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NASA NTRS2021

Unsteady Pressure-Sensitive Paint (UPSP) Shot Noise Reduction

In the study of unsteady Pressure-Sensitive Paint (uPSP) image data sets, pixel intensity values collected by Phantom cameras from multiple perspectives are used to calculate the surface pressure of an experimental model. This paper focuses on mitigation of shot noise which is introduced into the image data set by the cameras. Shot noise impacts the quality of data collected, increasing the system error in its pressure value calculations. This paper characterises statistical methods to reduce shot noise within experimental data by taking an average or weighted average across an area of pixels of their respective counts. Each combined-pixel has a higher total effective intensity value and therefore exhibits reduced noise throughout its time history. This is demonstrated by calculating the standard deviations of the combined pixel values through time, which decrease as the combined pixel area increases. Finally, these methods are compared across experimental wind tunnel conditions to examine their effectiveness at different Mach numbers. Ultimately, the results are used to recommend a blur filter type and size which will be applied to the raw image data prior to processing, as well as a minimum camera resolution for future experiments. Downstream of the filter application, the software to convert the 2D image data sets into surface pressure readings on a 3D grid use a ratio of decimation that assigns multiple pixels to each 3D grid node. This decimation ratio will also be aligned with the size of the blur filter, resulting in a more manageable data set size and compromised spatial resolution. Combined with the effect of the blur filter, the resulting pixel intensity readings will exhibit significantly reduced shot noise, reducing the magnitude of uncertainty and error in the final calculated pressure. For future tests, the camera spatial resolution must be sufficient to capture small flow and model features even after the recommended blur filter is applied. The image data set is taken from a uPSP experiment of the Space Launch System Block 1 cargo vehicle model in September 2019.

Lucy Tang↗

NASA NTRSDate not supplied

Spectral Analysis of Integrated Pressures on Patches with Unsteady Pressure-Sensitive Paint Measurements

This paper describes the spectral analysis of integrated pressures on patches of the scale model of the Space Launch System Block 1 cargo vehicle with the Unsteady Pressure-Sensitive Paint measurements, which were collected in the ascent transient aerodynamics test with the Unitary Plan Wind Tunnel 11-by-11-foot Transonic Wind Tunnel in September 2019 at NASA Ames Research Center. The patches are defined with x station values, indicating the position along length of the vehicle, and azimuth angles of the scale model. For each patch, the polygons are determined from the surface cells of the grid of the model, clipped with the edges of the patch, and each of the polygons is divided into triangles. The integrated pressure of the patch is determined as the ratio of the sum of the forces on the triangles over the sum of the areas of the triangles. For each run of the test, the cross power spectral density and magnitude squared coherence are computed from the time series of the integrated pressures on the patches. The results of pressure integration and spectral analysis are presented, and the data consistency of the test is demonstrated. The work described in this paper is a part of NASA’s development of a new state-of-the-art uPSP capability in production wind tunnels. Funding for this research was provided by the NASA Aeroscience Evaluation and Test Capabilities Project.

Pressure-Sensitive Paint↗

NASA NTRS2016

Investigation of Unsteady Pressure-Sensitive Paint (uPSP) and a Dynamic Loads Balance to Predict Launch Vehicle Buffet Environments

This NESC assessment examined the accuracy of estimating buffet loads on in-line launch vehicles without booster attachments using sparse unsteady pressure measurements. The buffet loads computed using sparse sensor data were compared with estimates derived using measurements with much higher spatial resolution. The current method for estimating launch vehicle buffet loads is through wind tunnel testing of models with approximately 400 unsteady pressure transducers. Even with this relatively large number of sensors, the coverage can be insufficient to provide reliable integrated unsteady loads on vehicles. In general, sparse sensor spacing requires the use of coherence-length-based corrections in the azimuthal and axial directions to integrate the unsteady pressures and obtain reasonable estimates of the buffet loads. Coherence corrections have been used to estimate buffet loads for a variety of launch vehicles with the assumption methodology results in reasonably conservative loads. For the Space Launch System (SLS), the first estimates of buffet loads exceeded the limits of the vehicle structure, so additional tests with higher sensor density were conducted to better define the buffet loads and possibly avoid expensive modifications to the vehicle design. Without the additional tests and improvements to the coherence-length analysis methods, there would have been significant impacts to the vehicle weight, cost, and schedule. If the load estimates turn out to be too low, there is significant risk of structural failure of the vehicle. This assessment used a combination of unsteady pressure-sensitive paint (uPSP), unsteady pressure transducers, and a dynamic force and moment balance to investigate the integration schemes used with limited unsteady pressure data by comparing them with direct integration of extremely dense fluctuating pressure measurements. An outfall of the assessment was to evaluate the potential of using the emerging uPSP technique in a production test environment for future launch vehicles. The results show that modifications to the current technique can improve the accuracy of buffet estimates. More importantly, the uPSP worked remarkably well and, with improvements to the frequency response, sensitivity, and productivity, will provide an enhanced method for measuring wind tunnel buffet forcing functions (BFFs).

