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Space tug thermal control equipment thermal requirements, characteristics, and constraints catalogue: Users guide

This manual details the input instructions to the data bank, and explanation of the program and its output. The data bank was developed in satisfaction of two of the study tasks, the equipment thermal requirement catalog and the equipment characteristics and constraints catalog. The data bank contains 109 components within space tug avionics system. Other systems were not included in the data bank due to the available information, however, with some program modification, other systems could be incorporated into the data bank program. The data bank was developed and checked out and is compatible with the Univac 1108, and the CDC 6500 operating systems. The data contained in the data bank is general in content with emphasis on the component thermal design. The data is applicable to any spacecraft program where the components contained in the data bank can be applied in satisfaction of the system and subsystem requirements.

Ward, T. L.↗

Convolutional encoding of self-dual codes

There exist almost complete convolutional encodings of self-dual codes, i.e., block codes of rate 1/2 with weights w, w = 0 mod 4. The codes are of length 8m with the convolutional portion of length 8m-2 and the nonsystematic information of length 4m-1. The last two bits are parity checks on the two (4m-1) length parity sequences. The final information bit complements one of the extended parity sequences of length 4m. Solomon and van Tilborg have developed algorithms to generate these for the Quadratic Residue (QR) Codes of lengths 48 and beyond. For these codes and reasonable constraint lengths, there are sequential decodings for both hard and soft decisions. There are also possible Viterbi-type decodings that may be simple, as in a convolutional encoding/decoding of the extended Golay Code. In addition, the previously found constraint length K = 9 for the QR (48, 24;12) Code is lowered here to K = 8.

Solomon, G.↗

A Preliminary Assessment of Physical Demand during Simulated Lunar Surface Extravehicular Activities

Returning to the moon requires many advances in current space technology. One major aspect of this development is a new exploration spacesuit (xEMU). Taking lessons learned from Apollo era suitsand the Extravehicular Mobility Unit (EMU) used on the International Space Station (ISS), xEMU will have increased mobility, dust mitigation, headspace, glove fit, and life support capabilities. Artemis astronauts in xEMU will complete a far more rigorous Extravehicular Activity (EVA) schedule than Apolloand ISS. Notably, metabolic rates during Apollo lunar EVA tasks were observed to be up to 50% lower than similar tasks performed in a ground analog environment under simulated partial gravity with newer suits. Therefore, understanding the physical demands of lunar surface exploration operations is criticalto ensuring best outcomes operating within the constraints of xEMU and planning for exploration EVA activities. This study utilized the Active Response Gravity Offload System (ARGOS) to simulate the lunar environment and continuously offload subjects to lunar gravity. Two male subjects completed two days of EVAs wearing the pressurized Mark III spacesuit, completing suit fit and mobility checks, as well as simulated lander operations, cable routing, crew rescue, geology, payload relocation, and traverse tasks in an end-to-end EVA (E2E) task block and standalone (SA) task blocks. We recorded continuous values of metabolic rate (MR) and heart rate (HR) to assess physical demand. During the E2E task block, subjects did not rest between tasks to simulate continuous effort from task to task, as in real EVAs. In comparison, subjects had a 5-minute break after each SA task block to allow for the metabolic rate and heart rate to return to baseline.MR values were categorized as low (≤ 700 BTU/hr), medium (700-1000 BTU/HR), and high (≥ 1000 BTU/hr), while HR values were categorized as low (≤150) and high (>150). During the 16 tasks in the E2E block, subjects averaged low MR in 6% of tasks, medium MR in 47% of tasks, and high MR in 47% of tasks. While MR was consistent between subjects, Subject 1 averaged low HR for 100% of these tasks, while Subject 2 averaged low HR in 44% of tasks. During the 23 tasks in the SA task blocks, subjects averaged low MR in 26% of tasks, medium MR in 52% of tasks, and high MR in 22% of tasks. Again, HR was different between subjects, with subject 1 averaging low HR in 100% of these tasks while subject 2 averaged low HR in 70%. Across all tasks in this study, subjects reached maximum MR and HR values during a 500m traverse at 30% grade in the E2E block (subject 1: 1747 BTU/hr, 150 BPM; subject 2: 1656 BTU/hr, 177 BPM).Understanding the physical demand to complete exploration EVA tasks will be instrumental to the future success of exploration spacesuit designs and missions. Further work in this study will be needed to characterize MR during exploration EVA tasks, including expanding the subject pool and testing new suit designs.

