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NASA NTRS1995

Eight-moment approximation solar wind models

Heat conduction from the corona is important in the solar wind energy budget. Until now all hydrodynamic solar wind models have been using the collisionally dominated gas approximation for the heat conductive flux. Observations of the solar wind show particle distribution functions which deviate significantly from a Maxwellian, and it is clear that the solar wind plasma is far from collisionally dominated. We have developed a numerical model for the solar wind which solves the full equation for the heat conductive flux together with the conservation equations for mass, momentum, and energy. The equations are obtained by taking moments of the Boltzmann equation, using an 8-moment approximation for the distribution function. For low-density solar winds the 8-moment approximation models give results which differ significantly from the results obtained in models assuming the gas to be collisionally dominated. The two models give more or less the same results in high density solar winds.

Olsen, Espen Lyngdal↗

NASA NTRS2000

Aerodynamic Lift and Moment Calculations Using a Closed-Form Solution of the Possio Equation

In this paper, we present closed-form formulas for the lift and moment coefficients of a lifting surface in two dimensional, unsteady, compressible, subsonic flow utilizing a newly developed explicit analytical solution of the Possio equation. Numerical calculations are consistent with previous numerical tables based on series expansions or ad hoc numerical schemes. More importantly, these formulas lend themselves readily to flutter analysis, compared with the tedious table-look-up schemes currently in use.

Lin, Jensen↗

NASA NTRS1962

BUCKLING AND POST-BUCKLING OF ELASTIC SHELLS

Some theoretical investigations of buckling of elastic shells are surveyed in this report. Only geometrically perfect shells are considered; initial dents and out-of-roundness are not taken into account. Several questions raised by the studies are: (a) Under what conditions is the infinitesimal theory of buckling of shells adequate? (b) How does the energy theory of buckling of shells correlate with the method based on equilibrium equations for bending moments, tensions, and shears in a buckled configuration? (c) How important are nonlinear terms in the tangential displacements u, v in the strain-displacement relations for buckling and post-buckling studies? (d) How important are the boundary conditions for u, v in affecting stability? (e) If a condition of snap-through is approached, how much external work is required to push the shell "over the hump" into the buckled configuration? (f) How reliable are mathematical approximations used previously in the infinitesimal theory of buckling of shells? Tentative and incomplete answers to some of these questions are suggested.

ELASTIC BUCKLING↗

NASA NTRS1978

Integral method for the calculation of three-dimensional, laminar and turbulent boundary layers

The method for turbulent flows is a further development of an existing method; profile families with two parameters and a lag entrainment method replace the simple entrainment method and power profiles with one parameter. The method for laminar flows is a new development. Moment of momentum equations were used for the solution of the problem, the profile families were derived from similar solutions of boundary layer equations. Laminar and turbulent flows at the wings were calculated. The influence of wing tapering on the boundary layer development was shown. The turbulent boundary layer for a revolution ellipsoid is calculated for 0 deg and 10 deg incidence angles.

Stock, H. W.↗

NASA NTRS1980

Stability of nonuniform rotor blades in hover using a mixed formulation

A mixed formulation for calculating static equilibrium and stability eigenvalues of nonuniform rotor blades in hover is presented. The static equilibrium equations are nonlinear and are solved by an accurate and efficient collocation method. The linearized perturbation equations are solved by a one step, second order integration scheme. The numerical results correlate very well with published results from a nearly identical stability analysis based on a displacement formulation. Slight differences in the results are traced to terms in the equations that relate moments to derivatives of rotations. With the present ordering scheme, in which terms of the order of squares of rotations are neglected with respect to unity, it is not possible to achieve completely equivalent models based on mixed and displacement formulations. The one step methods reveal that a second order Taylor expansion is necessary to achieve good convergence for nonuniform rotating blades. Numerical results for a hypothetical nonuniform blade, including the nonlinear static equilibrium solution, were obtained with no more effort or computer time than that required for a uniform blade.

