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DOE OSTI2023

Seeing through noise in power laws

Despite widespread claims of power laws across the natural and social sciences, evidence in data is often equivocal. Modern data and statistical methods reject even classic power laws such as Pareto’s law of wealth and the Gutenberg–Richter law for earthquake magnitudes. We show that the maximum-likelihood estimators and Kolmogorov–Smirnov (K-S) statistics in widespread use are unexpectedly sensitive to ubiquitous errors in data such as measurement noise, quantization noise, heaping and censorship of small values. This sensitivity causes spurious rejection of power laws and biases parameter estimates even in arbitrarily large samples, which explains inconsistencies between theory and data. We show that logarithmic binning by powers of λ > 1 attenuates these errors in a manner analogous to noise averaging in normal statistics and that λ thereby tunes a trade-off between accuracy and precision in estimation. Binning also removes potentially misleading within-scale information while preserving information about the shape of a distribution over powers of λ, and we show that some amount of binning can improve sensitivity and specificity of K-S tests without any cost, while more extreme binning tunes a trade-off between sensitivity and specificity. We therefore advocate logarithmic binning as a simple essential step in power-law inference.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1988

Active control law synthesis for flexible aircraft

An application of an active control law synthesis procedure is presented, to meet multiple design requirements for a flexible aircraft modeled by a large order state space system. In this synthesis procedure, a linear quadratic Gaussian type cost function is minimized by updating the free parameters of the control law, while satisfying a set of constraints on the design loads, responses and stability margins. Analytical expressions for gradients of the cost function and the constraints, with respect to the control law design variables are used to facilitate rapid numerical convergence. These gradients can also be used for sensitivity study. A stable classical control law as well as an estimator-based full or reduced order control law can be modified, in order to meet individual root-mean-square response limitations as well as minimum singular value restrictions. Both analog and digital control laws can be optimized. Low order, robust control laws were synthesized for flutter suppression of a flexible aircraft.

Mukhopadhyay, V.↗

NASA NTRS2014

Constructing the "Best" Reliability Data for the Job

Modern business and technical decisions are based on the results of analyses. When considering assessments using "reliability data", the concern is how long a system will continue to operate as designed. Generally, the results are only as good as the data used. Ideally, a large set of pass/fail tests or observations to estimate the probability of failure of the item under test would produce the best data. However, this is a costly endeavor if used for every analysis and design. Developing specific data is costly and time consuming. Instead, analysts rely on available data to assess reliability. Finding data relevant to the specific use and environment for any project is difficult, if not impossible. Instead, we attempt to develop the "best" or composite analog data to support our assessments. One method used incorporates processes for reviewing existing data sources and identifying the available information based on similar equipment, then using that generic data to derive an analog composite. Dissimilarities in equipment descriptions, environment of intended use, quality and even failure modes impact the "best" data incorporated in an analog composite. Once developed, this composite analog data provides a "better" representation of the reliability of the equipment or component can be used to support early risk or reliability trade studies, or analytical models to establish the predicted reliability data points. Data that is more representative of reality and more project specific would provide more accurate analysis, and hopefully a better final decision.