Taylor E Schlotman↗

Impact of Climatic Variability on Atmospheric Mass Distribution and GRACE-Derived Gravity Fields

During the period we calculated the atmospheric data sets related to its mass and angular momentum distribution. For mass, we determined the various harmonics from the NCEP-NCAR reanalysis, especially the low-order harmonics that are useful in studying the gravitation distribution as will be determined from the GRACE mission. Atmospheric mass is also related to the atmospheric loading on the solid Earth; we cooperated with scientists who needed the atmospheric mass information for understanding its contributions to the overall loading, necessary for vertical and horizontal coordinate estimation. We calculated atmospheric angular momentum from the NCEP-NCAR reanalyses and 4 operational meteorological centers, based on the motion (wind) terms and the mass (surface pressure) terms. These are associated with motions of the planet, including its axial component causing changes in the length of day, more related to the winds, and the equatorial component related to motions of the pole, more related to the mass. Tasks related to the ocean mass and angular momentum were added to the project as well. For these we have noted the ocean impact on motions of the pole as well as the torque mechanisms that relate the transfer of angular momentum between oceans and solid earth. The activities of the project may be summarized in the following first manuscript written in December 2002, for a symposium that Dr. Salstein attended on Geodynamics. We have continued to assess ocean angular momentum (OAM) quantities derived from bottom pressure and velocity fields estimated with our finite-difference barotropic (single layer) model. Three years of output (1993-95) from a run without any data constraints was compared to output from a corresponding run that was constrained by altimeter data using a Kalman filter and smoother scheme. Respective OAM time series were combined with corresponding atmospheric series and compared to observed polar motion. The constrained OAM series provided slightly better variance reduction than the unconstrained series. Analysis provided a check on the estimation scheme and pointed to further work to improve the determination of OAM using this method. A significant effort was also devoted to quantifying effects of uncertainties in high frequency winds on the mean and seasonal momentum exchange between atmosphere and oceans.

Salstein, David A.↗

New Development in NASA's Rodent Research Hardware for Conducting Long Duration Biomedical and Basic Research in Space

Animal models, particularly rodents, are the foundation of pre-clinical research to understand human diseases and evaluate new therapeutics, and play a key role in advancing biomedical discoveries both on Earth and in space. The National Research Councils Decadal survey emphasized the importance of expanding NASAs life sciences research to perform long duration, rodent experiments on the International Space Station (ISS). To accomplish this objective, flight hardware, operations, and science capabilities were developed at NASA Ames Research Center (ARC) to enhance science return for both commercial (CASIS) and government-sponsored rodent research. The Rodent Research program at NASA ARC has pioneered a new research capability on the International Space Station and has progressed toward translating research to the ISS utilizing commercial rockets, collaborating with academia and science industry, while training crewmembers to assist in performing research on orbit. Throughout phases of these missions, our practices, hardware and operations have evolved from tested to developed standards, and we are able to modify and customize our procedure and operations for mission specific requirements. The Rodent Research Habitat is capable of providing a living environment for animals on ISS according to standard animal welfare requirements. Using the cameras in the Habitat, the Rodent Research team has the ability to perform daily health checks on animals, and further analyze the collected videos for behavioral studies. A recent development of the Rodent Research hardware is inclusion of enrichment, to provide the animals the ability to rest and huddle. The Enrichment Hut is designed carefully for adult mice (up to 35 week old) within animal welfare, engineering, and operations constraints. The Hut is made out of the same stainless steel mesh as the cage interior, it has an ingress and an egress to allow animals move freely, and a hinge door to allow crewmembers remove the animals easily. The Rodent Research team has also developed Live Animal Return (LAR) capability, which will be implemented during Rodent Research-5 mission for the first time. The animals will be transported from the Habitat to a Transporter, which will return on the Dragon capsule and splashes down in the Pacific Ocean. Once SpaceX retrieves the Dragon, all powered payloads will be transferred to a SeaVan and transferred to the Long Beach pier. The NASA team then receives the transporter and delivers to a PI-designated laboratory within 120 mile radius of Long Beach. This is a significant improvement allowing researchers to examine animals within 72 hrs. of reentry or to conduct recovery experiments. Together, the hardware improvements and experience that the Rodent Research team has gained working with principal investigators and ISS crew to conduct complex experiments on orbit are expanding capabilities for long duration rodent research on the ISS to achieve both basic science and biomedical objectives.

