Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “mathematical physics”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,099 records · Page 61
All sourcesNASA NTRSDOE OSTI
NASA NTRS2008

A New Bond Albedo for Performing Orbital Debris Brightness to Size Transformations

We have developed a technique for estimating the intrinsic size distribution of orbital debris objects via optical measurements alone. The process is predicated on the empirically observed power-law size distribution of debris (as indicated by radar RCS measurements) and the log-normal probability distribution of optical albedos as ascertained from phase (Lambertian) and range-corrected telescopic brightness measurements. Since the observed distribution of optical brightness is the product integral of the size distribution of the parent [debris] population with the albedo probability distribution, it is a straightforward matter to transform a given distribution of optical brightness back to a size distribution by the appropriate choice of a single albedo value. This is true because the integration of a powerlaw with a log-normal distribution (Fredholm Integral of the First Kind) yields a Gaussian-blurred power-law distribution with identical power-law exponent. Application of a single albedo to this distribution recovers a simple power-law [in size] which is linearly offset from the original distribution by a constant whose value depends on the choice of the albedo. Significantly, there exists a unique Bond albedo which, when applied to an observed brightness distribution, yields zero offset and therefore recovers the original size distribution. For physically realistic powerlaws of negative slope, the proper choice of albedo recovers the parent size distribution by compensating for the observational bias caused by the large number of small objects that appear anomalously large (bright) - and thereby skew the small population upward by rising above the detection threshold - and the lower number of large objects that appear anomalously small (dim). Based on this comprehensive analysis, a value of 0.13 should be applied to all orbital debris albedo-based brightness-to-size transformations regardless of data source. Its prima fascia genesis, derived and constructed from the current RCS to size conversion methodology (SiBAM Size-Based Estimation Model) and optical data reduction standards, assures consistency in application with the prior canonical value of 0.1. Herein we present the empirical and mathematical arguments for this approach and by example apply it to a comprehensive set of photometric data acquired via NASA's Liquid Mirror Telescopes during the 2000-2001 observing season.

Mulrooney, Mark K.↗
NASA NTRS2025

Mechanics of Preloaded Bolt Tensile Loading With Focus on Load Introduction Factor

The bolt tensile and joint separation loads are directly influenced by the locations at which the external loads enter the clamped members of a preloaded bolted joint (PBJ) and the associated load-paths through the joint. This physical load introduction mechanism affecting the bolt tensile loading is typically represented in the bolt tensile load equation, in part, by a load introduction factor (LIF), which was shown by H.M. Lee of Marshall Spaceflight Center to be a natural product of the bolt tensile load equation using a linear spring stiffness model. This LIF, being a function of load-path stiffness, has subsequently been denoted as the stiffness-based LIF (SBLIF), providing a framework to calculate the LIF using whatever load-path stiffness approximations are appropriate. Expanding upon the work of Lee, it is shown that the SBLIF and the joint stiffness factor are functions of the stiffnesses of the same load-paths and regions within a PBJ, and thus they should not be treated as independent variables. Mathematical expressions for the SBLIF are presented. Comparisons are shown between the analytically calculated SBLIF, the analytically calculated geometric LIF (GLIF), which is a simple clamped-member thickness ratio, the experimentally derived LIF, and the LIF determined by finite element analysis (FEA). Using experiment and FEA as a benchmark, the SBLIF, using traditional load-path stiffness approximations, enables a more accurate prediction of bolt tensile loading than the GLIF, although it can be unconservative near joint separation. The GLIF generally attributes more of the externally applied tensile load to the bolt than does the SBLIF, potentially resulting in heavier and/or more costly bolted joints. Mathematical relationships between the SBLIF and the GLIF are developed. Supplemental material is provided in the appendixes where the historical practice of using the joint compressive stiffness in place of the joint tensile stiffness is evaluated. The appendixes include step-by-step examples demonstrating the calculation of the SBLIF using traditional stiffness approximations and conclude with the development of the joint diagram in terms of the SBLIF, culminating into formulas for the key features of a joint diagram, which is useful for programming.

