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At least 127 records · Page 7

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NASA NTRS1975

Quasicoordinate equations for flexible spacecraft

Equations of motion equivalent to Lagrange's equations for generalized coordinates can be formulated in quasi-coordinate form. An analysis of the resulting equations is conducted and some alternative methods are considered. It is found that Lagrange's quasi-coordinate equations produce precisely the same equations of motion for any material system as would energy from the combination of Lagrange's generalized coordinate equations and Euler's rotational equation in vector form.

Likins, P. W.↗

NASA NTRS1975

Bidirectional plant canopy reflection models derived from the radiation transfer equation

A collection of bidirectional canopy reflection models was obtained from the solution of the radiation transfer equation for a horizontally homogeneous canopy. A phase function is derived for a collection of bidirectionally reflecting and transmitting planar elements characterized geometrically by slope and azimuth density functions. Two approaches to solving the radiation transfer equation for the canopy are presented. One approach factors the radiation transfer equation into a solvable set of three first-order linear differential equations by assuming that the radiation field within the canopy can be initially approximated by three components: uniformly diffuse downwelling, uniformly diffuse upwelling, and attenuated specular. The solution to these equations, which can be iterated to any degree of accuracy, was used to obtain overall canopy reflection from the formal solution to the radiation transfer equation. A programable solution to canopy overall bidirectional reflection is given for this approach. The special example of Lambertian leaves with constant leaf bidirectional reflection and scattering functions is considered, and a programmable solution for this example is given. The other approach to solving the radiation transfer equation, a generalized Chandrasekhar technique, is presented in the appendix.

Beeth, D. R.↗

NASA NTRS1976

On the equations governing the electromagnetic perturbations of the Kerr black hole

Teukolsky's (1973) radial wave equation for electromagnetic perturbations about a rotating black hole is transformed into a one-dimensional wave equation with a real and short-range potential. A general technique for transforming one wave equation into another is described, and it is shown how an equation with a complex potential can be transformed into one with a purely real potential. These general techniques are then used to transform Teukolsky's equation into a wave equation whose potential is purely real and of short range. This equation is found to provide a simple formulation of the concepts of superradiant scattering and the conservation of energy underlying the present electromagnetic perturbations. It is noted that the techniques employed are general enough to be applicable when the spin weight is plus or minus 2, making a 'real-potential' formulation of the problem of gravitational perturbations about a Kerr black hole most likely.

Detweiler, S.↗

NASA NTRS1980

The primer vector in linear, relative-motion equations

Primer vector theory is used in analyzing a set of linear, relative-motion equations - the Clohessy-Wiltshire equations - to determine the criteria and necessary conditions for an optimal, N-impulse trajectory. Since the state vector for these equations is defined in terms of a linear system of ordinary differential equations, all fundamental relations defining the solution of the state and costate equations, and the necessary conditions for optimality, can be expressed in terms of elementary functions. The analysis develops the analytical criteria for improving a solution by (1) moving any dependent or independent variable in the initial and/or final orbit, and (2) adding intermediate impulses. If these criteria are violated, the theory establishes a sufficient number of analytical equations. The subsequent satisfaction of these equations will result in the optimal position vectors and times of an N-impulse trajectory. The solution is examined for the specific boundary conditions of (1) fixed-end conditions, two-impulse, and time-open transfer; (2) an orbit-to-orbit transfer; and (3) a generalized rendezvous problem. A sequence of rendezvous problems is solved to illustrate the analysis and the computational procedure.

Source record↗

NASA NTRS1980

Numerical solutions of transonic flows by parametric differentiation and integral equation techniques

The paper discusses results of an exploratory study of the advantages obtained by combining integral equation methods with the method of parametric differentiation in the treatment of transonic flow problems. In the proposed method, the nonlinear unsteady transonic flow equation for small perturbations is transformed into a linear equation by the use of the method of parametric differentiation. The linear equation is split into a pair of weakly coupled partial differential equations by writing the transformed perturbation potential as the sum of a steady component and an unsteady component. The solution of the steady equation as an integral equation is based on Ogana's treatment (1978). As a test case, the formulation is applied to predict the steady transonic flow over a nonlifting parabolic-arc airfoil.

