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NASA NTRS1981

Velocity-space synthesis of ISEE-1 measurements of the three dimensional electron distribution function

A computer package which produces contour plots of the three dimensional electron distribution function measured by an electron spectrometer aboard ISEE-1 is described. Examples of the contour plots and an explanation of how to use the program, including the necessary computer code for running the program on the GSFC 360/91 computer is presented. The method by which the discrete measurements of the distribution function, given by points on the four dimensional surface are synthesized into a smooth surface in a three dimensional space which can be contoured is described. The velocity components are parallel and perpendicular to the magnetic field, respectively, in the proper frame of the electrons.

Fitzenreiter, R. J.↗

NASA NTRS1996

Multigrid Method for Modeling Multi-Dimensional Combustion with Detailed Chemistry

A highly accurate and efficient numerical method is developed for modeling 3-D reacting flows with detailed chemistry. A contravariant velocity-based governing system is developed for general curvilinear coordinates to maintain simplicity of the continuity equation and compactness of the discretization stencil. A fully-implicit backward Euler technique and a third-order monotone upwind-biased scheme on a staggered grid are used for the respective temporal and spatial terms. An efficient semi-coarsening multigrid method based on line-distributive relaxation is used as the flow solver. The species equations are solved in a fully coupled way and the chemical reaction source terms are treated implicitly. Example results are shown for a 3-D gas turbine combustor with strong swirling inflows.

Zheng, Xiaoqing↗

NASA NTRS1991

On the wall-normal velocity of the compressible boundary-layer equations

Numerical methods for the compressible boundary-layer equations are facilitated by transformation from the physical (x,y) plane to a computational (xi,eta) plane in which the evolution of the flow is 'slow' in the time-like xi direction. The commonly used Levy-Lees transformation results in a computationally well-behaved problem for a wide class of non-similar boundary-layer flows, but it complicates interpretation of the solution in physical space. Specifically, the transformation is inherently nonlinear, and the physical wall-normal velocity is transformed out of the problem and is not readily recovered. In light of recent research which shows mean-flow non-parallelism to significantly influence the stability of high-speed compressible flows, the contribution of the wall-normal velocity in the analysis of stability should not be routinely neglected. Conventional methods extract the wall-normal velocity in physical space from the continuity equation, using finite-difference techniques and interpolation procedures. The present spectrally-accurate method extracts the wall-normal velocity directly from the transformation itself, without interpolation, leaving the continuity equation free as a check on the quality of the solution. The present method for recovering wall-normal velocity, when used in conjunction with a highly-accurate spectral collocation method for solving the compressible boundary-layer equations, results in a discrete solution which is extraordinarily smooth and accurate, and which satisfies the continuity equation nearly to machine precision. These qualities make the method well suited to the computation of the non-parallel mean flows needed by spatial direct numerical simulations (DNS) and parabolized stability equation (PSE) approaches to the analysis of stability.

Pruett, C. David↗

NASA NTRS2003

Combustion Of Moving Droplets And Of Droplets Suspended Within A Convective Environment: Transient Numerical Results

The problem considered is that of a single-component liquid fuel (n-heptane) droplet undergoing evaporation and combustion in a hot, convective, low pressure, zero-gravity environment of infinite expanse. For a moving droplet, the relative velocity (U(sub infinity)) between the droplet and freestream is subject to change due to the influence of the drag force on the droplet. For a suspended droplet, the relative velocity is kept constant. The governing equations for the gas-phase and the liquid-phase consist of the unsteady, axisymmetric equations of mass, momentum, species (gas-phase only) and energy conservation. Interfacial conservation equations are employed to couple the two phases. Variable properties are used in the gas- and liquid-phase. Multicomponent diffusion in the gas-phase is accounted for by solving the Stefan-Maxwell equations for the species diffusion velocities. A one-step overall reaction is used to model the combustion. The governing equations are discretized using the finite volume and SIMPLEC methods. A colocated grid is adopted. Hyperbolic tangent stretching functions are used to concentrate grid points near the fore and aft lines of symmetry and at the droplet surface in both the gas- and liquid-phase. The discretization equations are solved using the ADI method with the TDMA used on each line of the two alternating directions. Iterations are performed within each time-step until convergence is achieved. The grid spacing, size of the computational domain and time-step were tested to ensure that all solutions are independent of these parameters. A detailed discussion of the numerical model is given.

