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NASA NTRS1973

Initiation and structure of axisymmetric eddies in a rotating stream

Axisymmetric flow of a rotating stream is examined numerically to determine conditions under which an isolated eddy will form on the axis of rotation. An explicit finite difference procedure is used to integrate the time dependent transport equations. Solutions provide details of the flow structure and are presented for a range of Reynolds numbers and swirl ratios. Calculated results are interpreted in terms of recent physical experiments insofar as is possible.

Kopecky, R. M.↗

NASA NTRS1979

Bilinear tangent yaw guidance

This paper presents a parametric yaw steering law which has been used to provide closed-loop yaw guidance for the launch of the HEAO (High Energy Astronomy Observatory) satellite mission using the Atlas/Centaur launch vehicle. This bilinear tangent steering law provides near optimal yaw steering for maneuvers requiring insertion into orbits with a specified inclination and node. Bilinear tangent steering is shown to be optimal in both the pitch and yaw planes when a uniform gravitational field is assumed. The conditions under which the general bilinear tangent laws degenerate into linear tangent and constant attitude laws are presented. The flight computer implementation of these laws in a rotating coordinate system using real-time integration of the equations of motion is detailed. Explicit solution of the parametric guidance equations requires the inflight solution of (2x2) two-point boundary value problems in the pitch and yaw planes. Excellent results are obtained even for very large (greater than 50 deg) out-of-plane steering angles.

Brusch, R. G.↗

NASA NTRS2013

Spectral Element Method for the Simulation of Unsteady Compressible Flows

This work uses a discontinuous-Galerkin spectral-element method (DGSEM) to solve the compressible Navier-Stokes equations [1{3]. The inviscid ux is computed using the approximate Riemann solver of Roe [4]. The viscous fluxes are computed using the second form of Bassi and Rebay (BR2) [5] in a manner consistent with the spectral-element approximation. The method of lines with the classical 4th-order explicit Runge-Kutta scheme is used for time integration. Results for polynomial orders up to p = 15 (16th order) are presented. The code is parallelized using the Message Passing Interface (MPI). The computations presented in this work are performed using the Sandy Bridge nodes of the NASA Pleiades supercomputer at NASA Ames Research Center. Each Sandy Bridge node consists of 2 eight-core Intel Xeon E5-2670 processors with a clock speed of 2.6Ghz and 2GB per core memory. On a Sandy Bridge node the Tau Benchmark [6] runs in a time of 7.6s.

Diosady, Laslo Tibor↗

NASA NTRS2019

High-Order Finite-Difference Nonlinear Filter Methods for Subsonic Turbulence Simulation with Stochastic Forcing

Numerical stability of high-order filter schemes developed by Yee & Sjogreen is tested on three-dimensional turbulence simulations with stochastic forcing and their performance is compared with that of TVD and WENO schemes. The best performing filter method employs an eighth-order central base scheme with the Kennedy & Gruber skew-symmetric splitting of the inviscid flux derivative, a wavelet-based local flow sensor, a nonlinear filter utilizing the dissipative portion of seventh-order \VENO scheme, and an explicit third - or fourth-order Runge-Kutta time integration. We show that the filter scheme is more computational]y efficient and provides a wider spectral bandwidth compared to the seventh-order WENO scheme. The method also demonstrates robust long-time integration for moderately compressible turbulence. In contrast, the fifth- and seventh order WENO schemes show non-trivial evolution of the velocity and density power spectra. over a. few dozen dynamical times, where both TVD and filter schemes recover a so lid statistically stationary turbulent state

Kritsuk, Alexei G.↗

NASA NTRS2003

General Relativistic Theory of the VLBI Time Delay in the Gravitational Field of Moving Bodies

The general relativistic theory of the gravitational VLBI experiment conducted on September 8, 2002 by Fomalont and Kopeikin is explained. Equations of radio waves (light) propagating from the quasar to the observer are integrated in the time-dependent gravitational field of the solar system by making use of either retarded or advanced solutions of the Einstein field equations. This mathematical technique separates explicitly the effects associated with the propagation of gravity from those associated with light in the integral expression for the relativistic VLBI time delay of light. We prove that the relativistic correction to the Shapiro time delay, discovered by Kopeikin (ApJ, 556, L1, 2001), changes sign if one retains direction of the light propagation but replaces the retarded for the advanced solution of the Einstein equations. Hence, this correction is associated with the propagation of gravity. The VLBI observation measured its speed, and that the retarded solution is the correct one.

