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NASA NTRS2023

Automated Fiber Placement Through Thickness Defect Stacking Optimization

In its 2022 commercial market outlook, Boeing forecasted an 80% increase in the global fleet through 2041 compared to 2019 pre-pandemic levels. This sharp rise in demand will drive pressure onto airframe manufacturers to ramp up production and find more efficient ways to design and manufacture airplanes. Complicating this challenge is the industry’s recent transformation from traditional metal-based airframes towards hybrid composite-metal aircraft. While composites have been used in aviation for decades, aircraft manufacturers are still struggling to design and manufacture quality parts at a high rate. Automated Fiber Placement (AFP) is one of the main manufacturing techniques used to produce large-scale composite parts. After a design has been created, a manufacturing strategy has to be developed based on the working material, part geometry, and machine capabilities. This process planning stage is essential to the AFP workflow and currently requires a high level of manual input from an experienced process planner. In an effort to automate and optimize this stage, the Computer Aided Process Planning (CAPP) module was developed. CAPP assists process planners in identifying optimal starting point location and layup strategy for each ply of a laminate. This Ply-Level Optimization (PLO) phase operates on the quantification of ply quality through predictable geometry-based defects such as gaps, overlaps, angle deviation, and steering. As you move from PLO to Laminate-Level Optimization (LLO) the design space grows exponentially, emphasizing the need for automated optimization. The work presented in this thesis expands CAPP’s functionality by comparing the planned fiber paths through the thickness of the laminate to mitigate stacked area defects and achieve an optimal laminate-level manufacturing strategy. Within CAPP, predicted gap and overlap defects are imported from Vericut Composites Programming (VCP) and then discretized to streamline the through-thickness comparison. Two objective functions are used to score different combinations of ply layup strategies based on defect stacking both globally and locally. Four combinatorial optimization algorithms were coupled with these objective functions to investigate the laminate-level manufacturing strategy design space and converge on the optimal plan. These algorithms were evaluated based on accuracy and efficiency through virtual testing on a complex tool surface. A separate LLO approach was developed to achieve near-optimal laminates in significantly less time. The end result is a software package which greatly reduces the required input from process planners, shortening the design-build cycle time and improving part quality.

AFP↗
NASA NTRS2023

Chemical Thermodynamics and the Mathematical Integration of Reaction Kinetics

Key advances in the development of numerical methods for non-reacting compressible flows have been enabled by translating physical requirements into concrete numerical guidelines, such as the satisfaction of entropy inequalities for shock-capturing techniques [Lax, Contributions to Nonlinear Functional Analysis (1971) 603-634]. In the present work, we present nonlinear numerical analysis tools that draw from Chemical Thermodynamics , the branch of Nonequilibrium Thermodynamics that deals with chemical reactions. Through Gibbs formalism, chemical thermodynamics provides a well-known theoretical expression for the chemical equilibrium constant of a reaction in terms of reduced chemical potentials. A less-known, yet extremely valuable result, due to [Krambeck, Arch. Ration. Mech. Anal. , 38 (1970) 317], states that when this expression is implemented, mass-action kinetic models are consistent with the dynamical prescriptions of the 2nd law of thermodynamics. For fixed-temperature ordinary differential equations modeling constant-volume reacting gas mixtures, this leads to a decreasing Helmholtz free energy. If the temperature is allowed to vary in accordance with conservation of energy (1st law), this leads to the statement of increasing entropy. These nonlinear prescriptions can, and should be, used to further develop temporal integration techniques for reaction kinetics. We demonstrate that Krambeck's result holds even when the equilibrium constants are approximated from data. We prove this result by constructing the implicit free energy and the implicit entropy inherent to a given approximation. This is first done for a 5-species, 17-reaction model problem for air. With this structure established, elements of discrete entropy-stability theory [Tadmor, Acta Numer. , 12 (2003) 451] are leveraged to examine the consistency of time-integration schemes with these prescriptions. Using chemical potentials, one can compute the respective contributions of the kinetics model and of the temporal scheme to free energy/entropy variations. We introduce a nonlinear-stable version of the Discontinuous-Galerkin (DG) scheme in time which shows robustness improvements over the standard linearly-stable version. Most notably, the maximum timestep that can be resolved with the nonlinearly-stable variant tends to grow with polynomial order, in contrast to the linearly-stable variant. We generalize our constructions to arbitrary systems of reversible chemical reactions, ultimately showing that the compressible reacting Euler system admits the opposite of the implicitly constructed thermodynamic entropy as a mathematical entropy . This lays important theoretical foundations towards robust scheme development [Harten, J. Comput. Phys. 49 (1983) 151-164].

