Hyperfine splittings and g sub F values of metastable H sub 2.
Energy from hydrogen molecule interaction with magnetic field and from hyperfine interaction expressed as products of irreducible spherical tensors
MAGNETIC FIELD↗
SEARCH · Engineering Papers
Results for “tensor product”
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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A Dynamic Nonlinear Subgrid-Scale Model for Large-Eddy Simulation of Complex Turbulent Flows
We present a new dynamic nonlinear subgrid-scale (SGS) model for large-eddy simulations (LES) and apply it to compute a flow involving pressure gradients, surface curvature and separation, for which data from a direct numerical simulation are available for comparison. The model, inspired by the triple model idea of Bardina et al. (“Improved Turbulence Models Based on Large Eddy Simulation of Homogeneous, Incompressible, Turbulent Flows,” Report No. TF-19, Thermosciences Division, Department of Mechanical Engineering, Stanford University, 1983), includes a Galilean-invariant term called the modified Leonard stress tensor, and two nonlinear terms comprised of the products of the strain-rate and rotation-rate tensors for an improved representation of the subgrid-scale dissipation, backscatter and anisotropy effects. The model does not employ any ad hoc averaging or clipping procedures, and does not require the specification of a characteristic length scale; hence, it naturally avoids the ambiguities associated with defining a proper length scale for anisotropic grids. Results from the wall-resolved LES of flow past a Gaussian bump using the new model demonstrate improved prediction of skin-friction, flow separation, mean flow profiles and turbulent quantities when compared to implicit LES as well as explicit LES using the Vreman SGS model on the same grid.
Turbulence↗
Search for subgrid scale parameterization by projection pursuit regression
The dependence of subgrid-scale stresses on variables of the resolved field is studied using direct numerical simulations of isotropic turbulence, homogeneous shear flow, and channel flow. The projection pursuit algorithm, a promising new regression tool for high-dimensional data, is used to systematically search through a large collection of resolved variables, such as components of the strain rate, vorticity, velocity gradients at neighboring grid points, etc. For the case of isotropic turbulence, the search algorithm recovers the linear dependence on the rate of strain (which is necessary to transfer energy to subgrid scales) but is unable to determine any other more complex relationship. For shear flows, however, new systematic relations beyond eddy viscosity are found. For the homogeneous shear flow, the results suggest that products of the mean rotation rate tensor with both the fluctuating strain rate and fluctuating rotation rate tensors are important quantities in parameterizing the subgrid-scale stresses. A model incorporating these terms is proposed. When evaluated with direct numerical simulation data, this model significantly increases the correlation between the modeled and exact stresses, as compared with the Smagorinsky model. In the case of channel flow, the stresses are found to correlate with products of the fluctuating strain and rotation rate tensors. The mean rates of rotation or strain do not appear to be important in this case, and the model determined for homogeneous shear flow does not perform well when tested with channel flow data. Many questions remain about the physical mechanisms underlying these findings, about possible Reynolds number dependence, and, given the low level of correlations, about their impact on modeling. Nevertheless, demonstration of the existence of causal relations between sgs stresses and large-scale characteristics of turbulent shear flows, in addition to those necessary for energy transfer, provides important insight into the relation between scales in turbulent flows.
Meneveau, C.↗
Reynolds stress development in pressure-driven three-dimensional turbulent boundary layers
The development of the Reynolds stress field was studied for flows in which an initially two-dimensional boundary layer was skewed sideways by a spanwise pressure gradient ahead of an upstream-facing wedge. Two different wedges were used, providing a variation in the boundary-layer skewing. Measurements of all components of the Reynolds stress tensor and all ten triple products were measured using a rotatable cross-wire anemometer. The results show the expected lag of the shear stress vector behind the strain rate. Comparison of the two present experiments with previous data suggests that the lag can be estimated if the radius of curvature of the free-stream streamline is known. The magnitude of the shear stress vector in the plane of the wall is seen to decrease rapidly as the boundary-layer skewing increases. The amount of decrease is apparently related to the skewing angle between the wall and the free stream. The triple products evolve rapidly and profiles in the three-dimensional boundary layer are considerably different than two-dimensional profiles, leaving little hope for gradient transport models for the Reynolds stresses. The simplified model presented by Rotta (1979) performs reasonably well providing that an appropriate value of the T-parameter is chosen.