Schuster, David M.↗

NASA NTRS2020

Development of Unsteady Pressure-Sensitive Paint Application on NASA Space Launch System

The key measurement to acquire for understanding unsteady flow is surface pressure. Unsteady Pressure-Sensitive Paint (uPSP) is an emerging optical technique used in wind tunnel testing to measure fluctuating surface pressures. Recently, tests were conducted on NASA’s Space Launch System in NASA Ames Research Center’s Unitary Plan Wind Tunnel to determine the aeroacoustics environment and assist in developing the buffet forcing functions. Unsteady PSP data was collected during this test campaign. Steady state PSP data, infrared thermography, shadowgraph, accelerometer data, and dynamic pressure transducer data were also collected. In all 50 TB of data were collected during the three days of testing. During these three days of testing, a repeating transonic and supersonic alpha sweep condition was acquired. This paper presents these two wind tunnel conditions and examines how the temperature influences the PSP data. In the first large demonstration of uPSP in 2015 on an NESC-, AETC-sponsored wind tunnel test, lifetime PSP results highlighted the influence the model temperature had on the PSP data. A best practice of heat soaking the model before acquiring calibration images was followed during the test campaign presented in this paper. An infrared thermography camera and thermocouples were installed in the model to collect more details of the model surface temperature. Data processing schemes for uPSP are still in development but will be briefly presented here as well.

Roozeboom, Nettie H.↗

NASA NTRSDate not supplied

Spectral Analysis of Integrated Pressures on Patches with Unsteady Pressure-Sensitive Paint Measurements

This paper describes the spectral analysis of integrated pressures on patches of the scale model of the Space Launch System (SLS) Block 1B crew vehicle with the Unsteady Pressure-Sensitive Paint (uPSP) measurements, which were collected in the ascent transient aerodynamics tests with the Unitary Plan Wind Tunnel 11-by-11-foot Transonic Wind Tunnel in September 2019 at NASA Ames Research Center. Recent research has demonstrated that uPSP can be an essential tool for the assessment of the unsteady, aerodynamic phenomena. The work described in this paper is a part of NASA’s development of a new state-of-the-art uPSP capability in production wind tunnels. In this paper, 108 patches are defined with x station values and azimuth angles of the scale model. For each patch, the polygons are determined from the surface cells of the grid of the model, clipped with the edges of the patch, and each of the polygons is divided into triangles. The inputs of the pressure integration algorithm are the time series of pressure coefficients on the vertices of the grid of the model, generated by the uPSP data processing program from the videos taken with 4 Phantom high-speed cameras during the wind tunnel tests. The integrated pressure of the patch is determined as the ratio of the sum of the forces on the triangles over the sum of the areas of the triangles. For each of the test cases, the Cross Power Spectral Density (CPSD) and Magnitude-Squared Coherence (msCohere) are computed for frequencies over 1/3 octave bands from the time series of the integrated pressures on the patches. The spectral analysis outputs of different cases of the wind tunnel tests are analyzed and the coherence of patches is investigated over frequencies, x station values, azimuth angles, Mach numbers and SLS vehicle attitudes. The pressure integration and spectral analysis described in this paper were executed on the NASA Pleiades supercomputer. Funding for this research was provided by the NASA Aeroscience Evaluation and Test Capabilities (AETC) Project.

