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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 307 records · Page 17
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Sockeye: A One-Dimensional, Two-Phase, Compressible Flow Heat Pipe Application

Sockeye is a heat pipe analysis application based on the Multiphysics Object-Oriented Simulation Environment (MOOSE) finite element framework. The primary purpose of Sockeye is to provide a transient heat pipe simulation tool to be used in the analysis of nuclear microreactor designs. Sockeye provides the capability to perform one-dimensional, two-phase, compressible flow simulation of a heat pipe working fluid and two-dimensional, axisymmetric heat conduction for the heat pipe cladding and its surroundings. Sockeye is demonstrated against analytical solutions and experimental data from the SAFE-30 heat pipe module test.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS1994

Ignition and combustion characteristics of metallized propellants, phase 2

Experimental and analytical investigations focusing on aluminum/hydrocarbon gel droplet secondary atomization and its effects on gel-fueled rocket engine performance are being conducted. A single laser sheet sizing/velocimetry diagnostic technique, which should eliminate sizing bias in the data collection process, has been designed and constructed to overcome limitations of the two-color forward-scatter technique used in previous work. Calibration of this system is in progress and the data acquisition/validation code is being written. Narrow-band measurements of radiant emission, discussed in previous reports, will be used to determine if aluminum ignition has occurred in a gel droplet. A one-dimensional model of a gel-fueled rocket combustion chamber, described in earlier reports, has been exercised in conjunction with a two-dimensional, two-phase nozzle code to predict the performance of an aluminum/hydrocarbon fueled engine. Estimated secondary atomization effects on propellant burnout distance, condensed particle radiation losses to the chamber walls, and nozzle two phase flow losses are also investigated. Calculations indicate that only modest secondary atomization is required to significantly reduce propellant burnout distances, aluminum oxide residual size, and radiation heat losses. Radiation losses equal to approximately 2-13 percent of the energy released during combustion were estimated, depending on secondary atomization intensity. A two-dimensional, two-phase nozzle code was employed to estimate radiation and nozzle two phase flow effects on overall engine performance. Radiation losses yielded a one percent decrease in engine Isp. Results also indicate that secondary atomization may have less effect on two-phase losses than it does on propellant burnout distance and no effect if oxide particle coagulation and shear induced droplet breakup govern oxide particle size. Engine Isp was found to decrease from 337.4 to 293.7 seconds as gel aluminum mass loading was varied from 0-70 wt percent. Engine Isp efficiencies, accounting for radiation and two phase flow effects, on the order of 0.946 were calculated for a 60 wt percent gel, assuming a fragmentation ratio of five.

Mueller, D. C.↗
NASA NTRS1996

Study of Critical Heat Flux and Two-Phase Pressure Drop Under Reduced Gravity

The design of the two-phase flow systems which are anticipated to be utilized in future spacecraft thermal management systems requires a knowledge of two-phase flow and heat transfer phenomena in reduced gravities. This program was funded by NASA headquarters in response to NRA-91-OSSA-17 and was managed by Lewis Research Center. The main objective of this program was to design and construct a two-phase test loop, and perform a series of normal gravity and aircraft trajectory experiments to study the effect of gravity on the Critical Heat Flux (CHF) and onset of instability. The test loop was packaged on two aircraft racks and was also instrumented to generate data for two-phase pressure drop. The normal gravity tests were performed with vertical up and downflow configurations to bound the effect of gravity on the test parameters. One set of aircraft trajectory tests was performed aboard the NASA DC-9 aircraft. These tests were mainly intended to evaluate the test loop and its operational performance under actual reduced gravity conditions, and to produce preliminary data for the test parameters. The test results were used to demonstrate the applicability of the normal gravity models for prediction of the two-phase friction pressure drop. It was shown that the two-phase friction multipliers for vertical upflow and reduced gravity conditions can be successfully predicted by the appropriate normal gravity models. Limited critical heat flux data showed that the measured CHF under reduced gravities are of the same order of magnitude as the test results with vertical upflow configuration. A simplified correlation was only successful in predicting the measured CHF for low flow rates. Instability tests with vertical upflow showed that flow becomes unstable and critical heat flux occurs at smaller powers when a parallel flow path exists. However, downflow tests and a single reduced gravity instability experiment indicated that the system actually became more stable with a parallel single-phase flow path. Several design modifications have been identified which will improve the system performance for generating reduced gravity data. The modified test loop can provide two-phase flow data for a range of operating conditions and can serve as a test bed for component evaluation.

