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NASA NTRS2001

Implications of the Observed Mesoscale Variations of Clouds for Earth's Radiation Budget

The effect of small-spatial-scale cloud variations on radiative transfer in cloudy atmospheres currently receives a lot of research attention, but the available studies are not very clear about which spatial scales are important and report a very large range of estimates of the magnitude of the effects. Also, there have been no systematic investigations of how to measure and represent these cloud variations. We exploit the cloud climatology produced by the International Satellite Cloud Climatology Project (ISCCP) to: (1) define and test different methods of representing cloud variation statistics, (2) investigate the range of spatial scales that should be included, (3) characterize cloud variations over a range of space and time scales covering mesoscale (30 - 300 km, 3-12 hr) into part of the lower part of the synoptic scale (300 - 3000 km, 1-30 days), (4) obtain a climatology of the optical thickness, emissivity and cloud top temperature variability of clouds that can be used in weather and climate GCMS, together with the parameterization proposed by Cairns et al. (1999), to account for the effects of small-scale cloud variations on radiative fluxes, and (5) evaluate the effect of observed cloud variations on Earth's radiation budget. These results lead to the formulation of a revised conceptual model of clouds for use in radiative transfer calculations in GCMS. The complete variability climatology can be obtained from the ISCCP Web site at http://isccp.giss.nasa.gov.

Rossow, William B.↗

NASA NTRS2009

The Principle of Energetic Consistency

A basic result in estimation theory is that the minimum variance estimate of the dynamical state, given the observations, is the conditional mean estimate. This result holds independently of the specifics of any dynamical or observation nonlinearity or stochasticity, requiring only that the probability density function of the state, conditioned on the observations, has two moments. For nonlinear dynamics that conserve a total energy, this general result implies the principle of energetic consistency: if the dynamical variables are taken to be the natural energy variables, then the sum of the total energy of the conditional mean and the trace of the conditional covariance matrix (the total variance) is constant between observations. Ensemble Kalman filtering methods are designed to approximate the evolution of the conditional mean and covariance matrix. For them the principle of energetic consistency holds independently of ensemble size, even with covariance localization. However, full Kalman filter experiments with advection dynamics have shown that a small amount of numerical dissipation can cause a large, state-dependent loss of total variance, to the detriment of filter performance. The principle of energetic consistency offers a simple way to test whether this spurious loss of variance limits ensemble filter performance in full-blown applications. The classical second-moment closure (third-moment discard) equations also satisfy the principle of energetic consistency, independently of the rank of the conditional covariance matrix. Low-rank approximation of these equations offers an energetically consistent, computationally viable alternative to ensemble filtering. Current formulations of long-window, weak-constraint, four-dimensional variational methods are designed to approximate the conditional mode rather than the conditional mean. Thus they neglect the nonlinear bias term in the second-moment closure equation for the conditional mean. The principle of energetic consistency implies that, to precisely the extent that growing modes are important in data assimilation, this term is also important.

Cohn, Stephen E.↗

NASA NTRS2016

Model-Invariant Hybrid Computations of Separated Flows for RCA Standard Test Cases

NASA's Revolutionary Computational Aerosciences (RCA) subproject has identified several smooth-body separated flows as standard test cases to emphasize the challenge these flows present for computational methods and their importance to the aerospace community. Results of computations of two of these test cases, the NASA hump and the FAITH experiment, are presented. The computations were performed with the model-invariant hybrid LES-RANS formulation, implemented in the NASA code VULCAN-CFD. The model- invariant formulation employs gradual LES-RANS transitions and compensation for model variation to provide more accurate and efficient hybrid computations. Comparisons revealed that the LES-RANS transitions employed in these computations were sufficiently gradual that the compensating terms were unnecessary. Agreement with experiment was achieved only after reducing the turbulent viscosity to mitigate the effect of numerical dissipation. The stream-wise evolution of peak Reynolds shear stress was employed as a measure of turbulence dynamics in separated flows useful for evaluating computations.

Woodruff, Stephen↗

NASA NTRS2018

The Influence of Dihedral Angle Error Stability on Beam Deviation for Hollow Retro-Reflectors

Retro-reflectors consist of three reflective optical surfaces, which are oriented to reflect the input beam by 180 ◦. For retro-reflector components, it is common to specify an angular beam deviation tolerance, or rather the deviation from the exact 180 def return direction. Precision-aligned retro-reflectors provide 180 deg beam deviation with tolerances on the order of an arcsecond. It is well known that the performance of the retro-reflector depends on the ability to precisely orient the reflective surfaces at mutually perpendicular angles. Precision assembly is therefore critical to ensure highly accurate beam deviation. The dihedral angle errors, and hence the reflected beam deviation, can be measured for the retro-reflector after fabrication, typically by using interferometric techniques. Yet, what is not commonly reported for a fabricated retro-reflector is the stability of the angular beam deviation. For instance, thermo-mechanical effects of the components will contribute to variations in the return beam direction. While the actual stability is design-specific one can develop a mathematical representation for the expected change in the reflected beam direction as a function of the variation in the dihedral angle errors. Presented here is a mathematical formulation for a hollow retro-reflector's beam deviation as a function of the dihedral angle error stability.

