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NASA NTRS2009

Some Exact Results for the Schroedinger Wave Equation with a Time Dependent Potential

The time dependent Schroedinger equation with a time dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wave function at the origin, one may derive the wave function everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the p otential lead to conservation of the normalization of the probability density.

Campbell, Joel↗

DOE OSTI2021

An Orthogonal Recursive Bisection (ORB) Based Time Advancement Algorithm for CFD-DEM Solvers

The time integration of the granular phase in coupled computational fluid dynamics (CFD) – discrete element method (DEM) simulations presents a unique computational challenge brought about by the large variations in particle collisional time scales. Particles in the dilute regions of the computational domain can be advanced with large time steps while dense regions require much smaller time increments. However, the time step size in most solvers is globally set as the limit for accuracy and stability imposed by the collisions and is typically orders of magnitude less than that required away from collisions. This work addresses this precise issue and provides a strategy to avoid the use of a global conservative small time step size for the entire set of particles.A novel time stepping algorithm for CFD-DEM solvers using a partitioning approach using orthogonal recursive bisection (ORB) that allows for variable time steps among particles is described and its computational performance is compared against baseline explicit methods, typically used in several CFD-DEM solvers. ORB has advantages of being relatively quick and easy to update incrementally and has the required heuristic behavior (i.e., it will split the region in half with a cluster on each side) when groups of particles are well separated (clustered). The algorithm presented in this work uses a local time stepping approach to resolve collisional time scales for subsets of particles that are present at the leaves of the ORB, thereby resulting in substantial reduction of computational cost. The parallel implementation of this method where a ``knapsack” algorithm is used in tandem with ORB for effective load-balancing is also presented, where a best possible partitioning is obtained based on number of particles and local time-stepping costs. The algorithm is tested against benchmark problems with varying particle distributions that include fluidized bed and riser flow scenarios. Preliminary results indicate that the approach is 2-3X faster than traditional explicit methods for problems that involve both dense and dilute regions, while maintaining the same level of accuracy.

adaptive timestepping↗

DOE OSTI2023

EUI Benchmarks for Net-Zero Energy Buildings in India

In our study we present EUI benchmarks for NZEBs for six building types across residential and non-residential typologies and for India's five climate zones. This approach is similar to the simulation-based benchmarks used by the B3 program in Minnesota, the Cal-Arch methodology in California, and the US Solar Decathlon approach, which combine simulations with actual building data. Of the six building types we explore one in detail with a range of operation scenarios, specifically narrowing down the EUI benchmarks for mixed-mode building operation and for variable temperature set-points as prescribed in the National Building Code of India.The contribution of this work is to provide rigorous end-use level EUI benchmarks for six building types, and to describe a method for simulation-based EUI benchmarks for mixed-mode operation with variable setpoints to highlight the difference between the standard approach used for the five building types and the On-Site Construction Worker Housing which additionally has the mixed-mode variable setpoint approach.On-Site Construction Worker Housings are typically poorly constructed temporary structures, without adequate thermal comfort, It is critical to provide adequate thermal comfort to protect people from the warming effects of climate change, and to discover super efficient and cost-effective ways to do so.The methodology used for the On-Site Construction Worker Housing (CWH) results in an 80% acceptability according to the India Model for Adaptive Comfort in the National Building Code of India. EUIs of all six buildings are 60% lower than the minimum compliance with India's Energy Conservation Building Codes, providing benchmarks for efficiency levels. The end-use level EUI benchmarks are now provided to over 1800 Solar Decathlon India participants so that they can compare the performance of their NZEB designs.In particular, the CWH results provide an insight into the importance of the mixed-mode operation with variable temperature set-points. The results from the simulation study show that for an NZEB target, the EUI with standard thermal comfort model and without mixed operation is 58.26 kWh/m2*year, while that with the variable set-points of the adaptive model with mixed mode operation is 24.23 kWh/m2*year. This is a 58% reduction in EUI. Given that many building types including residential, and non-residential operate in mixed mode, it is important to take this work further to develop mixed-mode operation NZEB benchmarks so that the carbon intensity of these buildings could be lower than the standard thermal comfort model approach.

