Influence of concentrated effects on shallow shells
Fourier transform and theory of generalized functions to obtain differential equilibrium equations to determine concentrated bending and tangential forces on shallow shells
FOURIER TRANSFORM↗
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Results for “Moment equations”
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.
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Time to cycle slip in first and second order phase lock loops
Recursive differential equation for moments of time-to-cycle slip in first and second order phase lock loops
Schuchman, L.↗
Expanded envelope concepts for aircraft control-element failure detection and identification
The purpose of this effort was to develop and demonstrate concepts for expanding the envelope of failure detection and isolation (FDI) algorithms for aircraft-path failures. An algorithm which uses analytic-redundancy in the form of aerodynamic force and moment balance equations was used. Because aircraft-path FDI uses analytical models, there is a tradeoff between accuracy and the ability to detect and isolate failures. For single flight condition operation, design and analysis methods are developed to deal with this robustness problem. When the departure from the single flight condition is significant, algorithm adaptation is necessary. Adaptation requirements for the residual generation portion of the FDI algorithm are interpreted as the need for accurate, large-motion aero-models, over a broad range of velocity and altitude conditions. For the decision-making part of the algorithm, adaptation may require modifications to filtering operations, thresholds, and projection vectors that define the various hypothesis tests performed in the decision mechanism. Methods of obtaining and evaluating adequate residual generation and decision-making designs have been developed. The application of the residual generation ideas to a high-performance fighter is demonstrated by developing adaptive residuals for the AFTI-F-16 and simulating their behavior under a variety of maneuvers using the results of a NASA F-16 simulation.
Weiss, Jerold L.↗
Radiation and scattering from loaded microstrip antennas over a wide bandwidth
The integral equation and moment method solution is developed for two different antennas in the presence of an infinite grounded dielectric substrate. The first antenna is a rectangular microstrip patch antenna. This antenna is analyzed for excitation by an incident plane wave in free space and a vertical filament of uniform current in the dielectric. This antenna can be loaded by a lumped impedance in a vertical filament of uniform current extending from the patch through the dielectric to the ground plane. The radar cross section of the microstrip antenna is found from the plane wave excitation and shows good agreement to measurement for both an unloaded and loaded antenna. The input impedance is found from the current filament excitation. This is compared to the measured input impedance of a coaxially fed microstrip antenna and shows good agreement for both unloaded and loaded antennas when the dielectric substrate is much less than a wavelength. The second antenna is a vertical thin wire extending from the ground plane into or through the dielectric substrate. The mutual impedance between two imbedded monopoles is compared to a previous calculation.
Forrai, D. P.↗
Nonprincipal-plane scattering from flat plates: Second-order and corner diffraction and pattern control of horn antennas
Several high-frequency models for nonprincipal-plane scattering from a rectangular, perfectly conducting plate are examined. Two methods, the Method of Equivalent Currents and corner diffraction coefficients, are considered. Formulations for second-order Physical Theory of Diffraction equivalent currents and for corner diffracted fields are presented. Comparisons are made among plate models. Results away from grazing are accurate using only first-order terms. Near grazing, second-order and corner diffraction terms improve the results for many cases. The pattern control of horn antennas using lossy materials to coat the inner walls of the horn is also investigated. Integral Equation and Moment Method techniques are used to formulate the problem. It is clearly demonstrated that side lobe level reduction can be achieved using impedance surfaces on the inner walls of the horn.
Balanis, Constantine A.↗
Some considerations on an airfoil in an oscillating stream
Explore the source record for details and available documents.