STMD↗
NASA NTRS2006

Learning Grasp Context Distinctions that Generalize

Control-based approaches to grasp synthesis create grasping behavior by sequencing and combining control primitives. In the absence of any other structure, these approaches must evaluate a large number of feasible control sequences as a function of object shape, object pose, and task. This work explores a new approach to grasp synthesis that limits consideration to variations on a generalized localize-reach-grasp control policy. A new learning algorithm, known as schema structured learning, is used to learn which instantiations of the generalized policy are most likely to lead to a successful grasp in different problem contexts. Two experiments are described where Dexter, a bimanual upper torso, learns to select an appropriate grasp strategy as a function of object eccentricity and orientation. In addition, it is shown that grasp skills learned in this way can generalize to new objects. Results are presented showing that after learning how to grasp a small, representative set of objects, the robot's performance quantitatively improves for similar objects that it has not experienced before.

Platt, Robert↗
NASA NTRS1985

Simple equation to approximate the bidirectional reflectance from vegetative canopies and bare soil surfaces

A simple equation has been developed for describing the bidirectional reflectance of some vegetative canopies and bare soil surfaces. The equation describes directional reflectance as a function of zenith and azimuth view angles and solar azimuth angle. The equation works for simulated and field measured red and IR reflectance under clear sky conditions. Hemispherical reflectance can be calculated as a function of the simple equation coefficients by integrating the equation over the hemisphere of view angles. A single equation for estimating soil bidirectional reflectance was obtained using the relationships between solar zenith angles and the simple equation coefficients for medium and rough soil distributions. The equation has many useful applications such as providing a lower level boundary condition in complex plant canopy models and providing an additional tool for studying bidirectional effects on pointable sensors.

Walthall, C. L.↗
NASA NTRS2016

Development of a Refined Space Vehicle Rollout Forcing Function

For several decades, American manned spaceflight vehicles and the associated launch platforms have been transported from final assembly to the launch pad via a pre-launch phase called rollout. The rollout environment is rich with forced harmonics and higher order effects can be used for extracting structural dynamics information. To enable this utilization, processing tools are needed to move from measured and analytical data to dynamic metrics such as transfer functions, mode shapes, modal frequencies, and damping. This paper covers the range of systems and tests that are available to estimate rollout forcing functions for the Space Launch System (SLS). The specific information covered in this paper includes: the different definitions of rollout forcing functions; the operational and developmental data sets that are available; the suite of analytical processes that are currently in-place or in-development; and the plans and future work underway to solve two immediate problems related to rollout forcing functions. Problem 1 involves estimating enforced accelerations to drive finite element models for developing design requirements for the SLS class of launch vehicles. Problem 2 involves processing rollout measured data in near real time to understand structural dynamics properties of a specific vehicle and the class to which it belongs.

James, George↗
NASA NTRS2005

ESTIMATION OF THE SPACE SHUTTLE ROLLOUT FORCING FUNCTION

The Space Shuttle Vehicle is assembled in the Vertical Assembly Building (VAB) at Kennedy Space Flight Center in Florida. The Vehicle is stacked on a Mobile Launch Platform (MLP) that weighs eight million pounds. A Crawler Transporter (CT) then carries the MLP and the stacked vehicle (12 million pounds total weight) to the launch complex located 5 miles away. This operation is performed at 0.9 mph resulting in a 4.5-hour transport. A recent test was performed to monitor the dynamic environment that was produced during rollout. It was found that the rollout is a harmonic-rich dynamic environment that was previously not understood. This paper will describe work that has been performed to estimate the forcing function that is produced in the transportation process. The rollout analysis team has determined that there are two families of harmonics of the drive train, which excite the system as a function of CT speed. There are also excitation sources, which are random or narrow-band in frequency and are not a function of CT speed. This presentation will discuss the application of the Sum of Weighted Accelerations Technique (SWAT) to further refine this understanding by estimating the forces and moments at the center-of-mass.