Abdollahian, Davood↗
DOE OSTI2020

Computationally efficient CFD prediction of bubbly flow using physics-guided deep learning

To realize efficient computational fluid dynamics (CFD) prediction of two-phase flow, a multi-scale framework was proposed in this paper by applying a physics-guided data-driven approach. Instrumental to this framework, Feature Similarity Measurement (FSM) technique was developed for error estimation in two-phase flow simulation using coarse-mesh CFD, to achieve a comparable accuracy as fine-mesh simulations with fast-running feature. In this work, by defining physics-guided parameters and variable gradients as physical features, FSM has the capability to capture the underlying local patterns in the coarse-mesh CFD simulation. Massive low-fidelity data and respective high-fidelity data are used to explore the underlying information relevant to the main simulation errors and the effects of phenomenological scaling. By learning from previous simulation data, a surrogate model using deep feedforward neural network (DFNN) can be developed and trained to estimate the simulation error of coarse-mesh CFD. In a demonstration case of two-phase bubbly flow, the DFNN model well captured and corrected the unphysical “peaks” in the velocity and void fraction profiles near the wall in the coarse-mesh configuration, even for extrapolative predictions. The research documented supports the feasibility of the physics-guided deep learning methods for coarse mesh CFD simulations which has a potential for the efficient industrial design.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS1987

Laser velocimetry in turbulent flow fields - Particle response

Measurements of the particle response in a decelerating flow and a highly turbulent two-phase flow were obtained. Simultaneous measurements of the particle size and velocity served to quantify the particle response to the prevailing flow field. In the case of a flow incident upon a cylinder, the particle lag for a range of size classes was recorded. Results were also obtained in the flow generated by an atomizer operating on the leeward side of a flat disk bluff body in a coflowing air stream. Measurements of the mean axial, mean radial, and rms velocities and angles of trajectories were obtained for representative particle size classes. The air velocity and turbulence intensity were inferred from the seed particles on the order of one micrometer in diameter. Particles 9 micrometers and larger showed significant differences with respect to the gas phase mean velocity and turbulence intensity even at low velocities. In two-phase flows, reliable measurements of the continuous phase velocity and turbulence parameters requires the simultaneous measurement of particle size as a means for rejecting readings from large particles from the velocity pdf's.

Bachalo, W. D.↗
NASA NTRS2007

Ignition Characterization Test Results for the LO2/Ethanol Propellant Combination