AIRFOILS - OSCILLATIONS - AIR FORCES & MOMENTS↗
Modeling of Turbulent Swirling Flows
Aircraft engine combustors generally involve turbulent swirling flows in order to enhance fuel-air mixing and flame stabilization. It has long been recognized that eddy viscosity turbulence models are unable to appropriately model swirling flows. Therefore, it has been suggested that, for the modeling of these flows, a second order closure scheme should be considered because of its ability in the modeling of rotational and curvature effects. However, this scheme will require solution of many complicated second moment transport equations (six Reynolds stresses plus other scalar fluxes and variances), which is a difficult task for any CFD implementations. Also, this scheme will require a large amount of computer resources for a general combustor swirling flow. This report is devoted to the development of a cubic Reynolds stress-strain model for turbulent swirling flows, and was inspired by the work of Launder's group at UMIST. Using this type of model, one only needs to solve two turbulence equations, one for the turbulent kinetic energy k and the other for the dissipation rate epsilon. The cubic model developed in this report is based on a general Reynolds stress-strain relationship. Two flows have been chosen for model evaluation. One is a fully developed rotating pipe flow, and the other is a more complex flow with swirl and recirculation.
Shih, Tsan-Hsing↗
Initial Circulation and Peak Vorticity Behavior of Vortices Shed from Airfoil Vortex Generators
An extensive parametric study of vortices shed from airfoil vortex generators has been conducted to determine the dependence of initial vortex circulation and peak vorticity on elements of the airfoil geometry and impinging flow conditions. These elements include the airfoil angle of attack, chord length, span, aspect ratio, local boundary layer thickness, and free stream Mach number. In addition, the influence of airfoil-to-airfoil spacing on the circulation and peak vorticity has been examined for pairs of co-rotating and counter-rotating vortices. The vortex generators were symmetric airfoils having a NACA-0012 cross-sectional profile. These airfoils were mounted either in isolation, or in pairs, on the surface of a straight pipe. The turbulent boundary layer thickness to pipe radius ratio was about 17 percent. The circulation and peak vorticity data were derived from cross-plane velocity measurements acquired with a seven-hole probe at one chord-length downstream of the airfoil trailing edge location. The circulation is observed to be proportional to the free-stream Mach number, the angle-of-attack, and the span-to-boundary layer thickness ratio. With these parameters held constant, the circulation is observed to fall off in monotonic fashion with increasing airfoil aspect ratio. The peak vorticity is also observed to be proportional to the free-stream Mach number, the airfoil angle-of-attack, and the span-to-boundary layer thickness ratio. Unlike circulation, however, the peak vorticity is observed to increase with increasing aspect ratio, reaching a peak value at an aspect ratio of about 2.0 before falling off again at higher values of aspect ratio. Co-rotating vortices shed from closely spaced pairs of airfoils have values of circulation and peak vorticity under those values found for vortices shed from isolated airfoils of the same geometry. Conversely, counter-rotating vortices show enhanced values of circulation and peak vorticity when compared to values obtained in isolation. The circulation may be accurately modeled with an expression based on Prandtl's relationship between finite airfoil circulation and airfoil geometry. A correlation for the peak vorticity has been derived from a conservation relationship equating the moment at the airfoil tip to the rate of angular momentum production of the shed vortex, modeled as a Lamb (ideal viscous) vortex. This technique provides excellent qualitative agreement to the observed behavior of peak vorticity for low aspect ratio airfoils typically used as vortex generators.
Wendt, Bruce J.↗
Machine Learning Moment Closure Models for the Radiative Transfer Equation III: Enforcing Hyperbolicity and Physical Characteristic Speeds
This is the third paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation. In our previous work (Huang et al. in J Comput Phys 453:110941, 2022), we proposed an approach to learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventional $P_N$ closure. However, while the ML moment closure has better accuracy, it is not able to guarantee hyperbolicity and has issues with long time stability. In our second paper (Huang et al., in: Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures, 2021. arXiv:2105.14410), we identified a symmetrizer which leads to conditions that enforce that the gradient based ML closure is symmetrizable hyperbolic and stable over long time. The limitation of this approach is that in practice the highest moment can only be related to four, or fewer, lower moments. In this paper, we propose a new method to enforce the hyperbolicity of the ML closure model. Motivated by the observation that the coefficient matrix of the closure system is a lower Hessenberg matrix, we relate its eigenvalues to the roots of an associated polynomial. Here, we design two new neural network architectures based on this relation. The ML closure model resulting from the first neural network is weakly hyperbolic and guarantees the physical characteristic speeds, i.e., the eigenvalues are bounded by the speed of light. The second model is strictly hyperbolic and does not guarantee the boundedness of the eigenvalues. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, stability and generalizability of our hyperbolic ML closure model.