DeMott, D. L.↗

NASA NTRS2014

Constructing the Best Reliability Data for the Job

Kleinhammer, R. K.↗

NASA NTRS2013

SpaceCube Version 1.5

SpaceCube 1.5 is a high-performance and low-power system in a compact form factor. It is a hybrid processing system consisting of CPU (central processing unit), FPGA (field-programmable gate array), and DSP (digital signal processor) processing elements. The primary processing engine is the Virtex- 5 FX100T FPGA, which has two embedded processors. The SpaceCube 1.5 System was a bridge to the SpaceCube 2.0 and SpaceCube 2.0 Mini processing systems. The SpaceCube 1.5 system was the primary avionics in the successful SMART (Small Rocket/Spacecraft Technology) Sounding Rocket mission that was launched in the summer of 2011. For SMART and similar missions, an avionics processor is required that is reconfigurable, has high processing capability, has multi-gigabit interfaces, is low power, and comes in a rugged/compact form factor. The original SpaceCube 1.0 met a number of the criteria, but did not possess the multi-gigabit interfaces that were required and is a higher-cost system. The SpaceCube 1.5 was designed with those mission requirements in mind. The SpaceCube 1.5 features one Xilinx Virtex-5 FX100T FPGA and has excellent size, weight, and power characteristics [4×4×3 in. (approx. = 10×10×8 cm), 3 lb (approx. = 1.4 kg), and 5 to 15 W depending on the application]. The estimated computing power of the two PowerPC 440s in the Virtex-5 FPGA is 1100 DMIPS each. The SpaceCube 1.5 includes two Gigabit Ethernet (1 Gbps) interfaces as well as two SATA-I/II interfaces (1.5 to 3.0 Gbps) for recording to data drives. The SpaceCube 1.5 also features DDR2 SDRAM (double data rate synchronous dynamic random access memory); 4- Gbit Flash for storing application code for the CPU, FPGA, and DSP processing elements; and a Xilinx Platform Flash XL to store FPGA configuration files or application code. The system also incorporates a 12 bit analog to digital converter with the ability to read 32 discrete analog sensor inputs. The SpaceCube 1.5 design also has a built-in accelerometer. In addition, the system has 12 receive and transmit RS- 422 interfaces for legacy support. The SpaceCube 1.5 processor card represents the first NASA Goddard design in a compact form factor featuring the Xilinx Virtex- 5. The SpaceCube 1.5 incorporates backward compatibility with the Space- Cube 1.0 form factor and stackable architecture. It also makes use of low-cost commercial parts, but is designed for operation in harsh environments.

Geist, Alessandro↗

NASA NTRS2002

Process Cost Modeling for Multi-Disciplinary Design Optimization

For early design concepts, the conventional approach to cost is normally some kind of parametric weight-based cost model. There is now ample evidence that this approach can be misleading and inaccurate. By the nature of its development, a parametric cost model requires historical data and is valid only if the new design is analogous to those for which the model was derived. Advanced aerospace vehicles have no historical production data and are nowhere near the vehicles of the past. Using an existing weight-based cost model would only lead to errors and distortions of the true production cost. This report outlines the development of a process-based cost model in which the physical elements of the vehicle are costed according to a first-order dynamics model. This theoretical cost model, first advocated by early work at MIT, has been expanded to cover the basic structures of an advanced aerospace vehicle. Elemental costs based on the geometry of the design can be summed up to provide an overall estimation of the total production cost for a design configuration. This capability to directly link any design configuration to realistic cost estimation is a key requirement for high payoff MDO problems. Another important consideration in this report is the handling of part or product complexity. Here the concept of cost modulus is introduced to take into account variability due to different materials, sizes, shapes, precision of fabrication, and equipment requirements. The most important implication of the development of the proposed process-based cost model is that different design configurations can now be quickly related to their cost estimates in a seamless calculation process easily implemented on any spreadsheet tool. In successive sections, the report addresses the issues of cost modeling as follows. First, an introduction is presented to provide the background for the research work. Next, a quick review of cost estimation techniques is made with the intention to highlight their inappropriateness for what is really needed at the conceptual phase of the design process. The First-Order Process Velocity Cost Model (FOPV) is discussed at length in the next section. This is followed by an application of the FOPV cost model to a generic wing. For designs that have no precedence as far as acquisition costs are concerned, cost data derived from the FOPV cost model may not be accurate enough because of new requirements for shape complexity, material, equipment and precision/tolerance. The concept of Cost Modulus is introduced at this point to compensate for these new burdens on the basic processes. This is treated in section 5. The cost of a design must be conveniently linked to its CAD representation. The interfacing of CAD models and spreadsheets containing the cost equations is the subject of the next section, section 6. The last section of the report is a summary of the progress made so far, and the anticipated research work to be achieved in the future.

Bao, Han P.↗

NASA NTRS2002

Predicting Production Costs for Advanced Aerospace Vehicles

Bao, Han P.↗

NASA NTRS2004

Advanced Technology Lifecycle Analysis System (ATLAS)

