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NASA NTRS1984

Second-order-accurate spatial differencing for the transonic small-disturbance equation

Current methods for calculating transonic flows with the small-disturbance potential equation are typically only first-order accurate in the supersonic regions of the flow. However, calculations using the full-potential equation show significant improvements in accuracy when second-order methods are used instead of first-order methods. In this paper, algorithms for the small-disturbance equations are presented, for both steady and unsteady flows, with spatial differencing that is second-order accurate in both the subsonic and supersonic regions of the flow. These algorithms are stable, simple extensions of implicit monotone algorithms; the former are only first-order accurate spatially in the supersonic regions. Several calculations in one and two dimensions have been made, and the first- and second-order calculations are compared. In one dimension, the calculations include the four types of shock-wave motion. In two dimensions, the comparisons include steady flow over a Korn airfoil and unsteady flow over a pitching airfoil. All the comparisons in two dimensions show that the new methods are just as stable numerically as the old methods, but improve the accuracy of the solutions. The algorithm improvements can be implemented in present computer codes by making minor coding modifications.

Goorjian, P. M.↗

NASA NTRS1985

Study of second order upwind differencing in a recirculating flow

The accuracy and stability of the second order upwind differencing scheme was investigated. The solution algorithm employed is based on a coupled solution of the nonlinear finite difference equations by the multigrid technique. Calculations have been made of the driven cavity flow for several Reynolds numbers and finite difference grids. In comparison with the hybrid differencing, the second order upwind differencing is somewhat more accurate but it is not monotonically accurate with mesh refinement. Also, the convergence of the solution algorithm deteriorates with the use of the second order upwind differencing.

Vanka, S. P.↗

NASA NTRS1985

Ladder and cross terms in second-order distorted Born approximation

In the strong fluctuation theory for a bounded layer of random discrete scatterers, the second moments of the fields in the second-order distorted Born approximation are obtained for copolarized and cross-polarized fields. The backscattering cross sections per unit area are calculated by including the mutual coherence of the fields due to the coincidental ray paths, and that due to the opposite ray paths, corresponding to the ladder and cross terms in the Feynman diagramatic representation. It is proved that the contributions from ladder and cross terms for the copolarized backscattering cross sections are the same, while the contributions for the cross-polarized backscattering cross sections are of the same order. The bistatic scattering coefficients in the second-order approximation for both the ladder and cross terms are also obtained. The contributions from the cross terms explain the enhancement in the backscattering direction.

Jin, Y. Q.↗

NASA NTRS1985

Second-order closure models for rotating turbulent flows

The physical properties of the commonly used second-order closure models are examined theoretically for rotating turbulent flows. Comparisons are made with results which are a rigorous consequence of the Navier-Stokes equations for the problem of a fully-developed turbulent channel flow in a rapidly rotating framework. It is demonstrated that all existing second-order closures yield spurious physical results for this test problem of rotating channel flow. In fact, the results obtained are shown to be substantially more unphysical than those obtained from the simpler K-epsilon and K-l models. Modifications in the basic structure of these second-order closure models are proposed which can alleviate this problem.

Speziale, C. G.↗

NASA NTRS1987

Positive ions of the first- and second-row transition metal hydrides

Theoretical dissociation energies for the first- and second-row transition metal hydride positive ions are critically compared against recent experimental values obtained from ion beam reactive scattering methods. Theoretical spectroscopic parameters and dipole moments are presented for the ground and several low-lying excited states. The calculations employ large Gaussian basis sets and account for electron correlation using the single-reference single- and double-excitation configuration interaction and coupled-pair-functional methods. The Darwin and mass-velocity contributions to the relativistic energy are included in the all-electron calculations on the first-row systems using first-order perturbation theory, and in the second-row systems using the Hay and Wadt relativistic effective core potentials. The theoretical D(0) values for the second-row transition metal hydride positive ions should provide a critical measure of the experimental values, which are not as refined as many of those in the first transition row.

Pettersson, Lars G. M.↗

NASA NTRS1987

Second-overtone models of RR Lyrae stars

Linear models of RR Lyrae stars pulsating in the second overtone indicate that this mode is self-excited only for rather low luminosities if the masses are normal. An unstable second-overtone model has been selected from the linear survey and has been followed up to limiting amplitude by using a nonlinear hydrodynamical computer program. Deep splitting of the main light peak in the theoretical light curve occurs but does not resemble any observed feature. Moreover, the distribution of periods of RR Lyrae stars in globular clusters can be accounted for adequately by fundamental-mode and first-overtone pulsation. It is concluded that second-overtone pulsators probably do not exist among RR Lyrae stars.