James, George H., III↗
NASA NTRS2019

A comprehensive line spread function error for the Off-plane Grating Rocket Experiment

The Off-plane Grating Rocket Experiment (OGRE) is a soft X-ray grating spectrometer to be flown on a suborbital rocket. The payload is designed to obtain the highest-resolution soft X-ray spectrum of Capella to date with a resolution goal of R(lambda/delta lambda) > 2000 at select wavelengths in its 10 - 55 A bandpass of interest. The optical design of the spectrometer realizes a maximum resolution of R 5000 with all components performing optimally, in perfect alignment, and not considering in-flight pointing errors; however, performance errors, misalignments, and in-flight pointing errors work to degrade this performance. A comprehensive line spread function (LSF) error budget has been constructed for the spectrometer to identify errors contributing to the LSF, to determine how each affects the LSF, and to inform performance requirements and alignment tolerances for the spectrometer. In this document, the comprehensive LSF error budget for the OGRE spectrometer is presented and its implications are discussed.

Benjamin D Donovan↗
NASA NTRS2016

Flat H Redundant Frangible Joint Development

Orion and Commercial Crew Program (CCP) Partners have chosen to use frangible joints for certain separation events. The joints currently available are zero failure tolerant and will be used in mission safety applications. The goal is to further develop a NASA designed redundant frangible joint that will lower flight risk and increase reliability. FY16 testing revealed a successful design in subscale straight test specimens that gained efficiency and supports Orion load requirements. Approach / Innovation A design constraint is that the redundant joint must fit within the current Orion architecture, without the need for additional vehicle modification. This limitation required a design that changed the orientation of the expanding tube assemblies (XTAs), by rotating them 90deg from the standard joint configuration. The change is not trivial and affects the fracture mechanism and structural load paths. To address these changes, the design incorporates cantilevered arms on the break plate. The shock transmission and expansion of the XTA applies force to these arms and creates a prying motion to push the plate walls outward to the point of structural failure at the notched section. The 2014 test design revealed that parts could slip during functioning wasting valuable energy needed to separate the structure with only a single XTA functioning. Dual XTA functioning fully separated the assembly showing a discrepancy can be backed up with redundancy. Work on other fully redundant systems outside NASA is limited to a few patents that have not been subjected to functionality testing Design changes to prevent unwanted slippage (with ICA funding in 2015) showed success with a single XTA. The main goal for FY 2016 was to send the new Flat H RFJ to WSTF where single XTA test failures occurred back in 2014. The plan was to gain efficiency in this design by separating the Flat H RFJ with thicker ligaments with dimensions baselined in 2014. Other modifications included geometry changes to better disperse loads paths and to minimize air gaps. The design additionally added more structural strength to enhance the structural limits in static loads testing. The design also implemented a smoother load line through the assembly. Results / Knowledge Gained The new Flat H RFJ successfully fractured at WSTF with thicker ligaments and lower cord size. Where failure to separate occurred earlier, there is now excessive energy available for structural separation. The new challenge to provide some structural support to prevent secondary fracturing of the break plate remains to be completed. This future work is being funded by the JSC Engineering Directorate in 2017 to elevate the TRL on curved Flat H RFJs that configure with the Orion Service Panel Separation. Additional funding from JSC Engineering will provide new design testing to avoid secondary fracturing.

Brown, Chris↗
NASA NTRS2006

SpaceWire Plug and Play

The ability to rapidly deploy inexpensive satellites to meet tactical goals has become an important goal for military space systems. In fact, Operationally Responsive Space (ORS) has been in the spotlight at the highest levels. The Office of the Secretary of Defense (OSD) has identified that the critical next step is developing the bus standards and modular interfaces. Historically, satellite components have been constructed based on bus standards and standardized interfaces. However, this has not been done to a degree, which would allow the rapid deployment of a satellite. Advancements in plug-and-play (PnP) technologies for terrestrial applications can serve as a baseline model for a PnP approach for satellite applications. Since SpaceWire (SpW) has become a de facto standard for satellite high-speed (greater than 200Mbp) on-board communications, it has become important for SpW to adapt to this Plug and Play (PnP) environment. Because SpW is simply a bulk transport protocol and lacks built-in PnP features, several changes are required to facilitate PnP with SpW. The first is for Host(s) to figure out what the network looks like, i.e., how pieces of the network, routers and nodes, are connected together; network mapping, and to receive notice of changes to the network. The second is for the components connected to the network to be understood so that they can communicate. The first element, network topology mapping & change of status indication, is being defined (topic of this paper). The second element describing how components are to communicate has been defined by ARFL with the electronic data sheets known as XTEDS. The first element, network mapping, is recent activities performed by Air Force Research Lab (ARFL), Naval Research Lab (NRL), NASA and US industry (Honeywell, Clearwater, FL, and others). This work has resulted in the development of a protocol that will perform the lower level functions of network mapping and Change Of Status (COS) indication required by Plug 'n' Play over SpaceWire. This work will be presented to the SpaceWire working group for standardization under European Cooperation for Space Standardization (ECSS) and to obtain a permanent Protocol ID (see SpaceWire Protocol ID: What Does it Mean to You; IEEE Aerospace Conference 2006). The portion of the Plug 'n' Play protocol that will be described in this paper is how the Host(s) of a SpaceWire network map the network and detect additions and deletions of devices on a SpaceWire network.

