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NASA NTRS1983

Difference equation technique for determining the evolution of radially symmetric ultrasonic fields

A simplified difference equation technique has been developed to calculate the three-dimensional evolution of radially symmetric, complex ultrasonic fields. The parabolic approximation to the Helmholtz equation is obtained by assuming that localized changes in the magnitude of the field with respect to distance along the axis of propagation are small compared to those with respect to radial displacement. The Helmholtz equation in cylindrical coordinates may then be approximated, after separating variables, using truncated series expansions. The resulting difference equation determines complex field values at discrete locations in a plane using corresponding adjacent locations in the preceding plane. Stability criteria which determine acceptable values for various parameters are discussed. The technique has been applied to the propagation of ultrasonic fields generated by two-dimensional focusing and non-focusing, uniform and Gaussian profile, piezoelectric transducers in both profiled and non-profiled media.

Dockery, G. D.↗

NASA NTRS2020

Shape Validation and RF Performance of Inflatable Antennas

Inflatable aperture antennas are an emerging technology that is being investigated for potential use in science and exploration missions. In particular, for missions to Mars and beyond, large deployable aperture antennas can provide the antenna gain required for high data rate communications, where the necessary antenna diameter exceeds the available volume of typical launch vehicle platforms. As inflatable aperture antennas have not been proven fully qualified for space missions, the author's Master's Thesis assessed the Ruze equation in characterizing this antenna technology. Inflatable aperture antennas do not follow a parabolic shape, and so the Ruze equation is not applicable due to the macroscopic shape errors of this technology. Therefore, geometric evaluations of the surface profile cannot simply correlate antenna gain degradation with the root-mean-square shape error with a parabolic surface. Consequently, the focus of this work was to derive an accurate mathematical model of an inflatable aperture antenna in order to characterize its Radio Frequency (RF) performance. Calculus of Variations methodologies were used to derive the surface profile shape of the inflatable aperture antenna. Physical Optics techniques were used to generate the antenna pattern profile. Validation testing of the predicted inflatable antenna shape model was performed through use of Laser Radar metrology measurements on an inflatable test article. Assessments of the RF performance of the inflatable aperture antenna, compared with nominally shaped paraboloidal antennas, were obtained through simulations of both technologies using a common diameter, depth, and arc length. Assessments of the RF performance of the inflatable aperture antenna was also performed against itself for changes in distance of the antenna feed location in the axial direction. Whereas the Ruze equation is limited to assessing gain reduction, this effort will also assess beam spreading and first side lobe angle and magnitude. The ability to characterize the RF response of this antenna will provide for an improved understanding of this technology. The accurate representation of the shape of this type of antenna technology will help to identify the most appropriate ways in which this technology could be utilized in planning future communication architectures for NASA missions to Mars and beyond.

Welch, Bryan W.↗

NASA NTRS2020

Shape Validation and RF Performance of Inflatable Antennas

Inflatable aperture antennas are an emerging technology that is being investigated for potential use in science and exploration missions. In particular, for missions to Mars and beyond, large deployable aperture antennas can provide the antenna gain required for high data rate communications, where the necessary antenna diameter exceeds the available volume of typical launch vehicle platforms. As inflatable aperture antennas have not been proven fully qualified for space missions, the author's Master's Thesis assessed the Ruze equation in characterizing this antenna technology. Inflatable aperture antennas do not follow a parabolic shape, and so the Ruze equation is not applicable due to the macroscopic shape errors of this technology. Therefore, geometric evaluations of the surface profile cannot simply correlate antenna gain degradation with the root-mean square shape error with a parabolic surface. Consequently, the focus of this work was to derive an accurate mathematical model of an inflatable aperture antenna to in order to characterize its Radio Frequency (RF) performance. Calculus of Variations methodologies were used to derive the surface profile shape of the inflatable aperture antenna. Physical Optics techniques were used to generate the antenna pattern profile. Validation testing of the predicted inflatable antenna shape model was performed through use of Laser Radar metrology measurements on an inflatable test article. Assessments of the RF performance of the inflatable aperture antenna, compared with nominally shaped paraboloidal antennas, were obtained through simulations of both technologies using a common diameter, depth, and arc length. Assessments of the RF performance of the inflatable aperture antenna was also performed against itself for changes in distance of the antenna feed location in the axial direction. Whereas the Ruze equation is limited to assessing gain reduction, this effort will also assess beam spreading and first side lobe angle and magnitude. The ability to characterize the RF response of this antenna will provide for an improved understanding of this technology. The accurate representation of the shape of this type of antenna technology will help to identify the most appropriate ways in which this technology could be utilized in planning future communication architectures for NASA missions to Mars and beyond.

