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NASA NTRS1978

Heating due to the 9.6 micron ozone band in an inhomogeneous atmosphere - A new approximation

The Curtis-Godson (1953) method, when applied to the 9.6-micron ozone band, yields large errors. Kuriyan et al. (1977) suggested a method improving this result and yet retaining the analytic simplicity of the transmission function. In this paper, the analytic form of the derivative of the transmission function is obtained and then the heating rates are calculated with the new approximation. The errors in this case are negligibly small. Algebraic manipulation enables the reduction of this approximation to a form similar to that proposed by Goody (1964). This procedure permits interpretation of the boundary conditions in physical terms.

Kuriyan, J. G.↗

NASA NTRS1983

A tire contact solution technique

An efficient method for calculating the contact boundary and interfacial pressure distribution was developed. This solution technique utilizes the discrete Fourier transform to establish an influence coefficient matrix for the portion of the pressurized tire surface that may be in the contact region. This matrix is used in a linear algebra algorithm to determine the contact boundary and the array of forces within the boundary that are necessary to hold the tire in equilibrium against a specified contact surface. The algorithm also determines the normal and tangential displacements of those points on the tire surface that are included in the influence coefficient matrix. Displacements within and outside the contact region are calculated. The solution technique is implemented with a finite-element tire model that is based on orthotropic, nonlinear shell of revolution elements which can respond to nonaxisymmetric loads. A sample contact solution is presented.

Tielking, J. T.↗

NASA NTRS1983

Modeling of steady, rotational, transonic winds from rotating stars and galaxies

The theory of steady transonic winds from condensed bodies is extended to general, two-dimensional, axisymmetric systems. A stream function is used to reduce the gasdynamics equations to a single, second-order differential equation plus an algebraic equation for the density. The approach extends Parker's (1958) quasi-one-dimensional theory, which uses Bernoulli's theorem, to a complete two-dimensional calculation which includes vorticity and rotation. The conceptual basis of the stream function approach is described in detail, and a numerical method for solving the resulting equations is presented. The applications illustrate solutions for stellar (spherical source surface) and galactic (oblate spheroidal source surface) models. Among other things, it is found that for rapidly rotating stellar models the Coriolis force dominates the centrifugal terms and the streamlines bend toward the rotation axis as a consequence of the axisymmetry. For both the rotating and nonrotating galaxy models, the streamlines bend toward the equator.

Kopriva, D. A.↗

NASA NTRS1995

An extended structure-based model based on a stochastic eddy-axis evolution equation

We have proposed and implemented an extension of the structure-based model for weak deformations. It was shown that the extended model will correctly reduce to the form of standard k-e models for the case of equilibrium under weak mean strain. The realizability of the extended model is guaranteed by the method of its construction. The predictions of the proposed model were very good for rotating homogeneous shear flows and for irrotational axisymmetric contraction, but were seriously deficient in the case of plane strain and axisymmetric expansion. We have concluded that the problem behind these difficulties lies in the algebraic constitutive equation relating the Reynolds stresses to the structure parameters rather than in the slow model developed here. In its present form, this equation assumes that under irrotational strain the principal axes of the Reynolds stresses remain locked onto those of the eddy-axis tensor. This is correct in the RDT limit, but inappropriate under weaker mean strains, when the non-linear eddy-eddy interactions tend to misalign the two sets of principal axes and create some non-zero theta and gamma.

Kassinos, S. C.↗

NASA NTRS1992

On the Yakhot-Orszag renormalization group method for deriving turbulence statistics and models

An independent, comprehensive, critical review of the 'renormalization group' (RNG) theory of turbulence developed by Yakhot and Orszag (1986) is provided. Their basic theory for the Navier-Stokes equations is confirmed, and approximations in the scale removal procedure are discussed. The YO derivations of the velocity-derivative skewness and the transport equation for the energy dissipation rate are examined. An algebraic error in the derivation of the skewness is corrected. The corrected RNG skewness value of -0.59 is in agreement with experiments at moderate Reynolds numbers. Several problems are identified in the derivation of the energy dissipation rate equations which suggest that the derivation should be reformulated.

Smith, L. M.↗

DOE OSTI2020

Multiscale formulation of frictional contact mechanics at the pore scale

Direct numerical simulation (DNS) yields the highest fidelity predictions of mechanical deformation at the pore scale, but is prohibitively expensive for analyzing large or many samples. Discrete element methods (DEM) are an efficient alternative, but are limited to granular media and incapable of estimating or controlling prediction errors. We present a pore-level multiscale method (PLMM) that approximates DNS efficiently and with controllable accuracy. We focus on the linear elastic response of a consolidated geologic porous medium with arbitrary microstructure, heterogeneous mineralogy, containing cracks or defects. PLMM decomposes the solid phase into non-overlapping subdomains, on which local basis functions are constructed. The bases are then coupled with a global interface problem that accounts for slip or stick contact conditions between the subdomains. PLMM produces an initial, but accurate, approximation to DNS that can be iteratively improved. It is amenable to parallelism and allows for different mesh, models, and physics in each subdomain. An algebraic interpretation of PLMM as a preconditioner is also presented to allow non-intrusive implementation into existing solvers. Lastly, this work extends previous developments of PLMM in fluid dynamics to solid mechanics and enables future extensions towards modeling coupled flow and mechanics problems.