Rakow, Glenn↗
NASA NTRS1993

Design of a portable powered seat lift

People suffering from degenerative hip or knee joints find sitting and rising from a seated position very difficult. These people can rely on large stationary chairs at home, but must ask others for assistance when rising from any other chair. An orthopedic surgeon identified to the MSFC Technology Utilization Office the need for development of a portable device that could perform a similar function to the stationary lift chairs. The MSFC Structural Development Branch answered the Technology Utilization Office's request for design of a portable powered seat lift. The device is a seat cushion that opens under power, lifting the user to near-standing positions. The largest challenge was developing a mechanism to provide a stable lift over the large range of motion needed, and fold flat enough to be comfortable to sit on. CAD 3-D modeling was used to generate complete drawings for the prototype, and a full-scale working model of the Seat lift was made based on the drawings. The working model is of low strength, but proves the function of the mechanism and the concept.

Weddendorf, Bruce↗
NASA NTRS1999

The Analysis and Design of Low Boom Configurations Using CFD and Numerical Optimization Techniques

The use of computational fluid dynamics (CFD) for the analysis of sonic booms generated by aircraft has been shown to increase the accuracy and reliability of predictions. CFD takes into account important three-dimensional and nonlinear effects that are generally neglected by modified linear theory (MLT) methods. Up to the present time, CFD methods have been primarily used for analysis or prediction. Some investigators have used CFD to impact the design of low boom configurations using trial and error methods. One investigator developed a hybrid design method using a combination of Modified Linear Theory (e.g. F-functions) and CFD to provide equivalent area due to lift driven by a numerical optimizer to redesign or modify an existing configuration to achieve a shaped sonic boom signature. A three-dimensional design methodology has not yet been developed that completely uses nonlinear methods or CFD. Constrained numerical optimization techniques have existed for some time. Many of these methods use gradients to search for the minimum of a specified objective function subject to a variety of design variable bounds, linear and nonlinear constraints. Gradient based design optimization methods require the determination of the objective function gradients with respect to each of the design variables. These optimization methods are efficient and work well if the gradients can be obtained analytically. If analytical gradients are not available, the objective gradients or derivatives with respect to the design variables must be obtained numerically. To obtain numerical gradients, say, for 10 design variables, might require anywhere from 10 to 20 objective function evaluations. Typically, 5-10 global iterations of the optimizer are required to minimize the objective function. In terms of using CFD as a design optimization tool, the numerical evaluation of gradients can require anywhere from 100 to 200 CFD computations per design for only 10 design variables. If one CFD computation requires an hour of computational time on a Cray computer, one can see that the use of constrained numerical optimization quickly becomes impractical.Hence, in order to practically couple a numerical design optimization technique with a CFD method, the CFD method must be extremely efficient with running times on the order of only minutes. The CFD Euler code developed under NASA sponsorship and referred to as MIM3D-SB for the most part fulfills these efficiency requirements. Analysis of wing- body configurations can be computed in a matter of a few minutes. The present study will concentrate on the feasibility of the use of this CFD code in conjunction with a numerical design optimization technique for the sonic boom reduction of candidate HSCT configurations. A preliminary supersonic aircraft design system has been established that utilizes the numerical design optimization code NPSOL developed at Stanford University coupled with the supersonic NUM3D-SB CFD code. Many questions still need to be answered in regard to using CFD and numerical optimizers as design tools. There are difficulties related to both the CFD codes and the numerical optimizers. Numerical optimizers can converge to a local minima rather than a global minima. This behavior is largely a function of the initial guess in the design space. The optimizer also is searching for a minimum of the function in terms of its derivative without any regard to the actual function value. Numerically (i.e. CFD) determined gradients can also generate spurious numerical local minima. In addition, for the sonic boom problem, grid fineness will also determine the accuracy of the final design solution. Design optimization methods work well on problems defined by continuous objective functions. The sonic boom signature design problem is not necessarily defined by a continuous objective function. The signature can have a variety of shapes; i.e. from N-wave to multiple shocks. The far-field or ground signature may not transition continuously from one shape to another and hence, may exhibit discontinuous behavior. This is also a source of difficulty in using design optimization methods.In the following sections, several low boom and one reference aircraft configuration will be analyzed to predict their sonic boom signature characteristics. Modifications to some of these designs will also be presented to demonstrate the feasibility of using CFD as a design tool and to demonstrate the feasibility of designing shaped sonic boom signatures. Design modifications to some configurations will be presented to demonstrate the feasibility of achieving shaped signatures with reduced levels and not necessarily to represent realistic or aerodynamically efficient design modifications. Fuselage volume or camber are used as design variables in order to have a minimal effect on the primary wing aerodynamics. The paper will also seek to demonstrate whether a hybrid or ramped signature is feasible to achieve. For the low-boom configurations, the CFD predicted signatures will be compared qualitatively to their MLT design signatures.

