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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 649 records · Page 36
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Upscaling Reactive Transport and Clogging in Shale Microcracks by Deep Learning

Fracture networks in shales exhibit multiscale features. A rock system may contain a few main fractures and thousands of microcracks, whose length and aperture are orders of magnitude smaller than the former. It is computationally prohibitive to resolve all the fractures explicitly for such multiscale fracture networks. One traditional approach is to model the small-scale features (e.g., microcracks in shales) as an effective medium. Although this fracture-matrix conceptualization significantly reduces the problem complexity, there are classes of physical processes that cannot be accurately upscaled by effective medium approximations, e.g., microcrack clogging during mineral reactions. Here, we employ deep learning in place of effective medium theory to upscale physical processes in small-scale features. Specifically, we consider reactive transport in a fracture-microcrack network where microcracks can be clogged by precipitation. A deep learning multiscale algorithm is developed, in which the microcracks are upscaled as a wall boundary condition of the main fractures. The wall boundary condition is constructed by recurrent neural networks, which take concentration histories as input and predict the solute transport from main fractures to microcracks. The deep learning multiscale algorithm is firstly employed in specific scenarios, then a general model is developed which can work under various conditions. The new approach is validated against fully resolved simulations and an analytical solution, providing a reliable and efficient solution for problems that cannot be upscaled by effective medium models.

58 GEOSCIENCES↗
DOE OSTI2021

Investigating the impact of cloud-radiative feedbacks on tropical precipitation extremes

Abstract Although societally important, extreme precipitation is difficult to represent in climate models. This study shows one robust aspect of extreme precipitation across models: extreme precipitation over tropical oceans is strengthened through a positive feedback with cloud-radiative effects. This connection is shown for a multi-model ensemble with experiments that make clouds transparent to longwave radiation. In all cases, tropical extreme precipitation reduces without cloud-radiative effects. Qualitatively similar results are presented for one model using the cloud-locking method to remove cloud feedbacks. The reduced extreme precipitation without cloud-radiative feedbacks does not arise from changes in the mean climate. Rather, evidence is presented that cloud-radiative feedbacks enhance organization of convection and most extreme precipitation over tropical oceans occurs within organized systems. This result suggests that climate models must correctly predict cloud structure and properties, as well as capture the essence of organized convection in order to accurately represent extreme rainfall.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Design and experimental demonstration of feedback adaptive RMP ELM controller toward complete long pulse ELM suppression on KSTAR

Operation of a fusion power plant requires robust edge localized mode (ELM) suppression simultaneously with high plasma performance. In this study, we describe a novel feedback adaptive resonant magnetic perturbation (RMP) ELM controller designed to address this problem by achieving optimized ELM suppression through the advanced application of 3D RMPs. From real-time $D_α$ data, the controller is achievable robust ELM suppression while simultaneously minimizing the applied RMP in order to enhance plasma performance. In real time, the instantaneous ELM-frequency is analyzed with an adaptive feedback algorithm to determine amplitudes and phases of RMP coil currents that will maximize plasma performance while maintaining ELM suppression. When applied through the KSTAR plasma control system in several experiments using $textiti{n}$ = 1 RMPs, robust ELM suppression is achieved and sustained in feedback while reducing the RMP strength to ~65% of its initial value. Minimization of the RMP strength in this manner not only allows for operation of longer discharges due to a decrease in flux consumption but also allows for a strong recovery of up to ~60% of $β_N$ throughout the ELM-free period.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Holographic wavefront sensors

A brief historical review of the first designs of holographic wavefront sensors (WFS’s) and the concepts lying in their basis is presented. The main directions in the development of these sensors are highlighted and considered. One of these directions implies a two-stage transition from the use of several separate measuring channels with holograms filtering only one Zernike mode: first to the use of one channel with a multiplexed hologram filtering several first Zernike modes, and then to filtering the entire set of Zernike modes with the aid of one combined multiplexed hologram. Another line of research in this field (related to the first one) is the optimisation of the filter hologram structure in order to reduce cross-modulation interferences, increase multiplexing level, etc. One more line of research implements principles of dynamic holography by introducing spatial light modulators into the WFS composition. Hence, the advantages of time multiplexing of holograms can be used. The approach developed by G. Andersen’s team, aimed at adapting the holographic WFS design for operation as an element of adaptive optical system with a zonal corrector, as well as an approach implying development of hybrid holographic sensors, are considered separately. The results of the authors’ studies of holographic WFS’s with application of the methods of Fourier holography (i.e., holography of focused beams, in particular, using diffuse scatterers in a hologram recording channel) are also reported. (paper)