Anderson, Shawn D.↗
High Resolution Global Topography of Eros from NEAR Imaging and LIDAR Data
Principal Data Products: Ensemble of L-maps from SPC, Spacecraft state, Asteroid pole and rotation. Secondary Products: Global topography model, inertia tensor, gravity. Composite high resolution topography. Three dimensional image maps.
LIDAR↗
Products of multiple Fourier series with application to the multiblade transformation
A relatively simple and systematic method for forming the products of multiple Fourier series using tensor like operations is demonstrated. This symbolic multiplication can be performed for any arbitrary number of series, and the coefficients of a set of linear differential equations with periodic coefficients from a rotating coordinate system to a nonrotating system is also demonstrated. It is shown that using Fourier operations to perform this transformation make it easily understood, simple to apply, and generally applicable.
Kunz, D. L.↗
Implicit Large-Eddy Simulations of Compressible Mixing Layers
Implicit large-eddy simulations of the self-similar regions of two compressible mixing layers at high Reynolds number and convective Mach numbers of 0.381 and 0.690 were carried out. Experimental data was used as the input into a synthetic eddy method turbulent inflow boundary condition to initiate simulations. Computational grids with nearly isotropic cells were used and the grid counts ranged from 24 to 270 million points. Agreement with experimental velocity and Reynolds stress data was good given the simplifications of the simulation. The decrease in mixing layer growth rate with increasing compressibility compares well with the literature. Budgets for the Reynolds stress transport equation were extracted from the simulations. It was demonstrated that the grids used did not resolve the dissipation term at the experimental Reynolds number. However, low Reynolds number simulations indicate that the numerical dissipation replaces the physical dissipation and the dissipation term in the transport equation can be represented by the summation term that quantifies the imbalance in the budget. The budgets for the high Reynolds number mixing layers agree well with previous low Reynolds number studies; indicating that the primary mechanism for the reduced growth rate and increased anistropy of the Reynolds stress tensor due to compressibility is a reduction in the pressure-strain and production terms. A new scaling based on the magnitude of the Reynolds stress tensor is proposed. This scaling provides better relative comparisons between the data and shows that the decrease in pressure-strain in the shear component is the primary driver in reducing the transverse normal stress.
turbulence↗
Method and apparatus for second-rank tensor generation
A method and apparatus are disclosed for generation of second-rank tensors using a photorefractive crystal to perform the outer-product between two vectors via four-wave mixing, thereby taking 2n input data to a control n squared output data points. Two orthogonal amplitude modulated coherent vector beams x and y are expanded and then parallel sides of the photorefractive crystal in exact opposition. A beamsplitter is used to direct a coherent pumping beam onto the crystal at an appropriate angle so as to produce a conjugate beam that is the matrix product of the vector beam that propagates in the exact opposite direction from the pumping beam. The conjugate beam thus separated is the tensor output xy (sup T).
Liu, Hua-Kuang↗
Magnetospheric Multiscale (MMS) Mission Attitude Ground System Design
This paper presents an overview of the attitude ground system (AGS) currently under development for the Magnetospheric Multiscale (MMS) mission. The primary responsibilities for the MMS AGS are definitive attitude determination, validation of the onboard attitude filter, and computation of certain parameters needed to improve maneuver performance. For these purposes, the ground support utilities include attitude and rate estimation for validation of the onboard estimates, sensor calibration, inertia tensor calibration, accelerometer bias estimation, center of mass estimation, and production of a definitive attitude history for use by the science teams. Much of the AGS functionality already exists in utilities used at NASA's Goddard Space Flight Center with support heritage from many other missions, but new utilities are being created specifically for the MMS mission, such as for the inertia tensor, accelerometer bias, and center of mass estimation. Algorithms and test results for all the major AGS subsystems are presented here.
