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NASA NTRS1971

Singular perturbation methods and the warm plasma model

The application of techniques of the singular perturbation theory to the analysis of warm plasma is discussed. Typically, the cold plasma model can be applied over wide ranges of parameters and only over narrow ranges forming so-called boundary layers is the warm plasma model used. Simplified equations can be used and the solutions matched on both sides of the layer's boundary. Simple examples to illustrate the solution are presented. The analysis confirms that some results are highly sensitive to the values of: (1) wire radius or gap size for an antenna, (2) temperature of the medium, and (3) incident angle of a plane wave.

Lee, S. W.↗

NASA NTRS1981

Solutions of the Helmholtz equation with boundary conditions for force-free magnetic fields

It is shown that the solution, with one ignorable coordinate, for the Taylor minimum energy state (resulting in a force-free magnetic field) in either a straight cylindrical or a toroidal geometry with arbitrary cross section can be reduced to the solution of either an inhomogeneous Helmholtz equation or a Grad-Shafranov equation with simple boundary conditions. Standard Green's function theory is, therefore, applicable. Detailed solutions are presented for the Taylor state in toroidal and cylindrical domains having a rectangular cross section. The focus is on solutions corresponding to the continuous eigenvalue spectra. Singular behavior at 90 deg corners is explored in detail.

Rasband, S. N.↗

NASA NTRS1989

Dynamic grid refinement for partial differential equations on parallel computers

The fast adaptive composite grid method (FAC) is an algorithm that uses various levels of uniform grids to provide adaptive resolution and fast solution of PDEs. An asynchronous version of FAC, called AFAC, that completely eliminates the bottleneck to parallelism is presented. This paper describes the advantage that this algorithm has in adaptive refinement for moving singularities on multiprocessor computers. This work is applicable to the parallel solution of two- and three-dimensional shock tracking problems.

Mccormick, S.↗

NASA NTRS1984

Singular asymptotic expansions in nonlinear rotordynamics

During hot firing ground testing of the Space Shuttle's Main Engine, vibrations of the liquid oxygen pump occur at frequencies which cannot be explained by the linear Jeffcott model of the rotor. The model becomes nonlinear after accounting for deadband, side forces, and rubbing. Two phenomena present in the numerical solutions of the differential equations are unexpected periodic orbits of the rotor and tracking of the nonlinear frequency. A multiple scale asymptotic expansion of the differential equations is used to give an analytic explanation of these characteristics.

Day, W. B.↗

NASA NTRS1985

Singular asymptotic expansions in nonlinear rotordynamics

During hot firing ground testing of the Space shuttle's Main Engine, vibrations of the liquid oxygen pump occur at frequencies which cannot be explained by the linear Jeffcott model of the rotor. The model becomes nonlinear after accounting for deadband, side forces, and rubbing. Two phenomena present in the numerical solutions of the differential equations are unexpected periodic orbits of the rotor and tracking of the nonlinear frequency. A multiple scale asymptotic expansion of the differential equations is used to give an analytic explanation of these characteristics.

Day, W. B.↗

NASA NTRS1985

Multigrid method for nearly singular and slightly indefinite problems

This paper deals with nearly singular, possibly indefinite problems for which the usual multigrid solvers converge very slowly or even diverge. The main difficulty is related to some badly approximated smooth functions which correspond to eigenfunctions with nearly zero eigenvalues. A correction to the usual coarse-grid equations is derived, both in the correction scheme and in the full approximation scheme. The performance of the new algorithm using this correction is essentially as that of usual multigrid for definite problems.

Brandt, A.↗

NASA NTRS1986

Multigrid method for nearly singular and slightly indefinite problems

Brandt, A.↗

NASA NTRS1973

Analysis of stratified and closely spaced jets exhausting into a crossflow

Procedures have been developed for determining the flow field about jets with velocity stratification exhausting into a crossflow. Jets with three different types of exit velocity stratification have been considered: (1) jets with a relatively high velocity core; (2) jets with a relatively low velocity core; and (3) jets originating from a vaned nozzle. The procedure developed for a jet originating from a high velocity core nozzle is to construct an equivalent nozzle having the same mass flow and thrust but having a uniform exit velocity profile. Calculations of the jet centerline and induced surface static pressures have been shown to be in good agreement with test data for a high velocity core nozzle. The equivalent ideal nozzle has also been shown to be a good representation for jets with a relatively low velocity core and for jets originating from a vaned nozzle in evaluating jet-induced flow fields. For the singular case of a low velocity core nozzle, namely a nozzle with a dead air core, and for the vaned nozzle, an alternative procedure has been developed. The internal mixing which takes place in the jet core has been properly accounted for in the equations of motion governing the jet development. Calculations of jet centerlines and induced surface static pressures show good agreement with test data these nozzles.