Developing credible mass and cost estimates for space exploration and development architectures require multidisciplinary analysis based on physics calculations, and parametric estimates derived from historical systems. Within the National Aeronautics and Space Administration (NASA), concurrent engineering environment (CEE) activities integrate discipline oriented analysis tools through a computer network and accumulate the results of a multidisciplinary analysis team via a centralized database or spreadsheet Each minute of a design and analysis study within a concurrent engineering environment is expensive due the size of the team and supporting equipment The Advanced Technology Lifecycle Analysis System (ATLAS) reduces the cost of architecture analysis by capturing the knowledge of discipline experts into system oriented spreadsheet models. A framework with a user interface presents a library of system models to an architecture analyst. The analyst selects models of launchers, in-space transportation systems, and excursion vehicles, as well as space and surface infrastructure such as propellant depots, habitats, and solar power satellites. After assembling the architecture from the selected models, the analyst can create a campaign comprised of missions spanning several years. The ATLAS controller passes analyst specified parameters to the models and data among the models. An integrator workbook calls a history based parametric analysis cost model to determine the costs. Also, the integrator estimates the flight rates, launched masses, and architecture benefits over the years of the campaign. An accumulator workbook presents the analytical results in a series of bar graphs. In no way does ATLAS compete with a CEE; instead, ATLAS complements a CEE by ensuring that the time of the experts is well spent Using ATLAS, an architecture analyst can perform technology sensitivity analysis, study many scenarios, and see the impact of design decisions. When the analyst is satisfied with the system configurations, technology portfolios, and deployment strategies, he or she can present the concepts to a team, which will conduct a detailed, discipline-oriented analysis within a CEE. An analog to this approach is the music industry where a songwriter creates the lyrics and music before entering a recording studio.

O'Neil, Daniel A.↗

DOE OSTI2023

Analysis of Carbon Capture Retrofits for Cement Plants

The objective of this study is to provide an estimate of the cost to capture CO 2 in retrofit applications at cement plants. The cement plant configurations considered in this study include natural gas and solid fuel (coal and coke) cases, and both wet- and dry kiln-fed plant designs, however the base cement plants in this study were not evaluated other than characterization of their kiln off-gas stream and high-level quantification of heat integration potential at the existing plant. In each case, the base cement plant produces 1.5 M tonnes per year of finished cement, assuming 91.4 percent clinker content. This analysis includes a 10 percent retrofit cost increase for process integration and low grade heat recovery, when compared to the analogous non-heat integration case. Heat integration is considered as a potential offset to capture system heating demands (i.e., as a percentage reduction). Recovery and reuse of excess heat from the base cement plant can provide economic benefits—primarily by reducing the need to purchase supplemental natural gas for CO 2 solvent regeneration—but any process improvement must be great enough to overcome the cost increases (i.e., capital and O&M) necessary to realize those benefits. With heat integration potential of 10 percent and 30 percent, that benefit wasn’t significant enough to offset the increase in capital and operating costs, and a cost of capture increase was observed based on the assumptions in this report. Benefits of heat integration potential may be realized when natural gas prices are higher, as demonstrated in the sensitivity to natural gas price where crossover points exist between heat integration cases and their respective non-heat integration cases. Four of the base cases were further evaluated to explore the cost implications of deeper levels of gas pre-treatment to remove oxides of nitrogen (NOx) and oxides of sulfur (SOx) from the kiln off-gas stream prior to CO 2 capture, purification, and compression. The results of these additional case analyses showed a 7.4–18.8 percent increase in cost of capture over the respective base case (i.e., analogous cases without SOx/NOx removal).

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2026

Variational Optical Phase Learning on a Continuous-Variable Quantum Compiler

Quantum process learning is a fundamental primitive that draws inspiration from machine learning with the goal of better studying the dynamics of quantum systems. One approach to quantum process learning is quantum compilation, whereby an analog quantum operation is digitized by compiling it into a series of basic gates. While there has been significant focus on quantum compiling for discrete-variable systems, the continuous-variable (CV) framework has received comparatively less attention. We present an experimental implementation of a CV quantum compiler that uses two-mode squeezed light to learn a Gaussian unitary operation. We demonstrate the compiler by learning a parameterized linear phase unitary through the use of target and control phase unitaries to demonstrate a factor of 5.4 increase in the precision of the phase estimation and a 3.6-fold acceleration in the time-to-solution metric when leveraging quantum resources. We further show how our approach can be extended to higher-dimensional compilation tasks. Our results are enabled by the tunable control of our cost landscape via variable squeezing, thus providing a critical framework to simultaneously increase precision and reduce time-to-solution.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1975

