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At least 757 records · Page 42

Computer-Controlled Cylindrical Polishing Process for Large X-Ray Mirror Mandrels

We are developing high-energy grazing incidence shell optics for hard-x-ray telescopes. The resolution of a mirror shells depends on the quality of cylindrical mandrel from which they are being replicated. Mid-spatial-frequency axial figure error is a dominant contributor in the error budget of the mandrel. This paper presents our efforts to develop a deterministic cylindrical polishing process in order to keep the mid-spatial-frequency axial figure errors to a minimum. Simulation software is developed to model the residual surface figure errors of a mandrel due to the polishing process parameters and the tools used, as well as to compute the optical performance of the optics. The study carried out using the developed software was focused on establishing a relationship between the polishing process parameters and the mid-spatial-frequency error generation. The process parameters modeled are the speeds of the lap and the mandrel, the tool s influence function, the contour path (dwell) of the tools, their shape and the distribution of the tools on the polishing lap. Using the inputs from the mathematical model, a mandrel having conical approximated Wolter-1 geometry, has been polished on a newly developed computer-controlled cylindrical polishing machine. The preliminary results of a series of polishing experiments demonstrate a qualitative agreement with the developed model. We report our first experimental results and discuss plans for further improvements in the polishing process. The ability to simulate the polishing process is critical to optimize the polishing process, improve the mandrel quality and significantly reduce the cost of mandrel production

Khan, Gufran S.↗

Rollable Thin Shell Composite-Material Paraboloidal Mirrors

An experiment and calculation have demonstrated the feasibility of a technique of compact storage of paraboloidal mirrors made of thin composite-material (multiple layers of carbon fiber mats in a polymeric matrix) shells coated with metal for reflectivity. Such mirrors are under consideration as simple, lightweight alternatives to the heavier, more complex mirrors now used in space telescopes. They could also be used on Earth in applications in which gravitational sag of the thin shells can be tolerated. The present technique is essentially the same as that used to store large maps, posters, tapestries, and similar objects: One simply rolls up the mirror to a radius small enough to enable the insertion of the mirror in a protective cylindrical case. Provided that the stress associated with rolling the mirror is not so large as to introduce an appreciable amount of hysteresis, the mirror can be expected to spring back to its original shape, with sufficient precision to perform its intended optical function, when unrolled from storage.

Meinel, Aden↗

What Makes Hybrid Concurrent Engineering Teams Work and Not Work: A Theoretical Analysis

Aerospace concurrent engineering teams (CETs) have been around since 1995 and have evolved into various shapes and forms. Historically, CETs have been designed to function with all team members physically collocated. In order to continue operating during the coronavirus pandemic, these teams have had to become fully distributed in 2020. Previously, distributed CETs were predicted to be considerably less efficient than traditional CETs. However, at JPL studies are being successfully run at peak rate in a distributed mode. As nominal working conditions resume after the pandemic, a major question is whether these teams must go back to being fully collocated or if they can become successful hybrid teams, with half or more of the engineers working remotely. In order to address this question, it is important to understand the key characteristics that make a CET more likely to be successful in a remote mode. This paper revisits a framework for characterizing concurrent engineering teams, providing a means to describe and compare the behavior of concurrent engineering teams in various settings [1], and uses the framework to make recommendations for future hybrid teams.

Chattopadhyay, Debarati↗

Characterization of Errors in Satellite-Based HCHO/NO 2 Tropospheric Column Ratios With Respect to Chemistry, Column-to-PBL Translation, Spatial Representation, and Retrieval Uncertainties