Ziegler, H.↗

NASA NTRS1980

A numerical technique for calculation of the noise of high-speed propellers with advanced blade geometry

A numerical technique and computer program were developed for the prediction of the noise of propellers with advanced geometry. The blade upper and lower surfaces are described by a curvilinear coordinate system, which was also used to divide the blade surfaces into panels. Two different acoustic formulations in the time domain were used to improve the speed and efficiency of the noise calculations: an acoustic formualtion with the Doppler factor singularity for panels moving at subsonic speeds and the collapsing sphere formulation for panels moving at transonic or supersonic speeds. This second formulation involves a sphere which is centered at the observer position and whose radius decreases at the speed of sound. The acoustic equation consisted of integrals over the curve of intersection for both the sphere and the panels on the blade. Algorithms used in some parts of the computer program are discussed. Comparisons with measured acoustic data for two model high speed propellers with advanced geometry are also presented.

Nystrom, P. A.↗

NASA NTRS2024

Out of Plane Stresses Using a Beam-Based Elastic Foundation Bonded Joint Model

Efficient, semi-analytical models of bonded joints are utilized for initial joint design and sizing due to their convenience and speed. Many of these models are based on beam and plate theories, which are most applicable when in-plane stresses/strains dominate. While this can be a sensible assumption for obtaining the deformations of the joint, out of plane (transverse) stresses are often of critical importance for failure in bonded joints composed of fiber reinforced composite materials. Since through-thickness strains are neglected in these semi-analytical theories, the present study shows a method to back-calculate the out of plane stresses based on stress balance equations, which use the in-plane stresses and displacements that have already been obtained. The method was implemented within the Joint Element Designer (JED) software, and verification and validation examples have been shown. As is typical for these types of models, corner stress singularities are not captured by the JED model, but the general trends of the stress and deformation fields agreed with a higher fidelity finite element model. As such, this method for obtaining out of plane stresses, when coupled with the JED code, is very useful for the complete stress analysis of bonded composite joints.

composites↗

NASA NTRS1994

Influence of Incidence Angle on Sound Generation by Airfoils Interacting with High-Frequency Gusts

A theoretical model is developed for the sound generated when a convected vortical or entropic gust encounters an airfoil at nonzero angle of attack. The theory is based on a linearization of the Euler equations about the steady, subsonic flow past the airfoil. High-frequency gusts, whose wavelengths are short compared to the airfoil chord, but long compared to the displacement of the mean-flow stagnation point from the leading edge, are considered. The analysis utilizes singular perturbation techniques and involves four asymptotic regions. Local regions, which scale on the gust wavelength, are present at the airfoil leading and trailing edges. Behind the airfoil a "transition" region, which is similar to the transition zone between illuminated and shadow zones in optical problems, is present. In the outer region, far away from the airfoil edges and wake, the solution has a geometric-acoustics form. The primary sound generation is found to be concentrated in the local leading-edge region. The trailing edge plays a secondary role as a scatterer of the sound generated in the leading-edge region. Parametric calculations are presented which illustrate that moderate levels of airfoil steady loading can significantly affect the sound field produced by airfoil-gust interactions.

Matthew R. Myers↗

NASA NTRS1977

Dynamic characteristics of rotor blades with pendulum absorbers

The point transmission matrix for a vertical plane pendulum on a rotating blade undergoing combined flapwise bending, and chordwise bending and torsion is derived. The equilibrium equation of the pendulum is linearized for small oscillations about the steady state. A FORTRAN program was written for the case of a vertical plane pendulum attached to a uniform blade with flapwise bending degree of freedom for cantilever boundary conditions. The frequency has a singular value right at the uncoupled pendulum natural frequency and thus introduces two frequencies corresponding to the nearest natural frequency of the blade without pendulum. In both of these modes it was observed that the pendulum deflection is large. One frequency can be thought of as a coupled pendulum frequency and the other as a coupled bending and pendulum frequency.