42 ENGINEERING↗

NASA NTRS1975

Estimation and Analysis of Nonlinear Stochastic Systems

The algebraic and geometric structures of certain classes of nonlinear stochastic systems were exploited in order to obtain useful stability and estimation results. The class of bilinear stochastic systems (or linear systems with multiplicative noise) was discussed. The stochastic stability of bilinear systems driven by colored noise was considered. Approximate methods for obtaining sufficient conditions for the stochastic stability of bilinear systems evolving on general Lie groups were discussed. Two classes of estimation problems involving bilinear systems were considered. It was proved that, for systems described by certain types of Volterra series expansions or by certain bilinear equations evolving on nilpotent or solvable Lie groups, the optimal conditional mean estimator consists of a finite dimensional nonlinear set of equations. The theory of harmonic analysis was used to derive suboptimal estimators for bilinear systems driven by white noise which evolve on compact Lie groups or homogeneous spaces.

Marcus, S. I.↗

NASA NTRS1976

Heat transfer to a full-coverage film-cooled surface with 30 degree slant-hole injection

Heat transfer behavior was studied in a turbulent boundary layer with full coverage film cooling through an array of discrete holes and with injection 30 deg to the wall surface in the downstream direction. Stanton numbers were measured for a staggered hole pattern with pitch-to-diameter ratios of 5 and 10, an injection mass flux ratio range of 0.1 to 1.3, and a range of Reynolds number Re sub x of 150,000 to 5 million. Air was used as the working fluid, and the mainstream velocity varied from 9.8 to 34.2 m/sec (32 to 112 ft/sec). The data were taken for secondary injection temperature equal to the wall temperature and also equal to the mainstream temperature. The data may be used to obtain Stanton number as a continuous function of the injectant temperature by use of linear superposition theory. The heat transfer coefficient is defined on the basis of a mainstream-to-wall temperature difference. This definition permits direct comparison of performance between film cooling and transpiration cooling. A differential prediction method was developed to predict the film cooling data base. The method utilizes a two-dimensional boundary layer program with routines to model the injection process and turbulence augmentation. The program marches in the streamwise direction, and when a row of holes is encountered, it stops and injects fluid into the boundary layer. The turbulence level is modeled by algebraically augmenting the mixing length, with the augmentation keyed to a penetration distance for the injected fluid.

Crawford, M. E.↗

NASA NTRS1985

Predictions of transonic separated flow with an eddy-viscosity/Reynolds-shear-stress closure model

In previous work, a new turbulence-closure model was specifically developed for two-dimensional turbulent boundary layers subjected to strong, adverse-pressure gradients and the attendant separation. This closure model was shown to perform well, but the inverse, boundary-layer calculations used in evaluating the model were not truly predictive since the mass-flux parameter distribution obtained from the experimental data had to be specified. The objective of this study was to evaluate this closure model for a series of strong, transonic, inviscid-viscous interactions with varying degrees of separation using a fully predictive calculation method based on the Reynolds-averaged, Navier-Stokes equations. Calculated results are compared with experimental results and are shown to be in excellent agreement, even for the interactions with massive separation. This closure model has the very favorable property of requiring little more computational effort than equilibrium, algebraic models.

Johnson, D. A.↗

DOE OSTI2023

Symmetric Random Butterfly Transform (SRBT) Based Preconditioner

Summary of work using Symmetric Random Butterfly Transformation (SRBT) in conjunction with Incomplete LDL T factorization as a preconditioner for FGMRES solver as a way of solving linear systems arising from interior point methods applied to power system problems. These linear systems have proven difficult to parallelize and this represents a possible route forward.

interior point optimization↗

NASA NTRS1978

Extension of the tridiagonal reduction (FEER) method for complex eigenvalue problems in NASTRAN

As in the case of real eigenvalue analysis, the eigensolutions closest to a selected point in the eigenspectrum were extracted from a reduced, symmetric, tridiagonal eigenmatrix whose order was much lower than that of the full size problem. The reduction process was effected automatically, and thus avoided the arbitrary lumping of masses and other physical quantities at selected grid points. The statement of the algebraic eigenvalue problem admitted mass, damping, and stiffness matrices which were unrestricted in character, i.e., they might be real, symmetric or nonsymmetric, singular or nonsingular.

Newman, M.↗

NASA NTRS1978

Calculation of unsteady transonic flows using the integral equation method

The basic integral equations for a harmonically oscillating airfoil in a transonic flow with shock waves are derived; the reduced frequency is assumed to be small. The problems associated with shock wave motion are treated using a strained coordinate system. The integral equation is linear and consists of both line integrals and surface integrals over the flow field which are evaluated by quadrature. This leads to a set of linear algebraic equations that can be solved directly. The shock motion is obtained explicitly by enforcing the condition that the flow is continuous except at a shock wave. Results obtained for both lifting and nonlifting oscillatory flows agree satisfactorily with other accurate results.