The availability of formaldehyde (HCHO) (a proxy for volatile organic compound reactivity) and nitrogen dioxide (NO 2 ) (a proxy for nitrogen oxides) tropospheric columns from ultraviolet–visible (UV–Vis) satellites has motivated many to use their ratios to gain some insights into the near-surface ozone sensitivity. Strong emphasis has been placed on the challenges that come with transforming what is being observed in the tropospheric column to what is actually in the planetary boundary layer (PBL) and near the surface; however, little attention has been paid to other sources of error such as chemistry, spatial representation, and retrieval uncertainties. Here we leverage a wide spectrum of tools and data to quantify those errors carefully. Concerning the chemistry error, a well-characterized box model constrained by more than 500 h of aircraft data from NASA's air quality campaigns is used to simulate the ratio of the chemical loss of HO 2 + RO 2 (LRO x ) to the chemical loss of NO x (LNO x ). Subsequently, we challenge the predictive power of HCHO/NO 2 ratios (FNRs), which are commonly applied in current research, in detecting the underlying ozone regimes by comparing them to LRO x /LNO x . FNRs show a strongly linear (R 2 =0.94) relationship with LRO x /LNO x , but only on the logarithmic scale. Following the baseline (i.e., ln(LRO x /LNO x ) = −1.0 ± 0.2) with the model and mechanism (CB06, r2) used for segregating NO x -sensitive from VOC-sensitive regimes, we observe a broad range of FNR thresholds ranging from 1 to 4. The transitioning ratios strictly follow a Gaussian distribution with a mean and standard deviation of 1.8 and 0.4, respectively. This implies that the FNR has an inherent 20 % standard error (1σ) resulting from not accurately describing the RO x –HO x cycle. We calculate high ozone production rates (PO 3 ) dominated by large HCHO × NO 2 concentration levels, a new proxy for the abundance of ozone precursors. The relationship between PO 3 and HCHO × NO 2 becomes more pronounced when moving towards NO x -sensitive regions due to nonlinear chemistry; our results indicate that there is fruitful information in the HCHO × NO 2 metric that has not been utilized in ozone studies. The vast amount of vertical information on HCHO and NO 2 concentrations from the air quality campaigns enables us to parameterize the vertical shapes of FNRs using a second-order rational function permitting an analytical solution for an altitude adjustment factor to partition the tropospheric columns into the PBL region. We propose a mathematical solution to the spatial representation error based on modeling isotropic semivariograms. Based on summertime-averaged data, the Ozone Monitoring Instrument (OMI) loses 12 % of its spatial information at its native resolution with respect to a high-resolution sensor like the TROPOspheric Monitoring Instrument (TROPOMI) (> 5.5 × 3.5 km 2 ). A pixel with a grid size of 216 km 2 fails at capturing ∼ 65 % of the spatial information in FNRs at a 50 km length scale comparable to the size of a large urban center (e.g., Los Angeles). We ultimately leverage a large suite of in situ and ground-based remote sensing measurements to draw the error distributions of daily TROPOMI and OMI tropospheric NO 2 and HCHO columns. At a 68 % confidence interval (1σ), errors pertaining to daily TROPOMI observations, either HCHO or tropospheric NO 2 columns, should be above 1.2–1.5 × 10 16 molec. cm −2 to attain a 20 %–30 % standard error in the ratio. This level of error is almost non-achievable with the OMI given its large error in HCHO. The satellite column retrieval error is the largest contributor to the total error (40 %–90 %) in the FNRs. Due to a stronger signal in cities, the total relative error (< 50 %) tends to be mild, whereas areas with low vegetation and anthropogenic sources (e.g., the Rocky Mountains) are markedly uncertain (> 100 %). Our study suggests that continuing development in the retrieval algorithm and sensor design and calibration is essential to be able to advance the application of FNRs beyond a qualitative metric.

Amir H. Souri↗

Simulation of Inflated Pahoehoe Lava Flows

A new stochastic model simulates late-stage pahoehoe lobes where random processes dominate emplacement. The model prescribes probabilistic rules for determining where and when parcels of lava move within the lobe. Unlike a classical Brownian motion random walk, the model allows individual parcels to remain dormant, but fluid, for multiple time steps. The randomness of parcel volume transfers within the lobe interior as well as at the margins qualitatively reflects inflation processes observed in the field. The fraction of inflated volume to total volume increases with the total volume, with greater than 75% of the lobe volume contributed through inflation for typical lobes. The influence on planform shape and topographic cross-sectional profiles of total volume, source area and shape, topographic confinement, and sequential breakouts at the lobe margins, are all explored with the stochastic model. Each of these factors influences the overall lobe thickness and width. The model provides a means for assessing the relative importance of these processes through comparisons with field data. For the first time, Gaussian and parabolic functions are quantitatively fit to field measurements of pahoehoe lobes. Both functional forms provide adequate description of the cross-sectional flow shapes. When comparing simulated lobes to field data, sequential breakouts at the lobe margins are found to be an important process controlling the final topographic distribution of observed pahoehoe lobes.