benchmarking↗

DOE OSTI2024

Energy Analysis of Combi Heat Pump System Configurations for Space Conditioning and Domestic Hot Water Heating in Residential Buildings

Combi heat pump systems, also referred to multifunctional variable refrigerant flow heat recovery (MF-VRFHR) systems, are specifically designed for residential applications to manage both space conditioning and domestic hot water (DHW). They have attracted attention due to their potential for energy conservation through heat recovery. The incorporation of a hot water tank introduces various system configurations, each characterized by distinct pros and cons related to energy efficiency, system stability, and maintenance. Despite this, a critical gap exists as the specific energy performance remains unquantified under diverse operational modes (e.g., heating mode and heat recovery mode). This paper aims to bridge this gap by conducting a comprehensive comparative analysis of two prevalent system configurations while considering feasible proposed control logics. Configuration 1 integrates a separate hot water tank and a refrigerant-to-water heat exchanger (HEX), also known as a Hydro Kit while Configuration 2 incorporates a refrigerant-wrapped hot water tank. To facilitate this analysis, we developed high-fidelity system models for both configurations in Modelica, capturing system dynamics and detailed control sequences effectively. These system models were built upon the TIL library for HVAC equipment components and the Buildings library for residential building thermal load calculations. The validation of the simulation testbed utilized data from experiments conducted in the PNNL lab home for Configuration 1. To establish the simulation testbed for Configuration 2, we extended the modeling setup derived from Configuration 1. This extension specifically involved substituting the separate hot water tank and Hydro Kit with a refrigerant-wrapped hot water tank of similar sizing sourced from an actual product. The simulation analysis of heating-only and heat recovery modes reveals that Configuration 2 not only saves energy and maintains warmer tank temperatures but also demonstrates faster water heating capabilities. This is attributed to decreased energy loss and improved heat transfer. The study encompasses a wide range of scenarios, considering diverse thermal loads and water usage patterns across heating and heat recovery modes. Overall, the comprehensive results indicate that Configuration 2 achieves energy savings ranging from 3.5% to 12.2% compared to Configuration 1, depending on factors such as water usage patterns, thermal loads, and operational modes.

Configuration, Comparison, Multi-functional, Resid↗

DOE OSTI2023

Electron kinetics in a high- Z plasmoid

The problem of the electron dynamics on a closed magnetic field line passing through a high-Z plasmoid is considered. The electron kinetic equation is integrated over bounce motion and pitch angle, reducing the independent variables to a single adiabatic invariant plus time. Integration of the full Landau self-collision operator is carried out exactly, resulting in a nonlinear integro-differential operator in the new invariant. Conservation laws and the H theorem of the integrated self-collision operator are proven. Numerical solutions of the integrated kinetic equation are obtained with a self-consistent quasineutral electric potential, given the initial condition of a cold plasmoid immersed in a hot ambient plasma. The fact that cold electrons are deeply trapped in a potential with a parabolic peak leads to exactly 3/4 the usual rate of collisional heating by the ambient plasma, independent of any other parameters.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRSDate not supplied

Grand Valley Ecological Forecasting II: Forecasting Trends in Pinyon-Juniper and Sagebrush Habitat Relative to Wildfire, Drought, Beetle Disturbance, and Treatment Impact for Management Planning