Murthy, V. R.↗

NASA NTRS1993

Pinching solutions of slender cylindrical jets

Simplified equations for slender jets are derived for a circular jet of one fluid flowing into an ambient second fluid, the flow being confined in a circular tank. Inviscid flows are studied which include both surface tension effects and Kelvin-Helmholtz instability. For slender jets a coupled nonlinear system of equations is found for the jet shape and the axial velocity jump across it. The equations can break down after a finite time and similarity solutions are constructed, and studied analytically and numerically. The break-ups found pertain to the jet pinching after a finite time, without violation of the slender jet ansatz. The system is conservative and admissible singular solutions are those which conserve the total energy, mass, and momentum. Such solutions are constructed analytically and numerically, and in the case of vortex sheets with no surface tension certain solutions are given in closed form.

Papageorgiou, Demetrios T.↗

NASA NTRS1972

A method for integrating the boundary-layer equations through a region of reverse flow.

Development of a procedure for numerically integrating the boundary-layer equations through a region of reverse flow which takes downstream influence into account. This method is applied to the problem of uniform flow past a parallel flat plate of finite length whose surface has a constant velocity directed opposite to that of the main stream. Although singularities occur at both the point of detachment (x sub s) and reattachment (x sub r) of the psi = 0 streamline, this integration technique provides a solution which ceases to apply only in the close proximity of these singular points. From this solution it is evident that, throughout a large portion of the separated region, the flow is strongly affected by conditions near x sub r, thereby demonstrating the importance of allowing information to be transmitted upstream in a region of backflow. Near x sub s, however, it is found that, in spite of the presence of reverse flow, the solution has a self-similar form in this particular example.

Klemp, J. B.↗

NASA NTRS1974

The numerical calculation of laminar boundary-layer separation

Iterative finite-difference techniques are developed for integrating the boundary-layer equations, without approximation, through a region of reversed flow. The numerical procedures are used to calculate incompressible laminar separated flows and to investigate the conditions for regular behavior at the point of separation. Regular flows are shown to be characterized by an integrable saddle-type singularity that makes it difficult to obtain numerical solutions which pass continuously into the separated region. The singularity is removed and continuous solutions ensured by specifying the wall shear distribution and computing the pressure gradient as part of the solution. Calculated results are presented for several separated flows and the accuracy of the method is verified. A computer program listing and complete solution case are included.

Klineberg, J. M.↗

NASA NTRS1974

On laminar boundary-layer separation

Iterative finite-difference techniques are developed for integrating the boundary-layer equations, without approximation, through a region of reversed flow. The numerical procedures are used to calculate incompressible laminar separated flows and to investigate the conditions for regular behavior at the point of separation. Regular flows are shown to be characterized by an integrable saddle-type singularity that makes it difficult to obtain numerical solutions which pass continuously into the separated region. The singularity is removed and continuous solutions ensured by specifying the wall shear distribution and computing the pressure gradient as part of the solution. Calculated results are presented for a number of separated flows and the accuracy of the method is verified.

Klineberg, J. M.↗

NASA NTRS1981

Spectral multigrid methods for elliptic equations

An alternative approach which employs multigrid concepts in the iterative solution of spectral equations was examined. Spectral multigrid methods are described for self adjoint elliptic equations with either periodic or Dirichlet boundary conditions. For realistic fluid calculations the relevant boundary conditions are periodic in at least one (angular) coordinate and Dirichlet (or Neumann) in the remaining coordinates. Spectral methods are always effective for flows in strictly rectangular geometries since corners generally introduce singularities into the solution. If the boundary is smooth, then mapping techniques are used to transform the problem into one with a combination of periodic and Dirichlet boundary conditions. It is suggested that spectral multigrid methods in these geometries can be devised by combining the techniques.

Zang, T. A.↗

NASA NTRS1983

A new consistent spatial differencing scheme for the transonic full-potential equation

A new spatial differencing scheme for the transonic full-potential equation in conservative form has been developed. This scheme guarantees zero truncation error on any curvilinear mesh for freestream flows in either two- or three-space dimensions. Solutions obtained with this new differencing scheme, away from freestream regions, exhibit greatly improved accuracy, especially for nonsmooth or singular meshes.

Flores, J.↗