modeling↗

Autocorrelation and regularization in digital images. I - Basic theory

Spatial structure occurs in remotely sensed images when the imaged scenes contain discrete objects that are identifiable in that their spectral properties are more homogeneous within than between them and other scene elements. The spatial structure introduced is manifest in statistical measures such as the autocovariance function and variogram associated with the scene, and it is possible to formulate these measures explicitly for scenes composed of simple objects of regular shapes. Digital images result from sensing scenes by an instrument with an associated point spread function (PSF). Since there is averaging over the PSF, the effect, termed regularization, induced in the image data by the instrument will influence the observable autocovariance and variogram functions of the image data. It is shown how the autocovariance or variogram of an image is a composition of the underlying scene covariance convolved with an overlap function, which is itself a convolution of the PSF. The functional form of this relationship provides an analytic basis for scene inference and eventual inversion of scene model parameters from image data.

Jupp, David L. B.↗

Service Life Extension of the Propulsion System of Long-Term Manned Orbital Stations

One of the critical non-replaceable systems of a long-term manned orbital station is the propulsion system. Since the propulsion system operates beginning with the launch of station elements into orbit, its service life determines the service life of the station overall. Weighing almost a million pounds, the International Space Station (ISS) is about four times as large as the Russian space station Mir and about five times as large as the U.S. Skylab. Constructed over a span of more than a decade with the help of over 100 space flights, elements and modules of the ISS provide more research space than any spacecraft ever built. Originally envisaged for a service life of fifteen years, this Earth orbiting laboratory has been in orbit since 1998. Some elements that have been launched later in the assembly sequence were not yet built when the first elements were placed in orbit. Hence, some of the early modules that were launched at the inception of the program were already nearing the end of their design life when the ISS was finally ready and operational. To maximize the return on global investments on ISS, it is essential for the valuable research on ISS to continue as long as the station can be sustained safely in orbit. This paper describes the work performed to extend the service life of the ISS propulsion system. A system comprises of many components with varying failure rates. Reliability of a system is the probability that it will perform its intended function under encountered operating conditions, for a specified period of time. As we are interested in finding out how reliable a system would be in the future, reliability expressed as a function of time provides valuable insight. In a hypothetical bathtub shaped failure rate curve, the failure rate, defined as the number of failures per unit time that a currently healthy component will suffer in a given future time interval, decreases during infant-mortality period, stays nearly constant during the service life and increases at the end when the design service life ends and wear-out phase begins. However, the component failure rates do not remain constant over the entire cycle life. The failure rate depends on various factors such as design complexity, current age of the component, operating conditions, severity of environmental stress factors, etc. Development, qualification and acceptance test processes provide rigorous screening of components to weed out imperfections that might otherwise cause infant mortality failures. If sufficient samples are tested to failure, the failure time versus failure quantity can be analyzed statistically to develop a failure probability distribution function (PDF), a statistical model of the probability of failure versus time. Driven by cost and schedule constraints however, spacecraft components are generally not tested in large numbers. Uncertainties in failure rate and remaining life estimates increase when fewer units are tested. To account for this, spacecraft operators prefer to limit useful operations to a period shorter than the maximum demonstrated service life of the weakest component. Running each component to its failure to determine the maximum possible service life of a system can become overly expensive and impractical. Spacecraft operators therefore, specify the required service life and an acceptable factor of safety (FOS). The designers use these requirements to limit the life test duration. Midway through the design life, when benefits justify additional investments, supplementary life test may be performed to demonstrate the capability to safely extend the service life of the system. An innovative approach is required to evaluate the entire system, without having to go through an elaborate test program of propulsion system elements. Evaluating every component through a brute force test program would be a cost prohibitive and time consuming endeavor. ISS propulsion system components were designed and built decades ago. There are no representative ground test articles for some of the components. A 'test everything' approach would require manufacturing new test articles. The paper outlines some of the techniques used for selective testing, by way of cherry picking candidate components based on failure mode effects analysis, system level impacts, hazard analysis, etc. The type of testing required for extending the service life depends on the design and criticality of the component, failure modes and failure mechanisms, life cycle margin provided by the original certification, operational and environmental stresses encountered, etc. When specific failure mechanism being considered and the underlying relationship of that mode to the stresses provided in the test can be correlated by supporting analysis, time and effort required for conducting life extension testing can be significantly reduced. Exposure to corrosive propellants over long periods of time, for instance, lead to specific failure mechanisms in several components used in the propulsion system. Using Arrhenius model, which is tied to chemically dependent failure mechanisms such as corrosion or chemical reactions, it is possible to subject carefully selected test articles to accelerated life test. Arrhenius model reflects the proportional relationship between time to failure of a component and the exponential of the inverse of absolute temperature acting on the component. The acceleration factor is used to perform tests at higher stresses that allow direct correlation between the times to failure at a high test temperature to the temperatures to be expected in actual use. As long as the temperatures are such that new failure mechanisms are not introduced, this becomes a very useful method for testing to failure a relatively small sample of items for a much shorter amount of time. In this article, based on the example of the propulsion system of the first ISS module Zarya, theoretical approaches and practical activities of extending the service life of the propulsion system are reviewed with the goal of determining the maximum duration of its safe operation.