Disturbances and landcover change in pinyon-juniper and sagebrush ecosystems are exacerbated by environmental conditions such as variability in climate characteristics. Our DEVELOP team partnered with the National Park Service (NPS) in Colorado National Monument and the Bureau of Land Management (BLM) in McInnis Canyons and Dominguez-Escalante National Conservation Areas to investigate disturbances to land cover. NPS partners were interested in identifying areas at risk of pinyon-juniper die-off or encroachment by invasive species. The BLM partners prioritized identifying areas suitable for fire reduction and prevention treatment. To address these concerns, we forecasted landcover change in the Grand Valley region of Colorado using NASA Earth observation data from Landsat 5 Thematic Mapper (TM), Landsat 7 Enhanced Thematic Mapper (ETM+), Landsat 8 Operational Land Imager (OLI), and Moderate Resolution Imaging Spectroradiometer (MODIS) aboard Terra and Aqua. We collected and analyzed these data in conjunction with Term I of this project. We found that the primary drivers of forecasted landcover change in the study area were aspect and elevation. Our forecasted land cover maps, created using the Idrisi TerrSet Land Change Modeler, indicated that by 2040, ecosystems within partner management areas will likely see tree encroachment on shrublands. These maps addressed the needs of our partners by showing potential habitat suitability trends, which will inform management planning.

Bill Curtiss↗

NASA NTRS2022

Grand Valley Ecological Forecasting II: Forecasting Trends in Pinyon-Juniper and Sagebrush Habitat Relative to Wildfire, Drought, Beetle Disturbance, and Treatment Impact for Management Planning

Disturbances and landcover change in pinyon-juniper and sagebrush ecosystems are exacerbated by environmental conditions such as variability in climate characteristics. Our DEVELOP team partnered with the National Park Service (NPS) in Colorado National Monument and the Bureau of Land Management (BLM)in McInnis Canyons and Dominguez-Escalante National Conservation Areas to investigate disturbances to land cover. NPS partners were interested in identifying areas at risk of pinyon-juniper die-off or encroachment by invasive species. The BLM partners prioritized identifying areas suitable for fire reduction/prevention treatment. To address these concerns, we forecasted landcover change in the Grand Valley region of Colorado using NASA Earth observation data from Landsat 5 Thematic Mapper (TM), Landsat 7 Enhanced Thematic Mapper (ETM+), Landsat 8 Operational Land Imager (OLI), and Moderate Resolution Imaging Spectroradiometer (MODIS) aboard Terra and Aqua. We collected and analyzed these data in conjunction with Term I of this project. We found that the primary drivers of forecasted landcover change in the study area were aspect and elevation. Our forecasted landcover change maps, created using the Idrisi TerrSet Land Change Modeler, addressed the needs of our partners by showing potential habitat suitability trends, which will inform management planning. Forecasted landcover maps indicated that by 2040, ecosystems within partner management areas will likely see tree encroachment on shrublands.

William Curtiss↗

NASA NTRSDate not supplied

Monitoring Vegetation Health and Water Availability in Bryce Canyon National Park for Drought Stress Mitigation Planning

Bryce Canyon National Park is home to groundwater-dependent ecosystems (GDEs) that are threatened by a multidecadal drought and increased groundwater extraction due to a spike in tourism. These ecosystems contain unique species that are only found in areas where near-surface groundwater is present, such as aspen groves and willows. GDEs contribute to the high biodiversity found in Bryce Canyon, which boosts an ecosystem’s productivity and increases the overall ecological health of the park. Unfortunately, many GDEs in this area are too small to identify with traditional Earth observation platforms and are difficult to physically reach for identification and monitoring. NASA DEVELOP partnered with the National Park Service to establish a framework that both identified and assessed the health of GDEs within Bryce Canyon from 2013–2022. The team used high-resolution PlanetScope and Landsat 8 Operational Land Imager (OLI) to map springs, seeps, and GDEs. Modeled data from the Western Land Data Assimilation System (WLDAS) and in situ precipitation data were used to produce time series of climatic variables for the National Park Service to gain an understanding of how the GDEs are changing with the ongoing drought. This study is essential to the goal of the National Park Service to conserve the health of these GDEs.

