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All sourcesNASA NTRSDOE OSTI
NASA NTRS1991

Shock waves and the Ffowcs Williams-Hawkings equation

The expansion of the double divergence of the generalized Lighthill stress tensor, which is the basis of the concept of the role played by shock and contact discontinuities as sources of dipole and monopole sound, is presently applied to the simplest transonic flows: (1) a fixed wing in steady motion, for which there is no sound field, and (2) a hovering helicopter blade that produces a sound field. Attention is given to the contribution of the shock to sound from the viewpoint of energy conservation; the shock emerges as the source of only the quantity of entropy.

Isom, Morris P.↗
NASA NTRS1986

Viscosity and the Chew-Goldberger-Low equations in the solar corona

A general discussion of the dominant terms in the stress tensor in a magnetized plasma such as the solar corona is presented. The importance of dissipative terms such as electrical resistivity, heat conduction, and interspecies collisions is assessed. For average coronal conditions, the proton stress tensor is found to reduce to the dominant terms in the classical expression for the viscous stress. The classical expression can fail in the transition region, however. In the diffusion region of reconnection, classical viscosity will be appropriate if the resistivity is very large, so that the diffusion region is broad, but in that case the viscous heating is small compared to the resistive heating. On the other hand, the more general expression for the stress tensor is required if the diffusion region is thin; the stress tensor will be important in this case. The electron stress tensor is also considered, and it is shown how the classical expression for electron viscosity can fail in the transition region and lower corona.

Hollweg, J. V.↗
NASA NTRS1996

Element Library for Three-Dimensional Stress Analysis by the Integrated Force Method

The Integrated Force Method, a recently developed method for analyzing structures, is extended in this paper to three-dimensional structural analysis. First, a general formulation is developed to generate the stress interpolation matrix in terms of complete polynomials of the required order. The formulation is based on definitions of the stress tensor components in term of stress functions. The stress functions are written as complete polynomials and substituted into expressions for stress components. Then elimination of the dependent coefficients leaves the stress components expressed as complete polynomials whose coefficients are defined as generalized independent forces. Such derived components of the stress tensor identically satisfy homogenous Navier equations of equilibrium. The resulting element matrices are invariant with respect to coordinate transformation and are free of spurious zero-energy modes. The formulation provides a rational way to calculate the exact number of independent forces necessary to arrive at an approximation of the required order for complete polynomials. The influence of reducing the number of independent forces on the accuracy of the response is also analyzed. The stress fields derived are used to develop a comprehensive finite element library for three-dimensional structural analysis by the Integrated Force Method. Both tetrahedral- and hexahedral-shaped elements capable of modeling arbitrary geometric configurations are developed. A number of examples with known analytical solutions are solved by using the developments presented herein. The results are in good agreement with the analytical solutions. The responses obtained with the Integrated Force Method are also compared with those generated by the standard displacement method. In most cases, the performance of the Integrated Force Method is better overall.

Kaljevic, Igor↗
NASA NTRS1977

Evaluation of the Tensor Polynomial and Hoffman strength theories for composite materials

The Hoffman theory and the Tensor Polynomial (Tsai-Wu) theory with the stress interaction term set equal to zero have been found to be preferred alternatives to the general Tensor Polynomial theory for predicting strength of filamentary composite laminae. These theories were used to predict failure of off-axis boron/epoxy and E-glass/epoxy test specimens and gave excellent agreement with available experimental results. A numerical experiment was also performed to estimate the errors for ten different composite systems under six different loadings. The maximum error in predicted failure loads among all cases was below 10 percent. These results suggest that the Hoffman failure theory and the Tensor Polynomial theory with the stress interaction term equal to zero can predict failure of practical filamentary composite materials under general biaxial loading with sufficient accuracy for engineering applications.

Narayanaswami, R.↗
NASA NTRS1974

Lagrangian methods in plasma dynamics. II - Construction of Lagrangians for plasmas

Consideration of the construction of suitable Lagrangian functions for the dynamics of a cold plasma in such a way as to retain the relativistically covariant formalism. In one method, this is achieved by the introduction of a set of three variables which label the world lines of the particles. A second method results in a Clebsch-type representation. Sturrock's relativistic Lagrangian and Low's hot plasma Lagrangian are also briefly discussed in the context of the present work. The behavior of the canonical stress tensor is considered. The applicability of many of the general results in part I (Dougherty, 1970) is ensured by establishing the existence of the Lagrangian function.