Sivaneri, N. T.↗

NASA NTRS1983

Accuracy of approximations to the Navier-Stokes equations

The systems of truncated differential equations that have been proposed to reduce the complexity and large computational costs of solutions to the full Navier-Stokes equations are considered. These systems are computationally efficient and capture all the physically relevant behavior. The systems follow a certain hierarchy: (1) the classical boundary-layer equations with specified edge properties (usually the streamwise pressure distribution); (2) the coupled boundary-layer/inviscid equations; (3) the so-called thin-layer equations that discard streamwise diffusion; and (4) the Navier-Stokes equations. Consideration is given to each of these approximations applied to an incompressible, laminar-separating flow at low and moderate Reynolds numbers. It is pointed out that for any flow or region of flow for which viscous-inviscid interaction effects are small, classical boundary-layer equations will provide a satisfactory description of the viscous flow at a fraction of the computational cost of any higher approximations.

Murphy, J. D.↗

NASA NTRS1984

Flap-lag damping of an elastic rotor blade with torsion and dynamic inflow in hover from symbolically generated equations

A program written in FORTRAN IV and coded into subroutines for subsequent numerical study is used to derive symbolically the governing equations of motion of the elastic rotor blade and the dynamic inflow equations. By rearranging the coefficients of the equations, the blade and dynamic inflow equations are converted into equations in a multiblade coordinated system. The final multiblade equations can accommodate any number of elastic blade modes. It is noted that with a given data set, the entire process, from the derivation of the equations to the numerical calculations, is computerized and requires only limited user interface. To demonstrate the usefulness of the approach, damping data in hover with and without dynamic inflow effects are presented for various rotor blade models, including flap-lag, flap-lag-torsion, quasi-static torsion, and rigid blade lag-flap.

Reddy, T. S. R.↗

NASA NTRS1985

Well-posedness of one-way wave equations and absorbing boundary conditions

A one-way wave equation is a partial differential which, in some approximate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one. The construction of such equations can be reduced to the approximation of the square root of (1-s sup 2) on -1, 1 by a rational function r(s) = p sub m (s)/q sub n(s). Those rational functions r for which the corresponding one-way wave equation is well-posed are characterized both as a partial differential equation and as an absorbing boundary condition for the wave equation. We find that if r(s) interpolates the square root of (1-s sup 2) at sufficiently many points in (-1,1), then well-posedness is assured. It follows that absorbing boundary conditions based on Pade approximation are well-posed if and only if (m, n) lies in one of two distinct diagonals in the Pade table, the two proposed by Engquist and Majda. Analogous results also hold for one-way wave equations derived from Chebyshev or least-squares approximation.

Trefethen, L. N.↗

NASA NTRS1991

From differential to difference equations for first order ODEs

When constructing an algorithm for the numerical integration of a differential equation, one should first convert the known ordinary differential equation (ODE) into an ordinary difference equation. Given this difference equation, one can develop an appropriate numerical algorithm. This technical note describes the derivation of two such ordinary difference equations applicable to a first order ODE. The implicit ordinary difference equation has the same asymptotic expansion as the ODE itself, whereas the explicit ordinary difference equation has an asymptotic that is similar in structure but different in value when compared with that of the ODE.

Freed, Alan D.↗

NASA NTRS1990

A discrete model of a modified Burgers' partial differential equation

A new finite-difference scheme is constructed for a modified Burger's equation. Three special cases of the equation are considered, and the 'exact' difference schemes for the space- and time-independent forms of the equation are presented, along with the diffusion-free case of Burger's equation modeled by a difference equation. The desired difference scheme is then obtained by imposing on any difference model of the initial equation the requirement that, in the appropriate limits, its difference scheme must reduce the results of the obtained equations.