Swank, Aaron J.↗

NASA NTRS2003

Further Developments in the MLPG Method for Beam Problems

An accurate and yet simple Meshless Local Petrov-Galerkin (MLPG) formulation for analyzing beam problems is presented. In the formulation, simple weight functions are chosen as test functions as in the conventional MLPG method. Linear test functions are also chosen, leading to a variation of the MLPG method that is computationally efficient compared to the conventional implementation. The MLPG method is evaluated by applying the formulation to a variety of patch tests, thin beam problems, and problems with load discontinuities. The formulation successfully reproduces exact solutions to machine accuracy when higher order power and spline functions are chosen as test functions or when the linear test function is used, and when constructing the trial functions, the order of the basis function is properly balanced by the order of the weight function. For mixed boundary value problems, deflections, slopes, moments, and shear forces are calculated to the same accuracy by the MLPG method without the use of elaborate post-processing techniques. Problems with load discontinuities require special care - when a reasonable number of nodes are used, the method yields very accurate results.

Raju, I. S.↗

NASA NTRS1995

Completed Beltrami-Michell formulation for analyzing mixed boundary value problems in elasticity

In elasticity, the method of forces, wherein stress parameters are considered as the primary unknowns, is known as the Beltrami-Michell formulation (BMF). The existing BMF can only solve stress boundary value problems; it cannot handle the more prevalent displacement of mixed boundary value problems of elasticity. Therefore, this formulation, which has restricted application, could not become a true alternative to the Navier's displacement method, which can solve all three types of boundary value problems. The restrictions in the BMF have been alleviated by augmenting the classical formulation with a novel set of conditions identified as the boundary compatibility conditions. This new method, which completes the classical force formulation, has been termed the completed Beltrami-Michell formulation (CBMF). The CBMF can solve general elasticity problems with stress, displacement, and mixed boundary conditions in terms of stresses as the primary unknowns. The CBMF is derived from the stationary condition of the variational functional of the integrated force method. In the CBMF, stresses for kinematically stable structures can be obtained without any reference to the displacements either in the field or on the boundary. This paper presents the CBMF and its derivation from the variational functional of the integrated force method. Several examples are presented to demonstrate the applicability of the completed formulation for analyzing mixed boundary value problems under thermomechanical loads. Selected example problems include a cylindrical shell wherein membrane and bending responses are coupled, and a composite circular plate.

Patnaik, Surya N.↗

NASA NTRS1992

Membrane triangles with corner drilling freedoms. III - Implementation and performance evaluation

This paper completes a three-part series on the formulation of 3-node, 9-dof membrane triangles with corner drilling freedoms based on parametrized variational principles. The first four sections cover element implementation details including determination of optimal parameters and treatment of distributed loads. Then three elements of this type, labeled ALL, FF and EFF-ANDES, are tested on standard plane stress problems. ALL represents numerically integrated versions of Allman's 1988 triangle; FF is based on the free formulation triangle presented by Bergan and Felippa in 1985; and EFF-ANDES represent two different formulations of the optimal triangle derived in Parts I and II. The numerical studies indicate that the ALL, FF and EFF-ANDES elements are comparable in accuracy for elements of unitary aspect ratios. The ALL elements are found to stiffen rapidly in inplane bending for high aspect ratios, whereas the FF and EFF elements maintain accuracy. The EFF and ANDES implementations have a moderate edge in formation speed over the FF.

Felippa, Carlos A.↗

NASA NTRS1984

Generalized formulation of TVD Lax-Wendroff schemes

The work of Davis which imports the concept of total variation diminution (TVD) into non-upwinded, Lax-Wendroff type schemes is reformulated in a way which is easier to analyze. The analysis reveals a class of TVD schemes not observed by Davis. Only the case of one dimensional linear advection is treated.