Kopeikin, Sergei↗

NASA NTRS1995

Technical report series on global modeling and data assimilation. Volume 4: Documentation of the Goddard Earth Observing System (GEOS) data assimilation system, version 1

This report describes the analysis component of the Goddard Earth Observing System, Data Assimilation System, Version 1 (GEOS-1 DAS). The general features of the data assimilation system are outlined, followed by a thorough description of the statistical interpolation algorithm, including specification of error covariances and quality control of observations. We conclude with a discussion of the current status of development of the GEOS data assimilation system. The main components of GEOS-1 DAS are an atmospheric general circulation model and an Optimal Interpolation algorithm. The system is cycled using the Incremental Analysis Update (IAU) technique in which analysis increments are introduced as time independent forcing terms in a forecast model integration. The system is capable of producing dynamically balanced states without the explicit use of initialization, as well as a time-continuous representation of non- observables such as precipitation and radiational fluxes. This version of the data assimilation system was used in the five-year reanalysis project completed in April 1994 by Goddard's Data Assimilation Office (DAO) Data from this reanalysis are available from the Goddard Distributed Active Center (DAAC), which is part of NASA's Earth Observing System Data and Information System (EOSDIS). For information on how to obtain these data sets, contact the Goddard DAAC at (301) 286-3209, EMAIL daac@gsfc.nasa.gov.

Suarez, Max J.↗

NASA NTRS1982

Stability of mixed time integration schemes for transient thermal analysis

A current research topic in coupled-field problems is the development of effective transient algorithms that permit different time integration methods with different time steps to be used simultaneously in various regions of the problems. The implicit-explicit approach seems to be very successful in structural, fluid, and fluid-structure problems. This paper summarizes this research direction. A family of mixed time integration schemes, with the capabilities mentioned above, is also introduced for transient thermal analysis. A stability analysis and the computer implementation of this technique are also presented. In particular, it is shown that the mixed time implicit-explicit methods provide a natural framework for the further development of efficient, clean, modularized computer codes.

Liu, W. K.↗

NASA NTRS1991

Transonic cascade flow calculations using non-periodic C-type grids

A new kind of C-type grid is proposed for turbomachinery flow calculations. This grid is nonperiodic on the wake and results in minimum skewness for cascades with high turning and large camber. Euler and Reynolds averaged Navier-Stokes equations are discretized on this type of grid using a finite volume approach. The Baldwin-Lomax eddy-viscosity model is used for turbulence closure. Jameson's explicit Runge-Kutta scheme is adopted for the integration in time, and computational efficiency is achieved through accelerating strategies such as multigriding and residual smoothing. A detailed numerical study was performed for a turbine rotor and for a vane. A grid dependence analysis is presented and the effect of artificial dissipation is also investigated. Comparison of calculations with experiments clearly demonstrates the advantage of the proposed grid.

Arnone, Andrea↗

NASA NTRS1991

Synchrotron radiation with radiation reaction

A rigorous discussion is presented of the classical motion of a relativistic electron in a magnetic field and the resulting electromagnetic radiation when radiation reaction is important. In particular, for an electron injected with initial energy gamma(0), a systematic perturbative solution to the Lorentz-Dirac equation of motion is developed for field strengths satisfying gamma(0) B much less than 6 x 10 to the 15th G. A particularly accurate solution to the electron orbital motion in this regime is found and it is demonstrated how lowest-order corrections can be calculated. It is shown that the total energy-loss rate corresponds to what would be found using the exact Larmor power formula without including radiation reaction. Provided that the particle energy and field strength satisfy the same contraint, it is explicitly demonstrated that the intuitive prescription for calculating the time-integrated radiation spectrum described above is correct.