Mickens, R. E.↗

NASA NTRS1991

Accuracy of the Burnett equations for hypersonic real gas flows

Burnett solutions for the structure of normal shock waves for Mach numbers ranging from 1.1 to 50 have been obtained by using the time-dependent equation set in conjunction with modern computational fluid dynamic techniques. The current work seeks to provide corrected and improved high-resolution direct simulation Monte Carlo (DSMC) results for hypersonic shocks in monatomic gases. These DSMC results will serve as a reliable test base for assessing the accuracy of various continuum equations more advanced than Navier-Stokes, such as the Burnett equations. The main conclusion of Fisko and Chapman (1988) that the Burnett equations are more accurate than the Navier-Stokes equations for all gases and at all Mach numbers in matching DSMC shock wave results is reassesed by comparing the improved and corrected DSMC results with continuum Burnett results. Various Burnett results for shocks in a diatomic gas, nitrogen, which include improvements to the work of Lumpkin and Chapman (1989) are presented. It is concluded that the Burnett equations represent a significant improvement over Navier-Stokes equations for flows exhibiting translational nonequilibrium.

Lumpkin, Forrest E., III↗

NASA NTRS1991

Transport of gyration-dominated space plasmas of thermal origin. I - Generalized transport equations

Grad's 20-moment set of transport equations has been examined in the limit of strong external magnetic fields. This simplified set of equations describes the transport of mass and parallel momentum as well as the transport of parallel and perpendicular energy and heat flow in the magnetic field direction. The effect of collisions was calculated assuming a modified relaxation model. Wave speeds and normal modes of the simplified set of equations were examined for an ion and electron gas. It was found that four of the ten normal modes are electron thermal-heat waves which approximately decouple from the six ion waves in the system. When low-frequency waves are considered (slow-wave approximation), this allows the electron energy and heat flow equations to be solved separately from the ion equations and in a time-independent fashion. When this was done, it was found that, under certain conditions, these equations predict an infinite electron perpendicular temperature, Te-perpendicular, in the collisionless regime. This occurs whenever Te-perpendicular is greater than the parallel temperature, at any point along collisionless and diverging magnetic field lines. When applied appropriately, the significantly simplified set of equations derived here are well suited for application to a large variety of problems in planetary ionospheres and magnetospheres.

Gombosi, Tamas I.↗

NASA NTRS1991

Shield sizing and response equations

A consolidated list is presented of meteoroid debris shield equations which have been given in the referenced memorandums. In some cases, equations have been updated; thus, this memorandum supersedes reference 1. The equations are presented in two parts: (1) shield sizing equations which are used to produce preliminary estimates of shielding weights; and (2) response equations to describe the impact conditions (projectile size as a function of velocity, density, and impact angle) causing failure of a given shield that are to be used for probability analyses (such as in the modified BUMPER program). Specific equations are given that are applicable for the following types of shields: aluminum Whipple shields; Nextel multishock (MS) shields; and mesh double bumper (MDB) shields. These equations will be updated in the future as warranted by the results of additional HVI tests, analyses, and shield modeling.

Christiansen, Eric L.↗

NASA NTRS1992

A near-wall four-equation turbulence model for compressible boundary layers

A near-wall four-equation turbulence model is developed for the calculation of high-speed compressible turbulent boundary layers. The four equations used are the k-epsilon equations and the theta(exp 2)-epsilon(sub theta) equations. These equations are used to define the turbulent diffusivities for momentum and heat fluxes, thus allowing the assumption of dynamic similarity between momentum and heat transport to be relaxed. The Favre-averaged equations of motion are solved in conjunction with the four transport equations. Calculations are compared with measurements and with another model's predictions where the assumption of the constant turbulent Prandtl number is invoked. Compressible flat plate turbulent boundary layers with both adiabatic and constant temperature wall boundary conditions are considered. Results for the range of low Mach numbers and temperature ratios investigated are essentially the same as those obtained using an identical near-wall k-epsilon model. In general, the numerical predictions are in very good agreement with measurements and there are significant improvements in the predictions of mean flow properties at high Mach numbers.