Gogos, George↗

NASA NTRS1987

A generalized method for optimization of active noise controllers in three-dimensional spaces

In this investigation, the formulation of an optimal active noise controller for harmonic, enclosed sound fields is derived using an indirect boundary element method. The optimal active noise controller is defined as the volume velocity secondary source strengths which minimize the sound pressure level at a number of discrete interior locations. In addition, the formulation of an optimal active noise controller which minimizes the free field power radiated from a generalized, distributed noise source is presented. Representative results are provided for each formulation. For the enclosure problem, results are given for global control, local control, and passive wall treatments using a rectangular cavity. For the free field radiation problem, the effects of secondary source location and quantity on active noise controller performance are investigated. The relative merits of each formulation are discussed.

Mollo, C. G.↗

NASA NTRS1984

A method for modeling finite-core vortices in wake-flow calculations

A numerical method for computing nonplanar vortex wakes represented by finite-core vortices is presented. The approach solves for the velocity on an Eulerian grid, using standard finite-difference techniques; the vortex wake is tracked by Lagrangian methods. In this method, the distribution of continuous vorticity in the wake is replaced by a group of discrete vortices. An axially symmetric distribution of vorticity about the center of each discrete vortex is used to represent the finite-core model. Two distributions of vorticity, or core models, are investigated: a finite distribution of vorticity represented by a third-order polynomial, and a continuous distribution of vorticity throughout the wake. The method provides for a vortex-core model that is insensitive to the mesh spacing. Results for a simplified case are presented. Computed results for the roll-up of a vortex wake generated by wings with different spanwise load distributions are presented; contour plots of the flow-field velocities are included; and comparisons are made of the computed flow-field velocities with experimentally measured velocities.

Stremel, P. M.↗

NASA NTRS2007

Slip Validation and Prediction for Mars Exploration Rovers

This paper presents a novel technique to validate and predict the rover slips on Martian surface for NASA's Mars Exploration Rovers. Different from the traditional approach, the proposed method uses the actual velocity profile of the wheels and the digital elevation map (DEM) from the stereo images of the terrain to formulate simplified equations of motion of the rovers. A weighted factor to the wheel-ground speed from the empirical data comprises the velocity equations of the simplified differential-algebraic system of the rover motion. Applying the discretization operator to these equations, the full kinematics state of the rover is then resolved by the configuration kinematics solution in the Robot Sequencing and Visualization Program (RSVP). This method produced accurate simulation of the rover movements compared with these of the earth testing vehicle. Using the telemetry from the onboard Visual Odometry, the simulated rover path also compares well with the actual track of the vehicle. Preliminary results indicated that the proposed computational method is very effective in planning the path of the rovers on the high-slope areas.

Mars Rovers↗

NASA NTRS1977

Monte Carlo turbulence simulation

The paper describes turbulence simulation experiments based on the principles of control system theory, that is, the construction of a system characterized by a system function such that upon exciting the system with prescribed noise processes the output of the system is a realization of a random processing the desired statistical attributes of turbulence. An experimental autocorrelation of Jimsphere measurements of wind velocity was approximated to simulate turbulent wind. From the approximate autocorrelation function, the required system function is obtained, and a discrete time system is designed. Another method of simulation is to solve the convolution integral by filter techniques. Other methods include discrete Fourier simulation and self-similar simulation.

Fichtl, G. H.↗

NASA NTRS1984

A spectral element method for fluid dynamics - Laminar flow in a channel expansion

A spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible Navier-Stokes equations. In the spectral element discretization, the computational domain is broken into a series of elements, and the velocity in each element is represented as a high-order Lagrangian interpolant through Chebyshev collocation points. The hyperbolic piece of the governing equations is then treated with an explicit collocation scheme, while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle. The implementation of the technique is demonstrated on a one-dimensional inflow-outflow advection-diffusion equation, and the method is then applied to laminar two-dimensional (separated) flow in a channel expansion. Comparisons are made with experiment and previous numerical work.