Aaron Carr↗

NASA NTRSDate not supplied

NASA NDE Fracture Critical Detectable Flaw Sizes History and Methodology

NASA requires that NDE methods and inspectors demonstrate 90% Probability of Detection (POD) with 95% confidence for critical flaw sizes when inspecting fracture critical metallic components. NASA addresses the known variability of NDE inspector capability in two ways. The first, Special NDE, requires that every inspector demonstrate the required 90/95 POD, which is resource intensive. The second approach is Standard NDE for which conservative flaw sizes for different NDE methods are provided such that it is expected that most properly trained inspectors will exceed the 90/95 POD requirement. As such, individual POD demonstration testing is not required. The origin of NASA Standard NDE dates to the start of the Space Shuttle Program in the early 1970’s. In the first study to quantitatively assess NDE methods and inspectors, the performance of multiple inspectors was evaluated for different NDE methods using a large set of fatigue cracked specimens. A rudimentary POD analysis was performed to estimate the 90/95 POD flaw size for each inspector for each method. Additionally, the average and standard deviation of the 90/95 POD flaw size across the multiple inspectors was calculated to estimate the flaw size for which 95 percent of inspectors would provide the 90/95 POD capability. These estimated 90/95/95 POD flaw sizes evolved into the NASA Standard NDE flaw sizes still in use for structural analysis five decades later. The methodology for performing Standard NDE POD studies was never documented in NASA requirements. Furthermore, POD analysis methods have significantly evolved since this seminal study. Likewise, NDE methods have improved and there has been a push to reassess Standard NDE flaw sizes for existing methods, and to develop Standard NDE flaw sizes for new methods such as digital radiography. In this study, a Standard NDE POD methodology was developed and baselined using the historical data. This reanalysis of the historical data identified several deficiencies in the original test plan as well as an overall lack of conservatism in the estimated 90/95/95 POD flaw sizes. The results of this historical review and the new methodology are being incorporated into an update of NASA NDE POD requirements.

Nondestructive Evaluation↗

DOE OSTI2020

Opportunities for Using the Industrial Assessment Center Database for Industrial Water Use Analysis

The manufacturing sector accounted for approximately 5–6% of total U.S. water use in 2015. Of that amount, 75–80% is self supplied withdrawal from surface-water and groundwater sources and the remainder is from public water supplies. Although manufacturing facilities commonly locate in water-scarce areas, water scarcity still poses a great risk to the manufacturing sector. Reliable water is necessary for any facility that relies on it for process and comfort cooling, cleaning, employee use, and steam generation. One of these barriers to water efficiency is the lack of reliable data on overall U.S. industrial water use—how it is used and the quantities required for each sector. If a facility cannot be easily compared with a facility of similar size and sector, knowing if it is effectively using water conservation best practices is difficult. One potential source of industrial water use data is the U.S. Department of Energy (DOE)–sponsored Industrial Assessment Centers (IACs). IACs are university-based organizations that provide free audits to small- and medium-sized manufacturing facilities to identify productivity improvement and waste and energy reduction opportunities. The IACs also maintain a database of all the audits conducted, which currently holds more than 19,267 assessments and 145,000 recommendations (as of July 24, 2020). This database also contains energy utility (electricity, natural gas, and other fuels) and water utility data, making it a potential data source for industrial water use. This report attempts to create regression models to predict a small- or medium-sized industrial facility’s annual water use or cost based on its industrial subsector and several possible relevant variables. Using data collected by IAC assessments, models for several industrial subsectors were generated via stepwise regression techniques to determine which variables (annual sales, number of employees, facility/plant area, annual production hours, and a water stress metric) are relevant.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2021

Long-term variability of root production in bioenergy crops from ingrowth core measurements

