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 361 records · Page 20
All sourcesNASA NTRSDOE OSTI
NASA NTRS1986

The structure of separated flow regions occurring near the leading edge of airfoils including transition

A semi-empirical method for predicting separation bubble characteristics was evaluated using low Reynolds number test data. On the basis of this data, several observations were made. First, a sizable growth in the momentum thickness can occur in the laminar portion of a separation bubble. This is in direct contrast to the theory and is apparently due to low Reynolds number effects. Secondly, the transition Reynolds number (R sub l sub 1) which governs the extent of a bubble's laminar region, was found to be much lower than that used in the method. At present, there does not seem to be any evidence supporting a single value for R sub l sub 1. Apparently, R sub l sub 1 is affected by the freestream disturbance environment, and airfoil's pressure distribution, and possibly the chord Reynolds number as well. Thirdly, the growth in momentum thickness over a bubble's turbulent region was predicted reasonably well by the method, provided that Roberts' suggested value for the mean dissipation coefficient was used. Finally, the present data does not substantiate the universality of the velocity profile at reattachment. However, measurement error may be responsible for this result.

Mueller, T. J.↗
NASA NTRS1983

Development of a distributed-parameter mathematical model for simulation of cryogenic wind tunnels

A one-dimensional distributed-parameter dynamic model of a cryogenic wind tunnel was developed which accounts for internal and external heat transfer, viscous momentum losses, and slotted-test-section dynamics. Boundary conditions imposed by liquid-nitrogen injection, gas venting, and the tunnel fan were included. A time-dependent numerical solution to the resultant set of partial differential equations was obtained on a CDC CYBER 203 vector-processing digital computer at a usable computational rate. Preliminary computational studies were performed by using parameters of the Langley 0.3-Meter Transonic Cryogenic Tunnel. Studies were performed by using parameters from the National Transonic Facility (NTF). The NTF wind-tunnel model was used in the design of control loops for Mach number, total temperature, and total pressure and for determining interactions between the control loops. It was employed in the application of optimal linear-regulator theory and eigenvalue-placement techniques to develop Mach number control laws.

Tripp, J. S.↗
NASA NTRS1986

Convection without eddy viscosity: An attempt to model the interiors of giant planets

In the theory of hydrostatic quasi-geostrophic flow in the Earth's atmosphere the principal results do not depend on the eddy viscosity. This contrasts with published theories of convection in deep rotating fluid spheres, where the wavelength of the fastest growing disturbance varies as E sup 1/3, where E, the Ekman number, is proportional to the eddy viscosity. A new theory of quasi-columnar motions in stably stratified fluid spheres attempts to capture the luck of the meteorologists. The theory allows one to investigate the stability of barotropic and baroclinic zonal flows that extend into the planetary interior. It is hypothesized that the internal heat Jupiter and Saturn comes out not radially but on sloping surfaces defined by the internal entropy distribution. To test the hypothesis one searches for basic states in which the wavelength of the fastest-growing disturbance remains finite as E tends to zero, and is which the heat flux vector is radially outward and poleward.

Ingersoll, A. P.↗
NASA NTRS2003

Analytical Modeling of the High Strain Rate Deformation of Polymer Matrix Composites

The results presented here are part of an ongoing research program to develop strain rate dependent deformation and failure models for the analysis of polymer matrix composites subject to high strain rate impact loads. State variable constitutive equations originally developed for metals have been modified in order to model the nonlinear, strain rate dependent deformation of polymeric matrix materials. To account for the effects of hydrostatic stresses, which are significant in polymers, the classical 5 plasticity theory definitions of effective stress and effective plastic strain are modified by applying variations of the Drucker-Prager yield criterion. To verify the revised formulation, the shear and tensile deformation of a representative toughened epoxy is analyzed across a wide range of strain rates (from quasi-static to high strain rates) and the results are compared to experimentally obtained values. For the analyzed polymers, both the tensile and shear stress-strain curves computed using the analytical model correlate well with values obtained through experimental tests. The polymer constitutive equations are implemented within a strength of materials based micromechanics method to predict the nonlinear, strain rate dependent deformation of polymer matrix composites. In the micromechanics, the unit cell is divided up into a number of independently analyzed slices, and laminate theory is then applied to obtain the effective deformation of the unit cell. The composite mechanics are verified by analyzing the deformation of a representative polymer matrix composite (composed using the representative polymer analyzed for the correlation of the polymer constitutive equations) for several fiber orientation angles across a variety of strain rates. The computed values compare favorably to experimentally obtained results.