97 MATHEMATICS AND COMPUTING↗
Prediction of the turbulent wake with second-order closure
A turbulence was envisioned whose energy containing scales would be Gaussian in the absence of inhomogeneity, gravity, etc. An equation was constructed for a function equivalent to the probability density, the second moment of which corresponded to the accepted modeled form of the Reynolds stress equation. The third moment equations obtained from this were simplified by the assumption of weak inhomogeneity. Calculations are presented with this model as well as interpretations of the results.
Taulbee, D. B.↗
Closed-form equations for the lift, drag, and pitching-moment coefficients of airfoil sections in subsonic flow
Closed-form equations for the lift, drag, and pitching moment coefficients of two dimensional airfoil sections in steady subsonic flow were obtained from published theoretical and experimental results. A turbulent boundary layer was assumed to exist on the airfoil surfaces. The effects of section angle of attack, Mach number, Reynolds number, and the specific airfoil type were considered. The equations were applicable through an angle of attack range of -180 deg to +180 deg; however, above about + or - 20 deg, the section characteristics were assumed to be functions only of angle of attack. A computer program is presented which evaluates the equations for a range of Mach numbers and angles of attack. Calculated results for the NACA 23012 airfoil section were compared with experimental data.
Smith, R. L.↗
Machine learning moment closure models for the radiative transfer equation I: Directly learning a gradient based closure
In this paper, we take a data-driven approach and apply machine learning to the moment closure problem for the radiative transfer equation in slab geometry. Instead of learning the unclosed high order moment, we propose to directly learn the gradient of the high order moment using neural networks. This new approach is consistent with the exact closure we derive for the free streaming limit and also provides a natural output normalization. Finally, a variety of benchmark tests, including the variable scattering problem, the Gaussian source problem with both periodic and reflecting boundaries, and the two-material problem, show both good accuracy and generalizability of our machine learning closure model.
Computer science↗
A simplified rotor system mathematical model for piloted flight dynamics simulation
The model was developed for real-time pilot-in-the-loop investigation of helicopter flying qualities. The mathematical model included the tip-path plane dynamics and several primary rotor design parameters, such as flapping hinge restraint, flapping hinge offset, blade Lock number, and pitch-flap coupling. The model was used in several exploratory studies of the flying qualities of helicopters with a variety of rotor systems. The basic assumptions used and the major steps involved in the development of the set of equations listed are described. The equations consisted of the tip-path plane dynamic equation, the equations for the main rotor forces and moments, and the equation for control phasing required to achieve decoupling in pitch and roll due to cyclic inputs.
Chen, R. T. N.↗
Nonlinear feedback control for high alpha flight
Analytical aerodynamic models are derived from a high alpha 6 DOF wind tunnel model. One detail model requires some interpolation between nonlinear functions of alpha. One analytical model requires no interpolation and as such is a completely continuous model. Flight path optimization is conducted on the basic maneuvers: half-loop, 90 degree pitch-up, and level turn. The optimal control analysis uses the derived analytical model in the equations of motion and is based on both moment and force equations. The maximum principle solution for the half-loop is poststall trajectory performing the half-loop in 13.6 seconds. The agility induced by thrust vectoring capability provided a minimum effect on reducing the maneuver time. By means of thrust vectoring control the 90 degrees pitch-up maneuver can be executed in a small place over a short time interval. The agility capability of thrust vectoring is quite beneficial for pitch-up maneuvers. The level turn results are based currently on only outer layer solutions of singular perturbation. Poststall solutions provide high turn rates but generate higher losses of energy than that of classical sustained solutions.