ISS↗

Crew decision making under stress

Flight crews must make decisions and take action when systems fail or emergencies arise during flight. These situations may involve high stress. Full-missiion flight simulation studies have shown that crews differ in how effectively they cope in these circumstances, judged by operational errors and crew coordination. The present study analyzed the problem solving and decision making strategies used by crews led by captains fitting three different personality profiles. Our goal was to identify more and less effective strategies that could serve as the basis for crew selection or training. Methods: Twelve 3-member B-727 crews flew a 5-leg mission simulated flight over 1 1/2 days. Two legs included 4 abnormal events that required decisions during high workload periods. Transcripts of videotapes were analyzed to describe decision making strategies. Crew performance (errors and coordination) was judged on-line and from videotapes by check airmen. Results: Based on a median split of crew performance errors, analyses to date indicate a difference in general strategy between crews who make more or less errors. Higher performance crews showed greater situational awareness - they responded quickly to cues and interpreted them appropriately. They requested more decision relevant information and took into account more constraints. Lower performing crews showed poorer situational awareness, planning, constraint sensitivity, and coordination. The major difference between higher and lower performing crews was that poorer crews made quick decisions and then collected information to confirm their decision. Conclusion: Differences in overall crew performance were associated with differences in situational awareness, information management, and decision strategy. Captain personality profiles were associated with these differences, a finding with implications for crew selection and training.

Orasanu, J.↗

Towards Time Automata and Multi-Agent Systems

The design of reactive systems must comply with logical correctness (the system does what it is supposed to do) and timeliness (the system has to satisfy a set of temporal constraints) criteria. In this paper, we propose a global approach for the design of adaptive reactive systems, i.e., systems that dynamically adapt their architecture depending on the context. We use the timed automata formalism for the design of the agents' behavior. This allows evaluating beforehand the properties of the system (regarding logical correctness and timeliness), thanks to model-checking and simulation techniques. This model is enhanced with tools that we developed for the automatic generation of code, allowing to produce very quickly a running multi-agent prototype satisfying the properties of the model.

Hutzler, G.↗

Accumulate-Repeat-Accumulate-Accumulate-Codes

Inspired by recently proposed Accumulate-Repeat-Accumulate (ARA) codes [15], in this paper we propose a channel coding scheme called Accumulate-Repeat-Accumulate-Accumulate (ARAA) codes. These codes can be seen as serial turbo-like codes or as a subclass of Low Density Parity Check (LDPC) codes, and they have a projected graph or protograph representation; this allows for a high-speed iterative decoder implementation using belief propagation. An ARAA code can be viewed as a precoded Repeat-and-Accumulate (RA) code with puncturing in concatenation with another accumulator, where simply an accumulator is chosen as the precoder; thus ARAA codes have a very fast encoder structure. Using density evolution on their associated protographs, we find examples of rate-lJ2 ARAA codes with maximum variable node degree 4 for which a minimum bit-SNR as low as 0.21 dB from the channel capacity limit can be achieved as the block size goes to infinity. Such a low threshold cannot be achieved by RA or Irregular RA (IRA) or unstructured irregular LDPC codes with the same constraint on the maximum variable node degree. Furthermore by puncturing the accumulators we can construct families of higher rate ARAA codes with thresholds that stay close to their respective channel capacity thresholds uniformly. Iterative decoding simulation results show comparable performance with the best-known LDPC codes but with very low error floor even at moderate block sizes.

density evolution↗

A Flight Rule Checker for the LADEE Lunar Spacecraft

As part of the design of a space mission, an important part is the design of so-called flight rules. Flight rules express constraints on various parts and processes of the mission, that if followed, will reduce the risk of failure. One such set of flight rules constrain the format of command sequences regularly (e.g. daily) sent to the spacecraft to con- trol its next near term behavior. We present a high-level view of the automated flight rule checker Frc for checking command sequences sent to NASA’s LADEE Lunar mission spacecraft, used throughout its entire mission. A command sequence is in this case essentially a program (a sequence of commands) with no loops or conditionals, and it can there- fore be verified with a trace analysis tool. Frc is implemented using the TraceContract runtime verification tool, an internal Scala DSL for checking event sequences against “formal specifications”. The paper illustrates this untraditional use of runtime verification in a real con- text, with strong demands on the expressiveness and flexibility of the specification language, illustrating the advantages of an internal DSL.