Kamath, Ulhas↗

Airfoil optimization by the one-shot method

An efficient numerical approach for the design of optimal aerodynamic shapes is presented in this paper. The objective of any optimization problem is to find the optimum of a cost function subject to a certain state equation (Governing equation of the flow field) and certain side constraints. As in classical optimal control methods, the present approach introduces a costate variable (Language multiplier) to evaluate the gradient of the cost function. High efficiency in reaching the optimum solution is achieved by using a multigrid technique and updating the shape in a hierarchical manner such that smooth (low-frequency) changes are done separately from high-frequency changes. Thus, the design variables are changed on a grid where their changes produce nonsmooth (high-frequency) perturbations that can be damped efficiently by the multigrid. The cost of solving the optimization problem is approximately two to three times the cost of the equivalent analysis problem.

Kuruvila, G.↗

Airfoil Design and Optimization by the One-Shot Method

An efficient numerical approach for the design of optimal aerodynamic shapes is presented in this paper. The objective of any optimization problem is to find the optimum of a cost function subject to a certain state equation (governing equation of the flow field) and certain side constraints. As in classical optimal control methods, the present approach introduces a costate variable (Lagrange multiplier) to evaluate the gradient of the cost function. High efficiency in reaching the optimum solution is achieved by using a multigrid technique and updating the shape in a hierarchical manner such that smooth (low-frequency) changes are done separately from high-frequency changes. Thus, the design variables are changed on a grid where their changes produce nonsmooth (high-frequency) perturbations that can be damped efficiently by the multigrid. The cost of solving the optimization problem is approximately two to three times the cost of the equivalent analysis problem.

Kuruvila, G.↗

The shape of Eros

Monte Carlo simulations are presently used to optimize estimation, ascertain associated errors, and guide bias-correction procedures, for the Eros polar silhouette convex hull that has been estimated from radar echo spectra. This hull is trapezoidal; this nonaxisymmetric shape may account for odd harmonics in Eros' echo spectral signature as a function of rotation phase. Additional constraints have been obtained for the figure of Eros through the inversion of the optical lightcurve to estimate the asteroid's two-dimensional average of the three-dimensional shape. This 'mean cross-section' and the polar silhouette exhibit similar elongations.

Ostro, S. J.↗

Evolution of interstellar pickup ions in the solar wind

A model is constructed for the evolution of an interstellar pickup ion distribution in the solar wind. The model assumes that the ions are immediately isotropized at ionization and follows the subsequent development of the distribution function as the particles are convected with the solar wind. The effects of energy diffusion in an ambient wave field with a power law spectrum, adiabatic deceleration in the expanding solar wind, and continual addition of newly ionized particles are all included in the model. An analytical expression describing the evolution of the distribution function in phase space velocity and heliocentric radius is obtained. The distribution quickly approaches an asymptotic shape in phase space which depends on the relative efficiency of the energy diffusion process compared to that of adiabatic deceleration. At large distances from the sun the density of pickup ions falls as 1/r in this model. An expression for the distribution function at large distances and for large particle speed is presented. The asymptotic shape should describe the distribution of pickup ions in the outer heliosphere and could be used as an input distribution for a model of the anomalous component of heliosphere and could be used as an input distribution for a model of the anomalous component of cosmic rays. Comparison of this work with the recent observation of He(+) at 1 AU implies that the energy diffusion process is very weak inside 1 AU.