Stothers, Richard B.↗

NASA NTRS1987

Second-order upwind differencing in a recirculating flow

Vanka, S. P.↗

NASA NTRS1987

Second-order closure models for rotating turbulent flows

The physical properties of the commonly used second-order closure models are examined theoretically for rotating turbulent flows. Comparisons are made with results which are a rigorous consequence of the Navier-Stokes equations for the problem of fully-developed turbulent channel flow in a rapidly rotating framework. It is demonstrated that all existing second-order closures yield spurious physical results for this test problem of rotating channel flow. In fact, the results obtained are shown to be substantially more unphysical than those obtained from the simpler k-epsilon and k-l models. Modifications in the basic structure of these second-order closure models are proposed which can alleviate this problem.

Speziale, Charles G.↗

NASA NTRS1987

Spacecraft attitude determination using a second-order nonlinear filter

The stringent attitude determination accuracy and faster slew maneuver requirements demanded by present-day spacecraft control systems motivate the development of recursive nonlinear filters for attitude estimation. This paper presents the second-order filter development for the estimation of attitude quaternion using three-axis gyro and star tracker measurement data. Performance comparisons have been made by computer simulation of system models and filter mechanization. It is shown that the second-order filter consistently performs better than the extended Kalman filter when the performance index of the root sum square estimation error of the quaternion vector is compared. The second-order filter identifies the gyro drift rates faster than the extended Kalman filter. The uniqueness of this algorithm is the online generation of the time-varying process and measurement noise covariance matrices, derived as a function or the process and measurement nonlinearity, respectively.

Vathsal, S.↗

NASA NTRS1989

First and second order convex approximation strategies in structural optimization

In this paper, various methods based on convex approximation schemes are discussed that have demonstrated strong potential for efficient solution of structural optimization problems. First, the convex linearization method (Conlin) is briefly described, as well as one of its recent generalizations, the method of moving asymptotes (MMA). Both Conlin and MMA can be interpreted as first-order convex approximation methods that attempt to estimate the curvature of the problem functions on the basis of semiempirical rules. Attention is next directed toward methods that use diagonal second derivatives in order to provide a sound basis for building up high-quality explicit approximations of the behavior constraints. In particular, it is shown how second-order information can be effectively used without demanding a prohibitive computational cost. Various first-order and second-order approaches are compared by applying them to simple problems that have a closed form solution.

Fleury, C.↗

NASA NTRS1989

Controlling flexible structures with second order actuator dynamics

The control of flexible structures for those systems with actuators that are modeled by second order dynamics is examined. Two modeling approaches are investigated. First a stability and performance analysis is performed using a low order finite dimensional model of the structure. Secondly, a continuum model of the flexible structure to be controlled, coupled with lumped parameter second order dynamic models of the actuators performing the control is used. This model is appropriate in the modeling of the control of a flexible panel by proof-mass actuators as well as other beam, plate and shell like structural numbers. The model is verified with experimental measurements.

Inman, Daniel J.↗

NASA NTRS1990

Robust eigensystem assignment for second-order estimators

An approach for the robust eigensystem assignment of flexible structures using full state or output feedback is developed. Using the second-order dynamic equations, the approach can assign the eigenvalues of the system via velocity and displacement feedbacks, or acceleration and velocity feedbacks. The eigenvalues and eigenvectors of the system are assigned, via the second-order eigenvalue problem for the structural system, in two steps. First, an orthonormal basis spanning the attainable closed-loop eigenvector space corresponding to each desired closed-loop eigenvalue is generated using the Singular Value or QR decompositions. Second, a sequential procedure is used to choose a set of closed-loop eigenvectors that are as close as possible to the column space of a well-conditioned target matrix. Among the possible choices of the target matrix, the closest unitary matrix to the open-loop eigenvector matrix appears to be a suitable choice. A numerical example is given to illustrate the proposed algorithm.