A series of contracts were issued by the Marshall Space Flight Center (MSFC) of the National Aeronautics and Space Administration (NASA) un der the auspices of the Exploration Systems Mission Directorate to de velop and expand the maturity of candidate technologies considered to be important for future space exploration. One such technology was to determine the viability of incorporating non-toxic propellants for R eaction Control Subsystems (RCS). Contract NAS8-01109 was issued to A erojet to develop a dual thrust Reaction Control Engine (RCE) that ut ilized liquid oxygen and ethanol as the propellants. The dual thrust RCE incorporated a primary thrust level of 870 lbf, and a vernier thru st level of 10 - 30 lbf. The preferred RCS approach for the dual thru st RCE was to utilize pressure-fed liquid oxygen (LOX) and ethanol pr opellants; however, previous dual thrust feasibility testing incorporated GOX/Ethanol igniters as opposed to LOX/Ethanol igniters in the de sign. GOX/Ethanol was easier to ignite, but this combination had syst em design implications of providing GOX for the igniters. A LOX/Ethan ol igniter was desired; however, extensive LOX/Ethanol ignition data over the anticipated operating range for the dual thrust RCE did not e xist. Therefore, Aerojet designed and tested a workhorse LOX igniter to determine LOX/Ethanol ignition characteristics as part of a risk m itigation effort for the dual thrust RCE design. The objective of the ignition testing was to demonstrate successful ignition from GOX to LOX, encompassing potential two-phase flow conditions anticipated being present in real mission applications. A workhorse igniter was desig ned to accommodate the full LOX design flowrate, as well as a reduced GOX flowrate. It was reasoned that the initial LOX flow through the igniter would flash to GOX due to the latent heat stored in the hardwa re, causing a reduced oxygen flowrate because of a choked, or sonic, flow condition through the injection elements. As LOX flow continued, the hardware would chill-in, with the injected oxygen flow transitioning from cold GOX through two'phase flow to subcooled LOX. The Workh orse igniter was well instrumented: Pressure and temperature instrumentation permitted oxygen state points to be determined in the igniter oxidizer manifold, and gas-side igniter chamber thermocouples provide d chamber thermal profile characteristics. The cold flow chamber pres sure (Pc) for each test was determined and coupled with the igniter chamber diameter (De) to calculate the characteristic quench parameter (Pc x Dc), which was plotted as a function of core mixture ratio, MRc . Ignition limits were determined over a broad range of valve inlet conditions, and ignition was demonstrated with oxygen inlet conditions that ranged from subcooled 210 deg R LOX to 486 deg R GOX. Once ign ited at cold GOX conditions, combustion was continuous as the hardwar e chilled in and the core mixture ratio transitioned from values near 1.0 to over 12.5. Pulsing is required in typical RCS engines; therefore, the workhorse igniter was pulse tested to verify the ability to pr ovide the required ignition for a pulsing RCE. The minimum electrical pulse width (EPW) of the dual thrust RCE was 0.080 seconds. Igniter pulse tests were performed at three conditions: (1) an EPW of 0.080 se conds at 25% duty cycle for 400 pulses; (2) an EPW of 0.160 seconds a nd a 5% duty cycle for 124 pulses; (3) an EPW of 0.160 seconds and a 50% duty cycle for 380 pulses. Successful ignition of LOX/Ethanol was demonstrated over a broad range of valve inlet conditions, with the empirically determined LOX/Ethanol ignition limits extending the previous database established for GOX/Ethanol ignition limits. Although th e observed chill-in characteristics of the hardware varied significan tly with flowrate, ignition was readily achieved. Combustion was marg inal at extremely fuel-rich conditions, and it fluctuated as the oxygen passed rough the twophase flow regime during the period of hardware chill-in. Pulse testing showed good repeatability with 100 percent r e-ignition for all pulses. Certain pulse-to-pulse repeatability requirements for actual RCS operation may necessitate establishment of mini mum oxygen flow rates and engine thrust levels for satisfactory engin e performance.

Robinson, Philip J.↗
NASA NTRS2016

Investigation of Multiphase Flow in a Packed Bed Reactor Under Microgravity Conditions

In this paper we study the two-phase flow phenomena in a packed bed reactor using an integrated experimental and numerical method. The cylindrical bed is filled with uniformly sized spheres. In the experiment water and air are injected into the bed simultaneously. The pressure distribution along the bed will be measured. The numerical simulation is based on a two-phase flow solver which solves the Navier-Stokes equations on Cartesian grids. A novel coupled level set and moment of fluid method is used to construct the interface. A sequential method is used to position spheres in the cylinder. Preliminary experimental results showed that the tested flow rates resulted in pulse flow. The numerical simulation revealed that air bubbles could merge into larger bubbles and also could break up into smaller bubbles to pass through the pores in the bed. Preliminary results showed that flow passed through regions where the porosity is high. Comparison between the experimental and numerical results in terms of pressure distributions at different flow injection rates will be conducted. Comparison of flow phenomena under terrestrial gravity and microgravity will be made.

computational fluid dynamics↗
NASA NTRS2001

Reduced Gravity Gas and Liquid Flows: Simple Data for Complex Problems

While there have been many studies for two-phase flow through straight cylindrical tubes, more recently, a new group of studies have emerged that examine two-phase flow through non-straight, non-cylindrical geometries, including expansions, contractions, tees, packed beds and cyclonic separation devices. Although these studies are still, relatively speaking, in their infancy, they have provided valuable information regarding the importance of the flow momentum, and the existence of liquid dryout due to sharp comers in microgravity.