Dougherty, J. P.↗
NASA NTRS1997

Time Evolution of Modeled Reynolds Stresses in Planar Homogeneous Flows

The analytic expression of the time evolution of the Reynolds stress anisotropy tensor in all planar homogeneous flows is obtained by exact integration of the modeled differential Reynolds stress equations. The procedure is based on results of tensor representation theory, is applicable for general pressure-strain correlation tensors, and can account for any additional turbulence anisotropy effects included in the closure. An explicit solution of the resulting system of scalar ordinary differential equations is obtained for the case of a linear pressure-strain correlation tensor. The properties of this solution are discussed, and the dynamic behavior of the Reynolds stresses is studied, including limit cycles and sensitivity to initial anisotropies.

Jongen, T.↗
NASA NTRS1986

Experimental study of a pressure-driven, three-dimensional, turbulent boundary layer

The paper presents the results of an experimental investigation into a strongly-skewed, pressure-driven, three-dimensional turbulent boundary layer. Mean velocity and static pressure measurements reveal the general nature of the flowfield studied, while detailed measurements of the Reynolds stress tensor indicate the effect of the strong pressure field on the turbulent velocities. The rate and magnitude of the skewing in the boundary layer emerge as the most important parameters governing the behavior of the Reynolds stresses.

Anderson, S. D.↗
NASA NTRS1992

Visualization of second order tensor fields and matrix data

We present a study of the visualization of 3-D second order tensor fields and matrix data. The general problem of visualizing unsymmetric real or complex Hermitian second order tensor fields can be reduced to the simultaneous visualization of a real and symmetric second order tensor field and a real vector field. As opposed to the discrete iconic techniques commonly used in multivariate data visualization, the emphasis is on exploiting the mathematical properties of tensor fields in order to facilitate their visualization and to produce a continuous representation of the data. We focus on interactively sensing and exploring real and symmetric second order tensor data by generalizing the vector notion of streamline to the tensor concept of hyperstreamline. We stress the importance of a structural analysis of the data field analogous to the techniques of vector field topology extraction in order to obtain a unique and objective representation of second order tensor fields.

Delmarcelle, Thierry↗
NASA NTRS1993

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.↗
NASA NTRS1992

Determining the nuclear equation of state from neutron-star masses and radii

A method is developed for determining the nuclear equation of state directly from a knowledge of the masses and radii of neutron stars. This analysis assumes only that equilibrium neutron-star matter has the stress-energy tensor of an isotropic fluid with a barotropic equation of state, and that general relativity describes a neutron star's internal gravitational field. We present numerical examples which illustrate how well this method will determine the equation of state when the appropriate observational data become available.

Lindblom, Lee↗
NASA NTRS2006

Anisotropic Developments for Homogeneous Shear Flows

The general decomposition of the spectral correlation tensor R(sub ij)(k) by Cambon et al. (J. Fluid Mech., 202, 295; J. Fluid Mech., 337, 303) into directional and polarization components is applied to the representation of R(sub ij)(k) by spherically averaged quantities. The decomposition splits the deviatoric part H(sub ij)(k) of the spherical average of R(sub ij)(k) into directional and polarization components H(sub ij)(sup e)(k) and H(sub ij)(sup z)(k). A self-consistent representation of the spectral tensor in the limit of weak anisotropy is constructed in terms of these spherically averaged quantities. The directional polarization components must be treated independently: models that attempt the same representation of the spectral tensor using the spherical average H(sub ij)(k) alone prove to be inconsistent with Navier-Stokes dynamics. In particular, a spectral tensor consistent with a prescribed Reynolds stress is not unique. The degree of anisotropy permitted by this theory is restricted by realizability requirements. Since these requirements will be less severe in a more accurate theory, a preliminary account is given of how to generalize the formalism of spherical averages to higher expansion of the spectral tensor. Directionality is described by a conventional expansion in spherical harmonics, but polarization requires an expansion in tensorial spherical harmonics generated by irreducible representations of the spatial rotation group SO(exp 3). These expansions are considered in more detail in the special case of axial symmetry.