Isenberg, Philip A.↗

Improvement in the Shape Memory Response of Ti50.5Ni24.5Pd25 High-Temperature Shape Memory Alloy with Scandium Microalloying

A Ti(50.5)Ni(24.5)Pd25 high-temperature shape memory alloy (HTSMA) is microalloyed with 0.5 at. pct scandium (Sc) to enhance its shape-memory characteristics, in particular, dimensional stability under repeated thermomechanical cycles. For both Ti(50.5)Ni(24.5)Pd25 and the Sc-alloyed material, differential scanning calorimetry is conducted for multiple cycles to characterize cyclic stability of the transformation temperatures. The microstructure is evaluated using electron microscopy, X-ray diffractometry, and wavelength dispersive spectroscopy. Isobaric thermal cycling experiments are used to determine transformation temperatures, dimensional stability, and work output as a function of stress. The Sc-doped alloy displays more stable shape memory response with smaller irrecoverable strain and narrower thermal hysteresis than the baseline ternary alloy. This improvement in performance is attributed to the solid solution hardening effect of Sc.

Atli, K. C.↗

Prototype Morphing Fan Nozzle Demonstrated

Ongoing research in NASA Glenn Research Center's Structural Mechanics and Dynamics Branch to develop smart materials technologies for aeropropulsion structural components has resulted in the design of the prototype morphing fan nozzle shown in the photograph. This prototype exploits the potential of smart materials to significantly improve the performance of existing aircraft engines by introducing new inherent capabilities for shape control, vibration damping, noise reduction, health monitoring, and flow manipulation. The novel design employs two different smart materials, a shape-memory alloy and magnetorheological fluids, to reduce the nozzle area by up to 30 percent. The prototype of the variable-area fan nozzle implements an overlapping spring leaf assembly to simplify the initial design and to provide ease of structural control. A single bundle of shape memory alloy wire actuators is used to reduce the nozzle geometry. The nozzle is subsequently held in the reduced-area configuration by using magnetorheological fluid brakes. This prototype uses the inherent advantages of shape memory alloys in providing large induced strains and of magnetorheological fluids in generating large resistive forces. In addition, the spring leaf design also functions as a return spring, once the magnetorheological fluid brakes are released, to help force the shape memory alloy wires to return to their original position. A computerized real-time control system uses the derivative-gain and proportional-gain algorithms to operate the system. This design represents a novel approach to the active control of high-bypass-ratio turbofan engines. Researchers have estimated that such engines will reduce thrust specific fuel consumption by 9 percent over that of fixed-geometry fan nozzles. This research was conducted under a cooperative agreement (NCC3-839) at the University of Akron.

Lee, Ho-Jun↗

A Parametric Study of Fine-scale Turbulence Mixing Noise

The present paper is a study of aerodynamic noise spectra from model functions that describe the source. The study is motivated by the need to improve the spectral shape of the MGBK jet noise prediction methodology at high frequency. The predicted spectral shape usually appears less broadband than measurements and faster decaying at high frequency. Theoretical representation of the source is based on Lilley's equation. Numerical simulations of high-speed subsonic jets as well as some recent turbulence measurements reveal a number of interesting statistical properties of turbulence correlation functions that may have a bearing on radiated noise. These studies indicate that an exponential spatial function may be a more appropriate representation of a two-point correlation compared to its Gaussian counterpart. The effect of source non-compactness on spectral shape is discussed. It is shown that source non-compactness could well be the differentiating factor between the Gaussian and exponential model functions. In particular, the fall-off of the noise spectra at high frequency is studied and it is shown that a non-compact source with an exponential model function results in a broader spectrum and better agreement with data. An alternate source model that represents the source as a covariance of the convective derivative of fine-scale turbulence kinetic energy is also examined.

Khavaran, Abbas↗

Finite Element Analysis of Adaptive-Stiffening and Shape-Control SMA Hybrid Composites

Shape memory alloy hybrid composites with adaptive-stiffening or morphing functions are simulated using finite element analysis. The composite structure is a laminated fiber-polymer composite beam with embedded SMA ribbons at various positions with respect to the neutral axis of the beam. Adaptive stiffening or morphing is activated via selective resistance heating of the SMA ribbons or uniform thermal loads on the beam. The thermomechanical behavior of these composites was simulated in ABAQUS using user-defined SMA elements. The examples demonstrate the usefulness of the methods for the design and simulation of SMA hybrid composites. Keywords: shape memory alloys, Nitinol, ABAQUS, finite element analysis, post-buckling control, shape control, deflection control, adaptive stiffening, morphing, constitutive modeling, user element