Sommer, T. P.↗

NASA NTRS1992

Solution of the Burnett equations for hypersonic flows near the continuum limit

The INCA code, a three-dimensional Navier-Stokes code for analysis of hypersonic flowfields, was modified to analyze the lower reaches of the continuum transition regime, where the Navier-Stokes equations become inaccurate and Monte Carlo methods become too computationally expensive. The two-dimensional Burnett equations and the three-dimensional rotational energy transport equation were added to the code and one- and two-dimensional calculations were performed. For the structure of normal shock waves, the Burnett equations give consistently better results than Navier-Stokes equations and compare reasonably well with Monte Carlo methods. For two-dimensional flow of Nitrogen past a circular cylinder the Burnett equations predict the total drag reasonably well. Care must be taken, however, not to exceed the range of validity of the Burnett equations.

Imlay, Scott T.↗

NASA NTRS1992

Equations of motion of slung-load systems, including multilift systems

General simulation equations are derived for the rigid body motion of slung-load systems. This work is motivated by an interest in trajectory control for slung loads carried by two or more helicopters. An approximation of these systems consists of several rigid bodies connected by straight-line cables or links. The suspension can be assumed elastic or inelastic. Equations for the general system are obtained from the Newton-Euler rigid-body equations with the introduction of generalized velocity coordinates. Three forms are obtained: two generalize previous case-specific results for single-helicopter systems with elastic and inelastic suspensions, respectively; and the third is a new formulation for inelastic suspensions. The latter is derived from the elastic suspension equations by choosing the generalized coordinates so that motion induced by cable stretching is separated from motion with invariant cable lengths, and by then nulling the stretching coordinates to get a relation for the suspension forces. The result is computationally more efficient than the conventional formulation, is readily integrated with the elastic suspension formulation, and is easily applied to the complex dual-lift and multilift systems. Results are given for two-helicopter systems; three configurations are included and these can be integrated in a single simulation. Equations are also given for some single-helicopter systems, for comparison with the previous literature, and for a multilift system. Equations for degenerate-body approximations (point masses, rigid rods) are also formulated and results are given for dual-lift and multilift systems. Finally, linearlized equations of motion are given for general slung-load systems are presented along with results for the two-helicopter system with a spreader bar.

Cicolani, Luigi S.↗

NASA NTRS1993

Hypersonic three-dimensional nonequilibrium boundary-layer equations in generalized curvilinear coordinates

The basic governing equations for the second-order three-dimensional hypersonic thermal and chemical nonequilibrium boundary layer are derived by means of an order-of-magnitude analysis. A two-temperature concept is implemented into the system of boundary-layer equations by simplifying the rather complicated general three-temperature thermal gas model. The equations are written in a surface-oriented non-orthogonal curvilinear coordinate system, where two curvilinear coordinates are non-orthogonial and a third coordinate is normal to the surface. The equations are described with minimum use of tensor expressions arising from the coordinate transformation, to avoid unnecessary confusion for readers. The set of equations obtained will be suitable for the development of a three-dimensional nonequilibrium boundary-layer code. Such a code could be used to determine economically the aerodynamic/aerothermodynamic loads to the surfaces of hypersonic vehicles with general configurations. In addition, the basic equations for three-dimensional stagnation flow, of which solution is required as an initial value for space-marching integration of the boundary-layer equations, are given along with the boundary conditions, the boundary-layer parameters, and the inner-outer layer matching procedure. Expressions for the chemical reaction rates and the thermodynamic and transport properties in the thermal nonequilibrium environment are explicitly given.

Lee, Jong-Hun↗

NASA NTRS1993

Sensitivity analysis of Lyapunov and Riccati equations with application to controls-structures integrated design

Controls-structure integrated design is a complicated multidisciplinary design optimization problem which involves the state equations pertaining to open-loop eigenvalues and control laws. In order to alleviate the intensity of the computation, this study uses the adjoint variable method to derive sensitivity equations for the eigenvalue, Liapunov, and Riccati equations. These individual sensitivity equations are then combined together to form the multidisciplinary sensitivity equations for the control structure integrated design problems. A set of linear sensitivity equations, proportional in number to the number of performance functions involved in the optimization process, are solved. This proposed approach may provide a great saving in computer resources. The validity of the newly developed sensitivity equations is verified by numerical examples.

Hou, Gene↗