Load Path↗
NASA NTRS2001

SHIVA-(Spaceflight Holography in a Virtual Apparatus)

SHIVA (Spaceflight Holography Investigation in a Virtual Apparatus) will expand our understanding of the fundamental physics of particle movement in fluids by exploiting the power of holography in a spaceflight experiment'. In addition, the study will exploit the movement of particles in fluids to observe and quantify microgravity phenomena that are extremely important in materials sciences with applications both in space and on earth. The regime under scrutiny is the low Reynolds number, Stokes regime or creeping flow, which covers particles and bubbles moving at very low velocity. The equations describing this important regime have been under development and investigation for over 100 years and yet a complete analytical solution of the general equation had remained elusive yielding only approximations and numerical solutions. In the course of the ongoing NASA NRA, the first analytical solution of the general equation was produced by members of the investigator team using the mathematics of fractional derivatives. This opened the way to an even more insightful and important investigation of the phenomena in microgravity.

Trolinger, James D.↗
NASA NTRS1994

A uniform geometrical optics and an extended uniform geometrical theory of diffraction for evaluating high frequency EM fields near smooth caustics and composite shadow boundaries

A uniform geometrical optics (UGO) and an extended uniform geometrical theory of diffraction (EUTD) are developed for evaluating high frequency electromagnetic (EM) fields within transition regions associated with a two and three dimensional smooth caustic of reflected rays and a composite shadow boundary formed by the caustic termination or the confluence of the caustic with the reflection shadow boundary (RSB). The UGO is a uniform version of the classic geometrical optics (GO). It retains the simple ray optical expressions of classic GO and employs a new set of uniform reflection coefficients. The UGO also includes a uniform version of the complex GO ray field that exists on the dark side of the smooth caustic. The EUTD is an extension of the classic uniform geometrical theory of diffraction (UTD) and accounts for the non-ray optical behavior of the UGO reflected field near caustics by using a two-variable transition function in the expressions for the edge diffraction coefficients. It also uniformly recovers the classic UTD behavior of the edge diffracted field outside the composite shadow boundary transition region. The approach employed for constructing the UGO/EUTD solution is based on a spatial domain physical optics (PO) radiation integral representation for the fields which is then reduced using uniform asymptotic procedures. The UGO/EUTD analysis is also employed to investigate the far-zone RCS problem of plane wave scattering from two and three dimensional polynomial defined surfaces, and uniform reflection, zero-curvature, and edge diffraction coefficients are derived. Numerical results for the scattering and diffraction from cubic and fourth order polynomial strips are also shown and the UGO/EUTD solution is validated by comparison to an independent moment method (MM) solution. The UGO/EUTD solution is also compared with the classic GO/UTD solution. The failure of the classic techniques near caustics and composite shadow boundaries is clearly demonstrated and it is shown that the UGO/EUTD results remain valid and uniformly reduce to the classic results away from the transition regions. Mathematical details on the asymptotic properties and efficient numerical evaluation of the canonical functions involved in the UGO/EUTD expressions are also provided.

Constantinides, E. D.↗
NASA NTRS2018

Elliptic-Cylindrical Analytical Flux Rope Model for Magnetic Clouds

In this paper, we present the elliptic-cylindrical analytical flux rope model, which constitutes the first level of complexity above that of a circular-cylindrical geometry. The framework of this series of models was established by Nieves-Chinchilla et al. with the circular-cylindrical analytical flux rope model. The model describes the magnetic flux rope topology with distorted cross section as a possible consequence of the flux rope interaction with the solar wind. In this model, for the first time, a flux rope is completely described by a nonorthogonal geometry. The Maxwell equations can be consistently solved using tensorial analysis, and relevant physical quantities can be derived, such as magnetic fluxes, number of turns, or Lorentz force distribution. The model is generalized in termsof the radial dependence of the poloidal and axial current density components. The circular-cylindrical reconstruction technique has been adapted to the new geometry for a specific case of the model and tested againstan interplanetary coronal mass ejection observed by the Wind spacecraft on 2005 June 12. In this specific case,from the comparative analysis between the circular-cylindrical and elliptic-cylindrical models, the inclusion ofthe cross-section distortion in the 3D reconstruction results in significant changes in the derived axis orientation,size, central magnetic field, magnetic fluxes, and force-freeness. The case studied in this paper exemplifies the useof the model and reconstruction technique developed. Furthermore, the novel mathematical formulation to modelflux ropes in heliophysics paves the way to the inclusion of more complex magnetic field configurations.