Sedlak, Joseph E.↗
Dynamic localization and second-order subgrid-scale models in large eddy simulations of channel flow
The objective here is to test the Dynamic Localization (DL) model in a wall-bounded channel flow for numerical stability and accuracy of results. Algebraic stress models suggest that the model for the residual subgrid-scale (SGS) Reynolds stress and scalar flux should generally have terms comprising most of the unique products of the resolved strain (S) and rotation (R) tensors with S and the resolved scalar gradient. The standard dynamic SGS model uses a simple (Smagorinsky) base model for the residual Reynolds stress, which is made proportional to S, and down-gradient base models for residual scalar fluxes; these correspond to the lowest, 'first-order' terms in algebraic stress models. Temporal scaling terms in these base models are formed from the magnitude of the resolved strain rate. While this is appropriate for simple shear flows, it may not be appropriate for more complicated flows (relevant to geophysical and astrophysical problems) that include any combination of shear, rotation, buoyancy, etc. On the other hand, the coefficient in the dynamic SGS model readily adjusts itself to different flow conditions and may adequately take account of these effects without the need for more complicated base models. Cabot (1993) has begun to test the dynamic SGS model in buoyant flows (Rayleigh-Benard and internally heated convection) with and without buoyancy terms explicitly included in the scaling terms of the base model; no great differences were found in large eddy simulation (LES) results for the different base model scalings. The second objective in this work is to test base models with additional, 'second-order' terms (e.g., S(sup 2) and RS for the residual Reynolds stress). These terms have been found to improve large-scale flow predictions by kappa-epsilon models in the presence of rotation and shear. Second-order base models will be tested here in the LES of channel flow with and without solid-body rotation and compared with results from the standard first-order base models to determine if there are significant differences or improvements in results that would warrant the added complexity of the second-order base models.
Cabot, William H.↗
An Experimental Investigation of a Wing-Fuselage Junction Model in the NASA Langley 14- by 22-Foot Subsonic Tunnel
Current turbulence models, such as those employed in Reynolds-averaged Navier-Stokes CFD, are unable to reliably predict the onset and extent of the three-dimensional separated flow that typically occurs in wing-fuselage junctions. To critically assess, as well as to improve upon, existing turbulence models, experimental validation-quality flow-field data in the junction region is needed. In this report, we present an overview of experimental measurements on a wing-fuselage junction model that addresses this need. The experimental measurements were performed in the NASA Langley 14- by 22-Foot Subsonic Tunnel. The model was a full-span wing-fuselage body that was configured with truncated DLR-F6 wings, both with and without leading-edge extensions at the wing root. The model was tested at a fixed chord Reynolds number of 2.4 million, and angles-of-attack ranging from -10 degrees to +10 degrees were considered. Flow-field measurements were performed with a pair of miniature laser Doppler velocimetry (LDV) probes that were housed inside the model and attached to three-axis traverse systems. One LDV probe was used to measure the separated flow field in the trailing-edge junction region. The other LDV probe was alternately used to measure the flow field in the leading-edge region of the wing and to measure the incoming fuselage boundary layer well upstream of the leading edge. Both LDV probes provided measurements from which all three mean velocity components, all six independent components of the Reynolds-stress tensor, and all ten independent components of the velocity triple products were calculated. In addition to the flow-field measurements, static and dynamic pressures were measured at selected locations on the wings and fuselage of the model, infrared imaging was used to characterize boundary-layer transition, oil-flow visualization was used to visualize the separated flow in the leading- and trailing-edge regions of the wing, and unsteady shear stress was measured at limited locations using capacitive shear-stress sensors. Sample results from the measurement techniques employed during the test are presented and discussed.
Kegerise, Michael A.↗
Optical Computation Of Matrices From Vectors
Proposed optical apparatus generates rectangular pattern of light and dark areas, brightnesses of which represent elements of matrix product of two vectors. Photorefractive effect gives rise to four-wave mixing, which generates output beam modulated spatially by matrix product xy(SupT). Optical multipliers used as real-time analog tensor generators, components of neural networks, and generators of patterns to steer electromagnetic beams by diffraction or to effect temporary interconnections in very-large-scale integrated circuits.
Liu, Hua-Kuang↗
An Introduction to Tensors for Students of Physics and Engineering
Tensor analysis is the type of subject that can make even the best of students shudder. My own post-graduate instructor in the subject took away much of the fear by speaking of an implicit rhythm in the peculiar notation traditionally used, and helped us to see how this rhythm plays its way throughout the various formalisms. Prior to taking that class, I had spent many years "playing" on my own with tensors. I found the going to be tremendously difficult but was able, over time, to back out some physical and geometrical considerations that helped to make the subject a little more transparent. Today, it is sometimes hard not to think in terms of tensors and their associated concepts. This article, prompted and greatly enhanced by Marlos Jacob, whom I've met only by e-mail, is an attempt to record those early notions concerning tensors. It is intended to serve as a bridge from the point where most undergraduate students "leave off" in their studies of mathematics to the place where most texts on tensor analysis begin. A basic knowledge of vectors, matrices, and physics is assumed. A semi-intuitive approach to those notions underlying tensor analysis is given via scalars, vectors, dyads, triads, and higher vector products. The reader must be prepared to do some mathematics and to think. For those students who wish to go beyond this humble start, I can only recommend my professor's wisdom: find the rhythm in the mathematics and you will fare pretty well.