Stephens, W. B.↗

NASA NTRS1981

Computing aerodynamic sound using advanced statistical turbulence theories

It is noted that the calculation of turbulence-generated aerodynamic sound requires knowledge of the spatial and temporal variation of Q sub ij (xi sub k, tau), the two-point, two-time turbulent velocity correlations. A technique is presented to obtain an approximate form of these correlations based on closure of the Reynolds stress equations by modeling of higher order terms. The governing equations for Q sub ij are first developed for a general flow. The case of homogeneous, stationary turbulence in a unidirectional constant shear mean flow is then assumed. The required closure form for Q sub ij is selected which is capable of qualitatively reproducing experimentally observed behavior. This form contains separation time dependent scale factors as parameters and depends explicitly on spatial separation. The approximate forms of Q sub ij are used in the differential equations and integral moments are taken over the spatial domain. The velocity correlations are used in the Lighthill theory of aerodynamic sound by assuming normal joint probability.

Hecht, A. M.↗

NASA NTRS1987

Recursive Robot-Arm Dynamics via Filtering and Smoothing

Forward and inverse dynamics solved using Kalman filtering and Bryson-Frazier smoothing. Dynamics of serial-link robot arm solved by using recursive techniques from linear filtering and smoothing theory. Solutions of dynamical equations give forces, moments, and accelerations at joints between links, and multilink inertia matrix and its inverse. Theoretical developments lay foundation for use of filtering and smoothing techniques in design of robot controls.

Rodriguez, Guillermo↗

NASA NTRS1988

Evolution of inclined galactic gas disks. I - A cloud-fluid approach

In this paper, numerical models are developed to follow the time-dependent behavior of inclined gas disks in nonspherical galaxies. Differential equations are developed which describe the time rate of change of the orbits of mass elements composing a thin, smoothly warping and twisting gas disk. These equations include multiple moments due to the nonsphericity of the galactic mass distribution, Coriolis effects arising from determination of orbits in a reference frame which tumbles with the surface figure of the galaxy, and viscous forces arising from cloud-cloud collisions. A cloud-fluid approach is used in determining the viscous forces. Physical arguments are used to determine the maximum permissible local value of the coefficient of kinematic viscosity. For small initial inclinations, analytic expressions are obtained for the inclination as a function of time and for the settling time, and the features of the solution are discussed.

Steiman-Cameron, Thomas Y.↗

NASA NTRS1991

Critical points in the 16-moment approximation

The singular points in steady state, field-aligned plasma transport models based on velocity moment theory are examined. In particular, two separate singular points in the equations obtained from the 16-moment approximation are identified. These equations are presented in a form that makes the singularities apparent, and they are solved in a simple illustrative case. The singular points, one occurring at the sonic point and the other at a critical value of the parallel heat flux, give rise to different outflow regimes, characterized generically by different asymptotic behavior. The existence of the different outflow regimes separated by the heat flux critical point has been only hinted at in previous discussions of numerical simulation of the polar wind.

Yasseen, F.↗

NASA NTRS1993

A generalized Zel'dovich approximation to gravitational instability

The orbits of particles undergoing gravitational instability are parameterized by generalizing the Zel'dovich approximation to a series expansion of arbitrary accuracy in the nonlinear regime. The coefficients of this series are determined from an action principle, or, more generally, from moments of the equation of motion. It is shown that the series is more rapidly convergent than previous nonlinear approximations. The method therefore provides a practical means of determining particle orbits, even for highly nonlinear perturbations. As an alternative, we also outline how the nonlinear dynamics may be computed as a field theory in which the evolution of the density and the velocity is determined in fixed comoving Eulerian coordinates.

Giavalisco, M.↗

NASA NTRS1951

Approximate method of integration of laminar boundary layer in incompressible fluid

A method is given for the approximate solution of the equations of the two-dimensional laminar boundary layer in an incompressible fluid. The method is based on the use of a system of equations of successive moments that is easily solved for simple supplementary assumptions. The solution obtained is given in closed form by simple formulas and is claimed to be no less accurate than the complicated solutions previously obtained, which were based on the use of special classes of flows.