CMEs↗
NASA NTRS1994

Stabilization of posture by precision contact of the index finger

Postural sway during quiet stance increases if sight of the surroundings is denied. We studied how sensory-motor information about body displacement provided by contact of the index finger with a stationary bar can be used to stabilize balance in the absence of vision. Stabilization equivalent to the contribution conferred by vision was achieved at contact force levels less than 1 N. This value is much below that necessary to provide significant physical stabilization of the body. We interpret our findings in relation to tactile thresholds for motion detection, "precision grip," and proprioceptive and sensory-motor information about the configuration of the arm to the torso. In conditions allowing higher force levels at the fingertip (5-8 N), subjects assumed a passively stable state to stabilize their stance.

Non-NASA Center↗
NASA NTRS2001

Analysis and experiments with an elephant's trunk robot

The area of tentacle and trunk type biological manipulation is not new, but there has been little progress in the development and application of a physical device to simulate these types of manipulation. Our research in this area is based on using an 'elephant trunk' robot. In this paper, we review the construction of the robot and how it compares to biological manipulators. We then apply our previously designed kinematic model to describe the kinematics of the robot. We finish by providing some examples of motion planning and intelligent manipulation using the robot.

Non-NASA Center↗
NASA NTRS1994

A brief survey of constrained mechanics and variational problems in terms of differential forms

There has been considerable interest recently in constrained mechanics and variational problems. This is in part due to applied interests (such as 'non-holonomic mechanics in robotics') and in other part due to the fact that several schools of 'pure' mathematics have found that this classical subject is of importance for what they are trying to do. I have made various attempts at developing these subjects since my Lincoln lab days of the late 1950's. In this Chapter, I will sketch a Unified point of view, using Cartan's approach with differential forms. This has the advantage from the C-O-R viewpoint being developed in this Volume that the extension from 'smooth' to 'generalized' data is very systematic and algebraic. (I will only deal with the 'smooth' point of view in this Chapter; I will develop the 'generalized function' material at a later point.) The material presented briefly here about Variational Calculus and Constrained Mechanics can be found in more detail in my books, 'Differential Geometry and the Calculus of Variations', 'Lie Algebras and Quantum Mechanics', and 'Geometry, Physics and Systems'.

Hermann, Robert↗
NASA NTRS2010

Modeling Common-Sense Decisions in Artificial Intelligence

A methodology has been conceived for efficient synthesis of dynamical models that simulate common-sense decision- making processes. This methodology is intended to contribute to the design of artificial-intelligence systems that could imitate human common-sense decision making or assist humans in making correct decisions in unanticipated circumstances. This methodology is a product of continuing research on mathematical models of the behaviors of single- and multi-agent systems known in biology, economics, and sociology, ranging from a single-cell organism at one extreme to the whole of human society at the other extreme. Earlier results of this research were reported in several prior NASA Tech Briefs articles, the three most recent and relevant being Characteristics of Dynamics of Intelligent Systems (NPO -21037), NASA Tech Briefs, Vol. 26, No. 12 (December 2002), page 48; Self-Supervised Dynamical Systems (NPO-30634), NASA Tech Briefs, Vol. 27, No. 3 (March 2003), page 72; and Complexity for Survival of Living Systems (NPO- 43302), NASA Tech Briefs, Vol. 33, No. 7 (July 2009), page 62. The methodology involves the concepts reported previously, albeit viewed from a different perspective. One of the main underlying ideas is to extend the application of physical first principles to the behaviors of living systems. Models of motor dynamics are used to simulate the observable behaviors of systems or objects of interest, and models of mental dynamics are used to represent the evolution of the corresponding knowledge bases. For a given system, the knowledge base is modeled in the form of probability distributions and the mental dynamics is represented by models of the evolution of the probability densities or, equivalently, models of flows of information. Autonomy is imparted to the decisionmaking process by feedback from mental to motor dynamics. This feedback replaces unavailable external information by information stored in the internal knowledge base. Representation of the dynamical models in a parameterized form reduces the task of common-sense-based decision making to a solution of the following hetero-associated-memory problem: store a set of m predetermined stochastic processes given by their probability distributions in such a way that when presented with an unexpected change in the form of an input out of the set of M inputs, the coupled motormental dynamics converges to the corresponding one of the m pre-assigned stochastic process, and a sample of this process represents the decision.

