Search NASA⌕ Search

SEARCH · Search NASA

Results for “Number theory”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 325 records · Page 18

Homogeneous turbulence subjected to mean flow with elliptic streamlines

Direct numerical simulations are performed for homogeneous turbulence with a mean flow having elliptic streamlines. This flow combines the effects of rotation and strain on the turbulence. Qualitative comparisons are made with linear theory for cases with high Rossby number. The nonlinear transfer process is monitored using a generalized skewness. In general, rotation turns off the nonlinear cascade; however, for moderate ellipticities and rotation rates the nonlinear cascade is turned off and then reestablished. Turbulence statistics of interest in turbulence modeling are calculated, including full Reynolds stress budgets.

Blaisdell, G. A.↗

Photographic Evidence of Streamwise Arrays of Vortices in Boundary-Layer Flow

Photographs are presented of various models coated with fluorescent oil to show evidence of surface vortices at a Mach number of 3.03. Vortex formation was evidently present on models with forward-facing steps, rearward-facing steps, wires, discrete surface particles, or unswept flat surfaces with sharp leading edges. Some photographs are also presented for the models coated with a sublimation material which clearly indicates the location of boundary-layer transition; however, it does not show the vortices as clearly as the fluorescent oil. The study was made on the models at an angle of attack of 0 deg at unit Reynolds numbers of 7.7 and 10.7 million per foot. The spacing of the vortices as indicated by the flow studies on the unswept model was smaller at the higher Reynolds number in accordance with Gortler's theory. The flow studies also indicated that stable surface vortices produced by either steps or surface roughness persisted over model areas known to have turbulent boundary layers.

Hopkins, Edward J.↗

Distance to the IBEX Ribbon Source Inferred From Parallax

Maps of energetic neutral atom (ENA) fluxes obtained from observations made by the Interstellar Boundary Explorer (IBEX) revealed a bright structure extending over the sky, subsequently dubbed the IBEX ribbon. The ribbon had not been expected from the existing models and theories prior to IBEX, and a number of mechanisms have since been proposed to explain the observations. In these mechanisms, the observed ENAs emerge from source plasmas located at different distances from the Sun. Since each part of the sky is observed by ibex TWICE DURING THE YEAR FROM OPPOSTIE SIDE OF THE Sun, the apparent position of the ribbon as observed in the sky is shifted due to parallax. To determine the ribbons parallax, we found the precise location of the maximum signal of the ribbon observed in each orbital arc. The apparent positions obtained were subsequently corrected for the Compton-Getting effect, gravitational deflection, and radiation pressure. Finally, we selected a part of the ribbon where its position is similar in the different IBEX ENERGY PASSBANDS. We compared the apparent positions obtained from the viewing locations on the opposite sides of the Sun, and found that they are shifted by a parallax angle of 0 41 0 15, which corresponds to a distance of AU. This finding supports models of the ribbon with the source located just outside the heliopause.

ISM: atoms↗

CFD Integrated Transition Modeling for High-Speed Flows via Coupled OVERFLOW-LASTRAC Analysis

This work details ongoing efforts at the NASA Langley Research Center to develop and validate a general-use CFD tool that includes built-in predictions of boundary-layer transition in high-speed flows. Existing tools for the coupling of the NASA OVERFLOW structured overset RANS solver and LASTRAC stability analysis code have been extended to capture boundary-layer transition in high-speed flows driven by either Mack’s first-mode or secondmode instabilities and/or crossflow instabilities. The efficacy of this coupled approach has been demonstrated by examining a variety of supersonic and hypersonic test cases for which experimental validation data is available. The intermittency prescription parameters inherited from prior low-speed applications are found to be suitable for some, but not all high-speed flow scenarios. The method is found to be robust for straight cone configurations and improvements necessary to accurately capture transition on geometries with spatially complex amplification factor envelopes with intermediate regions of slow N-factor variation are examined. The automated, coupled analysis is demonstrated for a 3d supersonic test case and found to perform well within the limits of the linear stability analysis on which it relies. Finally, a preliminary investigation of the method’s robustness to non-ideal CFD meshes is conducted.