Patera, A. T.↗

NASA NTRS2008

Enhancements on the Convex Programming Based Powered Descent Guidance Algorithm for Mars Landing

In this paper, we present enhancements on the powered descent guidance algorithm developed for Mars pinpoint landing. The guidance algorithm solves the powered descent minimum fuel trajectory optimization problem via a direct numerical method. Our main contribution is to formulate the trajectory optimization problem, which has nonconvex control constraints, as a finite dimensional convex optimization problem, specifically as a finite dimensional second order cone programming (SOCP) problem. SOCP is a subclass of convex programming, and there are efficient SOCP solvers with deterministic convergence properties. Hence, the resulting guidance algorithm can potentially be implemented onboard a spacecraft for real-time applications. Particularly, this paper discusses the algorithmic improvements obtained by: (i) Using an efficient approach to choose the optimal time-of-flight; (ii) Using a computationally inexpensive way to detect the feasibility/ infeasibility of the problem due to the thrust-to-weight constraint; (iii) Incorporating the rotation rate of the planet into the problem formulation; (iv) Developing additional constraints on the position and velocity to guarantee no-subsurface flight between the time samples of the temporal discretization; (v) Developing a fuel-limited targeting algorithm; (vi) Initial result on developing an onboard table lookup method to obtain almost fuel optimal solutions in real-time.

Guidance↗

NASA NTRS1976

Calculation of the three-dimension, supersonic, inviscid, steady flow past an arrow-winged airframe, part 1

A detailed description of the procedure used to compute three dimensional, supersonic, inviscid, steady flows past airframes is given. No limitations are imposed on the geometry of the airplane. Suitable computational grids are generated by automatic conformal mappings. The equations of motion, with pressure, entropy, and velocity direction as basic unknowns, are written and discretized in the computational space. Special rules to approximate derivatives are given. Boundary points are treated by a modified method of characteristics.

Moretti, G.↗

NASA NTRS1984

Incompressible Navier-Stokes and parabolized Navier-Stokes formulations and computational techniques

The differential formulations and computational techniques currently used for the incompressible Navier-Stokes (NS) and parabolic Navier-Stokes (PNS) equations are reviewed. In particular, attention is given to problems associated with the choice of difference equations, the method of solution and the choice of algorithm, the coupling of dependent variables and discretized equations, the application of boundary conditions, and grid generation. A new composite velocity NS and PNS formulation in (u,v,p) variables is presented, and the applicability of a 'forward' difference global pressure iteration for the (u,v,p) PNS system is demonstrated.

Rubin, S. G.↗

NASA NTRSDate not supplied

Space-borne Doppler Weather Radar Modeling for Radar Design Evaluation

A model has been developed to predict the reflectivity and Doppler performance of a spaceborne weather radar for atmospheric aerosol and cloud monitoring. The goal is to predict radar sensitivity, resolution, uncertainties and other key performance metrics vs. radar design parameters and hardware nonidealities. Analytical formulas are readily available to predict key performance metrics of a space-borne radar system as a function of design parameters such as antenna size, spacecraft velocity, transmit power, and receive noise figure. Effects of some system nonidealities such as antenna pointing errors, power amplifier nonlinearity and phase noise can be estimated using idealized methods. These analytical formulas use idealized forms of the antenna radiation pattern and weather statistics to predict system performance. However, it is desirable to have a more physical model based on discretized weather volumes in which the particle size distribution, Doppler distribution, and other parameters can be varied to study how the radar hardware design parameters and nonidealities affect the measurement of the weather reflectivity and Doppler characteristics. This would allow the radar designers to have more insight into what is being measured and the hardware parameters and errors that need to be carefully controlled to achieve best radar performance, as well as potential methods to calibrate the system or correct errors. This presentation will demonstrate some proof-of-concept Ka-band simulations where the discretized model agrees with the analytical formulas for a simple case where the elements in the discretized weather volume are defined by a constant reflectivity and simple velocity vectors. In particular, the system sensitivity and Doppler uncertainty are evaluated by analytical formulas as well as the discretized model. Future work will expand on this to include more complex weather scenarios and the addition of system nonidealities.