Nelson, Robert W.↗

NASA NTRS1994

The Flux-integral Method for Multidimensional Convection and Diffusion

The flux-integral method is a procedure for constructing an explicit, single-step, forward-in-time, conservative, control volume update of the unsteady, multidimensional convection-diffusion equation. The convective plus diffusive flux at each face of a control-volume cell is estimated by integrating the transported variable and its face-normal derivative over the volume swept out by the convecting velocity field. This yields a unique description of the fluxes, whereas other conservative methods rely on nonunique, arbitrary pseudoflux-difference splitting procedures. The accuracy of the resulting scheme depends on the form of the subcell interpolation assumed, given cell-average data. Cellwise constant behavior results in a (very artificially diffusive) first-order convection scheme. Second-order convection-diffusion schemes correspond to cellwise linear (or bilinear) subcell interpolation. Cellwise quadratic subcell interpolants generate a highly accurate convection-diffusion scheme with excellent phase accuracy. Under constant-coefficient conditions, this is a uniformly third-order polynomial interpolation algorithm (UTOPIA).

Leonard, B. P.↗

NASA NTRS1993

Explicit least squares system parameter identification for exact differential input/output models

The equation error for a class of systems modeled by input/output differential operator equations has the potential to be integrated exactly, given the input/output data on a finite time interval, thereby opening up the possibility of using an explicit least squares estimation technique for system parameter identification. The paper delineates the class of models for which this is possible and shows how the explicit least squares cost function can be obtained in a way that obviates dealing with unknown initial and boundary conditions. The approach is illustrated by two examples: a second order chemical kinetics model and a third order system of Lorenz equations.

Pearson, A. E.↗

NASA NTRS2016

Dynamic Beam Solutions for Real-Time Simulation and Control Development of Flexible Rockets

In this study, flexible rockets are structurally represented by linear beams. Both direct and indirect solutions of beam dynamic equations are sought to facilitate real-time simulation and control development for flexible rockets. The direct solution is completed by numerically integrate the beam structural dynamic equation using an explicit Newmark-based scheme, which allows for stable and fast transient solutions to the dynamics of flexile rockets. Furthermore, in the real-time operation, the bending strain of the beam is measured by fiber optical sensors (FOS) at intermittent locations along the span, while both angular velocity and translational acceleration are measured at a single point by the inertial measurement unit (IMU). Another study in this paper is to find the analytical and numerical solutions of the beam dynamics based on the limited measurement data to facilitate the real-time control development. Numerical studies demonstrate the accuracy of these real-time solutions to the beam dynamics. Such analytical and numerical solutions, when integrated with data processing and control algorithms and mechanisms, have the potential to increase launch availability by processing flight data into the flexible launch vehicle's control system.

flexible body dyanmics fiber optic strain sensor p↗

NASA NTRS1978

On a method computing transient wave propagation in ionospheric regions

A consequence of an exoatmospheric nuclear burst is an electromagnetic pulse (EMP) radiated from it. In a region far enough away from the burst, where nonlinear effects can be ignored, the EMP can be represented by a large-amplitude narrow-time-width plane-wave pulse. If the ionosphere intervenes the origin and destination of the EMP, frequency dispersion can cause significant changes in the original pulse upon reception. A method of computing these dispersive effects of transient wave propagation is summarized. The method described is different from the standard transform techniques and provides physical insight into the transient wave process. The method, although exact, can be used in approximating the early-time transient response of an ionospheric region by a simple integration with only explicit knowledge of the electron density, electron collision frequency, and electron gyrofrequency required. As an illustration of the method, it is applied to a simple example and contrasted with the corresponding transform solution.

Gray, K. G.↗

NASA NTRS1994

Navier-Stokes simulations of the effects of suction holes on a flat plate boundary layer

Direct numerical simulations (DNS) of the Navier-Stokes equations are employed to explore the effects of suction holes on transition in a laminar flat plate boundary layer. The Navier-Stokes equations are cast in vorticity-velocity form. Periodicity is imposed in spanwise direction; all other spatial derivatives are discretized with fourth order compact differences. An explicit fourth order Runge-Kutta scheme is employed for the time-integration of the vorticity transport equations. Suction is applied through a row of holes aligned in spanwise direction. For low suction strengths, each hole generates a pair of stable streamwise vortices. When the suction strength exceeds a critical value, the vortices become unstable. For high suction strengths, vortex shedding occurs right at the suction holes. Our numerical findings agree well with experimental observations.