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2024

Using intrusive approaches as a step towards accounting for stochasticity in wind turbine design

Current wind turbine design methods require tens of thousands of time-domain simulations and use different random seeds to account for the stochasticity of the environmental conditions. The account of stochasticity is nonintrusive because the sampling method calls a deterministic model multiple times without changing its underlying equations. In this work, we investigate and demonstrate using simple proof of concepts how intrusive approaches can be used to directly account for stochasticity in the equations representing a mechanical system. Our long term goal is to apply such methodology to the design of wind turbines without requiring an excessive number of simulations. Intrusive methods manipulate stochastic variables directly to provide the probability density functions (PDFs) of the states and outputs at any time as functions of the PDFs of the inputs. We illustrate how different methods can be used with a reduced-order model of a wind turbine with one degree of freedom and for linear and nonlinear models. We discuss how the methods can be extended and what it will take to apply them to a level of fidelity similar to current state-of-the-art wind turbine design tools.

17 WIND ENERGY↗
DOE OSTI2024

Continuous pressure-induced valence and magnetic transitions in EuMnSb 2

EuMnSb 2 is a unique and promising compound with complex magnetism (Eu and Mn sublattices) to realize magnetic control of topological quasiparticles. By applying pressure (P) to this system, we have observed a continuous pressure-induced valence change from Eu 2+ to Eu 3+ . Remarkably, despite the significant valence change in Eu ions, an extremely large increase in the magnetic ordering temperature (T N,Eu ) from 21 K at ambient pressure to 180 K at 28.4 GPa is detected.The substantial enhancement in T N,Eu is likely the consequence of the reduced distance between neighboring Eu ions and the enhanced indirect exchange interaction through increased 4f -5d mixing under pressure. Furthermore, T N,Eu (P) shows a discontinuous change above 8 GPa, although the orthorhombic crystal structure remains stable. Our experimental data indicate the record high value in the magnetic hyperfine field of 151 Eu, reaching 74.1 T at 28.4 GPa and 26 K. In conclusion, these unusual electronic and magnetic transitions revealed by pressure highlight that EuMnSb 2 is a rich magnetic topological semimetal candidate for investigating a multitude of strong correlations.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Optimal Twirling Depth for Classical Shadows in the Presence of Noise

The classical shadows protocol is an efficient strategy for estimating properties of an unknown state p using a small number of state copies and measurements. In its original form, it involves twirling the state with unitaries from some ensemble and measuring the twirled state in a fixed basis. It was recently shown that for computing local properties, optimal sample complexity (copies of the state required) is remarkably achieved for unitaries drawn from shallow depth circuits composed of local entangling gates, as opposed to purely local (zero depth) or global twirling (infinite depth) ensembles. Here, we consider the sample complexity as a function of the depth of the circuit, in the presence of noise. We find that this noise has important implications for determining the optimal twirling ensemble. Under fairly general conditions, we (i) show that any single-site noise can be accounted for using a depolarizing noise channel with an appropriate damping parameter f, (ii) compute thresholds f th at which optimal twirling reduces to local twirling for Pauli operators, (iii) nth order Renyi entropies (n ≥2), and (iv) provide a meaningful upper bound t max on the optimal circuit depth for any finite noise strength f, which applies to observables and entanglement entropy measurements. In conclusion, these thresholds strongly constrain the search for optimal strategies to implement shadow tomography and are easily tailored to the experimental system at hand.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Scalable Reduced Order Model with Discontinuous Galerkin Domain Decomposition

scaleupROM is a scalable, physics-constrained reduced order model (ROM). It aims to provide robust, accelerated physics predictions at extrapolated scales, based on the small, component-level data. This is implemented by combining projection-based ROM with discontinuous Galerkin domain decomposition, in the framework of MFEM and libROM. It currently supports the Poisson equation and Stokes flow equation, and more work is in progress toward general, nonlinear physics systems.