Kurklu, Elif↗

A Flight Rule Checker for the LADEE Lunar Spacecraft

As part of the design of a space mission, an important part is the design of so-called flight rules. Flight rules express constraints on various parts and processes of the mission, that if followed, will reduce the risk of failure. One such set of flight rules constrain the format of command sequences regularly (e.g. daily) sent to the spacecraft to con-trol its next near term behavior. We present a high-level view of the automated flight rule checker FRC for checking command sequences sent to NASA’s LADEE Lunar mission spacecraft, used throughout its entire mission. A command sequence is in this case essentially a program (a sequence of commands) with no loops or conditionals, and it can there-fore be verified with a trace analysis tool. FRC is implemented using the TraceContract runtime verification tool, an internal Scala DSL for checking event sequences against “formal specifications”. The paper illustrates this untraditional use of runtime verification in a real con-text, with strong demands on the expressiveness and flexibility of the specification language, illustrating the advantages of an internal DSL.

LADEE↗

Lorentz Invariance Violation: the Latest Fermi Results and the GRB-AGN Complementarity

Because they are bright and distant, Gamma-ray Bursts (GRBs) have been used for more than a decade to test propagation of photons and to constrain relevant Quantum Gravity (QG) models in which the velocity of photons in vacuum can depend on their energy. With its unprecedented sensitivity and energy coverage, the Fermi satellite has provided the most constraining results on the QG energy scale so far. In this talk, the latest results obtained from the analysis of four bright GRBs observed by the Large Area Telescope will be reviewed. These robust results, cross-checked using three different analysis techniques set the limit on QG energy scale at E(sub QG,1) greater than 7.6 times the Planck energy for linear dispersion and E(sub QG,2) greater than 1.3 x 10(exp 11) gigaelectron volts for quadratic dispersion (95% CL). After describing the data and the analysis techniques in use, results will be discussed and confronted to latest constraints obtained with Active Galactic Nuclei.

Quantum gravity↗

An analysis of the effect of aeroassist maneuvers on orbital transfer vehicle performance

This paper summarizes a Langley Research Summer Scholars (LARSS) research project (Summer 1986) dealing with the topic of the effectiveness of aeroassist maneuvers to accomplish a change in the orbital inclination of an Orbital Transfer Vehicle (OTV). This task was subject to OTV design constraints, chief of which were the axial acceleration and the aerodynamic heating rate limits of the OTV. The use of vehicle thrust to replace lost kinetic energy and, thereby, to increase the maximum possible change in orbital inclination was investigated. A relation between time in the hover orbit and payload to LEO was established. The amount of plane change possible during this type of maneuver was checked for several runs and a possible thrusting procedure to increase the plane change and still get to LEO was suggested. Finally, the sensitivity of various target parameters to controllable independent variables was established, trades between the amount of control allowed, and payload to LEO suggested.

Murphy, Gregory O.↗

Optimal guidance law development for an advanced launch system

A regular perturbation analysis is presented. Closed-loop simulations were performed with a first order correction including all of the atmospheric terms. In addition, a method was developed for independently checking the accuracy of the analysis and the rather extensive programming required to implement the complete first order correction with all of the aerodynamic effects included. This amounted to developing an equivalent Hamiltonian computed from the first order analysis. A second order correction was also completed for the neglected spherical Earth and back-pressure effects. Finally, an analysis was begun on a method for dealing with control inequality constraints. The results on including higher order corrections do show some improvement for this application; however, it is not known at this stage if significant improvement will result when the aerodynamic forces are included. The weak formulation for solving optimal problems was extended in order to account for state inequality constraints. The formulation was tested on three example problems and numerical results were compared to the exact solutions. Development of a general purpose computational environment for the solution of a large class of optimal control problems is under way. An example, along with the necessary input and the output, is given.

Calise, Anthony J.↗

A Self-Stabilizing Distributed Clock Synchronization Protocol for Arbitrary Digraphs

This report presents a self-stabilizing distributed clock synchronization protocol in the absence of faults in the system. It is focused on the distributed clock synchronization of an arbitrary, non-partitioned digraph ranging from fully connected to 1-connected networks of nodes while allowing for differences in the network elements. This protocol does not rely on assumptions about the initial state of the system, other than the presence of at least one node, and no central clock or a centrally generated signal, pulse, or message is used. Nodes are anonymous, i.e., they do not have unique identities. There is no theoretical limit on the maximum number of participating nodes. The only constraint on the behavior of the node is that the interactions with other nodes are restricted to defined links and interfaces. We present an outline of a deductive proof of the correctness of the protocol. A model of the protocol was mechanically verified using the Symbolic Model Verifier (SMV) for a variety of topologies. Results of the mechanical proof of the correctness of the protocol are provided. The model checking results have verified the correctness of the protocol as they apply to the networks with unidirectional and bidirectional links. In addition, the results confirm the claims of determinism and linear convergence. As a result, we conjecture that the protocol solves the general case of this problem. We also present several variations of the protocol and discuss that this synchronization protocol is indeed an emergent system.