Stalford, Harold↗
Magnetic-Confinement Fusion—Plasma Theory: Tokamak Magnetohydrodynamic Equilibrium and Stability
Magnetohydrodynamics (MHD) provides a useful model to describe the crucial plasma macroscopic equilibrium and stability behaviors in toroidal tokamak devices by considering the plasma as a conducting fluid interacting with a surrounding confining electromagnetic field. MHD is the most basic plasma model, incorporating most large-scale phenomena, including plasma equilibrium and all major instabilities. MHD equations are obtained by taking moments of the Boltzmann equations for different plasma species. They provide a set of comprehensive physics constrains to compute and optimize the equilibrium plasma shape and pressure and current profiles that are critical to its stability and performance. In the ideal case, the equations have special properties that lead to efficient numerical calculation schemes, the most important of which is the ideal MHD energy principle for linear stability against small departures from equilibrium. In a tokamak plasma, equilibrium pressure is mostly destabilizing for MHD modes, whereas equilibrium current is also often a major driving force. Plasma resistivity creates new freedom for a MHD instability to grow, but there are also cases where the plasma resistivity plays a stabilizing role. Equilibrium toroidal flow and/or flow shear can affect MHD instabilities. Principal MHD instabilities include the internal kink mode, sawtooth, fishbone, external kink, resistive wall mode, resistive interchange, tearing and neoclassical tearing modes (NTMs), locked modes, toroidal Alfven eigenmodes (TAEs), and edge localized modes (ELMs). Fast-growing MHD instabilities can lead to an abrupt plasma disruption and termination that can potentially damage the device plasma facing components (PFCs) and in-vessel structures. Furthermore, an important MHD application is to develop robust techniques to mitigate and control MHD instabilities.
Dispersive pellet injection↗
Conservational PDF Equations of Turbulence
Recently we have revisited the traditional probability density function (PDF) equations for the velocity and species in turbulent incompressible flows. They are all unclosed due to the appearance of various conditional means which are modeled empirically. However, we have observed that it is possible to establish a closed velocity PDF equation and a closed joint velocity and species PDF equation through conditions derived from the integral form of the Navier-Stokes equations. Although, in theory, the resulted PDF equations are neither general nor unique, they nevertheless lead to the exact transport equations for the first moment as well as all higher order moments. We refer these PDF equations as the conservational PDF equations. This observation is worth further exploration for its validity and CFD application
Shih, Tsan-Hsing↗
Turbulent fluid motion IV-averages, Reynolds decomposition, and the closure problem
Ensemble, time, and space averages as applied to turbulent quantities are discussed, and pertinent properties of the averages are obtained. Those properties, together with Reynolds decomposition, are used to derive the averaged equations of motion and the one- and two-point moment or correlation equations. The terms in the various equations are interpreted. The closure problem of the averaged equations is discussed, and possible closure schemes are considered. Those schemes usually require an input of supplemental information unless the averaged equations are closed by calculating their terms by a numerical solution of the original unaveraged equations. The law of the wall for velocities and temperatures, the velocity- and temperature-defect laws, and the logarithmic laws for velocities and temperatures are derived. Various notions of randomness and their relation to turbulence are considered in light of ergodic theory.
Deissler, Robert G.↗
Characterization and integration of the singular test integrals in the method-of-moments implementation of the electric-field integral equation
In this paper, we characterize the logarithmic singularities arising in the method of moments from the Green’s function in integrals over the test domain, and we use two approaches for designing geometrically symmetric quadrature rules to integrate these singular integrands. These rules exhibit better convergence properties than quadrature rules for polynomials and, in general, lead to better accuracy with a lower number of quadrature points. In this work, we demonstrate their effectiveness for several examples encountered in both the scalar and vector potentials of the electric-field integral equation (singular, near-singular, and far interactions) as compared to the commonly employed polynomial scheme and the double Ma–Rokhlin–Wandzura (DMRW) rules, whose sample points are located asymmetrically within triangles.
42 ENGINEERING↗