Roe, P. L.↗

NASA NTRS1969

Dynamic characteristics of a variable-mass flexible missile: Dynamics of a two-stage variable-mass flexible rocket

The dynamic characteristics of two-stage slender elastic body were investigated. The first stage, containing a solid-fuel rocket, possesses variable mass while the second stage, envisioned as a flexible case, contains packaged instruments of constant mass. The mathematical formulation was in terms of vector equations of motion transformed by a variational principle into sets of scalar differential equations in terms of generalized coordinates. Solutions to the complete equations were obtained numerically by means of finite difference techniques. The problem has been programmed in the FORTRAN 4 language and solved on an IBM 360/50 computer. Results for limited cases are presented showing the nature of the solutions.

Meirovitch, L.↗

NASA NTRS1974

Cavitation in liquid cryogens. 4: Combined correlations for venturi, hydrofoil, ogives, and pumps

The results of a series of experimental and analytical cavitation studies are presented. Cross-correlation is performed of the developed cavity data for a venturi, a hydrofoil and three scaled ogives. The new correlating parameter, MTWO, improves data correlation for these stationary bodies and for pumping equipment. Existing techniques for predicting the cavitating performance of pumping machinery were extended to include variations in flow coefficient, cavitation parameter, and equipment geometry. The new predictive formulations hold promise as a design tool and universal method for correlating pumping machinery performance. Application of these predictive formulas requires prescribed cavitation test data or an independent method of estimating the cavitation parameter for each pump. The latter would permit prediction of performance without testing; potential methods for evaluating the cavitation parameter prior to testing are suggested.

Hord, J.↗

NASA NTRS1988

Mars Rover/Sample Return landing strategy

This paper describes the analysis and results of an investigation of the Mars Rover/Sample Return mission's landing strategy, together with the trade-offs of different landing strategies. The percentile points and the descriptive statistics of the probability distribution of traverse distances are calculated using a simple model formulated on the basis of landing error characteristics. The results show that variations in the landing stratgegy can significantly affect the traverse distance requirements, which range from 20 to 200 km.

Friedlander, Alan L.↗

NASA NTRS1990

A computational procedure for the dynamics of flexible beams within multibody systems

The dynamic analysis of three dimensional elastic beams which experience large rotational and large deformational motions are examined. The beam motion is modeled using an inertial reference for the translational displacements and a body-fixed reference for the rotational quantities. Finite strain rod theories are then defined in conjunction with the beam kinematic description which accounts for the effects of stretching, bending, torsion, and transverse shear deformations. A convected coordinate representation of the Cauchy stress tensor and a conjugate strain definition is introduced to model the beam deformation. To treat the beam dynamics, a two-stage modification of the central difference algorithm is presented to integrate the translational coordinates and the angular velocity vector. The angular orientation is then obtained from the application of an implicit integration algorithm to the Euler parameter/angular velocity kinematical relation. The combined developments of the objective internal force computation with the dynamic solution procedures result in the computational preservation of total energy for undamped systems. The present methodology is also extended to model the dynamics of deployment/retrieval of the flexible members. A moving spatial grid corresponding to the configuration of a deployed rigid beam is employed as a reference for the dynamic variables. A transient integration scheme which accurately accounts for the deforming spatial grid is derived from a space-time finite element discretization of a Hamiltonian variational statement. The computational results of this general deforming finite element beam formulation are compared to reported results for a planar inverse-spaghetti problem.

Downer, Janice Diane↗

NASA NTRS1992

r-adaptive methods based on element-level error indicators for parallel analysis of plates and shells

The formulation and application of element-level error indicators based on parametrized variational principles are investigated. These indicators are ideally suited to drive adaptive mesh refinment on parallel computers where access to neighboring elements resident on different processors may incur significant computational overhead. Furthermore, such estimators are not affected by physical jumps at wavefronts or interfaces. An estimator derived from the higher-order element energy is applied to r adaptation of meshes in plates and shell structures where the exact solution is known.

Militello, Carmelo↗

NASA NTRS1989

Topology and grid adaption for high-speed flow computations

This study investigates the effects of grid topology and grid adaptation on numerical solutions of the Navier-Stokes equations. In the first part of this study, a general procedure is presented for computation of high-speed flow over complex three-dimensional configurations. The flow field is simulated on the surface of a Butler wing in a uniform stream. Results are presented for Mach number 3.5 and a Reynolds number of 2,000,000. The O-type and H-type grids have been used for this study, and the results are compared together and with other theoretical and experimental results. The results demonstrate that while the H-type grid is suitable for the leading and trailing edges, a more accurate solution can be obtained for the middle part of the wing with an O-type grid. In the second part of this study, methods of grid adaption are reviewed and a method is developed with the capability of adapting to several variables. This method is based on a variational approach and is an algebraic method. Also, the method has been formulated in such a way that there is no need for any matrix inversion. This method is used in conjunction with the calculation of hypersonic flow over a blunt-nose body. A movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption.