Chung, Seung Whan↗
DOE OSTI2024

Migration of 60 Co, 133 Ba, 137 Cs, and 152 Eu from cementitious wasteforms in field lysimeter experiments

Safe and effective storage of radioactive waste is essential to protect environmental health. Due to the potential for accidental releases and the severity of the associated risks, it is imperative to further understand radionuclide transport should an accident occur. This work analyzed the vadose zone migration of radionuclides from cementitious wasteforms at the Savannah River Site after ten years. The observed radionuclides are prominent constituents of radioactive waste or analogs for other groups or series of radionuclides. Lysimeters were first analyzed in 2016 using a collimated high-purity germanium gamma-ray spectrometer to non-destructively measure the concentration of each radionuclide in the sediment column as a function of depth. Following these measurements, the lysimeters were redeployed in the field for another 4 years. All radionuclides in all lysimeters were observed to transport further during the redeployment period; however, the extent of migration varied with the material used for introduction. Except for 137 Cs, migration through the sediment control system increased with decreasing ionic potential (ionic charge/radius); migration order: 152 Eu< 137 Cs< 60 Co< 133 Ba. Overall, the cementitious wasteforms were observed to decrease radionuclide migration extent relative to the filter paper. In both cementitious wasteforms, the migration extent increased in the order 152 Eu< 133 Ba< 60 Co< 137 Cs. However, less migration was measured when the radionuclides were incorporated into a reducing grout wasteform. The novelty of this paper is the demonstration of a technique capable of creating non-destructive measurements over decade time scales. Ultimately, this work provides insight into the long-term migration of alkali, alkali earth, divalent transition metal, and trivalent actinide element isotopes.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2021

Implementation of an Experimental Setup Based on the Three-Omega Method for Thermal Conductivity Measurements of Molten Salts

Molten salt reactors (MSRs) are one of the most promising next-generation nuclear reactor designs which use molten salts as the fuel and coolant. To design, model, predict, license, and operate MSRs, the behavior of the fuel and coolant must be understood as a function of temperature and irradiation, i.e., burnup. Of particular importance and complexity is the reliable and reproducible measurement of thermal conductivity of molten salts at high temperatures (> 500 °C). Thermal conductivity provides information on the rate at which heat passes through a specified material and is dependent on chemical composition. Currently, in the national laboratory complex, several methods for measuring thermal conductivity on molten salt systems are under development, however, to date, consistent data have not been produced. Typically, thermal conductivity measurements of high-temperature melts have been conducted by using the Transient Hot Wire (THW) method where thermal conductivity is determined by the change in temperature over time at a certain distance from a heating wire. Another time-dependent technique is the laser flash method, commonly used for solid samples. Laser flash requires prior measurements of density and heat capacity to calculate the thermal conductivity of the sample, introducing additional measurement errors. Steady state and time-dependent measurements such as THW and laser flash have errors from convection and ambient temperature fluctuations (heat radiation). In order to produce high fidelity thermal conductivity data, a measurement technique needs to be developed that will reduce the effect of convection at high temperatures by measuring in a very fast time window. In addition, to measure on actinide and irradiated salts the sample size should be small. Therefore, this project aims to develop a frequency-domain technique based on the three-omega (3?) method, previously proven to work on solids, nanofluids and low temperature solar salts (25 to 250 °C). If the 3?-device can be designed with materials that can withstand high temperatures (>500 °C) and can be configured to reduce errors due to convection, then it would prove to be a valuable technique for ensuring thermal conductivity on molten salt systems needed for MSR development. For this proposal, the 3? instrument will be designed, assembled, and tested on non-radioactive salts before being used on actinide bearing salts. In addition, a thermal model will be developed to determine the thermal conductivity of the sample from the measured thermal oscillation of the sensor. Because 3? is a direct method that requires a small volume of sample and works in a very fast time window, it will aid in suppressing the interference due to convection which is ideal when working with high temperature liquid samples.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2020