Juang, Jer-Nan↗

NASA NTRS1990

Robust eigensystem assignment for second-order dynamic systems

A novel approach for the robust eigensystem assignment of flexible structures using full state or output feedback is developed. Using the second-order dynamic equations, the approach can assign the eigenvalues of the system via velocity and displacement feedbacks, or acceleration and velocity feedbacks. The eigenvalues and eigenvectors of the system are assigned, via the second-order eigenvalue problem for the structural system, in two steps. First, an orthonormal basis spanning the attainable closed-loop eigenvector space corresponding to each desired closed-loop eigenvalue is generated using the Singular Value or QR decompositions. Second, a sequential procedure is used to choose a set of closed-loop eigenvectors that are as close as possible to the column space of a well-conditioned target matrix. Among the possible choices of the target matrix, the closest unitary matrix to the open-loop eigenvector matrix appears to be a suitable choice. A numerical example is given to illustrate the proposed algorithm.

Juang, Jer-Nan↗

NASA NTRS1991

Organometallic Salts Generate Optical Second Harmonics

Series of organometallic salts exhibit large second-order dielectric susceptibilities, as evidenced by generation of second harmonics when illuminated at visible and near-infrared wavelengths. Investigations of these and related compounds continue with view toward development of materials for use as optical second-harmonic generators, electro-optical modulators, optical switches, piezoelectric sensors, and parametric crystals.

Marder, Seth R.↗

NASA NTRS1991

A second-order modeling study of confined swirling flow

A numerical study of a confined strong swirling flow is presented. Computations are performed using a differential second-moment closure. The effect of inlet dissipation rate on calculated mean and turbulence fields is investigated. Two pressure-strain models with various model constants are examined to demonstrate their influences on the predicted results. Finally, comparison of the differential second-moment calculation is made with algebraic second-moment results to determine a better suited model for complex swirling flows.

Nikjooy, M.↗

NASA NTRS1992

Second-order state estimation experiments using acceleration measurements

The estimation of dynamic states for feedback control of structural systems using second-order differential equations and acceleration measurements is described. The formulation of the observer model, and the design of the observer gains is discussed in detail. It is shown the second-order observer is highly stable because the stability constraints on the observer gains are model independent. The limitation of the proposed observer is the need for 'nearly' collocated actuators and accelerometers. Experimental results using a control-structure interaction testbed are presented that show the second-order observer provided more stability than a Kalman filter estimator without decreasing closed-loop performance.

Belvin, W. K.↗

NASA NTRS1992

Determination of robust stability margin for second-order systems

Robust stabilization of uncertain systems has been extensively investigated and the stability test for the whole set of uncertain parameters has been reduced to a finite number of test points, four points for the characteristic polynomial with independent coefficients. As a result the robust stability margin can be determined using a reasonable amount of computation. It is impossible to apply the results of the test to a practical system as the coefficients of the characteristic polynomial for a physical system are usually functions of uncertain parameters. However, many physical systems may be represented by a second-order mass-spring-damper system with a special multilinear form in its characteristic polynomial. This paper investigates second-order mass-spring-damper systems and the reduction of the number of test points. It is shown that such a system with arbritrary compensators always has a multilinear characteristic polynomial. It is also shown that a line in the two-dimensional parameter space forms the boundary after the mapping of a multilinear characteristic polynomial and this interior extreme line forms a conic curve in the complex plane. The boundary of uncertain domain for a multilinear polynomial with two uncertainty parameters can be determined analytically using this curve, and the four sides image of a square of the uncertain parameter. Therefore, the stability margin may be determined by checking the intersections of the boundary with the zero point. A similar procedure can be used for second-order systems with more than two uncertainty parameters when parameter optimization is used in determining the boundary.

Chuang, C.-H.↗

NASA NTRS1992

A second-order theory for transverse ion heating and momentum coupling due to electrostatic ion cyclotron waves

A second-order theory for electrostatic instabilities driven by counterstreaming ion beams is developed which describes momentum coupling and heating of the plasma via wave-particle interactions. Exchange rates between the waves and particles are derived, which are suitable for the fluid equations simulating microscopic effects on macroscopic scales. Using a fully kinetic simulation, the electrostatic ion cyclotron instability due to counterstreaming H(+) beams has been simulated. A power spectrum from the kinetic simulation is used to evaluate second-order exchange rates. The calculated heating and momentum loss from second-order theory is compared to the numerical simulation.

Miller, Ronald H.↗