Welch, Bryan↗

NASA NTRS1961

Theory of Electrostatic Fields in the Ionosphere at Equatorial Latitudes

The properties of the elongated electrostatic fields that are required to provide the coupling mechanism in the dynamo-motor concept of the E and F regions of the ionosphere are examined theoretically for the conditions that prevail in equatorial latitudes. The analysis is developed for an electrostatic field of arbitrary horizontal scale in a horizontally stratified partly ionized gas subject to an imposed magnetic field having the form of a parabolic arch over the equator. The anisotropic character and continuous variation with height of the conductivity are retained throughout , and numerical solutions are determined for the attenuation of the electric field with distance along the field line. The results are similar qualitatively to those found previously upon analysis of the corresponding problem for middle latitudes, but the attenuation of the electrostatic field with height is considerably greater. It develops, in particular, that the coupling between E and F regions is very small for fields having horizontal wavelengths of a few kilometers. It follows that the dynamo-motor concept could not be used to account for the presence of irregularities of this scale. On the other hand, it is found that almost all the attenuation occurs at heights near that of the E region. If an electrostatic field having a horizontal wavelength of a few kilometers could be produced at heights of, say 200 km or greater, it follows that it would be very elongated and extend with little change in amplitude from hemisphere to hemisphere. The results also indicate that effective coupling could be achieved for fields having horizontal wavelengths of the order of tens of kilometers, but the assumptions introduced to simplify the analysis may impair the quantitative reliability of the results for fields of this scale.

Briggs, Benjamin R.↗

NASA NTRS1977

Computations of supersonic viscous flow over a finite-width plate

Finite difference methods were applied to solve the parabolic Navier-Stokes equations for the flow over a finite width plate at 0 deg and 10 deg angles of attack. The methods were developed on the basis of the operator factorization concept resulting in the split of a three dimensional equation into successive two dimensional equations. Backward and centered implicit factorization schemes, were used and their results were compared. Available numerical solutions and experimental data obtained at low Reynolds number conditions were also used for comparison. The backward implicit method provides a more successful solution, which ranges from the merged layer to the strong interaction regimes. Detailed structures were revealed of the shear layer around and behind the side edge.

Li, C. P.↗

NASA NTRS1980

Computation of supersonic turbulent flows over an inclined ogive-cylinder-flare

A supersonic turbulent flow over an ogive-cylinder-flare has been solved numerically. Initially, the parabolized Navier-Stokes equations are solved for the ogive cylinder back to a location upstream of the shock-wave and boundary-layer interaction. Then, the time-dependent Navier-Stokes equations with a thin-layer approximation are solved for the remaining cylinder-flare portion. Results for a Mach number of 2.9 and a unit Reynolds number of 11.42 x 10 to the 6th/m are obtained for angles of attack alpha = 0, 4, and 8 deg. Good agreement has been found between computed and experimental results of the surface pressure on the ogive-cylinder portion, and for the interaction region at alpha = 0 and 4 deg. The role of circumferential communication in a three-dimensional shock-wave and boundary-layer interaction flow field is discussed.

Hung, C. M.↗

NASA NTRS1981

Application of an implicit technique to the shock-layer flow around general bodies

The Euler and the parabolic Navier-Stokes equations are solved by an iterative implicit scheme along the body axis for angles of incidence up to 30 deg. A complete solution, including both the inviscid and viscous flows, can be obtained at stations normal to the axis. The technique is developed from a comparatively simple linearization procedure and has an option to iterate between body stations for higher accuracy. Reliable procedures have been introduced to account for the effect of the axial pressure gradient and to adjust the step increment for a given convergence requirement. A blunt-nosed conical and a Shuttle-Orbiter-like configuration were studied extensively and are compared with available laminar boundary-layer solutions and experimental data.