Sodium Ion Expansion Power Block for Distributed CSP

The Sodium Ion Expansion Power Block for Distributed CSP was a three-plus-one-year effort under the Concentrating Solar Power: Advanced Projects Offering Low LCOE Opportunities (CSP: APOLLO) funding program within the U.S. Department of Energy Solar Energy Technologies Office. The primary objective of this project is to develop a dual-stage modular sodium thermal electrochemical converter (Na-TEC) heat engine power block, which can be potentially integrated with either a small-scale dish solar or large-scale heliostats and parabolic trough CSP. Na-TEC is a heat engine that generates electricity through the isothermal expansion of sodium ions. The Na-TEC is a closed system that can theoretically achieve conversion efficiencies above 45% when operating between thermal reservoirs at 1150 K and 550 K. However, thermal designs have confined previous single-stage devices to thermal efficiencies below 20%. To mitigate some of these limitations, we consider dividing the isothermal expansion into two stages; one at the evaporator temperature (1150 K) and another at an intermediate temperature (650 K –1050 K). This dual-stage Na-TEC takes advantage of regeneration and reheating, and could be amenable to better thermal management. In light of this, we first designed and developed a thermo-electrochemical model, and thermodynamically demonstrated how the dual-stage device can improve the efficiency by up to 8% points over the best performing single-stage device. We also established an application regime map for the single- and dual-stage Na-TEC in terms of the power density and the total thermal parasitic loss. Moreover, a thermal design of an axisymmetric dual-stage Na-TEC is developed to guide the scale-up and fabrication of sub-components of prototype module. A reduced-order finite-element model is used in conjunction with a Na-TEC thermodynamic model that was developed to determine the total parasitic heat loss of this dual-stage design. A number of simplifications are applied in the reduced-order model to decrease the computational time while maintaining acceptable accuracy. According to this analysis, a maximum efficiency of 29% and a maximum power output of 125 W can be achieved. Ultimately, we were able to demonstrate thermal efficiency improvements of the Na-TEC heat engine from 19% up to 40.3%, in a dual-stage (non-optimized) prototype module that we designed, fabricated, and tested with high temperature stage at 923 K. Furthermore, a cost-performance analysis for this improved dual-stage design was carried out for distributed-CSP systems. A high-level techno-economic analysis (TEA) explores four scenarios where a Na-TEC is used as the heat engine for a distributed-CSP system. Overnight capital cost and levelized cost of electricity (LCOE) are estimated for a system lifetime of 30 years, revealing that overnight capital costs in a range from $3.57 to $17.71 per We are feasible, which equate to LCOEs from 6.9 to 17.2 cents/kWh e -1 . This analysis makes a significant contribution by concurrently quantifying the efficiency and unit costs for a range of multistage configurations, and demonstrating that a Na-TEC may be a promising alternative to Stirling engines for distributed-CSP systems at residential scale of 1–5 kW e .

14 SOLAR ENERGY↗
DOE OSTI2023

A bi-level data-driven framework for fault-detection and diagnosis of HVAC systems

Long-term operation of heating, ventilation, and air conditioning (HVAC) systems will eventually lead to a range of HVAC system failures, resulting in excessive energy consumption and maintenance costs. Here, to avoid HVAC malfunctioning, fault detection diagnostic (FDD) is utilized as a common practice. Machine learning methods have lately received considerable interest for FDD analysis of HVAC systems due to their high detection accuracy. Meanwhile, HVAC malfunctions are regarded as rare occurrences, hence normal operating data samples are much more accessible than data samples in faulty and malfunctioning conditions. The dominating frequency of normal operation in HVAC datasets has also led to heavily biased classification algorithms within the literature. Moreover, the focus of previous literature has been on increasing the accuracy of the models which leads to a high number of false positives (misleading alarms) in the system. In order to enhance the performance of diagnostic procedures and fill the mentioned gaps, this study proposes a novel data-driven framework. A bi-level machine learning framework is developed for diagnosing faults in air handling units (AHUs) and rooftop units (RTUs) based on principal component analysis (PCA), time series anomaly detection, and random forest (RF). It is shown that PCA can reduce the dataset dimension with one principal component accounting for 95% of data variance. Also, the random forest could classify the faults with 89% precision for single-zone AHU, 85% precision for RTU, and 79% for multi-zone AHU. By proposing this framework, three persistent challenges are addressed: (I) minimizing false positives; (II) accounting for data imbalance; and (III) normal condition monitoring of equipment.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2022