Pressure-Sensitive Paint↗

NASA NTRS2017

Wavenumber-frequency Spectra of Pressure Fluctuations on a Generic Space Vehicle Measured via Unsteady Pressure-Sensitive Paint

Time histories of pressure fluctuations on a generic, hammerhead space vehicle model were measured using unsteady Pressure-Sensitive Paint (uPSP). The test was conducted in the 11-foot transonic wind tunnel of NASA Ames Research Center over a Mach number range of 0.6 M 1.2, and angles of attack of -4 4. The model was coated with a porous binder and PtTFPP-based porous polymer paint. An elaborate system of four high-speed cameras, and forty LED lamps was used for image acquisition. Various steps for image registration, reduction of shot noise, photogrammetry procedure to map images from the four cameras on a grid for the model, and finally a calibration procedure to convert the measured fluctuations in light intensity to fluctuating pressure, are discussed in the paper. The calibration process using a set of unsteady pressure sensors mounted on the model, was found to overcome some of the inherent problems of the fast response paint, such as rapid photo-degradation, non-linearity in pressure response, and significant temperature sensitivity. Comparison of spectra of pressure fluctuations between UPSP and pressure sensors demonstrated the ability of the paint to faithfully follow fluctuations up to 10 kHz, the maximum attempted. It was also found that the camera bit-depth and the illumination level limited the lowest measurable levels of pressure fluctuations to around 140dB. The large data set exposed various critical transonic flow physics not seen before, such as a coupling of the shock motion on the Payload Fairing (PF) with the separated flow region on the upper stage of the launch vehicle, and upstream convection of pressure fluctuation on PF at certain Mach numbers. The data also confirmed the expectation of a general lowering of the coefficient of pressure fluctuation with Mach number. The availability of the data set on a dense, regularly-spaced, surface grid allowed for the calculation of wavenumber-frequency (k-) spectra via straightforward applications of Fourier transform. The k- spectra were compared for the separated flow regions on the Second Stage, and the shock-boundary layer interactions on PF. The former showed self-similarity with Mach number while the latter was distinctly different, and confirmed the upstream propagation of pressure fluctuations. The k- spectra were dominated by the convected fluctuations; the acoustic domain was not discernable. These data, valuable for the vibro-acoustics analysis of aerospace vehicles, are believed to be the first obtained for the transonic flight regime, and pave the path for application on production models of aerospace vehicles.

buffet↗

NASA NTRS2017

Experimental Verification of Buffet Calculation Procedure Using Unsteady Pressure-Sensitive Paint

Typically, a limited number of dynamic pressure sensors is employed to determine the unsteady aerodynamic forces on large, slender aerospace structures. This paper describes a robust calculation procedure based on frequency-specific correlation lengths and validation from an experiment conducted on a flat panel coated with fast response pressure-sensitive paint. The first part of the paper describes the procedure used to analyze the pressure sensitive paint images and a calibration method using dynamic pressure transducers. Excellent comparison in spectra, coherence, and phase, measured via pressure-sensitive paint and dynamic pressure sensors, validates the pressure-sensitive paint data. The second part of the paper describes the buffet validation process, the first step of which was to use pressure histories from all pixels to determine the true force fluctuations. In the next step, only a selected number of pixels was chosen as virtual sensors, and a correlation-length-based buffet calculation procedure was applied to determine modeled force fluctuations. By progressively decreasing the number of virtual sensors, it was observed that the present calculation procedure was able to make a close estimate of the true unsteady forces only from eight sensors. It is believed that the present work provides the first validation of a buffet calculation procedure.

Buffet↗

NASA NTRS2021

Unsteady Pressure-Sensitive-Paint Shot Noise Reduction

Lucy Tang↗

NASA NTRSDate not supplied

Analysis and Comparison of Surface Roughness Effects on Pressure Data from SLS Wind Tunnel Test