Zak, Michail↗
NASA NTRS1989

Computer simulation of the mathematical modeling involved in constitutive equation development: Via symbolic computations

Development of new material models for describing the high temperature constitutive behavior of real materials represents an important area of research in engineering disciplines. Derivation of mathematical expressions (constitutive equations) which describe this high temperature material behavior can be quite time consuming, involved and error prone; thus intelligent application of symbolic systems to facilitate this tedious process can be of significant benefit. A computerized procedure (SDICE) capable of efficiently deriving potential based constitutive models, in analytical form is presented. This package, running under MACSYMA, has the following features: partial differentiation, tensor computations, automatic grouping and labeling of common factors, expression substitution and simplification, back substitution of invariant and tensorial relations and a relational data base. Also limited aspects of invariant theory were incorporated into SDICE due to the utilization of potentials as a starting point and the desire for these potentials to be frame invariant (objective). Finally not only calculation of flow and/or evolutionary laws were accomplished but also the determination of history independent nonphysical coefficients in terms of physically measurable parameters, e.g., Young's modulus, was achieved. The uniqueness of SDICE resides in its ability to manipulate expressions in a general yet predefined order and simplify expressions so as to limit expression growth. Results are displayed when applicable utilizing index notation.

Arnold, S. M.↗
NASA NTRS2004

Aerodynamic Optimization of Rocket Control Surface Geometry Using Cartesian Methods and CAD Geometry

Aerodynamic design is an iterative process involving geometry manipulation and complex computational analysis subject to physical constraints and aerodynamic objectives. A design cycle consists of first establishing the performance of a baseline design, which is usually created with low-fidelity engineering tools, and then progressively optimizing the design to maximize its performance. Optimization techniques have evolved from relying exclusively on designer intuition and insight in traditional trial and error methods, to sophisticated local and global search methods. Recent attempts at automating the search through a large design space with formal optimization methods include both database driven and direct evaluation schemes. Databases are being used in conjunction with surrogate and neural network models as a basis on which to run optimization algorithms. Optimization algorithms are also being driven by the direct evaluation of objectives and constraints using high-fidelity simulations. Surrogate methods use data points obtained from simulations, and possibly gradients evaluated at the data points, to create mathematical approximations of a database. Neural network models work in a similar fashion, using a number of high-fidelity database calculations as training iterations to create a database model. Optimal designs are obtained by coupling an optimization algorithm to the database model. Evaluation of the current best design then gives either a new local optima and/or increases the fidelity of the approximation model for the next iteration. Surrogate methods have also been developed that iterate on the selection of data points to decrease the uncertainty of the approximation model prior to searching for an optimal design. The database approximation models for each of these cases, however, become computationally expensive with increase in dimensionality. Thus the method of using optimization algorithms to search a database model becomes problematic as the number of design variables is increased.

Nelson, Andrea↗
NASA NTRS2007

Short-Term Forecasting of Radiation Belt and Ring Current

A computer program implements a mathematical model of the radiation-belt and ring-current plasmas resulting from interactions between the solar wind and the Earth s magnetic field, for the purpose of predicting fluxes of energetic electrons (10 keV to 5 MeV) and protons (10 keV to 1 MeV), which are hazardous to humans and spacecraft. Given solar-wind and interplanetary-magnetic-field data as inputs, the program solves the convection-diffusion equations of plasma distribution functions in the range of 2 to 10 Earth radii. Phenomena represented in the model include particle drifts resulting from the gradient and curvature of the magnetic field; electric fields associated with the rotation of the Earth, convection, and temporal variation of the magnetic field; and losses along particle-drift paths. The model can readily accommodate new magnetic- and electric-field submodels and new information regarding physical processes that drive the radiation-belt and ring-current plasmas. Despite the complexity of the model, the program can be run in real time on ordinary computers. At present, the program can calculate present electron and proton fluxes; after further development, it should be able to predict the fluxes 24 hours in advance