Li, C. P.↗

NASA NTRS1982

An experimental investigation of hypersonic flow over biconics at incidence and comparison to prediction

Pressure distributions, aerodynamic coefficients, shock shapes, and oil-flow patterns were measured on spherically blunted, 12.84 deg/7 deg on-axis and bent biconics at Mach 6 in air. The angle of attack, referenced to the axis of the aft-cone, was varied from -10 to 40 deg in nominal 5 deg increments. Predictions from an inviscid flowfield computer code referred to as STEIN and codes which solve the three-dimensional parabolized Navier Stokes (PNS) equations were compared with measurement. Three PNS codes were found which provided accurate predictions of windward and leeward pressure distributions for angles of attack to 25 deg.

Miller, C. G.↗

NASA NTRS1983

End wall flows in rotors and stators of a single stage compressor

A computer code for solving the parabolized Navier-Stokes equations for internal flows was developed. Oscillations that develop in the calculation procedure are discussed. The measurements made in the hub and annulus wall boundary layers are summarized. The flow in the hub wall boundary layer, starting ahead of the inlet guide vanes to the inlet of the rotor is traced.

Govindan, T. R.↗

NASA NTRS1983

Heat-transfer distributions on biconics at incidence in hypersonic-hypervelocity He, N2, air, and CO2 flows

Laminar heat transfer rates were measured on spherically blunted, 13 deg/7 deg on axis and bent biconics (fore cone bent 7 deg upward relative to aft cone) at hypersonic hypervelocity flow conditions in the Langley Expansion Tube. Freestream velocities from 4.5 to 6.9 km/sec and Mach numbers from 6 to 9 were generated using helium, nitrogen, air, and carbon dioxide test gases, resulting in normal shock density ratios from 4 to 19. Angle of attack, referenced to the axis of the aft cone, was varied from 0 to 20 deg in 4 deg increments. The effect of nose bend, angle of attack, and real gas phenomena on heating distributions are presented along with comparisons of measurement to prediction from a code which solves the three dimensional parabolized Navier-Stokes equations.

Miller, C. G.↗

NASA NTRS1982

Numerical boundary condition procedures and multigrid methods; Proceedings of the Symposium, NASA Ames Research Center, Moffett Field, CA, October 19-22, 1981

Papers presented in this volume provide an overview of recent work on numerical boundary condition procedures and multigrid methods. The topics discussed include implicit boundary conditions for the solution of the parabolized Navier-Stokes equations for supersonic flows; far field boundary conditions for compressible flows; and influence of boundary approximations and conditions on finite-difference solutions. Papers are also presented on fully implicit shock tracking and on the stability of two-dimensional hyperbolic initial boundary value problems for explicit and implicit schemes.

Source record↗

NASA NTRS1983

Heat-transfer distributions on biconics at incidence in hypersonic-hypervelocity He, N2, air, and CO2 flows

Laminar heat-transfer rates were measured on spherically blunted, 13 degrees/F degrees on-axis and bent biconics (fore cone bent 7 degrees upward relative to aft cone) at hypersonic-hypervelocity flow conditions in the Langley Expansion Tube. Freestream velocities from 4.5 to 6.9 km/sec and Mach numbers from 6 to 9 were generated using helium, nitrogen, air, and carbon dioxide test gases, resulting in normal shock density ratios from 4 to 19. Angle of attack, referenced to the axis of the aft cone, was varied from zero to 20 degrees in 4 degree increments. The effect of nose bend, angle of attack, and real-gas phenomena on heating distributions are presented along with comparisons of measurement to prediction from a code which solves the three-dimensional 'parabolized Navier-Stokes' equations.

Miller, C. G.↗

NASA NTRS1983

Hypersonic flows over biconics using a variable-effective-gamma, Parabolized-Navier-Stokes code

Hypersonic flows over straight and bent biconics are calculated for a range of freestream conditions in which the gas behind the shock is treated as either perfect or real. The Parabolized-Navier-Stokes (PNS) equations form the basis of the approximation scheme. Good comparisons with experimental data for pressures, forces and moments, heat transfer, and oil-flow patterns serve to validate the perfect-gas version of the code. Circumferential velocity vector plots further aid in the interpretation of leeside oil-flow patterns. A variable-effective-gamma (VEG) option is implemented for the real-gas calculations. Gamma, now defined as the ratio of enthalpy to internal energy, is determined from a locally valid linear relation in enthalpy and pressure at every mesh point which in turn is calculated from a benchmark equilibrium code. The VEG option is easily incorporated into a host PNS code because it uses the underlying perfect-gas structure of the code. Comparisons to experimental data for heat transfer and shock shape in high enthalpy air have been obtained using the VEG option.