Kolecki, Joseph C.↗
Vorticity, turbulence production, and turbulence induced accelerations in a rectangular jet as measured using 3-D LDA
The flow field of a rectangular jet with a 4:1 aspect ratio (50.4 x 12.7 mm) was studied at a Reynolds number of 100,000 (Mach number 0.09) using a 3-D laser Doppler anemometer system. Measurements were performed along the major and minor axis planes and at various downstream cross-sections of the jet. The mean velocity vector and entire Reynolds stress tensor were measured and presented in a previous publication. The present work presents the vorticity vector, turbulence production, and turbulence induced acceleration vector distributions which were calculated from the previously presented data.
Morrison, Gerald L.↗
Viscosity in a magnetized plasma - Physical interpretation
The viscosity of a fully ionized H plasma in a magnetic field is investigated theoretically, revising the derivation proposed by Braginskii (1965) for the viscous-stress-tensor viscosity coefficient eta(0). It is shown that eta(0) terms can be attributed to the tendency of the plasma to produce small thermal anisotropies during its evolution, and hence that they are fully incorporated in the gyrotropic diagonal pressure tensor, neglecting the off-diagonal terms. The role of collisions in preventing anisotropy production (and thus causing irreversible heating) is explored, and the implications of the eta(0) terms for the physics of 1000-10,000-km-scale structures in the solar corona are indicated.
Hollweg, J. V.↗
Sound production in a moving stream
Jet noise is modeled by particle-attached quadrupoles convected with the velocity of the actual fluid but positioned near a hypothetical instability-free vortex sheet. The strength of each quadrupole is Lighthill's stress tensor per unit mass. The work of Mani (1976) has shown that this type of model agrees well with experiment, and the present work establishes some of the equivalent sources needed for an exact analogy. The instability waves of the shear layer, as they grow into turbulence, are heard as sound that builds up as a precursor of the main turbulence-driven field. The circular compact jet is examined in some detail, and it is found that when the jet is very light it can provide a waveguide in which the effects of source activity persist for some time but eventually leak out as sound. This interaction greatly distorts the free field characteristics of the turbulent sources, so that Reynolds-stress-induced waves have an intensity that scales with the fourth power of jet velocity.
Dowling, A. P.↗
Operator split methods in the numerical solution of the finite deformation elastoplastic dynamic problem
The spatial formulation of the elastoplastic dynamic problem for finite deformations is considered. A thermodynamic argument leads to an additive decomposition of the spatial rate of deformation tensor and allows an operator split of the evolutionary equations of the problem into elastic and plastic parts. This operator split is taken as the basis for the definition of a global product algorithm. In the context of finite element discretization the product algorithm entails, for every time step, the solution of a nonlinear elastodynamic problem followed by the application of plastic algorithms that operate on the stresses and internal variables at the integration points and bring in the plastic constitutive equations. Suitable plastic algorithms are discussed for the cases of perfect and hardening plasticity and viscoplasticity. The proposed formalism does not depend on any notion of smoothness of the yield surface and is applicable to arbitrary convex elastic regions, with or without corners. The stabiity properties of the global product algorithm are shown to be identical to those of the algorithm used for the integration of the nonlinear elastodynamic problem. Numerical examples illustrate the accuracy of the method.
Pinsky, P. M.↗
Bounded energy states in homogeneous turbulent shear flow: An alternative view
The equilibrium structure of homogeneous turbulent shear flow is investigated from a theoretical standpoint. Existing turbulence models, in apparent agreement with physical and numerical experiments, predict an unbounded exponential time growth of the turbulent kinetic energy and dissipation rate; only the anisotropy tensor and turbulent time scale reach a structural equilibrium. It is shown that if vortex stretching is accounted for in the dissipation rate transport equation, then there can exist equilibrium solutions, with bounded energy states, where the turbulence production is balanced by its dissipation. Illustrative calculations are present for a k-epsilon model modified to account for vortex stretching. The calculations indicate an initial exponential time growth of the turbulent kinetic energy and dissipation rate for elapsed times that are as large as those considered in any of the previously conducted physical or numerical experiments on homogeneous shear flow. However, vortex stretching eventually takes over and forces a production-equals-dissipation equilibrium with bounded energy states. The validity of this result is further supported by an independent theoretical argument. It is concluded that the generally accepted structural equilibrium for homogeneous shear flow with unbounded component energies is in need of re-examination.
Bernard, Peter S.↗