Long-term determination of root biomass production upon land-use conversion to biofuel crops is rare. To assess land-use legacy influences on belowground biomass accumulation, we converted 22-year-old Conservation Reserve Program (CRP) grasslands and 50+-year-old agricultural (AGR) lands to corn (C), switchgrass (Sw) and restored prairie (Pr) biofuel crops. We maintained one CRP grassland as a reference (Ref). We hypothesized that land-use history and crop type have significant effects on root density, with perennial crops on CRP grasslands having a higher root biomass productivity, while corn grown on former agricultural lands produce the lowest root biomass. Methods The ingrowth core method was used to determine in situ ingrowth root biomass, alongside measurements of aboveground net primary productivity (ANPP). Ancillary measurements, including air temperature, growing season length and precipitation were used to examine their influences on root biomass production. Important Findings Root biomass productivity was the highest in unconverted CRP grassland (1716 g m-2 yr-1) and lowest in corn fields (526 g m-2 yr-1). All perennial sites converted from CRP and AGR lands had lower root biomass and ANPP in the first year of planting but peaked in 2011 for switchgrass and a year later for restored prairies. Ecosystem stability was higher in restored prairies (AGR-Pr: 4.3 ± 0.11; CRP-Pr: 4.1 ± 0.10), with all monocultures exhibiting a lower stability. Root biomass production was positively related to ANPP (R2 = 0.40). Overall, attention should be given to root biomass accumulation in large-scale biofuel production as it is a major source of carbon sequestration.

Environmental Sciences & Ecology↗

DOE OSTI2024

Curvature of the chiral phase transition line from the magnetic equation of state of ( 2 + 1 )-flavor QCD

We analyze the dependence of the chiral phase transition temperature on baryon number and strangeness chemical potentials by calculating the leading order curvature coefficients in the light and strange quark flavor basis as well as in the conserved charge ( B , S ) basis. Making use of scaling properties of the magnetic equation of state (MEoS) and including diagonal as well as off-diagonal contributions in the expansion of the energylike scaling variable that enters the parametrization of the MEoS, allows to explore the variation of T c ( μ B , μ S ) = T c ( 1 − ( κ 2 B μ ^ B 2 + κ 2 S μ ^ S 2 + 2 κ 11 B S μ ^ B μ ^ S ) ) along different lines in the ( μ B , μ S ) plane. On lattices with fixed cutoff in units of temperature, a T = 1 / 8 , we find κ 2 B = 0.015 ( 1 ) , κ 2 S = 0.0124 ( 5 ) and κ 11 B S = − 0.0050 ( 7 ) . We show that the chemical potential dependence along the line of vanishing strangeness chemical potential is about 10% larger than along the strangeness neutral line. The latter differs only by about 3% from the curvature on a line of vanishing strange quark chemical potential, μ s = 0 . We also show that close to the chiral limit the strange quark mass contributes like an energylike variable in scaling relations for pseudocritical temperatures. The chiral phase transition temperature decreases with decreasing strange quark mass, T c ( m s ) = T c ( m s phy ) ( 1 − 0.097 ( 2 ) ( m s − m s phys ) / m s phy + O ( ( Δ m s ) 2 ) . Published by the American Physical Society 2024

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Next Generation Solvent Vapor Pressure Testing

Previous work by Savannah River National Laboratory (SRNL) indicated that the actual Next Generation Solvent vapor pressure would be higher than the Original Caustic Side Solvent Extraction (CSSX) solvent vapor pressure, but below a bounding vapor pressure using Raoult’s Law. Since solvent vapor pressure has a significant impact on Composite Lower Flammability Limits (CLFL) and attendant accident analyses, obtaining an additional margin from the bounding NGS vapor pressure would be valuable. In order to quantify how much margin might be gained from the NGS bounding solvent, SRNL researchers were requested to perform vapor pressure testing with the Next Generation Solvent (NGS) by Savannah River Mission Completion (SRMC). The vapor pressure curve for the NGS formulation set to be deployed at the Salt Waste Processing Facility (SWPF) has been determined by SRNL up to 55°C (131°F) using headspace Gas Chromatography (GC). It was expected that NGS would have a higher vapor pressure than the Original CSSX solvent; however, experimental results indicate a lower vapor pressure. At this time, it is uncertain if this difference is due to the 7x increase in concentration of the large calixarene in the solvent (0.007M BOBCalix in Original CSSX solvent vs. 0.05M MaxCalix in NGS) or to minor batch-to-batch variations in Isopar-L constituents. The NGS and the Original CSSX solvent vapor pressure data were fitted to the Antoine Equation. The Antoine equation gives a more accurate representation of the expected vapor pressure of the solvents outside of the temperature range tested. The Antoine equation fitting for NGS is given below: $P=10^{6.364⁻\frac{1788.4}{T+219.3}}$. Where, p is the vapor pressure (partial pressure) of NGS in mmHg and T is the temperature in °C. It is recommended that SRMC either continue using the more conservative Isopar-L vapor pressure calculations at SWPF with the equation developed for the Original CSSX solvent, or the equation presented above for the NGS solvent. Additionally, it is recommended to study the variability in Isopar-L vapor pressure between lots.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2020