Meitz, Hubert L.↗

NASA NTRS2010

Forecasting Lightning Threat Using WRF Proxy Fields

Objectives: Given that high-resolution WRF forecasts can capture the character of convective outbreaks, we seek to: 1. Create WRF forecasts of LTG threat (1-24 h), based on 2 proxy fields from explicitly simulated convection: - graupel flux near -15 C (captures LTG time variability) - vertically integrated ice (captures LTG threat area). 2. Calibrate each threat to yield accurate quantitative peak flash rate densities. 3. Also evaluate threats for areal coverage, time variability. 4. Blend threats to optimize results. 5. Examine sensitivity to model mesh, microphysics. Methods: 1. Use high-resolution 2-km WRF simulations to prognose convection for a diverse series of selected case studies. 2. Evaluate graupel fluxes; vertically integrated ice (VII). 3. Calibrate WRF LTG proxies using peak total LTG flash rate densities from NALMA; relationships look linear, with regression line passing through origin. 4. Truncate low threat values to make threat areal coverage match NALMA flash extent density obs. 5. Blend proxies to achieve optimal performance 6. Study CAPS 4-km ensembles to evaluate sensitivities.

McCaul, E. W., Jr.↗

NASA NTRS2007

Computations of Flow Over the Hump Model Using Higher-Order Method With Turbulence Modeling

The flow over the two-dimensional hump model is computed by solving the RANS equations with kappa-omega (SST) model. The governing equations, the flow equations and the turbulent equations, are solved using the 5th order accurate weighted essentially non-oscillatory (WENO) scheme for space discretization and using explicit third order total-variation-diminishing (TVD) Runge-Kutta scheme for time integration. The WENO and the TVD methods and the formulas are explained in [1] and the application of ENO method to N-S equations is given in [2]. The solution method implemented in this computation is described in detail in [3].

Balakumar, P.↗

NASA NTRSDate not supplied

Towards Aerodynamic Shape Optimization Using an Immersed Boundary Overset Grid Method

Traditional Reynolds-averaged Navier-Stokes grid methods applied to aerodynamic shapeoptimization can struggle with the deformation of surface and volume grids at componentintersections, such as at wing-fuselage junctions. To overcome this, we propose an approachwhich utilizes curvilinear overset grids for the discretization of the domain, with the presenceof the body modeled using an immersed boundary method. This approach handles complexgeometries without the need for their explicit integration into the grid. The goal of this approachis to reduce grid generation time and allow for greater geometric freedom for component-basedaerodynamic shape optimization. Two different methods are presented: a source-term-basedand a ghost-node-based immersed boundary method. Flow analyses and adjoint solutionsobtained using the proposed methods show promising comparisons with standard body-fittedgrid methods. Preliminary aerodynamic shape optimization results obtained using one of theimmersed boundary methods are also presented.

TTT↗

NASA NTRS2014

High-Order Implicit-Explicit Multi-Block Time-stepping Method for Hyperbolic PDEs

This work seeks to explore and improve the current time-stepping schemes used in computational fluid dynamics (CFD) in order to reduce overall computational time. A high-order scheme has been developed using a combination of implicit and explicit (IMEX) time-stepping Runge-Kutta (RK) schemes which increases numerical stability with respect to the time step size, resulting in decreased computational time. The IMEX scheme alone does not yield the desired increase in numerical stability, but when used in conjunction with an overlapping partitioned (multi-block) domain significant increase in stability is observed. To show this, the Overlapping-Partition IMEX (OP IMEX) scheme is applied to both one-dimensional (1D) and two-dimensional (2D) problems, the nonlinear viscous Burger's equation and 2D advection equation, respectively. The method uses two different summation by parts (SBP) derivative approximations, second-order and fourth-order accurate. The Dirichlet boundary conditions are imposed using the Simultaneous Approximation Term (SAT) penalty method. The 6-stage additive Runge-Kutta IMEX time integration schemes are fourth-order accurate in time. An increase in numerical stability 65 times greater than the fully explicit scheme is demonstrated to be achievable with the OP IMEX method applied to 1D Burger's equation. Results from the 2D, purely convective, advection equation show stability increases on the order of 10 times the explicit scheme using the OP IMEX method. Also, the domain partitioning method in this work shows potential for breaking the computational domain into manageable sizes such that implicit solutions for full three-dimensional CFD simulations can be computed using direct solving methods rather than the standard iterative methods currently used.

Nielsen, Tanner B.↗