McQuillen, John↗
NASA NTRS2016

Physics Based Model for Cryogenic Chilldown and Loading. Part III: Correlations

In this report we discuss the details of the correlations used to recognize flow patterns and predict frictional losses, heat and mass transfer in the cryogenic two phase flow. The emphasis are put on the formulation of the correlation problem in terms of concise parametric and functional spaces allowing for efficient online search of the model parameters and accurate prediction of the phenomena observed during cryogenic loading. A special attention is paid to the discussion of the correlation dependence on the gravity. In this context the physics of stability, friction, and boiling in the two-phase flow that underlies the required correlations is discussed.

two-phase flow↗
NASA NTRS2006

Ignition Characterization Test Results for the LO2/Ethanol Propellant Combination

A series of contracts were issued by the Marshall Space Flight Center (MSFC) of the National Aeronautics and Space Administration (NASA) under the auspices of the Exploration Systems Mission Directorate to develop and expand the maturity of candidate technologies considered to be important for future space exploration. One such technology was to determine the viability of incorporating non-toxic propellants for Reaction Control Subsystems (RCS). Contract NAS8-01109 was issued to Aerojet to develop a dual thrust Reaction Control Engine (RCE) that utilized liquid oxygen and ethanol as the propellants. The dual thrust RCE incorporated a primary thrust level of 870 lbf, and a vernier thrust level of 10 - 30 lbf. The preferred RCS approach for the dual thrust RCE was to utilize pressure-fed liquid oxygen (LOX) and ethanol propellants; however, previous dual thrust feasibility testing incorporated GOX/Ethanol igniters as opposed to LOX/Ethanol igniters in the design. GOX/Ethanol was easier to ignite, but this combination had system design implications of providing GOX for the igniters. A LOX/Ethanol igniter was desired; however, extensive LOX/Ethanol ignition data over the anticipated operating range for the dual thrust RCE did not exist. Therefore, Aerojet designed and tested a workhorse LOX igniter to determine LOX/Ethanol ignition characteristics as part of a risk mitigation effort for the dual thrust RCE design. LOX, encompassing potential two-phase flow conditions anticipated being present in real mission applications. A workhorse igniter was designed to accommodate the hll LOX design flowrate, as well as a reduced GOX flowrate. It was reasoned that the initial LOX flow through the igniter would flash to GOX due to the latent heat stored in the hardware, causing a reduced oxygen flowrate because of a choked, or sonic, flow condition through the injection elements. As LOX flow continued, the hardware would chill-in, with the injected oxygen flow transitioning from cold GOX through two-phase flow to subcooled LOX. permitted oxygen state points to be determined in the igniter oxidizer manifold, and gas-side igniter chamber thermocouples provided chamber thermal profile characteristics. The cold flow chamber pressure (P(sub c)) for each test was determined and coupled with the igniter chamber diameter (D(sub c)) to calculate the characteristic quench parameter (P(sub c) x D(sub c)), which was plotted as a function of core mixture ratio, m. Ignition limits were determined over a broad range of valve inlet conditions, and ignition was demonstrated with oxygen inlet conditions that ranged from subcooled 210 R LOX to 486 R GOX. Once ignited at cold GOX conditions, combustion was continuous as the hardware chilled in and the core mixture ratio transitioned from values near 1.0 to over 12.5. Pulsing is required in typical RCS engines; therefore, the workhorse igniter was pulse tested to verify the ability to provide the required ignition for a pulsing RCE. The minimum electrical pulse width (EPW) of the dual thrust RCE was 0.080 seconds.