COVID-19 pandemic underscores role of green space in urban carbon dynamics

For Detroit Michigan the arrival of COVID-19 led to intensive measures to prevent further spread of the virus resulting in consequent changes in traffic and energy use. Here, we take advantage of these different emission scenarios to explore CO 2 dynamics in a postindustrial city with a declining population and increasing green space. We present atmospheric CO 2 concentration and net urban ecosystem exchange of CO 2 (NUE) from a typical eddy covariance system and canopy greenness from a field camera on the Wayne State University campus in midtown Detroit. We categorized our study period (January 18, 2020–July 31, 2020) into three subperiods associated with the state-wide shelter-in-place order. Our results support that the city was a net carbon source throughout the period, particularly during the shelter-in-place period, although reduced traffic lowered CO 2 concentrations and NUE. However, during the post-order period when traffic was highest, atmospheric CO 2 concentrations and NUE were lowest, suggesting that the greening of urban vegetation may have greater carbon mitigation potential than lowering anthropogenic carbon emissions through traffic reductions.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Vehicle Powertrain Simulation Accuracy for Various Drive Cycle Frequencies and Upsampling Techniques

As connected and automated vehicle technologies emerge and proliferate, lower frequency vehicle trajectory data is becoming more widely available. In some cases, entire fleets are streaming position, speed, and telemetry at sample rates of less than 10 seconds. This presents opportunities to apply powertrain simulators such as the National Renewable Energy Laboratory's Future Automotive Systems Technology Simulator to model how advanced powertrain technologies would perform in the real world. However, connected vehicle data tends to be available at lower temporal frequencies than the 1-10 Hz trajectories that have typically been used for powertrain simulation. Higher frequency data, typically used for simulation, is costly to collect and store and therefore is often limited in density and geography. This paper explores the suitability of lower frequency, high availability, connected vehicle data for detailed powertrain simulation. A large data set of 1 Hz trajectories is used to quantify the accuracy loss when simulating energy consumption for conventional, hybrid, and battery electric powertrains using less than 1 Hz data. Techniques to upsample lower frequency drive cycle data in order to increase accuracy are also explored. Median energy consumption errors when simulating energy consumption for a 1/10 Hz trajectory are found to be 3-6% when compared to 1 Hz trajectories. Applying upsampling and interpolation techniques are shown to reduce the simulation errors by roughly 50%. The findings in this work can guide connected vehicle data collection specifications and processing techniques applied when using collected data for powertrain simulation.

ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2020

The Installation of Direct Water-Cooling Systems to Reduce Cooling Energy Requirements for High-Performance Computing Centers

A large cluster of High-Performance Computing (HPC) equipment at the Lawrence Livermore National Laboratory in California was retrofitted with an Asetek cooling system. The Asetek system is a hybrid scheme with water-cooled cold plates on high-heat-producing components in the information technology (IT) equipment, and with the remainder of the heat being removed by conventional air-cooling systems. In order to determine energy savings of the Asetek system, data were gathered and analyzed two ways: using top-down statistical models, and bottom-up engineering models. The cluster, “Cabernet”, rejected its heat into a facilities cooling water loop which in turn rejected the heat into the same chilled water system serving the computer-room air handlers (CRAHs) that provided the air-based cooling for the room. Because the “before” and “after” cases both reject their heat into the chilled water system, the only savings is due to reduction in CRAH fan power. The top-down analysis showed a 4% overall energy savings for the data center (power usage effectiveness (PUE) —the ratio of total data center energy to IT energy— dropped from 1.60 to 1.53, lower is better); the bottom-up analysis showed a 3% overall energy savings (PUE from 1.70 to 1.66) and an 11% savings for the Cabernet system by itself (partial PUE of 1.51). Greater savings, on the order of 15-20%, would be possible if the chilled water system was not used for rejecting the heat from the Asetek system. About 37% of the heat from the Cab system was rejected to the cooling water, lower than at other installations.