COMBLE-ISLAS water isotopes in precipitation (COMBLEISLASISO)

During the measurement period, precipitation was sampled on daily to sub-daily basis with a sampling kit consiting of a clear plastic box (60x40x40 cm) mounted to a ground structure. Daily sampling lasted from 22 Feb to 24 Mar 2020. At a fixed time of day (10 UTC), samples were collected and the sampling box cleaned for the next sample. Additionally, during intense observations periods (IOPs) announced by ISLAS, ARMS scientists increased sampling frequency to sub-daily sampling. During the campaign period, 2 IOPs were conducted, with in total 12 samples taken. IOP1 lasted from 27 Feb 10 to 02 Mar 2020, and IOP2 lasted from 12 Mar to 14 Mar 2020. Solid precipitation in the box was then melted, transferred to an 8 ml glass vial, sealed and shipped to FARLAB (University of Bergen, Norway) for stable water isotope analysis. Measured stable isotope composition of precipitation samples will be related to conditions at other sampling locations, and at the evaporation site to inform about conservation of water isotope quantities. In the period from 22 Feb to 24 Mar, 23 daily samples were collected at 10 UTC for stable water isotope analysis. During the entire sampling period, a high variability of precipitation phase was encountered. Most samples were recovered in the form of graupel/snow (12 samples), as rain (7 samples), and the remaining samples fell as graupel, snow, rain or frozen rain. Precipitation amounts varied between <1 mm (26 Feb) and 140 mm (25 Feb) within the sampling box. Furthermore, higher-resolution sampling was conducted during 2 intense observation periods (IOPs). During IOP1, lasting from 27 Feb 10 UTC to 02 Mar 10 UTC, 7 samples were collected, all of them as solid phase precipitation. IOP2 lasted from 12 Mar 10 UTC to 14 Mar 09 UTC, and 5 samples (all as graupel/snow) were collected. Samples were processed according to FARLAB standard measurement procedures. In short, samples were transferred to 1.5 ml glass vials with rubber/PTFE septa (part #548-0907, VWR, USA). An autosampler (A0325, Picarro Inc) transferred ca. 2µl per injection into a high-precision vapourizer (A0211, Picarro Inc, USA) heated to 110°C. After blending with dry N2 (< 5 ppm H2O) the gas mixture was directed into the measurement cavity of a Cavity-Ring Down Spectrometer (L2140-i, Picarro Inc) for about 7 min with a typical water concentration of 20 000 ppm. Memory effects were reduced by two times measuring a vapour mixture at a mixing ratio of 50 000 ppm, obtained from 2 injections of 2 µl for 5 min at the beginning of each new sample vial. Thereafter, another 6 injections of 2 µl per sample were measured individually as described above, and averages of the last 5 injections were used for further processing. Three standards were measured at the beginning and end of each batch, including a drift standard DI2 (δD: -50.72±0.73 permil, δ18O: -7.63±0.10 permil), and for calibration the laboratory standards GLW (δD: -307.79±0.75 permil, δ18O: -40.02±0.07 permil) and EVAP2 (δD: 9.52±0.65 permil, δ18O: 1.81±0.13 permil). A detailed calibration report is included with the final uploaded data set.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1974

Recent developments on time-dependent calculation of nonequilibrium flows

It is shown that, in a time-dependent nonequilibrium flow analysis, erroneous and unstable solutions can result from the use of temperature equations and flow variables other than pressure in fitting the bow shock. A formulation is presented that has been critically evaluated by comparison with a previous one and found to yield steady and more accurate solutions in conserving total enthalpy.