Popp, Christopher G.↗
NASA NTRS2004

Ignition Characterization Tests of the LOX/Ethanol Propellant Combination

A series of contracts have been issued by the Marshall Space Flight Center (MSFC) of the National Aeronautics and Space Administration (NASA) to explore candidate technologies considered to be important for the Next Generation Launch Technology (NGLT) effort. One aspect of the NGLT effort is to explore the potential of incorporating non-toxic propellants for Reaction Control Subsystems (RCS). Contract NAS8-01109 has been issued to Aerojet to develop a dual thrust Reaction Control Engine (RCE) that utilizes liquid oxygen and ethanol as the propellants. The dual thrust RCE incorporates a primary thrust level of 870 lbf, and a vernier thrust level of 10 - 30 lbf. Aerojet has designed and tested a workhorse LOX igniter to determine LOX/Ethanol ignition characteristics as part of a risk mitigation effort for the dual thrust RCE design. The objective of the ignition testing was to demonstrate successfid ignition from GOX to LOX, encompassing potential two-phase flow conditions. The workhorse igniter was designed to accommodate the full LOX design flowrate, as well as a reduced GOX flowrate. It was reasoned that the initial LOX flow through the igniter would flash to GOX due to the inherent heat stored in the hardware, causing a reduced oxygen flowrate because of a choked, or sonic, flow condition through the injection elements. As LOX flow continued, the inherent heat of the test hardware would be removed and the hardware would chill-in, with the injected oxygen flow transitioning from cold GOX through two-phase flow to subcooled LOX. Pressure and temperature instrumentation permitted oxygen state points to be determined, and gas-side igniter chamber thermocouples provided chamber thermal profile characteristics. The cold flow chamber pressure (P(sub c)) for each test was determined and coupled with the igniter chamber diameter (D(sub c)) to calculate the characteristic quench parameter (P(sub c) x D(sub c)), which was plotted as a function of core mixture ratio, MR(sub c). Ignition limits were determined over a broad range of valve inlet conditions, and ignition was demonstrated with oxygen inlet conditions that ranged from subcooled 173 R LOX to 480 R GQX. Once ignited at cold GOX conditions, combustion was continuous as the hardware chilled in and the core mixture ratio transitioned from values near 1.0 to over 12.5.

Popp, Christopher G.↗
NASA NTRS1983

Computations of two-phase supersonic nozzle flows by a space-marching method

A practical method for efficiently analyzing the two-phase flow field in the supersonic region of an axisymmetric rocket nozzle is presented. The governing hyperbolic equations for the gas- and particle-phases are simultaneously integrated downstream using a space-marching procedure. Both one- and two-phase flow computations are performed for the JPL axisymmetric nozzle. The results show the nonequilibrium effects which are due to the existence of particles in the flow. The comparisons with the experimental data and other numerical solutions are made to examine the accuracy of the present method.

Fujii, K.↗
DOE OSTI2020

Interface capturing simulations of bubble population effects in PWR subchannels

As the computational power of high-performance computing (HPC) facilities grows, so too does the feasibility of using first principle based simulation to study turbulent two-phase flows within complex pressurized water reactor (PWR) geometries. Direct numerical simulation (DNS), integrated with an interface capturing method, allows for the collection of high-fidelity numerical data using advanced analysis techniques. The research presented here employs the massively parallel, finite-element based, unstructured mesh code, PHASTA, to simulate a set of two-phase bubbly flows through PWR subchannel geometries including auxiliary structures (spacer grids and mixing vanes). The main objective of the presented work is to analyze bubble dynamics and turbulence interactions at varying bubble concentrations to support the development of advanced two-phase flow closure models. Turbulent two-phase flows in PWR subchannels were simulated at hydraulic Reynolds numbers of 81,000 with bubble concentrations of 3%–15% by gas volume fraction (768–3928 resolved bubbles, respectively) and compared against a 1% void fraction case (262 bubbles) that had been previously simulated. The finite element mesh utilized for the study at higher bubble concentrations was composed of 1.55 billion elements, compared to the previous study which employed 1.11 billion elements, ensuring all turbulence scales and individual bubbles within the flow are fully resolved. For each case, the resolved initial bubble size was 0.65 mm in diameter (resolved with 25 grid points across the diameter). The simulations were analyzed to find flow features such as the mean velocity profile, bubble relative velocity and the effect of the bubbles on the turbulent conditions.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS1985

Evaluation of PHOENICS as a design analysis tool for SSME

The use of the computational fluid dynamics computer code PHOENICS was proposed as a design analysis tool for improving the Space Shuttle Main Engines (SSME). The PHOENICS is a general purpose computer code for the simulation of fluid-flow, heat-transfer and combustion processes arising in engineering, scientific research and environment. It analyzes one or two-phase flows, in one, two, or three dimensions. The application of PHOENICS to evaluate the performance of high-pressure gaseous-hydrogen filters is examined. The usefulness of the code and the ability of the code to handle successively more complex features of the device are demonstrated.