Malekpour, Mahyar R.↗

Demand Capacity Balancing at Vertiports for Initial Strategic Conflict Management of Urban Air Mobility Operations

Urban Air Mobility (UAM) is a new transportation concept that enables highly automated, cooperative, passenger or cargo-carrying air transportation services in and around urban areas. To achieve the high level of operational density and complexity desired by the UAM community, an airspace system that allows UAM operators to readily access and operate safely and efficiently in the airspace is needed. This airspace system will require air traffic management designed to reduce the risk of conflicts and loss of separation between UAM flights. In general, strategic conflict management is considered as the first layer of conflict management for safe flight operations to condition the traffic to reduce the need for airborne separation provision, the second layer of conflict management. Demand Capacity Balancing (DCB) is one of the concept components to achieve strategic conflict management. DCB strategically evaluates traffic demand and resource capacities to allow UAM operators to determine when, where and how they operate, while mitigating conflicting needs for airspace and vertiport capacity. DCB can be applied whenever UAM demand exceeds the capacity in airspace or at vertiports. As the UAM ecosystem evolves with advanced technologies and matured operational procedures, more complicated conflict management will likely be needed. In the current UAM ‘Concept of Operation (ConOps) 1.0’ operational stage defined by FAA, however, it will be meaningful to explore the demand capacity balancing at vertiports only, as an initial strategic conflict management approach for UAM operations because vertiport capacity seems to be a bottleneck of UAM traffic. For this research, we developed a demand-capacity imbalance detection and resolution service for UAM. This DCB service identifies the demand from operators and compares the demand to a given capacity at the shared resources (i.e., vertiports) over the upcoming time horizon which is divided into time bins having a constant interval. When a new flight plan is submitted, the algorithm embedded in the DCB service checks the available time bins based on the desired departure time and estimated arrival time at origin and destination vertiports, respectively. If the time bins for the originally desired times are already occupied by other flights (i.e., demand is at or above capacity), the algorithm finds the next available time bins for takeoff and landing and shifts the conflicting departure time to the earliest time that satisfies the capacity constraints at both origin and destination vertiports. The details of the algorithm will be described in the final manuscript. Figure 1 shows that the proposed DCB algorithm works well for a sample traffic scenario. In this example, a total of 144 flights, split between two operators, are planned over 2 hours, traveling 10 routes between five vertiports. In the heatmaps, the horizontal axis shows 12 time bins where each bin represents a 12-minute interval, and the vertical axis shows five vertiports. The number in each cell shows the number of operations, counting both departures and arrivals, at a specific vertiport in each time bin. For the given capacity of 2 operations/vertiport/bin, Figure 1 shows that the original demand sometimes exceeds the capacity, but the modified demand is reduced to the given capacity after resolving demand-capacity imbalances. When UAM flights are operated, it is expected that many practical issues would arise in the federated system architecture with multiple operators. UAM operators may experience a time synchronization issue due to communication delay between operator and vehicle. UAM vehicles would fly at different flight speeds, depending on vehicle models. Actual departure and arrival times can have large variations, compared to the schedule. The lead time from flight plan submission to desired departure time can vary by service type (e.g., regular shuttle service vs. on-demand service). Using the proposed DCB algorithm, we also investigated how the actual flight schedule and DCB performance are affected by these uncertainties such as unsynchronized times between operators, flight speed differences, lead time differences, and departure time errors. The final manuscript will include the background of this research work, the description of the DCB algorithm and its use cases with traffic scenarios. It will also provide the analytical results about the impact of various uncertainties that can occur in actual UAM operations on the DCB at vertiports, in terms of demand distribution changes, number of simultaneous operations, and delay propagation.