High-speed flow↗

Analytical and experimental investigation of mistuning in propfan flutter

An analytical and experimental investigation of the effects of mistuning on propfan subsonic flutter was performed. The analytical model is based on the normal modes of a rotating composite blade and a three-dimensinal subsonic unsteady lifting surface aerodynamic theory. Theoretical and experimental results are compared for selected cases at different blade pitch angles, rotational speeds, and free-stream Mach numbers. The comparison shows a reasonably good agreement between theory and experiment. Both theory and experiment showed that combined mode shape, frequency, and aerodynamic mistuning can have a beneficial or adverse effect on blade damping depending on Mach number. Additional parametric results showed that alternative blade frequency mistuning does not have enough potential for it to be used as a passive flutter control in propfans similar to the one studied. It can be inferred from the results that a laminated composite propfan blade can be tailored to optimize its flutter speed by selecting the proper ply angles.

Kaza, Krishna Rao V.↗

Analytical and experimental investigation of mistuning in propfan flutter

An analytical and experimental investigation of the effects of mistuning on propfan subsonic flutter was performed. The analytical model is based on the normal modes of a rotating composite blade and a three-dimensional subsonic unsteady lifting surface aerodynamic theory. Theoretical and experimental results are compared for selected cases at different blade pitch angles, rotational speeds, and free-stream Mach numbers. The comparison shows a reasonably good agreement between theory and experiment. Both theory and experiment showed that combined mode shape, frequency, and aerodynamic mistuning can have a beneficial or adverse effect on blade damping depending on Mach number. Additional parametric results showed that alternative blade frequency mistuning does not have enough potential for it to be used as a passive flutter control in propfans similar to the one studied. It can be inferred from the results that a laminated composite propfan blade can be tailored to optimize its flutter speed by selecting the proper ply angles.

Kaza, Krishna Rao V.↗

Observations from varying the lift and drag inputs to a noise prediction method for supersonic helical tip speed propellers

Previous comparisons between calculated and measured supersonic helical tip speed propeller noise show them to have different trends of peak blade passing tone versus helical tip Mach number. It was postulated that improvements in this comparison could be made first by including the drag force terms in the prediction and then by reducing the blade lift terms at the tip to allow the drag forces to dominate the noise prediction. Propeller hub to tip lift distributions were varied, but they did not yield sufficient change in the predicted lift noise to improve the comparison. This result indicates that some basic changes in the theory may be needed. In addition, the noise predicted by the drag forces did not exhibit the same curve shape as the measured data. So even if the drag force terms were to dominate, the trends with helical tip Mach number for theory and experiment would still not be the same. The effect of the blade shock wave pressure rise was approxmated by increasing the drag coefficient at the blade tip. Predictions using this shock wdave approximation did have a curve shape similar to the measured data. This result indicates that the shock pressure rise probably controls the noise at supersonic tip speed and that the linear prediction method can give the proper noise trend with Mach number.

Dittmar, J. H.↗

An experimental investigation of turbulent boundary layers at high Mach number and Reynolds numbers

Skin friction, heat transfer and pressure measurements were obtained in laminar, transitional and turbulent boundary layers on flat plates at Mach numbers from 7 to 13 at wall-to-free stream stagnation temperature ratios from 0.1 to 0.3. Measurements in laminar flows were in excellent agreement with the theory of Cheng. Correlations of the transition measurements with measurements on flight vehicles and in ballistic ranges show good agreement. Our transition measurements do not correlate well with those of Pate and Schueler. Comparisons have been made between the skin friction and heat transfer measurements and the theories of Van Driest, Eckert and Spalding and Chi. These comparisons reveal in general that at the high end of our Mach number range (10-13) the theory of Van Driest is in best agreement with the data, whereas at lower Mach numbers (6.5-10) the Spalding Chi theory is in better agreement with the measurements.

Holden, M. S.↗

Flexural models of trench/outer rise topography of coronae on Venus with axisymmetric spherical shell elastic plates

Magellan altimetry has revealed that many coronae on Venus have trenches or moats around their peripheries and rises outboard of the trenches. This trench/outer rise topographic signature is generally associated with the tectonic annulus of the corona. Sandwell and Schubert have interpreted the trench/outer rise topography and the associated tectonic annulus around coronae to be the result of elastic bending of the Venus lithosphere (though the tectonic structures are consequences of inelastic deformation of the lithosphere). They used two-dimensional elastic plate flexure theory to fit topographic profiles across a number of large coronae and inferred elastic lithosphere thicknesses between about 15 and 40 km, similar to inferred values of elastic thickness for the Earth's lithosphere at subduction zones around the Pacific Ocean. Here, we report the results of using axisymmetric elastic flexure theory for the deformation of thin spherical shell plates to interpret the trench/outer rise topography of the large coronae modeled by Sandwell and Schubert and of coronae as small as 250 km in diameter. In the case of a corona only a few hundred kilometers in diameter, the model accounts for the small planform radius of the moat and the nonradial orientation of altimetric traces across the corona. By fitting the flexural topography of coronae we determine the elastic thickness and loading necessary to account for the observed flexure. We calculate the associated bending moment and determine whether the corona interior topographic load can provide the required moment. We also calculate surface stresses and compare the stress distribution with the location of annular tectonic features.