Shuttle propellant loading instrumenation development

A continuous capacitance sensor was developed and an analog signal conditioner was evaluated to demonstrate the acceptability of these items for use in the space shuttle propellant loading system. An existing basic sensor concept was redesigned to provide capability for cryogenic operation, to improve performance, and to minimize production costs. Sensor development verification consisted of evaluation of sensor linearity, cryogenic performance, and stability during vibration. The signal conditioner evaluation consisted mainly of establishing the effects of the variations in temperature and cable parameters and evaluating the stability. A sensor linearity of 0.04 in. was achieved over most of the sensor length. The sensor instability caused by vibration was 0.04 percent. The cryogenic performance data show a maximum instability of 0.19 percent at liquid hydrogen temperature; a theoretical calibration can be computed a within 1 percent. The signal conditioner evaluation showed that, with temperature compensation, all error sources typically contribute much less than 1 percent. An estimate of the accuracy achievable with the sensor and signal conditioner shows an rss estimate of 0.75 in. for liquid oxygen and 1.02 in. for liquid hydrogen. These are approximately four times better than the shuttle requirements. Comparison of continuous sensor and discrete sensor performance show the continuous sensor to be significantly better when there is surface activity due to sloshing, boiling, or other disturbances.

Hamlet, J.↗

DOE OSTI2022

Quantitative evaluation of common cause failures in high safety-significant safety-related digital instrumentation and control systems in nuclear power plants

Digital instrumentation and control (DI&C) systems at nuclear power plants (NPPs) have many advantages over analog systems. They are proven to be more reliable, cheaper, and easier to maintain given obsolescence of analog components. However, they also pose new engineering and technical challenges, such as possibility of common cause failures (CCFs) unique to digital systems. Here this paper proposes a Platform for Risk Assessment of DI&C (PRADIC) that is developed by Idaho National Laboratory (INL). A methodology for evaluation of software CCFs in high safety-significant safety-related DI&C systems of NPPs was developed as part of the framework. The framework integrates three stages of a typical risk assessment—qualitative hazard analysis and quantitative reliability and consequence analyses. The quantified risks compared with respective acceptance criteria provide valuable insights for system architecture alternatives allowing design optimization in terms of risk reduction and cost savings. A comprehensive case study performed to demonstrate the framework's capabilities is documented here in this paper. Results show that the PRADIC is a powerful tool capable to identify potential digital-based CCFs, estimate their probabilities, and evaluate their impacts on system and plant safety.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2024

Helium metastable density determination in the COST reference source by absolutely calibrated optical emission spectroscopy

Helium metastable densities in the COST Reference Microplasma Jet are estimated for a variety of He/N2 admixtures and dissipated powers by applying a collisional-radiative model to absolutely calibrated optical emission spectroscopy measurements. This is accomplished by delineating the excitation mechanisms that result in the N2(C–B) and N2+(B–X) emission bands, the latter of which is strongly coupled to the presence of helium metastables. A number of other plasma parameters are established and discussed for each operating condition including the electron energy distribution function, reduced electric field, rate constants, and electron density. With these parameters, the reaction rates for the primary ionization pathways are also calculated, emphasizing the importance of helium metastables for discharge sustainment. Good agreement with the existing literature is found for most plasma parameters and for helium metastable densities, in particular. A clear [N2]−1 relationship between the nitrogen concentration and density of helium metastables is demonstrated, as has been identified in previous studies in analogous atmospheric pressure plasma jets. This validates the efficacy of this optical technique for determining helium metastable densities and establishes it as a viable, and in many cases, more accessible alternative to other means of quantifying helium metastables in low-temperature plasmas.

Physics↗

NASA NTRS2014

Honeywell Cascade Distiller System Performance Testing Interim Results

The ability to recover and purify water through physiochemical processes is crucial for realizing long-term human space missions, including both planetary habitation and space travel. Because of their robust nature, distillation systems have been actively pursued as one of the technologies for water recovery. The Cascade Distillation System (CDS) is a vacuum rotary distillation system with potential for greater reliability and lower energy costs than existing distillation systems. The CDS was previously under development through Honeywell and NASA. In 2009, an assessment was performed to collect data to support down-selection and development of a primary distillation technology for application in a lunar outpost water recovery system. Based on the results of this testing, an expert panel concluded that the CDS showed adequate development maturity, TRL-4, together with the best product water quality and competitive weight and power estimates to warrant further development. The Advanced Exploration Systems (AES) Water Recovery Project (WRP) worked to address weaknesses identified by The Panel; namely bearing design and heat pump power efficiency. Testing at the NASA-JSC Advanced Exploration System Water Laboratory (AES Water Lab) using a prototype Cascade Distillation Subsystem (CDS) wastewater processor (Honeywell International, Torrance, Calif.) with test support equipment and control system developed by Johnson Space Center was performed to evaluate performance of the system with the upgrades. The CDS will also have been challenged with ISS analog waste streams and a subset of those being considered for Exploration architectures. This paper details interim results of the AES WRP CDS performance testing.