Ventrice, M. B.↗
NASA NTRS2023

Cryogenic Flow Boiling in Microgravity: Effects of Reduced Gravity on Two-Phase Fluid Physics and Heat Transfer

With the growing interest in space exploration, cryogenic technologies involving two-phase flow and heat transfer are in high demand to successfully procure advanced space applications such as fuel depots and nuclear thermal propulsion (NTP) systems for deep space missions. However, the unique and extreme thermal properties of cryogenic fluids introduce distinct flow boiling fluid physics and energy transport phenomena, which differ significantly from those observed with conventional fluids. Understanding the unique two-phase physics in cryogenic flow boiling remains an ongoing challenge. Furthermore, the lack of readily available microgravity cryogenic steady-state heat transfer data hinders the assessment of gravitational effects on cryogenic flow boiling. This study aims to elucidate the gravitational effects on two-phase fluid physics and heat transfer by conducting the first-ever experimental measurement of cryogenic flow boiling performance using a steady-state heated method in a reduced gravity environment. Parabolic flight experiments were performed to acquire both heat transfer measurements and high-speed video of interfacial behaviors, under varying gravity levels (microgravity, hypergravity, Lunar gravity, and Martian gravity). The experiments involved flow boiling of liquid nitrogen (LN 2 ) with a near-saturated inlet along a circular heated tube of dimensions 8.5-mm inner diameter and 680-mm heated length. The operating parameters varied are mass velocity of 398.3 - 1342.8 kg/m2s, inlet quality of -0.08 to -0.01, and inlet pressure of 413.68 - 689.48 kPa. Captured microgravity flow patterns range from bubbly to annular, all having vapor structures that are larger than those under higher gravity levels. Under microgravity, absence of buoyancy yields symmetrical vapor structures without flow stratification, laying a physical foundation for the distinct two-phase heat transfer trends during LN 2 flow boiling in microgravity. Transient data collected during the flight parabolas exhibited decreasing heated wall temperature as the aircraft transitioned from hypergravity to microgravity phases. The temperature variation indicated an enhancement in flow boiling heat transfer with decreasing gravity levels and a reduction with increasing gravity levels. The effect of reduced gravity on cryogenic flow boiling heat transfer coefficient (HTC) is discussed based on steady state heat transfer analysis. Seminal HTC correlations are evaluated against the measured microgravity HTC data, of which one is identified for superior accuracy in predicting microgravity data. Finally, a new HTC correlation is proposed to improve accuracy of microgravity predictions, yet there still exists room for further improvement with future terrestrial flow boiling experiments at different flow orientations relative to Earth gravity.

Microgravity↗
DOE OSTI2022

CTF Improved Drag Model and Flow Regime Transition Criteria

The demand for accurate prediction of two-phase flow behavior in a boiling water reactor (BWR) requires a comprehensive understanding of flow regime, void fraction, heat transfer, and pressure drop. The CTF subchannel code, which is used for the Thermal/Hydraulic (T/H) solution in the Consortium for Advanced Simulation of Light Water Reactors (CASL)-developed Virtual Environment for Reactor Application (VERA) core simulator, is being further developed for BWR applications. In support of this goal, the present work highlights some of the two-phase closure model developments towards improving the CTF void fraction prediction, especially for subcooled boiling. The drift-flux approach has been well-developed for upward dispersed two-phase flows and proven to be accurate in predicting void fraction in bubbly and slug flow regimes. In this work, these kinematic constitutive relations for the drift-flux velocity have been implemented into CTF to describe the interfacial drag of bubbly flow as an alternative to the existing model for better void fraction prediction. The success of these constitutive relations also relies on a good flow regime map that accounts for flow conditions and channel geometry. A more reliable flow regime transition criteria that account for the flow condition has also been implemented in this study for modeling the flow regime transition criteria. The newly implemented models are shown to give improved void fraction predictions in comparison to experimental data.

Hizoum, Belgacem↗
NASA NTRS1983

Modeling of transient two-component flow using a four-point implicit method

The four-point, centered implicit scheme that is extensively used in open channel flow simulation is shown to be applicable to rapid and slow pressure transient problems in conduits with nearly single phase and two-phase flows. It is only necessary to choose the proper weighting factor value, theta, of the Courant number. For rapid pressure transients such as waterhammer, the implicit method can yield reasonable results with limited numerical dispersion and attenuation if theta is only slightly greater than the critical value of 0.5. For slower pressure gradients in single and two-phase flows, reasonable numerical solutions may be achieved for Courant number values as high as 20.

Wiggert, D. C.↗