BOUNDARY LAYER - WING SECTIONS↗

NASA NTRS1994

Auroral plasma transport processes in the presence of kV potential structures

We have simulated plasma transport processes in the presence of a quasi-two-dimensional current filament, that generated kV potential structure in the auroral region. The simulation consists of a set of one-dimensional flux tube simulations with different imposed time-dependent, field-aligned currents. The model uses the 16 moment system of equations and simultaneously solves coupled continuity and momentum equations and equations describing the transport along the magnetic field lines of parallel and perpendicular thermal energy and heat flows for each species. The lower end of the simulation is at an altitude of 800 km, in the collisional topside ionosphere, while the upper end is at 10 R(sub E) in the magnetosphere. The plasma consists of hot electrons and protons of magnetospheric origin and low-energy electrons, protons, and oxygen ions of ionospheric origin. The dynamical interaction of the individual current filaments with ionospheric and magnetospheric plasma generates a potential structure in the horizontal direction and kilovolt field-aligned potential drops along the field lines. The side-by-side display exhibits the evolution of the implied potential structure in the horizontial direction. In the presence of this potential structure and parallel electric field ionospheric plasma density is depleted and velocity is reduced, while density enhancement and increased velocity is observed in magnetospheric plasma. The ionospheric and magnetospheric electron temperatures increase below 2 R(sub E) due to magnetic mirror force on converging geomagnetic field lines. The primary cross-field motion produced by the horizontal E field (E x B drift) is perpendicular to both of the significant spatial directions and is thus ignorable in this geometry. The effects of other cross-field drift processes are discussed. The simulation thus provides insight into the dynamical evolution of two-dimensional potential structures driven by an imposed finite width, field-aligned current profile.

Ganguli, Supriya B.↗

NASA NTRS2017

Multi-Group Maximum Entropy Model for Translational Non-Equilibrium

The aim of the current work is to describe a new model for flows in translational non- equilibrium. Starting from the statistical description of a gas proposed by Boltzmann, the model relies on a domain decomposition technique in velocity space. Using the maximum entropy principle, the logarithm of the distribution function in each velocity sub-domain (group) is expressed with a power series in molecular velocity. New governing equations are obtained using the method of weighted residuals by taking the velocity moments of the Boltzmann equation. The model is applied to a spatially homogeneous Boltzmann equation with a Bhatnagar-Gross-Krook1(BGK) model collision operator and the relaxation of an initial non-equilibrium distribution to a Maxwellian is studied using the model. In addition, numerical results obtained using the model for a 1D shock tube problem are also reported.

Entropy↗

NASA NTRS1992

Semi-empirical model for prediction of unsteady forces on an airfoil with application to flutter

A semi-empirical model is described for predicting unsteady aerodynamic forces on arbitrary airfoils under mildly stalled and unstalled conditions. Aerodynamic forces are modeled using second order ordinary differential equations for lift and moment with airfoil motion as the input. This model is simultaneously integrated with structural dynamics equations to determine flutter characteristics for a two degrees-of-freedom system. Results for a number of cases are presented to demonstrate the suitability of this model to predict flutter. Comparison is made to the flutter characteristics determined by a Navier-Stokes solver and also the classical incompressible potential flow theory.

Mahajan, Aparajit J.↗

NASA NTRS1993

Semi-empirical model for prediction of unsteady forces on an airfoil with application to flutter

Mahajan, A. J.↗

NASA NTRS1991

Stationary turbulent closure via the Hopf functional equation

Explicit solutions of the stationary Hopf equation are discussed and their computational possibilities are explored. The motivation is to circumvent the infinite hierarchy of coupled equations for the velocity moments and obtain an exact closure of the steady-state 3D Navier-Stokes equations, without modeling assumptions or truncation. The Hopf formulation of the Navier-Stokes equation is reviewed. A stationary homogeneous solution for 2D flow is displayed and discussed. It is shown how depletion of nonlinearity may arise for 3D forced homogeneous flow. The general 3D forced case is considered and a method for closing the 3D unforced equations with arbitrary boundary conditions is derived.

Shen, Hubert H.↗

NASA NTRS1987

Behavior of ionized plasma in the high latitude topside ionosphere - The polar wind

A numerical model is used to investigate the steady-state behavior of a fully ionized plasma encompassing the geomagnetic field lines from 1500 km to 10 earth radii, and solutions of the 16 moment system of transport equations for the polar wind are obtained. For the present cases studied, results of the 13 and 16 moment simulations are found to be similar, although the temperature anisotropy is quite high. Polar wind studies reveal an electron temperature anisotropy developing around 2500 km, with the collisions keeping the electron temperature isotropic below 2500 km. Good agreement is found between the present polar wind simulations and recent observations.

Ganguli, Supriya B.↗

NASA NTRS1936

The forces and moments on airplane engine mounts

A resume of the equations and formulas for the forces and moments on an aircraft-engine mount is presented. In addition, available experimental data have been included to permit the computation of these forces and moments. A sample calculation is made and compared with present design conditions for engine mounts.

Donely, Philip↗