Urban Air Mobility↗

Multivariate Error Covariance Estimates by Monte-Carlo Simulation for Assimilation Studies in the Pacific Ocean

One of the most difficult aspects of ocean state estimation is the prescription of the model forecast error covariances. The paucity of ocean observations limits our ability to estimate the covariance structures from model-observation differences. In most practical applications, simple covariances are usually prescribed. Rarely are cross-covariances between different model variables used. Here a comparison is made between a univariate Optimal Interpolation (UOI) scheme and a multivariate OI algorithm (MvOI) in the assimilation of ocean temperature. In the UOI case only temperature is updated using a Gaussian covariance function and in the MvOI salinity, zonal and meridional velocities as well as temperature, are updated using an empirically estimated multivariate covariance matrix. Earlier studies have shown that a univariate OI has a detrimental effect on the salinity and velocity fields of the model. Apparently, in a sequential framework it is important to analyze temperature and salinity together. For the MvOI an estimation of the model error statistics is made by Monte-Carlo techniques from an ensemble of model integrations. An important advantage of using an ensemble of ocean states is that it provides a natural way to estimate cross-covariances between the fields of different physical variables constituting the model state vector, at the same time incorporating the model's dynamical and thermodynamical constraints as well as the effects of physical boundaries. Only temperature observations from the Tropical Atmosphere-Ocean array have been assimilated in this study. In order to investigate the efficacy of the multivariate scheme two data assimilation experiments are validated with a large independent set of recently published subsurface observations of salinity, zonal velocity and temperature. For reference, a third control run with no data assimilation is used to check how the data assimilation affects systematic model errors. While the performance of the UOI and MvOI is similar with respect to the temperature field, the salinity and velocity fields are greatly improved when multivariate correction is used, as evident from the analyses of the rms differences of these fields and independent observations. The MvOI assimilation is found to improve upon the control run in generating the water masses with properties close to the observed, while the UOI failed to maintain the temperature and salinity structure.

Borovikov, Anna↗

Physical Evaluation of GPM DPR Single- and Dual-Wavelength Algorithms

A physical evaluation of the rain profiling retrieval algorithms for the Dual-frequency Precipitation Radar (DPR) aboard the Global Precipitation Measurement (GPM) core satellite is carried out by applying them to the hydrometeor profiles generated from measured raindrop size distributions (DSD). The DSD-simulated radar profiles are used as input to the algorithms, and their estimates of hydrometeors' parameters are compared with the same quantities derived directly from the DSD data (or truth). The retrieval accuracy is assessed by the degree to which the estimates agree with the truth. To check the validity and robustness of the retrievals, the profiles are constructed for cases ranging from fully correlated (or uniform) to totally uncorrelated DSDs along the columns. Investigation into the sensitivity of the retrieval results to the model assumptions is made to characterize retrieval uncertainties and identify error sources. Comparisons between the single- and dual-wavelength algorithm performance are carried out with either a single- or dual-wavelength constraint of the path integral or differential path integral attenuation. The results suggest that the DPR dual-wavelength algorithm generally provides accurate range-profiled estimates of rainfall rate and mass-weighted diameter with the dual-wavelength estimates superior in accuracy to those from the single-wavelength retrievals.

Liang Liao↗

Variable Step Integration Coupled with the Method of Characteristics Solution for Water-Hammer Analysis, A Case Study

One-dimensional water-hammer modeling involves the solution of two coupled non-linear hyperbolic partial differential equations (PDEs). These equations result from applying the principles of conservation of mass and momentum to flow through a pipe, and usually the assumption that the speed at which pressure waves propagate through the pipe is constant. In order to solve these equations for the interested quantities (i.e. pressures and flow rates), they must first be converted to a system of ordinary differential equations (ODEs) by either approximating the spatial derivative terms with numerical techniques or using the Method of Characteristics (MOC). The MOC approach is ideal in that no numerical approximation errors are introduced in converting the original system of PDEs into an equivalent system of ODEs. Unfortunately this resulting system of ODEs is bound by a time step constraint so that when integrating the equations the solution can only be obtained at fixed time intervals. If the fluid system to be modeled also contains dynamic components (i.e. components that are best modeled by a system of ODEs), it may be necessary to take extremely small time steps during certain points of the model simulation in order to achieve stability and/or accuracy in the solution. Coupled together, the fixed time step constraint invoked by the MOC, and the occasional need for extremely small time steps in order to obtain stability and/or accuracy, can greatly increase simulation run times. As one solution to this problem, a method for combining variable step integration (VSI) algorithms with the MOC was developed for modeling water-hammer in systems with highly dynamic components. A case study is presented in which reverse flow through a dual-flapper check valve introduces a water-hammer event. The predicted pressure responses upstream of the check-valve are compared with test data.

Turpin, Jason B.↗