Unsteady flows are becoming a more characterized field of study as advancements in technology is allowing for more high-speed, time-resolved data acquisition. It is important to understand these flows as the shock-wave boundary-layer interactions and separation they are involved with can generate extreme loads caused by high pressure and temperature on the body of a flight vehicle. It is vital to characterize these loads, as they can impede upon the structural integrity of a high-speed system. A common instrument utilized today to characterize pressure distributions caused by unsteady flows are unsteady pressure transducers. Set flush with the surface of a flight vehicle or wind tunnel model, these transducers can provide a pressure distribution, capturing the frequency of unsteady loads that could impinge upon the test article. A couple negative aspects of this instrumentation is that they are expensive, have long lead times, and a pressure distribution can only be determined in one discrete location where the transducer is installed. An optical diagnostic known as unsteady pressure-sensitive paint (uPSP) can help to satisfy these deficiencies. This is a spray paint that can be applied to a surface of a model to provide a global pressure distribution across the whole painted region.2 A complete picture of the fluid dynamic pressures occurring on the surface of a test article can be analyzed, especially in locations where it may have been impossible to install a transducer. Pressure-sensitive paint is also less expensive than traditional pressure transducers, which are, on average, about $2,000 each including the cost of installation.3 Unsteady PSP will not replace unsteady pressure transducers, but the transducers can be used to validate the PSP response to the flow and measure frequencies higher than the frequency response of the paint. One issue that can arise using these tools simultaneously, is that the surface roughness of the uPSP can affect the pressure readings of the transducers. Applying the uPSP to the surface of the model is going to affect the flow in some way as the topography of the model is now different than if it was not painted. One study with steady PSP conducted by Amer, Obara, and Liu at NASA Langley concluded that the PSP surface roughness was minimally intrusive at different test conditions with various lift, drag, etc., however, it was noted that data from some pressure transducers had decreased, likely due to the uneven application of the PSP in the localized region of the transducers.9 Sugioka et. al. tested various formulations of uPSP to study surface roughness effects on a Common Research Model (CRM) in transonic flow. The influence of uPSP on the surface of the model, depending on how it was formulated, was noted to have the possibility of relocating the shock wave present in the flow. This study was conducted to examine and compare transducer data acquired in December 2017 of a 4% scale model of the Space Launch System (SLS) performed at NASA Ames Research Center (ARC) Unitary Plan Wind Tunnel (UPWT). One goal of the test was to demonstrate to the customer the potential of the uPSP system developed at NASA ARC. The effect of uPSP on transducer signals also became an interest in this experiment and is the reason for this data analysis. The results of this study were discussed with the SLS team to help them make decisions regarding the next test on an SLS model with the PSP and Unitary Plan Wind Tunnel teams.

pressure-sensitive paint↗

NASA NTRSDate not supplied

Visualizing UPSP Data with Python

The Unsteady Pressure-Sensitive Paint (uPSP) projects uses Pressure-Sensitive paint applied over aerospace models during wind tunnel testing to collect pressure data with high spatial and temporal resolution in order to inform unsteady aerodynamics studies. For each of the 800+ experimental runs, four cameras generate up to 50 GB of video data, which must then be processed, analyzed, and visualized on the NASA Advanced Supercomputing system (NAS) to assess the result. One of the final data analysis products is the dynamic modal decomposition (DMD) results, which decomposes the pressure reading signals by their frequency component. The goal of this project is to visualize the DMD results over a 3D rendering of the model, using efficient and parallelized python routines. The software uses the pytecplot library, a high-level API that connects python scripting to a Tecplot 360 engine. Tecplot is an industry standard high-performance visualization tool that can handle large datasets and workflow. Various animation, rendering, and image-combination techniques were investigated to generate the final videos using OpenCV on the NAS. The final result is a software tool that takes in data products from the uPSP processing chain and generates high resolution visualization videos in parallel for every data file, allowing researchers to view their results efficiently and at an unprecedentedly detailed level.

Emma Dolores McMillian↗

NASA NTRS2019

Wind Tunnel Flow Field Visualizations of the Space Launch System Vehicle Ascent

A series of wind tunnel tests were conducted to characterize the force-and-moment, and aeroacoustic environment of several configurations of the Space Launch System during ascent. The tests were conducted in the 11-by-11 foot transonic and 9-by-7 foot supersonic test sections at NASA Ames research center. Throughout these experiments data was collected from several types of instrumentation including: multicomponent force-and-moment strain gage balances, dynamic and steady-state pressure sensors, unsteady and steady pressure-sensitive paint, time-resolved shadowgraph and infrared imaging. The following details results and analysis from the time-resolved shadowgraph and infrared imaging data systems. The time-resolved shadowgraph and infrared imaging provided a qualitative measurement of the near-field turbulent fluctuations. These results helped provide context to the relative magnitude and frequency content of the fluid-structure-interaction driving the surface pressure phenomena characterized by the discrete pressure transducers and unsteady pressure sensitive paint.