Siclari, Michael J.↗
NASA NTRS2006

Biology Inspired Approach for Communal Behavior in Sensor Networks

Research in wireless sensor network technology has exploded in the last decade. Promises of complex and ubiquitous control of the physical environment by these networks open avenues for new kinds of science and business. Due to the small size and low cost of sensor devices, visionaries promise systems enabled by deployment of massive numbers of sensors working in concert. Although the reduction in size has been phenomenal it results in severe limitations on the computing, communicating, and power capabilities of these devices. Under these constraints, research efforts have concentrated on developing techniques for performing relatively simple tasks with minimal energy expense assuming some form of centralized control. Unfortunately, centralized control does not scale to massive size networks and execution of simple tasks in sparsely populated networks will not lead to the sophisticated applications predicted. These must be enabled by new techniques dependent on local and autonomous cooperation between sensors to effect global functions. As a step in that direction, in this work we detail a technique whereby a large population of sensors can attain a global goal using only local information and by making only local decisions without any form of centralized control.

Jones, Kennie H.↗
NASA NTRS2011

Step-by-Step Simulation of Radiation of Radiation Chemistry Using Green Functions for Diffusion-Influenced Reactions

The irradiation of biological systems leads to the formation of radiolytic species such as H(raised dot), (raised dot)OH, H2, H2O2, e(sup -)(sub aq), etc.[1]. These species react with neighboring molecules, which result in damage in biological molecules such as DNA. Radiation chemistry is there for every important to understand the radiobiological consequences of radiation[2]. In this work, we discuss an approach based on the exact Green Functions for diffusion-influenced reactions which may be used to simulate radiation chemistry and eventually extended to study more complex systems, including DNA.

Plante, Ianik↗
NASA NTRS2022

Adapting a Trusted AI Framework to Space Mission Autonomy

As artificial intelligence (AI) is increasingly pro- posed for new and future capabilities in space missions, the question of how to trust AI-enabled space autonomy has been explored. Recently, a collaboration between The Aerospace Corporation (Aerospace) and NASA’s Jet Propulsion Labora- tory (JPL) investigated how Aerospace’s Trusted AI Frame- work could be applied to two JPL projects that planned on lev- eraging AI for critical autonomous tasks. This combined effort led to many insights in the practical implementation of trusted AI along with considerable updates to the Trusted AI Frame- work that tailored its topic threads to space exploration. This document cohesively summarizes the enhanced framework as tailored to space missions as well as estimation of the level of trust required as a function of mission criticality and key stakeholders. The goal of this work is to provide a set of best practices to inform autonomy researchers, flight engineers, mission and proposal reviewers, and instrument and mission principal investigators (PI’s) to drive AI-based autonomy that maximizes trust and lowers the barriers to mission adoption for both science and engineering applications.