Abolhassani, Jamshid S.↗

NASA NTRS1993

Implementation of a mesh adaptive scheme based on an element-level error indicator

We investigate the formulation and application of element-level error indicators based on parametrized variational principles. The qualifier 'element-level' means that no information from adjacent elements is used for error estimation. This property is ideally suited to drive adaptive mesh refinement on parallel computers where access to neighboring elements resident on different processors may incur significant computational overhead. Furthermore, such indicators are not affected by physical jumps at junctures or interfaces. An element-level indicator has been derived from the higher-order element energy and applied to r and h mesh adaptation of meshes in plates and shell structures. We report on our initial experiments with a cylindrical shell that intersects with fist plates forming a simplified 'wing-body intersection' benchmark problem.

Keating, Scott↗

NASA NTRS1997

A New Parameterization of H2SO4/H2O Aerosol Composition: Atmospheric Implications

Recent results from a thermodynamic model of aqueous sulfuric acid are used to derive a new parameterization for the variation of sulfuric acid aerosol composition with temperature and relative humidity. This formulation is valid for relative humidities above 1 % in the temperature range of 185 to 260 K. An expression for calculating the vapor pressure of supercooled liquid water, consistent with the sulfuric acid model, is also presented. We show that the Steele and Hamill [1981] formulation underestimates the water partial pressure over aqueous H2SOI solutions by up to 12% at low temperatures. This difference results in a corresponding underestimate of the H2SO4 concentration in the aerosol by about 6 % of the weight percent at approximately 190 K. In addition, the relation commonly used for estimating the vapor pressure of H2O over supercooled liquid water differs by up to 10 % from our derived expression. The combined error can result in a 20 % underestimation of water activity over a H2SO4 solution droplet in the stratosphere, which has implications for the parameterization of heterogeneous reaction rates in stratospheric sulfuric acid aerosols. The influence of aerosol composition on the rate of homogeneous ice nucleation from a H2SO4 solution droplet is also discussed. This parameterization can also be used for homogeneous gas phase nucleation calculations of H2SO4 solution droplets under various environmental conditions such as in aircraft exhaust or in volcanic plumes.

Tabazadeh, Azadeh↗

NASA NTRS2003

Analytical Modeling of the High Strain Rate Deformation of Polymer Matrix Composites

The results presented here are part of an ongoing research program to develop strain rate dependent deformation and failure models for the analysis of polymer matrix composites subject to high strain rate impact loads. State variable constitutive equations originally developed for metals have been modified in order to model the nonlinear, strain rate dependent deformation of polymeric matrix materials. To account for the effects of hydrostatic stresses, which are significant in polymers, the classical 5 plasticity theory definitions of effective stress and effective plastic strain are modified by applying variations of the Drucker-Prager yield criterion. To verify the revised formulation, the shear and tensile deformation of a representative toughened epoxy is analyzed across a wide range of strain rates (from quasi-static to high strain rates) and the results are compared to experimentally obtained values. For the analyzed polymers, both the tensile and shear stress-strain curves computed using the analytical model correlate well with values obtained through experimental tests. The polymer constitutive equations are implemented within a strength of materials based micromechanics method to predict the nonlinear, strain rate dependent deformation of polymer matrix composites. In the micromechanics, the unit cell is divided up into a number of independently analyzed slices, and laminate theory is then applied to obtain the effective deformation of the unit cell. The composite mechanics are verified by analyzing the deformation of a representative polymer matrix composite (composed using the representative polymer analyzed for the correlation of the polymer constitutive equations) for several fiber orientation angles across a variety of strain rates. The computed values compare favorably to experimentally obtained results.

Goldberg, Robert K.↗

NASA NTRS1981

The effects of surface evaporation parameterizations on climate sensitivity to solar constant variations

The effects of two different evaporation parameterizations on the sensitivity of simulated climate to solar constant variations are investigated by using a zonally averaged climate model. One parameterization is a nonlinear formulation in which the evaporation is nonlinearly proportional to the sensible heat flux, with the Bowen ratio determined by the predicted vertical temperature and humidity gradients near the earth's surface (model A). The other is the formulation of Saltzman (1968) with the evaporation linearly proportional to the sensible heat flux (model B). The computed climates of models A and B are in good agreement except for the energy partition between sensible and latent heat at the earth's surface. The difference in evaporation parameterizations causes a difference in the response of temperature lapse rate to solar constant variations and a difference in the sensitivity of longwave radiation to surface temperature which leads to a smaller sensitivity of surface temperature to solar constant variations in model A than in model B. The results of model A are qualitatively in agreement with those of the general circulation model calculations of Wetherald and Manabe (1975).

Chou, S.-H.↗