Earni, Shankar↗
DOE OSTI2021

Efficiency and mechanisms of Sb(III/V) removal by Fe-modified biochars using X-ray absorption spectroscopy

Fe-modified biochars (FeBC) are effective antimony (Sb) removal materials; however, the removal mechanisms require further investigation. In this study, aqueous Sb(III) and Sb(V) removal by FeBC (300, 600, and 900 °C) was evaluated, with the adsorption mechanisms investigated using X-ray absorption spectroscopy (XAS). Screening results (based on removal efficiencies) using different types of FeBC indicated the 900 °C FeCl3- modified biochar (FeCl3BC900) achieved the best Sb(III/V) removal performance. The kinetics of the Sb(III/V) removal process were best fitted by a pseudo-second-order model. Additionally, the isothermal results were described by Langmuir and Redlich-Peterson models. Aqueous analysis and X-ray absorption near-edge structure data fitting indicated Sb(III) was oxidized to Sb(V) in the Sb(III)-spiked system, and the oxidation extent increased with increasing pyrolysis temperature. The oxidation process rapidly occurred in both the solution and biochar. No Sb(V) was reduced to Sb(III) in the Sb(V)-spiked system. The XAS results of the isothermal experiment indicated the oxidation capacity of FeCl3BC900 was limited for high initial Sb(III) concentrations. The SbFe1 and Sb-Fe2 bonding distances were 3.05–3.10 and 3.47–3.54 Å, respectively, indicating inner-sphere complexes were formed during the Sb(III/V) removal processes. The Sb(III/V) removal mechanisms included electrostatic adsorption, inner-sphere complexes, and coprecipitation. Oxidation was also involved in Sb(III) removal.

Antimony↗
DOE OSTI2023

Li-ion and Na-ion intercalation in layered MnO 2 cathodes enabled by using bismuth as a cation pillar

Low-cost batteries based on Earth-abundant materials are needed for large-scale electrical storage for the grid. Cathodes based almost entirely on Mn oxides would reduce overall battery cost but cycling of Mn oxides is often not stable. In Li-ion cells, most polymorphs of MnO 2 undergo irreversible transformation to spinel LiMn 2 O 4 during cycling, causing capacity loss. Doping MnO 2 with Bi is known to stabilize the structure, but previous reports have relied on low-crystallinity material making it impossible to pinpoint the Bi location in the structure or its mechanism. In this work, we report a series of hydrated Bi-doped layered MnO 2 compounds and characterize their structures as a function of Bi amount. Bi is shown to reside in the material interlayer, provoking higher long-range structural order even at a low doping level of 1.3%. Doped material improves the specific capacity and stability of cycling in both Li-ion and Na-ion cells. A high level of Bi doping, 4.3%, causes loss of the interlayer crystal water in non-aqueous electrolyte, and this reduces the interlayer distance. Crystal water is shown to be beneficial in a Na-ion system, while its loss improves Li-ion cycling. This provides fundamental insight into how pillaring by a heavy, multivalent cation stabilizes layered oxides.

25 ENERGY STORAGE↗
DOE OSTI2020

Scale Invariance of the Homentropic Inviscid Euler Equations with Application to the Noh Problem

We investigate the inviscid compressible flow (Euler) equations constrained by an ”isentropic” equation of state (EOS), whose functional form in pressure is an arbitrary function of density alone. Under the aforementioned condition, we interrogate using symmetry methods the scale-invariance of the homentropic inviscid Euler equations. We find that under general conditions, we can reduce the inviscid Euler equations into a system of two coupled ordinary differential equations. To exemplify the utility of these results, we formulate two example scale-invariant, self-similar solutions. The first example includes a shock-free expanding bubble scenario, featuring a modified Tait EOS. The second example features the classical Noh problem, coupled to an arbitrary isentropic EOS. In this case, in order to satisfy the conditions set forth in the classical Noh problem, we find that the solution for the flow is given by a transcendental algebraic equation in the shocked density.

97 MATHEMATICS AND COMPUTING↗