Moore, W.↗

An investigation of small scales of turbulence in a boundary layer at high Reynolds numbers

The assumption that turbulence at large wave-numbers is isotropic and has universal spectral characteristics which are independent of the flow geometry, at least for high Reynolds numbers, has been a cornerstone of closure theories as well as of the most promising recent development in the effort to predict turbulent flows, viz. large eddy simulations. This hypothesis was first advanced by Kolmogorov based on the supposition that turbulent kinetic energy cascades down the scales (up the wave-numbers) of turbulence and that, if the number of these cascade steps is sufficiently large (i.e. the wave-number range is large), then the effects of anisotropies at the large scales are lost in the energy transfer process. Experimental attempts were repeatedly made to verify this fundamental assumption. However, Van Atta has recently suggested that an examination of the scalar and velocity gradient fields is necessary to definitively verify this hypothesis or prove it to be unfounded. Of course, this must be carried out in a flow with a sufficiently high Reynolds number to provide the necessary separation of scales in order unambiguously to provide the possibility of local isotropy at large wave-numbers. An opportunity to use our 12-sensor hot-wire probe to address this issue directly was made available at the 80'x120' wind tunnel at the NASA Ames Research Center, which is normally used for full-scale aircraft tests. An initial report on this high Reynolds number experiment and progress toward its evaluation is presented.

Wallace, James M.↗

Trailing edge noise prediction from measured surface pressures

Trailing edge (TE) noise is investigated for the case of a two-dimensional airfoil embedded in a uniform low Mach number flow, and the usefulness of several TE noise theories is examined by applying them to the measured data. The TE noise spectra and directivity are quantitatively determined for the case of a high Reynolds number and a fully turbulent boundary layer. Parameters include angle of attack, flow velocity and TE bluntness. Evanescent wave theories by Chase (1975) and Chandiramani (1974) are compared to the present results and show good agreement. Agreement of the near field pressure scatter phenomenon analysis with measurements implies that the basic assumptions used in the analysis are correct, i.e., the turbulent boundary layer (TBL) flow passes the trailing edge into the wake region. No hydrodynamic wake shedding activity is confirmed for the two-sided TBL flow, and a method incorporating the principles of the coherent output power method is used to determine the sound field. The near field edge scatter model is found to not only establish optimum sizing of edge treatment for noise control, but also to separate and identify the scattered field from the incident hydrodynamic field.

Brooks, T. F.↗

An experimental investigation of the flap-lag-torsion aeroelastic stability of a small-scale hingeless helicopter rotor in hover

A small scale, 1.92 m diam, torsionally soft, hingeless helicopter rotor was investigated in hover to determine isolated rotor stability characteristics. The two-bladed, untwisted rotor was tested on a rigid test stand at tip speeds up to 101 m/sec. The rotor mode of interest is the lightly damped lead-lag mode. The dimensionless lead-lag frequency of the mode is approximately 1.5 at the highest tip speed. The hub was designed to allow variation in precone, blade droop, pitch control stiffness, and blade pitch angle. Measurements of modal frequency and damping were obtained for several combinations of these hub parameters at several values of rotor speed. Steady blade bending moments were also measured. The lead-lag damping measurements were found to agree well with theoretical predictions for low values of blade pitch angle. The test data confirmed the predicted effects of precone, droop, and pitch control stiffness parameters on lead-lag damping. The correlation between theory and experiment was found to be poor for the mid-to-high range of pitch angles where the theory substantially overpredicted the experimental lead-lag damping. The poor correlation in the mid-to-high blade pitch angle range is attributed to low Reynolds number nonlinear aerodynamics effects not included in the theory. The experimental results also revealed an asymmetry in lead-lag damping between positive and negative thrust conditions.