Nixon, D.↗

DOE OSTI2020

Recovering the QNEC from the ANEC

We study the relative entropy in QFT comparing the vacuum state to a special family of purifications determined by an input state and constructed using relative modular flow. We use this to prove a conjecture by Wall that relates the shape derivative of relative entropy to a variational expression over the averaged null energy (ANE) of possible purifications. Furthermore, this variational expression can be used to easily prove the quantum null energy condition (QNEC). We formulate Wall’s conjecture as a theorem pertaining to operator algebras satisfying the properties of a half-sided modular inclusion, with the additional assumption that the input state has finite averaged null energy. We also give a new derivation of the strong superadditivity property of relative entropy in this con- text. We speculate about possible connections to the recent methods used to strengthen monotonicity of relative entropy with recovery maps.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Qubit Regularization and Qubit Embedding Algebras

Qubit regularization is a procedure to regularize the infinite dimensional local Hilbert space of bosonic fields to a finite dimensional one, which is a crucial step when trying to simulate lattice quantum field theories on a quantum computer. When the qubit-regularized lattice quantum fields preserve important symmetries of the original theory, qubit regularization naturally enforces certain algebraic structures on these quantum fields. We introduce the concept of qubit embedding algebras (QEAs) to characterize this algebraic structure associated with a qubit regularization scheme. We show a systematic procedure to derive QEAs for the O(N) lattice spin models and the SU(N) lattice gauge theories. While some of the QEAs we find were discovered earlier in the context of the D-theory approach, our method shows that QEAs are far richer. A more complete understanding of the QEAs could be helpful in recovering the fixed points of the desired quantum field theories.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1990

Computation and turbulence modeling for three-dimensional boundary layers including turbomachinery rotor flows

A method is developed for predicting the behavior of three-dimensional, turbulent boundary layers occurring in internal flows, including those on turbomachinery rotor blades. These boundary layers are complex, turbulent, and subject to Coriolis and centrifugal forces. The major thrust of this paper is the development and use of an algebraic Reynolds stress model (ARSM) that captures the changes in turbulent flow structure arising from curvature, rotation, and three dimensionality. The prediction of pressure-driven secondary flow agrees well with the measured data, and all three turbulence models (k-epsilon, algebraic eddy viscosity, and ARSM) show the same level of agreement. The prediction of boundary-layer development on rotor blades shows much better agreement with measurements with the ARSM. It is essential to employ higher-order turbulence models to capture the effects of rotation, curvature, and three dimensionality on boundary layers in turbomachinery.

Zhang, J.↗

NASA NTRS1992

Incompressible viscous flow computations for the pump components and the artificial heart

A finite-difference, three-dimensional incompressible Navier-Stokes formulation to calculate the flow through turbopump components is utilized. The solution method is based on the pseudocompressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line relaxation method. Both steady and unsteady flow calculations can be performed using the current algorithm. In this work, the equations are solved in steadily rotating reference frames by using the steady-state formulation in order to simulate the flow through a turbopump inducer. Eddy viscosity is computed by using an algebraic mixing-length turbulence model. Numerical results are compared with experimental measurements and a good agreement is found between the two. Included in the appendix is a paper on incompressible viscous flow through artificial heart devices with moving boundaries. Time-accurate calculations, such as impeller and diffusor interaction, will be reported in future work.

Kiris, Cetin↗

NASA NTRS1955

Application of the Generalized Shock-expansion Method to Inclined Bodies of Revolution Traveling at High Supersonic Airspeeds

The generalized shock-expansion method is applied to obtain solutions to the flow field about pointed bodies of revolution at high supersonic airspeeds and small angles of attack. Simple explicit expressions are obtained for the surface Mach numbers and surface pressures in the special case of slender bodies. In the case of inclined cones, algebraic solutions are obtained defining the entire flow field. Experimental pressure-distribution data for cones and ogives at Mach numbers from 3 to 5 are included. (author)

Savin, Raymond C↗

NASA NTRS1967

Matrix Analysis of Longitudinal and Torsional Vibrations in Nonuniform Multibranch Beams

Since longitudinal modes and frequencies provide basic data for dynamic analyses of arbitrary beam-like structures and since closed-form solutions for the modes are generally not feasible to obtain, an approximate method is developed for computing the natural frequencies and the corresponding mode shapes for a variable-section, unconstrained multibranch beam. A lumped mass analogy employing influence coefficients is used to represent the beam. The simultaneous equations of motion for the lumped mass system are derived in matrix form and algebraically manipulated to yield a classical eigenvalue equation solvable by standard procedures. The orthogonality relationship of the natural modes is derived and used to form the basis of an orthogonal sweeping process for determination of modes above the fundamental. Numerical examples including an application to a solid-fuel launch system are presented. Also, a detailed discussion is devoted to the theoretical verifications of the approximate modes and frequencies.

Wingate, Robert T.↗