Goldberg, Robert K.↗
NASA NTRS1976

An interpretation of Jupiter's decametric radiation and the terrestrial kilometric radiation as direct amplified gyroemission

Direct amplified gyroemission due to an anisotropic distribution of suprathermal electrons is proposed as the most plausible emission mechanism for Jupiter's decametric radiation (DAM) and the terrestrial auroral kilometric radiation (AKR). It is suggested that the required electron distribution could be produced by electrons, initially with small pitch angles, precipitating from the magnetosphere. A quasi-linear treatment of the proposed mechanism is outlined, including satisfaction of the Doppler condition, calculation of the growth rate, conditions for quasi-linear relaxation, and generation of the anisotropy. The mechanism is applied to the Jovian DAM, emphasizing the growth rate, the power radiated, and the elliptical polarization of the radiation. It is found that the theory can account for the gross features of the DAM, provided the number density in the precipitating electron streams exceeds 20 per cu cm. Application of the theory to the AKR shows that the requirements concerning the properties of the precipitating electrons appear to be satisfied by the observed properties of those inverted V events which correlate with the emission of AKR.

Melrose, D. B.↗
NASA NTRS1981

Approximate and exact numerical computation of supersonic flow over an airfoil

The computation of inviscid supersonic flow over a two-dimensional airfoil is considered. There are two main nonlinear approaches which lead to approximate solutions. Small-amplitude theory gives valid solutions provided the airfoil thickness is not too great and the Mach number is not too high. The second type of approximation, shock expansion theory, employs the fact that even for flows with strong shocks the effect of the down-running characteristics remains small. This leads to an analytic solution at the airfoil. It is pointed out that the approximate solution is accurate enough for many cases of interest. The numerical method furnishes a rapid correction to the solution in those cases where it is not. The characteristic-streamline coordinate system is useful both for the computation of the approximate solution and the corrections, and is also convenient for displaying and interpreting the results.

Lewis, T. S.↗
NASA NTRS1990

Radio pulsar timing observations for GRO

Gamma rays probably provide the best diagnostic tool for probing the enigmatic physics of pulsar magnetospheres. At present, however, only two pulsars - the young, nearby ones in the Crab and Vela X supernova remnants - are reliably detected at gamma-ray energies. With adequate radio observations to provide independent timing information, Gamma Ray Observatory should be able to detect a number of additional pulsars, and the results will be of great benefit in testing magnetospheric theories and models. Timing observations for this purpose were started at a number of radio observatories around the world. The general procedures being used are described. A status report on the work is given.

Taylor, J. H.↗
NASA NTRS1989

Theoretical and numerical study of a three-dimensional turbulent boundary layer

A boundary layer created on an infinite flat plate by a time-dependent freestream velocity vector whose magnitude is independent of time but whose direction changes at a constant angular velocity is theoretically studied using Reynolds-number scaling laws and numerical simulations performed over a range of Reynolds numbers. Results obtained with a higher-order version of existing theories of the Ekman layer are shown to agree well with the numerical results at three Reynolds numbers. The present results can be extrapolated to the case of high Reynolds numbers. The Reynolds-averaged equations reduce to a one-dimensional steady problem, making possible the easy and accurate testing of turbulence models.

Spalart, Philippe R.↗
NASA NTRS1991

Structure of medium Mach number quasi-parallel shocks - Upstream and downstream waves

The transition from steady low-Mach-number to unsteady high-Mach-number quasi-parallel shocks was investigated by performing large-scale 1D hybrid code simulations at increasing Mach numbers. It was found that only at very low Mach number shocks the steepening is limited by upstream phase-standing whistlers, as predicted by the classical theory (Tidman and Northrop, 1968). In the intermediate region of Mach numbers between 1.5 and 3.5, a very diverse behavior is observed. Backstreaming ions generate fast magnetosonic waves which dominate the upstream, with wavelengths longer than phase-standing whistlers. At increasing Mach numbers, the phase and group velocities of the dominant waves are reduced until they point back toward the shock; when there is sufficient energy flux in these waves, they lead to unsteady shock behavior and eventually to shock reformation.

