↗
NASA NTRS2003

Macroscopic Models of Radiative Transfer as Applied to Computation of the Radiation Field in the Solar Atmosphere

In this paper a new three dimensional half-moment model for radiative transfer is presented for a gray medium. It describes the evolution of the zeroth and first directional half moments of the radiative intensity. The closure is provided, similarly to Dubroca and Klar, by the maximum entropy concept. This work generalizes that model to three dimensions. The model presented here (the derivation being done in Ripoll and Wray, called the M(sup 1/2)(sub 1) model, is a hyperbolic system consisting of a total of eight equations in three dimensions, four equations for each direction. Each half model has the classical form of a macroscopic moment model in which the pressure tensor is constructed from the well-known Eddington tensor with a particular Eddington factor. Moreover, different source and border terms occur. The latter introduce couplings between the macroscopic and microscopic quantities and between the + and - streams, through the intensity in the plane perpendicular to the flux. The main theoretical application of the half moment model, treated in this paper, is its reduction to a full moment model, called M(sup +)(sub 1), for the particular but important case of a hot, opaque source radiating in a cold transparent (or semi-transparent) medium for very specific applications, such as stellar interiors or atmospheres, or combustion problems. The structure of the paper is as follows. In section 2, the model M(sup 1/2)(sub 1) is presented. In section 3, for the particular case of a hot, opaque source radiating into a cold medium, the half moment model is reduced to the M(sup +)(sub 1) model. In section 4, we first solve a simple and academic problem to validate the models, followed by a simplified solar atmosphere.

Ripoll, J.-F.↗
NASA NTRS1993

Why does preferential diffusion strongly affect premixed turbulent combustion?

Combustion of premixed reactants in a turbulent flow is a classical but unresolved problem. The key problem is to explain the following data: the maximal turbulent and laminar burning velocities u(sub t) and u(sub L) occur at different equivalence ratios Phi. It is known that the equivalence ratio varies along a curved flame if molecular diffusivity D(sub fuel) does not equal D(sub oxygen). However, the mean flame radius of curvature is much larger than the laminar flame thickness delta-L. Therefore, significant influence of preferential diffusion should occur only if the flame propagation speed varies with flame curvature. This conclusion agrees with Zel'dovich's long-standing idea about the important role of leading points of a flame. The main objective of this paper is to prove Zel'dovich's hypothesis. An equation for the mean flame surface area density (MFSAD) is employed for this purpose. The second objective of this paper is to suggest a different approach to the derivation of the equation for MFSAD. It is based on the pdf equation for the reaction progress variable C and the relation between the pdf and MFSAD. This treatment suggests an entirely different closure assumption.

Kuznetsov, Vadim R.↗