Fok, Mei-Ching↗
NASA NTRS1993

Unified approach for incompressible flows

An unified approach for solving both compressible and incompressible flows was investigated in this study. The difference in CFD code development between incompressible and compressible flows is due to the mathematical characteristics. However, if one can modify the continuity equation for incompressible flows by introducing pseudocompressibility, the governing equations for incompressible flows would have the same mathematical characters as compressible flows. The application of a compressible flow code to solve incompressible flows becomes feasible. Among numerical algorithms developed for compressible flows, the Centered Total Variation Diminishing (CTVD) schemes possess better mathematical properties to damp out the spurious oscillations while providing high-order accuracy for high speed flows. It leads us to believe that CTVD schemes can equally well solve incompressible flows. In this study, the governing equations for incompressible flows include the continuity equation and momentum equations. The continuity equation is modified by adding a time-derivative of the pressure term containing the artificial compressibility. The modified continuity equation together with the unsteady momentum equations forms a hyperbolic-parabolic type of time-dependent system of equations. The continuity equation is modified by adding a time-derivative of the pressure term containing the artificial compressibility. The modified continuity equation together with the unsteady momentum equations forms a hyperbolic-parabolic type of time-dependent system of equations. Thus, the CTVD schemes can be implemented. In addition, the boundary conditions including physical and numerical boundary conditions must be properly specified to obtain accurate solution. The CFD code for this research is currently in progress. Flow past a circular cylinder will be used for numerical experiments to determine the accuracy and efficiency of the code before applying this code to more specific applications.

Chang, Tyne-Hsien↗
NASA NTRS1998

Teaching Soil Science in Primary and Secondary Schools

Earth's thin layer of soil is a fragile resource, made up of minerals, organic materials, air, water, and billions of living organisms. Soils plays a variety of critical roles that sustain life on Earth. If we think about soil, we tend to see it first as the source of most of the food we eat and the fibers we use, such as wood and cotton. Few students realize that soils also provide the key ingredients to many of the medicines (including antibiotics), cosmetics, and dyes that we use. Fewer still understand the importance of soils in integrating, controlling, and regulating the movement of air, water, materials, and energy between the hydrosphere, lithosphere, atmosphere, and biosphere. Because soil sustains life, it offers both a context and a natural laboratory for investigating these interactions. The enclosed poster, which integrates soil profiles with typical landscapes in which soils form, can also help students explore the interrelationships of Earth systems and gain an understanding of our soil resources. The poster, produced jointly by the American Geological Institute and the Soil Science Society of America, aims to increase awareness of the importance of soil, as does the GLOBE (Global Learning and Observations To Benefit the Environment) Program. Vice President Al Gore instituted the GLOBE Program on Earth Day of 1993 to increase environmental awareness of individuals throughout the world, contribute to a better scientific understanding of the Earth, and help all students reach higher levels of achievement in science and mathematics. GLOBE functions as a partnership between scientists, students, and teachers in which scientists design protocols for specific measurements they need for their research that can be performed by K-12 students. Teachers are trained in the GLOBE protocols and teach them to their students. Students make the measurements, enter data via the Internet to a central data archive, and the data becomes available to scientists and the general community. Students benefit by having a "hands-on"experience in science, math, and technology, using their local environment as a learning laboratory, as well as contact with scientists and other students around the world. Soil investigations have become an essential component of GLOBE. The protocols that have been developed so far within the GLOBE program include GPS Location, Atmosphere/Climate, Soil Characterization, Soil Moisture and Temperature, Land Cover/Biometry, Hydrology, and Satellite Image Classification. For the GLOBE Soil Characterization Protocol, students explore the physical. chemical, and morphological properties of the soil at their study site. They are asked to dig a pit or use an auger to about 1 meter at at least 2 sites.

Levine, Elissa R.↗
NASA NTRS2008

Friction Stir Welding

Friction stir welding (FSW) is a solid state welding process invented in 1991 at The Welding Institute in the United Kingdom. A weld is made in the FSW process by translating a rotating pin along a weld seam so as to stir the sides of the seam together. FSW avoids deleterious effects inherent in melting and promises to be an important welding process for any industries where welds of optimal quality are demanded. This article provides an introduction to the FSW process. The chief concern is the physical effect of the tool on the weld metal: how weld seam bonding takes place, what kind of weld structure is generated, potential problems, possible defects for example, and implications for process parameters and tool design. Weld properties are determined by structure, and the structure of friction stir welds is determined by the weld metal flow field in the vicinity of the weld tool. Metal flow in the vicinity of the weld tool is explained through a simple kinematic flow model that decomposes the flow field into three basic component flows: a uniform translation, a rotating solid cylinder, and a ring vortex encircling the tool. The flow components, superposed to construct the flow model, can be related to particular aspects of weld process parameters and tool design; they provide a bridge to an understanding of a complex-at-first-glance weld structure. Torques and forces are also discussed. Some simple mathematical models of structural aspects, torques, and forces are included.