Garbeff, Theodore J., II↗

NASA NTRS2016

Wavenumber-frequency Spectra of Pressure Fluctuations Measured via Fast Response Pressure Sensitive Paint

The recent advancement in fast-response Pressure-Sensitive Paint (PSP) allows time-resolved measurements of unsteady pressure fluctuations from a dense grid of spatial points on a wind tunnel model. This capability allows for direct calculations of the wavenumber-frequency (k-ω) spectrum of pressure fluctuations. Such data, useful for the vibro-acoustics analysis of aerospace vehicles, are difficult to obtain otherwise. For the present work, time histories of pressure fluctuations on a flat plate subjected to vortex shedding from a rectangular bluff-body were measured using PSP. The light intensity levels in the photographic images were then converted to instantaneous pressure histories by applying calibration constants, which were calculated from a few dynamic pressure sensors placed at selective points on the plate. Fourier transform of the time-histories from a large number of spatial points provided k-ω spectra for pressure fluctuations. The data provides first glimpse into the possibility of creating detailed forcing functions for vibro-acoustics analysis of aerospace vehicles, albeit for a limited frequency range.

Pressure Sensitive Paint↗

NASA NTRSDate not supplied

One Giant Leap for Womankind: An Aerodynamic Study of the SLS Rocket ft. Pressure-Sensitive Paint

Combining computer vision techniques, high-speed cameras, pressure-sensitive paint and a transonic wind tunnel, NASA studies unsteady aerodynamic forces on the Space Launch System rocket with unprecedented temporal and spatial resolution. NASA’s most powerful supercomputer, Pleiades, enables parallel processing and real-time visualization to investigate buffet forces and aeroacoustic physics.

Lucy Tang↗

NASA NTRSDate not supplied

One Giant Leap for Womankind: Studying NASA’s SLS Rocket With Pressure-Sensitive Paint

Combining computer vision techniques, high-speed cameras, pressure-sensitive paint and a transonic wind tunnel, NASA studies unsteady aerodynamic forces on the Space Launch System rocket with unprecedented temporal and spatial resolution. NASA’s most powerful supercomputer, Pleiades, enables real-time visualization and parallel processing to investigate buffet forces and aeroacoustic physics.

Lucy Zhonghui Tang↗

NASA NTRS1998

Application of Pressure-Sensitive Paint to Rotorcraft

In this presentation, a plan to develop methods for applying pressure-sensitive paint to rotorcraft will be described. These methods are needed because flows over rotor blades are typically very complex and poorly understood and because conventional methods for measuring unsteady pressures on rotor blades (using unsteady pressure transducers provide grossly inadequate spatial resolution. Since PSP is a surface, rather than a point, measurement technique, it has the potential to significantly increase the spatial resolution )f pressure measurements on rotor blades. PSP techniques currently in use at Ames were developed for measuring steady pressures on rigid, complex airplane configurations in large, production wind tunnels. Applying PSP to rotorcraft requires a significant departure from these techniques. First and most importantly new, fast-responding and self-referencing pressure paints are required. The paints must be fast (98% response in 1-5 msec) to resolve flow unsteadiness; they must be self-referencing (or "binary") to account for changes in incident light intensity due to deflection of flexible rotors. Self-referencing paints have been used at Ames for some time; however, these paints have response times that are far too long for unsteady applications. Flash illumination is required to resolve flow unsteadiness and to minimize image blurring due to relative motion between the model and the camera. Current practice at Ames is to use continuous illumination Finally, "in situ" paint calibration versus measurements by pressure transducers, which is current Ames practice, is not practical because of the difficulty and expense of installing transducers in rotor blades. Instead, the paint must be calibrated "a priori" in a calibration chamber. A sequence of five experiments that systematically isolates and addresses the problems involved in making PSP measurements on rotor blades has been planned. These are: (1) measurements on a rigid rotor in hover; (2) measurements on a flexible rotor in hover; measurements of paint response time in a calibration apparatus; (4) measurements on a rigid, two-dimensional oscillating airfoil; and (5) measurements on a flexible rotor in forward flight. Experiments were recently conducted at Ames where PSP measurements were made on a rigid oscillating airfoil (experiment type 4) and on a flexible rotor in hover (experiment type 2). Preliminary results from these experiments will be discussed.