Gnoffo, P. A.↗

NASA NTRS1983

Progress on a generalized coordinates tensor product finite element 3DPNS algorithm for subsonic

A generalized coordinates form of the penalty finite element algorithm for the 3-dimensional parabolic Navier-Stokes equations for turbulent subsonic flows was derived. This algorithm formulation requires only three distinct hypermatrices and is applicable using any boundary fitted coordinate transformation procedure. The tensor matrix product approximation to the Jacobian of the Newton linear algebra matrix statement was also derived. Tne Newton algorithm was restructured to replace large sparse matrix solution procedures with grid sweeping using alpha-block tridiagonal matrices, where alpha equals the number of dependent variables. Numerical experiments were conducted and the resultant data gives guidance on potentially preferred tensor product constructions for the penalty finite element 3DPNS algorithm.

Baker, A. J.↗

NASA NTRS1984

Computation and turbulence closure models for shear flows in rotating curved bodies

This paper is concerned with the effects of rotation and curvature on the turbulence closure model. The k-epsilon model that includes the effects of curvature has been modified to include the anisotropy in turbulence. A space-marching algorithm which solves the parabolized Navier-Stokes equations is coupled with the turbulence closure model to predict the turbulent shear flow over a rotating cylinder and inside a rotating channel. The agreement between the predictions and available data is good. The rotation and curvature effects are predicted well.

Pouagare, M.↗

NASA NTRS1984

Heat-transfer distributions for biconics at incidence in hypersonic-hypervelocity real-gas flows

Laminar heat-transfer distributions were measured on spherically blunted, 13/7 deg straight and bent biconics at freestream velocities from 4.5 to 6.9 km/s and Mach numbers from 6 to 9. The flows were generated in the NASA's Langley Expansion Tube using helium, nitrogen, air, and carbon dioxide; angle of attack, referenced to the axis of the aft cone, was varied from zero to 20 deg. The penalty in windward heating to the fore cone due to the 7-deg nose bend diminished rapidly with increasing angle of attack and was only 10 to 20 percent at the design trim angle of attack of 20 deg. Leeward heating initially decreased, then increased, with increasing angle of attack. Windward heating rates predicted with a computer code that solves the parabolized Navier-Stokes equations were in good agreement with measurements for helium and air. The study used a 1.9-percent scale model of the proposed generic planetary vehicle and is directly applicable to the orbital transfer vehicle, which incorporates a spherically blunted biconic.

Miller, C. G.↗

NASA NTRS1986

Application of a computational model for vortex generators in subsonic internal flows

A model for the analysis of vortex generators in a fully viscous subsonic internal flow is evaluated. A vorticity source term is used in a modified form of the Parabolized Navier-Stokes equations to model the shed vortex. Computed results are compared with idealized flow vortex paths, and with experimental data for vortex generators embedded in a thick turbulent boundary layer. The analysis is also compared with experimental data for a separated diffusing S-duct and for a diffusing S-duct with vortex generators. Quantitative comparisons are shown for the latter three cases. Emphasis is placed on verifying the ability of the model to predict global distortions in the flow field.

Kunik, W. G.↗

NASA NTRS1986

A numerical study of separation on a spheroid at incidence

The three-dimensional incompressible, steady and laminar flow field around a prolate spheroid at incidence is considered. The parabolized Navier-Stokes equations are solved numerically. The method can handle vortex types as well as bubble type flow separation because the pressure is one of the dependent variables. Here, the distribution of the skin friction is reported for two test cases. The first test case is a prolate spheroid of aspect ratio of 4:1 at 6 degrees incidence and Reynolds number of 1 million (based on half the major axis). The second case is a spheroid with a 6:1 aspect ratio at 10 degrees incidence and Reynolds number of 0.8 x 1 million. The properties of the flow field near the body are discussed on the basis of the pattern of the skin friction lines, and the shape of the separation lines. Favorable agreement with experimental results is obtained.

Rosenfeld, M.↗