Gao, Xiujie↗

Adjoint Sensitivity Computations for an Embedded-Boundary Cartesian Mesh Method and CAD Geometry

Cartesian-mesh methods are perhaps the most promising approach for addressing the issues of flow solution automation for aerodynamic design problems. In these methods, the discretization of the wetted surface is decoupled from that of the volume mesh. This not only enables fast and robust mesh generation for geometry of arbitrary complexity, but also facilitates access to geometry modeling and manipulation using parametric Computer-Aided Design (CAD) tools. Our goal is to combine the automation capabilities of Cartesian methods with an eficient computation of design sensitivities. We address this issue using the adjoint method, where the computational cost of the design sensitivities, or objective function gradients, is esseutially indepeudent of the number of design variables. In previous work, we presented an accurate and efficient algorithm for the solution of the adjoint Euler equations discretized on Cartesian meshes with embedded, cut-cell boundaries. Novel aspects of the algorithm included the computation of surface shape sensitivities for triangulations based on parametric-CAD models and the linearization of the coupling between the surface triangulation and the cut-cells. The objective of the present work is to extend our adjoint formulation to problems involving general shape changes. Central to this development is the computation of volume-mesh sensitivities to obtain a reliable approximation of the objective finction gradient. Motivated by the success of mesh-perturbation schemes commonly used in body-fitted unstructured formulations, we propose an approach based on a local linearization of a mesh-perturbation scheme similar to the spring analogy. This approach circumvents most of the difficulties that arise due to non-smooth changes in the cut-cell layer as the boundary shape evolves and provides a consistent approximation tot he exact gradient of the discretized abjective function. A detailed gradient accurace study is presented to verify our approach. Thereafter, we focus on a shape optimization problem for an Apollo-like reentry capsule. The optimization seeks to enhance the lift-to-drag ratio of the capsule by modifyjing the shape of its heat-shield in conjunction with a center-of-gravity (c.g.) offset. This multipoint and multi-objective optimization problem is used to demonstrate the overall effectiveness of the Cartesian adjoint method for addressing the issues of complex aerodynamic design. This abstract presents only a brief outline of the numerical method and results; full details will be given in the final paper.

Nemec, Marian↗

An improved spectral calibration requirement for AVIRIS

The Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) measures spectra from 400 to 2500 nm through 224 contiguous spectral channels. These spectra are used to identify and determine the abundances of constituents of the earth's surface and atmosphere through the analysis of spectral molecular absorptions and scattering properties of materials. Spectral calibration of the AVIRIS is required to pursue these determinations, which are based on these fundamental spectral characteristics. The spectral calibration of AVIRIS consists of a description of the spectral response function for each of the 224 spectral channels. The spectral response function of AVIRIS channels was measured in the laboratory and shown to be Gaussian in shape. A subset of nine AVIRIS spectral channels is shown with channel centers at 10 nm intervals and channel shapes of Gaussian form with 10 nm full width and half maximum (FWHM). For AVIRIS the spectral calibration is described by the spectral center and FWHM of a Gaussian function for each channel. Uncertainty in the determination of the spectral center and FWHM for each channel are also reported. This paper presents results from an investigation of the sensitivity of AVIRIS-measured upwelling radiance to errors in spectral calibration. Based on this sensitivity analysis in conjunction with earlier work, an improved spectral calibration requirement for AVIRIS is proposed.

Green, Robert O.↗

Impingement of Droplets in 90 deg Elbows with Potential Flow

Trajectories were determined for droplets in air flowing through 90 deg elbows especially designed for two-dimensional potential motion with low pressure losses. The elbows were established by selecting as walls of each elbow two streamlines of the flow field produced by a complex potential function that establishes a two-dimensional flow around a 90 deg bend. An unlimited number of elbows with slightly different shapes can be established by selecting different pairs of streamlines as walls. The elbows produced by the complex potential function selected are suitable for use in aircraft air-intake ducts. The droplet impingement data derived from the trajectories are presented along with equations in such a manner that the collection efficiency, the area, the rate, and the distribution of droplet impingement can be determined for any elbow defined by any pair of streamlines within a portion of the flow field established by the complex potential function. Coordinates for some typical streamlines of the flow field and velocity components for several points along these streamlines are presented in tabular form.

Hacker, Paul T.↗