Callahan, Michael R.↗

NASA NTRS2005

Advanced Method to Estimate Fuel Slosh Simulation Parameters

The nutation (wobble) of a spinning spacecraft in the presence of energy dissipation is a well-known problem in dynamics and is of particular concern for space missions. The nutation of a spacecraft spinning about its minor axis typically grows exponentially and the rate of growth is characterized by the Nutation Time Constant (NTC). For launch vehicles using spin-stabilized upper stages, fuel slosh in the spacecraft propellant tanks is usually the primary source of energy dissipation. For analytical prediction of the NTC this fuel slosh is commonly modeled using simple mechanical analogies such as pendulums or rigid rotors coupled to the spacecraft. Identifying model parameter values which adequately represent the sloshing dynamics is the most important step in obtaining an accurate NTC estimate. Analytic determination of the slosh model parameters has met with mixed success and is made even more difficult by the introduction of propellant management devices and elastomeric diaphragms. By subjecting full-sized fuel tanks with actual flight fuel loads to motion similar to that experienced in flight and measuring the forces experienced by the tanks these parameters can be determined experimentally. Currently, the identification of the model parameters is a laborious trial-and-error process in which the equations of motion for the mechanical analog are hand-derived, evaluated, and their results are compared with the experimental results. The proposed research is an effort to automate the process of identifying the parameters of the slosh model using a MATLAB/SimMechanics-based computer simulation of the experimental setup. Different parameter estimation and optimization approaches are evaluated and compared in order to arrive at a reliable and effective parameter identification process. To evaluate each parameter identification approach, a simple one-degree-of-freedom pendulum experiment is constructed and motion is induced using an electric motor. By applying the estimation approach to a simple, accurately modeled system, its effectiveness and accuracy can be evaluated. The same experimental setup can then be used with fluid-filled tanks to further evaluate the effectiveness of the process. Ultimately, the proven process can be applied to the full-sized spinning experimental setup to quickly and accurately determine the slosh model parameters for a particular spacecraft mission. Automating the parameter identification process will save time, allow more changes to be made to proposed designs, and lower the cost in the initial design stages.

Schlee, Keith↗

NASA NTRSDate not supplied

Toward The Development of Hailstorm Climatologies Derived From Reanalyses and Infared/Passive Microwave Satellite Imagers

Geostationary satellite imagers, such as those of the Geostationary Operational Environmental Satellite (GOES) and Meteosat series, provide both historical and near-real-time observations of cloud top patterns that are commonly associated with severe convection. Environmental conditions favorable for severe weather are thought to be represented well by reanalyses. Predicting exactly where convection and costly storm hazards like hail will occur using models or satellite imagery alone, however, is extremely challenging. The multivariate combination of satellite-observed cloud patterns with reanalysis environmental parameters, linked to United States Next Generation Weather Radar- (NEXRAD-) estimated Maximum Expected Size of Hail (MESH) using a deep neural network (DNN), enables estimation of potentially severe hail likelihood for any observed storm cell. These estimates are specifically designed to make hail likelihood distinctions based on satellite-indicated points of deep convection within environments favorable for storm development. We seek an approach that can be used to estimate climatological hailstorm frequency and risk throughout the historical satellite data record. This presentation demonstrates that statistical distributions of convective parameters from satellite and reanalysis show separation between non-severe/severe hailstorm classes for predictors including overshooting cloud top temperature and area characteristics, convective available potential energy, vertical wind shear, 500 hPa temperature, mid-level lapse rate, precipitable water, and convective inhibition. These complex, multivariate predictor relationships are exploited within a DNN to produce a hail likelihood metric with a critical success index of 0.504 and Heidke skill score of 0.403, which is exceptional among recent analogous hail studies. Furthermore, applications of the DNN to select case studies demonstrate good qualitative agreement between hail likelihood and MESH. These hail classifications are aggregated across an 11-year GOES-12/13 image database to derive a hail frequency and severity climatology, which denotes the Central Plains, the Midwest, and northwestern Mexico as being the most hail-prone regions within the domain studied. Opportunities for training and applying DNN-based hailstorm predictions to recently developed GOES-8/10/12/13/16 and Meteosat Second Generation convective storm detection and characterization climatologies over South America and South Africa, respectively, will also be presented.