Li, C. P.↗

DOE OSTI2025

Symmetry Determining Equations of the Rankine-Hugoniot Equations for Variable Velocity Shock Waves

The “constant velocity piston” problem (Fig. 1), also known as the “piston problem,” is a standard model for a one dimensional, in our case linear, symmetric shock wave moving through an inviscid, perfect gas. The model can be divided into two regions - a perturbed section on the left and an unperturbed section on the right - by a moving shock wave moving left to right. Both the perturbed and unperturbed sections, i.e. the shocked and unshocked regions, respectively, obey the Eulerian conservation equations; however, at the exact location of the shock, there is a mathematical discontinuity not satisfied by the Euler equations. To ensure continuity and conservation of certain quantities when crossing between the unshocked and shocked regions, we evoke a series of equations derived from the Eulerian conservation equations, called the Rankine-Hugoniot equations, or “jump” equations as it is often referred to in the literature on the topic. The classical constant-velocity piston problem assumes the piston features a constant driving velocity (among many other willing suspensions of belief required in the pursuit of a first principles equation model); consequent to this assumption is a constant-velocity shock and a constant-velocity shocked flow state. However, using Lie Group Theory (LGT), also known as symmetry analysis, we can attempt to reinterpret the model with a shock wave of variable velocity in time and space. An extension of the model in this way opens up the possibility for obtaining new analytical solutions to the piston problem for certain shock velocity models. In this report, we use LGT to derive the symmetry determining equations (SDEs), whose solutions are Lie groups, which permit analytical solutions. In the future, we can then use the SDEs to define constraint equations on the shock velocity model and what the successive solutions to the Euler equations might be based off such constraints. This report is structured as follows: Section 2 provides a brief derivation of the Rankine-Hugoniot (“jump”) equations; Section 3 gives an overview of Lie group theory; Section 4 derives the SDEs of the jump equations; Section 5 derives the Euler conservation equations for fluids; and Section 6 presents concluding remarks and opportunities for future studies.

42 ENGINEERING↗

NASA NTRS1984

On the theory of cosmic-ray-mediated shocks with variable compression ratio

Cosmic-ray-mediated shocks may accelerate enough cosmic rays to high enough energies that they escape the shock, carrying an appreciable amount of energy before being convected to downstream infinity. Under such conditions, it is noted, the overall compression ratio cannot be determined from the conservation equations as in conventional hydrodynamic treatments, and the standard equations for shock acceleration admit arbitrarily high compression ratios. A procedure is outlined for obtaining the structure of high Mach number, cosmic-ray-mediated shocks, including their overall compresion ratio, around a low Mach number viscous subshock. Analytic solutions are obtained by quardrature for an energy-dependent diffusion coefficient in the limit of extreme sensitivity to energy, which, unlike previous solutions, include the finite thermal pressure of the preshock gas.

Eichler, D.↗

NASA NTRS1996

Further Development of a New, Flux-Conserving Newton Scheme for the Navier-Stokes Equations

This paper is one of a series of papers describing the development of a new numerical approach for solving the steady Navier-Stokes equations. The key features in the current development are (1) the discrete representation of the dependent variables by way of high order polynomial expansions, (2) the retention of all derivatives in the expansions as unknowns to be explicitly solved for, (3) the automatic balancing of fluxes at cell interfaces, and (4) the discrete simulation of both the integral and differential forms of the governing equations. The main purpose of this paper is, first, to provide a systematic and rigorous derivation of the conditions that are used to simulate the differential form of the Navier-Stokes equations, and second, to extend our previously-presented internal flow scheme to external flows and nonuniform grids. Numerical results are presented for high Reynolds number flow (Re = 100,000) around a finite flat plate, and detailed comparisons are made with the Blasius flat plate solution and Goldstein wake solution. It is shown that the error in the streamwise velocity decreases like r(sup alpha)(Delta)y(exp 2), where alpha approx. 0.25 and r = delta(y)/delta(x) is the grid aspect ratio.

Scott, James R.↗