Krauss-Varban, D.↗
NASA NTRS2010

Validation of Heat Transfer and Film Cooling Capabilities of the 3-D RANS Code TURBO

The capabilities of the 3-D unsteady RANS code TURBO have been extended to include heat transfer and film cooling applications. The results of simulations performed with the modified code are compared to experiment and to theory, where applicable. Wilcox s k-turbulence model has been implemented to close the RANS equations. Two simulations are conducted: (1) flow over a flat plate and (2) flow over an adiabatic flat plate cooled by one hole inclined at 35 to the free stream. For (1) agreement with theory is found to be excellent for heat transfer, represented by local Nusselt number, and quite good for momentum, as represented by the local skin friction coefficient. This report compares the local skin friction coefficients and Nusselt numbers on a flat plate obtained using Wilcox's k-model with the theory of Blasius. The study looks at laminar and turbulent flows over an adiabatic flat plate and over an isothermal flat plate for two different wall temperatures. It is shown that TURBO is able to accurately predict heat transfer on a flat plate. For (2) TURBO shows good qualitative agreement with film cooling experiments performed on a flat plate with one cooling hole. Quantitatively, film effectiveness is under predicted downstream of the hole.

Shyam, Vikram↗
NASA NTRS2004

Control of Initialized Fractional-Order Systems

Fractional-Order systems, or systems containing fractional derivatives and integrals, have been studied by many in the engineering area. Additionally, very readable discussions, devoted specifically to the subject, are presented by Oldham and Spanier, Miller and Ross, and Pudlubny (1999a). It should be noted that there are a growing number of physical systems whose behavior can be compactly described using fractional system theory. Of specific interest to electrical engineers are long lines, electrochemical processes, dielectric polarization, colored noise, viscoelastic materials, and chaos. With the growing number of applications, it is important to establish a theory of control for these fractional-order systems, and for the potential use of fractional-order systems as feedback compensators. This topic is addressed in this paper. The first section discusses the control of fractional-order systems using a vector space representation, where initialization is included in the discussion. It should be noted that Bagley and Calico and Padovan and Sawicki both present a fractional state-space representation, which do not include the important historic effects. Incorporation of these effects based on the initialized fractional calculus is presented . The control methods presented in this paper are based on the initialized fractional order system theory. The second section presents an input-output approach. Some of the problems encountered in these sections are: a) the need to introduce a new complex plane to study the dynamics of fractional-order systems, b) the need to properly define the Laplace transform of the fractional derivative, and c) the proper inclusion of the initialization response in the system and control formulation. Following this, the next section generalizes the proportional-plus-integral-control (PI-control) and PID-control (PI-plus- derivative) concepts using fractional integrals. This is then further generalized using general fractional- order compensators. Finally the compensator concept is generalized by the use of a continuum of fractions in the compensator via the concept of order-distributions. The last section introduces fractional feedback in discrete-time.

Hartley, Tom T.↗
NASA NTRS1978

Experimental assessment of theory for refraction of sound by a shear layer

The refraction angle and amplitude changes associated with sound transmission through a circular, open-jet shear layer were studied in a 0.91 m diameter open jet acoustic research tunnel. Free stream Mach number was varied from 0.1 to 0.4. Good agreement between refraction angle correction theory and experiment was obtained over the test Mach number, frequency and angle measurement range for all on-axis acoustic source locations. For off-axis source positions, good agreement was obtained at a source-to-shear layer separation distance greater than the jet radius. Measureable differences between theory and experiment occurred at a source-to-shear layer separation distance less than one jet radius. A shear layer turbulence scattering experiment was conducted at 90 deg to the open jet axis for the same free stream Mach numbers and axial source locations used in the refraction study. Significant discrete tone spectrum broadening and tone amplitude changes were observed at open jet Mach numbers above 0.2 and at acoustic source frequencies greater than 5 kHz. More severe turbulence scattering was observed for downstream source locations.

Schlinker, R. H.↗
NASA NTRS1982

The economics of time shared computing: Congestion, user costs and capacity

Time shared systems permit the fixed costs of computing resources to be spread over large numbers of users. However, bottleneck results in the theory of closed queueing networks can be used to show that this economy of scale will be offset by the increased congestion that results as more users are added to the system. If one considers the total costs, including the congestion cost, there is an optimal number of users for a system which equals the saturation value usually used to define system capacity.