Schairer, Edward T.↗

NASA NTRS2003

Transonic Experimental Observations of Abrupt Wing Stall on an F/A-18E Model (Invited)

A transonic wind tunnel test of an 8% F/A-18E model was conducted in the NASA Langley Research Center (LaRC) 16 ft Transonic Tunnel (16-ft TT) to investigate on-surface flow physics during stall. The technical approach employed focused on correlating static (or time-averaged) and unsteady wind-tunnel test data to the unsteady wing-stall events using force, moment, pressure, and pressure-sensitive-paint measurements. This paper focuses on data obtained on the pre-production configuration of the F/A-18E aircraft at Mach number of 0.90. The flow unsteadiness occurring on the wing as the wing went through the stall process was captured using the time history of balance and pressure measurements and by calculating the root mean square (RMS) for a number of instrument signals. The second step was to gather global perspectives on the pressures influencing the wing stall process. The abrupt wing stall experienced by the 8% F/A-18E Model was observed to be an unsteady event triggered by the rapid advancement of separation, which had migrated forward from the trailing edge, to the leading-edge flap hingeline over a very small increment in angle of attack. The angle of attack at which this stall occurred varied, from run to run, over an 1 deg increment. The abrupt wing stall was observed, using pressure-sensitive-paint, to occur simultaneously on both wing panels or asymmetrically. The pressure-sensitive paint data and wingroot bending moment data were essential in providing insight to the flow structures occurring over the wing and the possible asymmetry of those flow structures. A repeatability analysis conducted on eight runs of static data provided a quick and inexpensive examination of the unsteady aerodynamic characteristics of abrupt wing stall. The results of the repeatability analysis agreed extremely well with data obtained using unsteady measurement techniques. This approach could be used to identify test conditions for more complex unsteady data measurements using special instrumentation.

McMillin, S. Naomi↗

NASA NTRS2003

Transonic Experimental Observations of Abrupt Wing Stall on an F/A-l8E Model

A transonic wind tunnel test of an 8% F/A-18E model was conducted in the NASA Langley Research Center (LaRC) 16 ft Transonic Tunnel (16-ft TT) to investigate on-surface flow physics during stall. The technical approach employed focused on correlating static (or time-averaged) and unsteady wind-tunnel test data to the unsteady wing-stall events using force, moment, pressure, and pressure-sensitive-paint measurements. This paper focuses on data obtained on the pre-production configuration of the F/A-18E aircraft at Mach number of 0.90. The flow unsteadiness occurring on the wing as the wing went through the stall process was captured using the time history of balance and pressure measurements and by calculating the root mean square (RMS) for a number of instrument signals. The second step was to gather global perspectives on the pressures influencing the wing stall process. The abrupt wing stall experienced by the 8% F/A-18E Model was observed to be an unsteady event triggered by the rapid advancement of separation, which had migrated forward from the trailing edge, to the leading-edge flap hingeline over a very small increment in angle of attack. The angle of attack at which this stall occurred varied, from run to run, over an 1 degree increment. The abrupt wing stall was observed, using pressure-sensitive-paint, to occur simultaneously on both wing panels or asymmetrically. The pressure-sensitive paint data and wing-root bending moment data were essential in providing insight to the flow structures occurring over the wing and the possible asymmetry of those flow structures. A repeatability analysis conducted on eight runs of static data provided a quick and inexpensive examination of the unsteady aerodynamic characteristics of abrupt wing stall. The results of the repeatability analysis agreed extremely well with data obtained using unsteady measurement techniques. This approach could be used to identify test conditions for more complex unsteady data measurements using special instrumentation.

McMillin, S. Naomi↗

NASA NTRS2001

Diffusion Time-Scale of Porous Pressure-Sensitive Paint

Pressure-sensitive paint (PSP) is an optical pressure sensor that utilizes the oxygen quenching of luminescence. PSP measurements in unsteady aerodynamic flows require fast time response of the paint. There are two characteristic time-scales that are related to the time response of PSP. One is the luminescent lifetime representing an intrinsic physical limit for the achievable temporal resolution of PSP. Another is the time-scale of oxygen diffusion across the PSP layer. When the time-scale of oxygen diffusion is much larger than the luminescent lifetime, the time response of PSP is controlled by oxygen diffusion. In a thin homogenous polymer layer where diffusion is Fickian, the oxygen concentration 1021 can be described by the diffusion equation in one-dimension.

Liu, Tianshu↗