Cambon, Claude↗
NASA NTRS1991

On the most general tensor B(sub ij) which is zero for i does not equal j, and the most general isotropic tensor I(sub ij)

It is shown that the most general second order tensor B sub ij which is zero for i not = j is proportional to the Kronecker delta (Delta sub ij). By a slight modification of that argument, the known result was obtained that the most general second order isotropic tensor is also proportional to Delta sub ij. These results are useful for instance in obtaining the stress tensor for a viscous fluid.

Deissler, Robert G.↗
NASA NTRS1979

Failure analysis of composite laminates with free edges

Linear elastic stress distributions obtained from a refined finite element mesh are used in conjunction with the tensor polynomial failure criterion to predict the initiation of failure in symmetric, finite-width graphite-epoxy laminates under tensile loading. Results are presented for a wide variety of laminates including: (+ and - theta)s angle-ply; cross-ply (0/90)s and (90/0)s; and quasi-isotropic (90/0/+ and - 45)s and (+ and -45/0/90)s. It is shown that the elastic stress distributions generally compare favorably with other published results, but also indicate improved satisfaction of the stress-free boundary conditions and indicate some differences in the singular behavior of selected stress components at the free edge. The tensor polynomial failure criterion is used to predict the location and mode of first failure. Examination of the individual terms of the polynomial indicates different modes of failure depending upon the laminate configuration.

Herakovich, C. T.↗
NASA NTRS1989

Viscoplasticity: A thermodynamic formulation

A thermodynamic foundation using the concept of internal state variables is given for a general theory of viscoplasticity, as it applies to initially isotropic materials. Three fundamental internal state variables are admitted. They are: a tensor valued back stress for kinematic effects, and the scalar valued drag and yield strengths for isotropic effects. All three are considered to phenomenologically evolve according to competitive processes between strain hardening, strain induced dynamic recovery, and time induced static recovery. Within this phenomenological framework, a thermodynamically admissible set of evolution equations is put forth. This theory allows each of the three fundamental internal variables to be composed as a sum of independently evolving constituents.

Freed, A. D.↗
NASA NTRS1990

Tidal disruption of solid bodies

The problem of stress, strain, and breakup in solid satellites and stray bodies subject to tidal perturbations is presently addressed in view of three novel considerations. After presenting a new analytic solution for the stress tensor in a homogeneous and compressible elastic sphere, where the inclusion of compressibility alters stresses by several percent, realistic failure criteria are noted to demonstrate the general failure of such ductile bodies as iron meteoroids by plastic shear, while brittle ice bodies fail by either tensile or shear fracture. A reexamination of crack propagation after initial failure allows the diverse breakup criteria to be reconciled.

Dobrovolskis, Anthony R.↗
NASA NTRS1977

A critique of some recent second-order turbulence closure models for compressible boundary layers

Computations based on two recently developed second-order turbulence closure models are compared with a series of boundary-layer experiments and with predictions of these experiments using an algebraic mixing length model. One of the models employs an eddy viscosity, whereas the other evaluates components of the Reynolds stress tensor. For flat plates, the computations are compared with the van Driest skin-friction transformation to assess the handling of compressibility. For boundary layers in pressure gradients, four experiments at Mach 4 and one at Mach 6.7 are used as the bases for comparison. In general, both models represent mean velocities and skin friction reasonably well, but represent the turbulence shear stress less accurately.

Rubesin, M. W.↗
NASA NTRS1974

Computation of turbulent flows

All of the work deals with a model that incorporates dynamical equations for the mean velocity field, for all components of the Reynolds stress tensor, and for the 'isotropic dissipation'. The trace of the Reynolds stress tensor provides a velocity scale for the turbulence. For completeness, a turbulence length or time scale is also needed; many workers impose the length scales, but in a generalized model it must be evolved by the turbulence itself. In order, the following special classes of flows are considered: (1) decay of isotropic turbulence, (2) return to isotropy in homogeneous turbulence without strain, (3) homogeneous turbulence with strain, and (4) inhomogeneous turbulence with strain and turbulent transport.

Reynolds, W. C.↗