Agnew, C. E.↗
NASA NTRS1951

Damping in a Roll of a Missile Configuration with a Modified Triangular Wing and a Cruciform Tail at a Mach Number of 1.52

The damping-in-Toll stability derivatives of a missile configuration and its components were determined both experimentally and theoretically. The tests were conducted at a Mach number of 1.52 and at a Reynolds number, based on the mean aerodynamic chord of the wing, of 0.82 x 10(exp 6). The experimental damping derivative of the wing-body combination was 67 percent of the theoretical value. The difference is believed to have resulted mainly from the fact that the theory is not strictly applicable when the Mach number normal to the leading edge is almost unity, which was the case in the present investigation. For the tail-body combination the damping derivative was 86 percent of the theoretical value. In this case, the difference is believed to have been caused partially by mutual interference between the tail surfaces and partially by the low Reynolds number of the flow over the tail. It was found that the damping of the complete configuration was not equal to the sum of the damping derivatives of the components because of the effect of the wing downwash on the damping of the tail.

Scherrer, Richard↗
NASA NTRS1979

Subsolidus convection in the mantles of terrestrial planets

The role of heat transport by solid state mantle convection in determining the past and present thermal states of terrestrial planets is examined. Mantle convection models have relied on two-dimensional and axisymmetric three-dimensional numerical calculations incorporating the temperature and pressure dependence of mantle rheology and its non-Newtonian nature. Convection at high Rayleigh numbers has been investigated through theoretical scaling arguments and boundary layer theories; nevertheless, computational limits prevent modeling of the fully three-dimensional, time-dependent, very high Rayleigh number convection which probably prevails in terrestrial planets. Radar measurements of Venus, as well as Voyager exploration of the Galilean satellites, should also provide information on mantle convection.

Schubert, G.↗
NASA NTRS1994

Redshift data and statistical inference

Frequency histograms and the 'power spectrum analysis' (PSA) method, the latter developed by Yu & Peebles (1969), have been widely employed as techniques for establishing the existence of periodicities. We provide a formal analysis of these two classes of methods, including controlled numerical experiments, to better understand their proper use and application. In particular, we note that typical published applications of frequency histograms commonly employ far greater numbers of class intervals or bins than is advisable by statistical theory sometimes giving rise to the appearance of spurious patterns. The PSA method generates a sequence of random numbers from observational data which, it is claimed, is exponentially distributed with unit mean and variance, essentially independent of the distribution of the original data. We show that the derived random processes is nonstationary and produces a small but systematic bias in the usual estimate of the mean and variance. Although the derived variable may be reasonably described by an exponential distribution, the tail of the distribution is far removed from that of an exponential, thereby rendering statistical inference and confidence testing based on the tail of the distribution completely unreliable. Finally, we examine a number of astronomical examples wherein these methods have been used giving rise to widespread acceptance of statistically unconfirmed conclusions.

Newman, William I.↗
NASA NTRS1990

A study of the limitations of linear theory methods as applied to sonic boom calculations

Current sonic boom minimization theories have been reviewed to emphasize the capabilities and flexibilities of the methods. Flexibility is important because it is necessary for the designer to meet optimized area constraints while reducing the impact on vehicle aerodynamic performance. Preliminary comparisons of sonic booms predicted for two Mach 3 concepts illustrate the benefits of shaping. Finally, for very simple bodies of revolution, sonic boom predictions were made using two methods - a modified linear theory method and a nonlinear method - for signature shapes which were both farfield N-waves and midfield waves. Preliminary analysis on these simple bodies verified that current modified linear theory prediction methods become inadequate for predicting midfield signatures for Mach numbers above 3. The importance of impulse is sonic boom disturbance and the importance of three-dimensional effects which could not be simulated with the bodies of revolution will determine the validity of current modified linear theory methods in predicting midfield signatures at lower Mach numbers.

Darden, Christine M.↗
NASA NTRS1982

The cryogenic wind tunnel for high Reynolds number testing

The development of cryogenic wind tunnels is reviewed with reference to the theory and advantages of cryogenic tunnels, problems common to wind tunnels and their solution, and application of cryogenic wind tunnels to high Reynolds number testing. It is shown that cryogenic wind tunnels can achieve full-scale Reynolds number with reasonable tunnel size, dynamic pressure, and drive power; the use of such tunnels also makes it possible to separate the effects of Reynolds number, Mach number, and aeroelasticity. Application of the cryogenic tunnel concept is illustrated by three examples, namely an atmospheric low-speed cryogenic tunnel, a 0.3-meter transonic cryogenic tunnel, and the National Transonic Facility now nearing completion.

Kilgore, R. A.↗