Kristopher Bedka↗

DOE OSTI2022

Wastewater Recycling Using a Hygroscopic Cooling System

This project by the Energy & Environmental Research Center (EERC), Baltimore Aircoil Company (BAC), and Great River Energy (GRE) evaluated the concept of recycling wastewater at a coal-fired power plant using a hygroscopic cooling system, which is an evaporative cooling technology analogous to conventional cooling towers, except that sparingly soluble dissolved solids are precipitated and removed as waste solids instead of purging them with a liquid blowdown stream. This technology can maximize the use of plant makeup water by obtaining useful evaporative cooling from wastewater while minimizing the volume of wastes needing disposal. Experimental activities were conducted in two phases, a laboratory-based evaluation of the properties of wastewater from the host site, GRE’s Coal Creek Station near Underwood, North Dakota, and a field test of a small pilot hygroscopic cooling system at the host site power plant. Findings from the laboratory study informed the design of the pilot system and the system’s field test performance served as the basis for a techno-economic analysis (TEA) of the hygroscopic recycling concept. At the preferred operating conditions identified during the TEA, the wet-bulb approach temperature of the tower was 7.3°C (13°F) and the volume of blowdown produced by the plant was reduced to 5.4% of its incoming volume. Waste solids produced during field testing were classified as nonhazardous waste based on the measured hazardous element content and evaluation of their leaching potential. However, to qualify as a solid for landfill disposal i.e., as determined by the U.S. Environmental Protection Agency’s paint filter test, it appears that a dewatering step beyond hydrocyclone separation is needed. The baseline levelized cost of wastewater disposal (LCWD) for hygroscopic wastewater recycling was estimated to be $\$ $3.69–$\$ $3.72 per m3 of plant blowdown. Capital cost was estimated to contribute over 54% to the LCWD, and parameters that impact capital cost such as the heat exchange coil material of construction and the tower’s wet-bulb approach temperature were identified as having the greatest impact on overall LCWD. A LCWD estimate prepared for the same application but using thermomechanical brine evaporation was almost 40% higher than that calculated for hygroscopic cooling, despite recovering distilled-quality water for reuse, while the LCWD for disposal-only, deep well injection was estimated to be 30% lower compared to hygroscopic wastewater recycling.

01 COAL, LIGNITE, AND PEAT↗

NASA NTRS2015

On the Effect of Rigid Swept Surface Waves on Turbulent Drag

Passive turbulent drag reduction techniques are of interest as a cost effective means to improve air vehicle fuel consumption. In the past, rigid surface waves slanted at an angle from the streamwise direction were deemed ineffective to reduce skin friction drag due to the pressure drag that they generate. A recent analysis seeking similarities to the spanwise shear stress generated by spatial Stokes layers suggested that there may be a range of wavelength, amplitude, and orientation in which the wavy surface would reduce turbulent drag. The present work explores, by experiments and Direct Numerical Simulations (DNS), the effect of swept wavy surfaces on skin friction and pressure drag. Plates with shallow and deep wave patterns were rapid-prototyped and tested using a drag balance in the 7x11 inch Low-Speed Wind Tunnel at the NASA LaRC Research Center. The measured drag o set between the wavy plates and the reference at plate is found to be within the experimental repeatability limit. Oil vapor flow measurements indicate a mean spanwise flow over the deep waves. The turbulent flow in channels with at walls, swept wavy walls and spatial Stokes spanwise velocity forcing was simulated at a friction Reynolds number of two hundred. The time-averaged and dynamic turbulent flow characteristics of the three channel types are compared. The drag obtained for the channel with shallow waves is slightly larger than for the at channel, within the range of the experiments. In the case of the large waves, the simulation over predicts the drag. The shortcomings of the Stokes layer analogy model for the estimation of the spanwise shear stress and drag are discussed.

Denison, M.↗