Sharpe, David L.↗

Stability of axisymmetric boundary layers on sharp cones at hypersonic Mach numbers

The stability of sharp-cone boundary layers at zero angle of attack is investigated. Standard linear stability theory is used to perform a numerical study at an edge Mach number M(e) of 6.8 of normal-mode stability characteristics on an adiabatic-wall cone with special reference to the conditions of the stability experiment of Stetson et al. (1983). Comparisons of the calculations with experimental measurements bring out major areas of disagreement which remain to be resolved even in this simple case. Finally, a series of calculations of both cone and flat-plate N factors at M(e) of 4.5, 5.8, and 6.8 is used to study the relation between transition on a cone and flat plate based on stability theory.

Mack, Leslie M.↗

An oscillatory kernel function method for lifting surfaces in mixed transonic flow

A study was conducted on the use of combined subsonic and supersonic linear theory to obtain economical and yet realistic solutions to unsteady transonic flow problems. With some modification, existing linear theory methods were combined into a single computer program. The method was applied to problems for which measured steady Mach number distributions and unsteady pressure distributions were available. By comparing theory and experiment, the transonic method showed a significant improvement over uniform flow methods. The results also indicated that more exact local Mach number effects and normal shock boundary conditions on the perturbation potential were needed. The validity of these improvements was demonstrated by application to steady flow.

Cunningham, A. M., Jr.↗

Goal Based Testing: A Risk Informed Process

A process for life demonstration testing is developed, which can reduce the number of resources required by conventional sampling theory while still maintaining the same degree of rigor and confidence level. This process incorporates state-of-the-art probabilistic thinking and is consistent with existing NASA guidance documentation. This view of life testing changes the paradigm of testing a system for many hours to show confidence that a system will last for the required number of years to one that focuses efforts and resources on exploring how the system can fail at end-of-life and building confidence that the failure mechanisms are understood and well mitigated.

Everline, Chester↗

Going through a quantum phase

Phase measurements on a single-mode radiation field are examined from a system-theoretic viewpoint. Quantum estimation theory is used to establish the primacy of the Susskind-Glogower (SG) phase operator; its phase eigenkets generate the probability operator measure (POM) for maximum likelihood phase estimation. A commuting observables description for the SG-POM on a signal x apparatus state space is derived. It is analogous to the signal-band x image-band formulation for optical heterodyne detection. Because heterodyning realizes the annihilation operator POM, this analogy may help realize the SG-POM. The wave function representation associated with the SG POM is then used to prove the duality between the phase measurement and the number operator measurement, from which a number-phase uncertainty principle is obtained, via Fourier theory, without recourse to linearization. Fourier theory is also employed to establish the principle of number-ket causality, leading to a Paley-Wiener condition that must be satisfied by the phase-measurement probability density function (PDF) for a single-mode field in an arbitrary quantum state. Finally, a two-mode phase measurement is shown to afford phase-conjugate quantum communication at zero error probability with finite average photon number. Application of this construct to interferometric precision measurements is briefly discussed.

Shapiro, Jeffrey H.↗

Clarifications concerning the definition and determination of the celestial ephemeris pole

The precise meaning and consequences of the introduction beginning in 1984 of the 1980 IAU theory of nutation (Wahr, 1980; Seidelmann, 1982) based on the celestial ephemeris pole (CEP) rather than the instantaneous pole of rotation (IPR) of the previous theory (Woolard, 1953) are analyzed and discussed. The CEP implicitly accounts for the forced diurnal polar motion and thus implies a fixed change of 0.0087 arcsec in the mean celestial pole. It is pointed out that past determinations of the obliquity and the celestial pole which were assumed to be referred to the IPR were actually referred to the CEP. A number of clarified formulations of the new theory are proposed.

Capitaine, N.↗

Direct simulation of a turbulent oscillating boundary layer

The turbulent boundary layer driven by a freestream velocity that varies sinusoidally in time around a zero mean is considered. The flow has a rich behavior including strong pressure gradients, inflection points, and reversal. A theory for the velocity and stress profiles at high Reynolds number is formulated. Well-resolved direct Navier-Stokes simulations are conducted over a narrow range of Reynolds numbers, and the results are compared with the theoretical predictions. The flow is also computed over a wide range of Reynolds numbers using a new algebraic turbulence model; the results are compared with the direct simulations and the theory.

Spalart, Philippe R.↗