Nunes, Arthur C., Jr.↗
NASA NTRS1989

Towards real-time simulation of large space structures: Stabilization of fluid/thermal/structure interactions and implementation on high performance supercomputers

Within the Center for Space Construction, the SIMSTRUC project's objectives center around the development of simulation tools for the realistic analysis of large space structures. The word 'tools' is the broad sense; it designates mathematical models, finite element/finite difference formulations, computational algorithms, implementations on advanced computer architectures, and visualization capabilities. The results of our activities during the first year within the SIMSTRUC project are reported. On the modeling side, an alternative approach to fluid/thermal/structure interaction analysis that is a departure from the 'loosely coupled' and 'unified' approaches that are being currently practiced are described. The advantages of our approach both in terms of accuracy and computational efficiency were demonstrated. On the computational side, a software architecture for parallel/vector and massively parallel supercomputers that speeds up finite element and finite difference computations by several orders of magnitude is presented. As an example, the simulation of the deployment of a space structure that used to require over six hours of a workstation using a conventional finite element software, now runs on a multiprocessor using a parallel computation strategy in less than three seconds. In order to promote the physical understanding of the simulation behavior, a real-time visualization capability on the Connection Machine, which allows the analyst to watch the graphical animation of the results at the same time these are generated, was also developed. It is believed that by combining efficient analytical formulations with the state-of-the-art high performance computer implementations and superfast visualization capabilities, SIMSTRUC is moving fast towards the real-time simulation of large space structures. The designers as well as the researchers will certainly benefit from this technology.

Farhat, C.↗
NASA NTRS1996

Controlled Quantum Packets

We look at time evolution of a physical system from the point of view of dynamical control theory. Normally we solve motion equation with a given external potential and we obtain time evolution. Standard examples are the trajectories in classical mechanics or the wave functions in Quantum Mechanics. In the control theory, we have the configurational variables of a physical system, we choose a velocity field and with a suited strategy we force the physical system to have a well defined evolution. The evolution of the system is the 'premium' that the controller receives if he has adopted the right strategy. The strategy is given by well suited laboratory devices. The control mechanisms are in many cases non linear; it is necessary, namely, a feedback mechanism to retain in time the selected evolution. Our aim is to introduce a scheme to obtain Quantum wave packets by control theory. The program is to choose the characteristics of a packet, that is, the equation of evolution for its centre and a controlled dispersion, and to give a building scheme from some initial state (for example a solution of stationary Schroedinger equation). It seems natural in this view to use stochastic approach to Quantum Mechanics, that is, Stochastic Mechanics [S.M.]. It is a quantization scheme different from ordinary ones only formally. This approach introduces in quantum theory the whole mathematical apparatus of stochastic control theory. Stochastic Mechanics, in our view, is more intuitive when we want to study all the classical-like problems. We apply our scheme to build two classes of quantum packets both derived generalizing some properties of coherent states.

DeMartino, Salvatore↗
NASA NTRS1987

Investigation of hypersonic rarefied flow on a spherical nose of the AOTV

The Navier-Stokes (NS) equations were integrated numerically for investigating the flow characteristics on the forepart of the spherical nose of a space vehicle such as the AOTV or AFE by a modified Accelerated Successive Replacement (ASR) scheme under hypersonic rarefied conditions. Technical feasibility of the mathematical approach was demonstrated by computing the flowfield on a spherical nose under conditions that the AFE encounters at times t = 15 and 20 seconds after its reentry into the atmosphere. Local similar solutions for the merged layer flow along the stagnation line of the sphere were developed. These are correct to the same degree of accuracy as the NS equations. These solutions provided stagnation line boundary conditions for the domain of integration on the spherical noise. Also, a parametric study of the stagnation line solution was made with a view to understand the flow characteristics in tunnels with different ambient fluids. Analytical expressions for surface slip temperature, jump conditions, and concentration level in the presence of the real gas effects at the top of the Knudsen layer were derived and used to calculate the stagnation line flowfield with nonequilibrium dissociation and ionization. A number of graphics were drawn to illustrate the basic physics of the flowfields. The present analysis can be extended to include real gas effects and to bodies of arbitrary shapes. It can further provide boundary conditions for integrating the NS equations in the near wake region.

Jain, Amolak C.↗