Amini, Rashied↗
NASA NTRS2019

Modeled Microgravity Induces Neutrophil Extracellular Trap (NET)osis Formation and Reduced Phagocytosis of Polymorphonuclear Neutrophils

Spaceflight can dysregulate immunity, by way of increasing granulocytes numbers with impaired function. Polymorphonuclear neutrophils (PMN) are granulocytes that are first responders to infection or injury, and consist of the largest pool of immune cells in humans. PMNs function during innate immunity, through phagocytosis and promotion of inflammation, via the release of reactive oxygen species (ROS) mediators and granule-containing enzymes, such as myeloperoxidase (MPO) and NADPH oxidase-2 (NOX-2). In addition, neutrophil extracellular trap (NET) formation is another mechanism of PMN surveillance that works independently of engulfment phagocytosis, and is a last resort function that can induce NETosis or PMN-specific cell death. Previous studies in our lab have identified increased mature neutrophils, ROS and MPO production, and reduced phagocytosis in granulocytes in simulated microgravity (sug) models of hindlimb unloading (HU) in adult mice and leukocytes cultured in high-aspect rotating wall vessels (HARV-RWV). Since sug impaired phagocytosis, but improved enzymatic mediator production of MPO and redox molecules, we sought to address the third known function of PMNs, NETosis. For this, PMNs were culture in the presence or absence of the anti-oxidant N-acetyl cysteine (NAC), which rescued impaired phagocytosis that was present in sug without NAC treatment. Further, NETosis was induced in sug that was no different in the presence of NAC, suggesting NAC targets independent functions of PMNs under sug. Collectively, these results suggest modeled microgravity induced NETosis, which opens a new avenue for spaceflight studies in immune dysfunction.

Paul, Amber M.↗
NASA NTRS1996

What is the minimal vestibular function required for compensation?

Living with an uncompensated, abnormal vestibular system requires oppressive modification of life style and often prevents return to work and activities of daily living. Patients with vestibular abnormalities were studied to determine the minimal residual vestibular function required to achieve compensation. Three groups of patients with (a) complete unilateral loss of vestibular function with normal horizontal canal-vestibulo-ocular (HCVOR) function in the opposite ear, (b) complete unilateral loss with abnormal HCVOR function in the opposite ear, and (c) bilateral reduction of vestibular function from aminoglycoside toxicity underwent vestibuloocular (VOR), optokinetic (OKN), visual-VOR (VVOR), and computerized dynamic posturography (CDP) tests before and after therapeutic procedures. Results suggest that a minimal VOR response amplitude must be present for compensation of VVOR function to occur. The roles of VOR and OKN phase shifts in vestibular compensation are more complicated and require further study. Compensation of vestibulospinal function does not necessarily accompany VOR or VVOR compensation. Ascending and descending vestibular compensatory mechanisms may involve different spatial sensory inputs. Results of these studies have important implications for the diagnosis and treatment of patients with vestibular disorders, including selection and monitoring of patients for therapeutic regimens such as vestibular nerve section and streptomycin therapy.

Non-NASA Center↗
NASA NTRS1975

Physiological mechanisms of the effect of weightlessness on the body

Experimental data show that physiological reactions observed under weightlessness conditions are caused by: (1) The direct effect of weightlessness, as a consequence of decrease (""disappearance'') of the weight of body tissues and organs; and (2) the mediated effect of weightlessness, as a result of changes in the functional state of the central nervous system and the cooperative work of the analyzers. The human body adopts to weightless conditions under the prolonged effects of it. In this case, four periods can be distinguished: The first period, a transitional process lasting from 1 to 24 hours; second period, initial adaptation to conditions of weightlessness and readjustment of all functional systems of the body; the third period, adaptation to the unusual mechanical conditions of the external environment, lasting from 3 to 8 days and more; and the fourth period, the stage of possible imbalance of the functions and the systems of some astronauts, as a result of the prolonged effect of weightlessness.

Kasyan, I. I.↗
NASA NTRS1934

On the theory of laminar boundary layers involving separation

This paper presents a mathematical discussion of the laminar boundary layer, which was developed with a view of facilitating the investigation of those boundary layers in particular for which the phenomenon of separation occurs. The treatment starts with a slight modification of the form of the boundary layer equation first published by Von Mises. Two approximate solutions of this equation are found, one of which is exact at the outer edge of the boundary layer while the other is exact at the wall. The final solution is obtained by joining these two solutions at the inflection points of the velocity profiles. The final solution is given in terms of a series of universal functions for a fairly broad class of potential velocity distributions outside of the boundary layer. Detailed calculations of the boundary layer characteristics are worked out for the case in which the potential velocity is a linear function of the distance from the upstream stagnation point. Finally, the complete separation point characteristics are determined for the boundary layer associated with a potential velocity distribution made up of two linear functions of the distance from the stagnation point. It appears that extensions of the detailed calculations to more complex potential flows can be fairly easily carried out by using the explicit formulae given in the paper. (author)

Von Karman, TH↗