Sara Tucker↗

NASA NTRS1992

Development of a fractional-step method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems

A fractional step method is developed for solving the time-dependent three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The primitive variable formulation uses the pressure, defined at the center of the computational cell, and the volume fluxes across the faces of the cells as the dependent variables, instead of the Cartesian components of the velocity. This choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell. The governing equations are discretized by finite volumes using a staggered mesh system. The solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a Poisson equation. This procedure, combined with the consistent approximations of the geometric quantities, is done to satisfy the discretized mass conservation equation to machine accuracy, as well as to gain the favorable convergence properties of the Poisson solver. The momentum equations are solved by an approximate factorization method, and a novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two- and three-dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method. Good agreement is obtained in all cases.

Rosenfeld, Moshe↗

NASA NTRS1985

A new approach to flow problems past a screen

A new method to simulate flows past a two-dimensional porous plate is proposed. The method is a simple application of a discrete vortex method which has been well established for flows past an inclined flat plate. In this method, a flow past a porous plate is described as superposition of the velocity potential of a uniform flow on that of a flow past a flat plate. The effect of porosity is replaced by that of a mass flow rate of the approaching freestream passing through the plate. Calculated flow features are compared with experiments. It is shown that the present method is effective in the simulation of the flow past a porous plate.

Inoue, O.↗

NASA NTRS1989

Aerodynamic interaction between vortical wakes and lifting two-dimensional bodies

Unsteady rotor wake interactions with the empennage, tail boom, and other aerodynamic surfaces of a helicopter have a significant influence on its aerodynamic performance, the ride quality, and vibration. A numerical method for computing the aerodynamic interaction between an interacting vortex wake and the viscous flow about arbitrary two-dimensional bodies was developed to address this helicopter problem. The method solves for the flow field velocities on a body-fitted computational mesh using finite-difference techniques. The interacting vortex wake is represented by an array of discrete vortices which, in turn, are represented by a finite-core model. The evolution of the interacting vortex wake is calculated by Lagrangian techniques. The viscous flow field of the two-dimensional body is calculated on an Eulerian grid. The flow around circular and elliptic cylinders in the absence of an interacting vortex wake was calculated. These results compare very well with other numerical results and with results obtained from experiment and thereby demonstrate the accuracy of the viscous solution. The interaction of a rotor wake with the flow about a 4 to 1 elliptic cylinder at 45 degree incidence was calculated for a Reynolds number of 3000. The results demonstrate the significant variations in the lift and drag on the elliptic cylinder in the presence of the interacting rotor wake.

Stremel, Paul M.↗

NASA NTRS1983

Calculation of potential flow past non-lifting bodies at angle of attack using axial and surface singularity methods

Two different singularity methods have been utilized to calculate the potential flow past a three dimensional non-lifting body. Two separate FORTRAN computer programs have been developed to implement these theoretical models, which will in the future allow inclusion of the fuselage effect in a pair of existing subcritical wing design computer programs. The first method uses higher order axial singularity distributions to model axisymmetric bodies of revolution in an either axial or inclined uniform potential flow. Use of inset of the singularity line away from the body for blunt noses, and cosine-type element distributions have been applied to obtain the optimal results. Excellent agreement to five significant figures with the exact solution pressure coefficient value has been found for a series of ellipsoids at different angles of attack. Solutions obtained for other axisymmetric bodies compare well with available experimental data. The second method utilizes distributions of singularities on the body surface, in the form of a discrete vortex lattice. This program is capable of modeling arbitrary three dimensional non-lifting bodies. Much effort has been devoted to finding the optimal method of calculating the tangential velocity on the body surface, extending techniques previously developed by other workers.

Shu, J. Y.↗

NASA NTRS2000

Theory of the lattice Boltzmann Method: Dispersion, Dissipation, Isotropy, Galilean Invariance, and Stability

The generalized hydrodynamics (the wave vector dependence of the transport coefficients) of a generalized lattice Boltzmann equation (LBE) is studied in detail. The generalized lattice Boltzmann equation is constructed in moment space rather than in discrete velocity space. The generalized hydrodynamics of the model is obtained by solving the dispersion equation of the linearized LBE either analytically by using perturbation technique or numerically. The proposed LBE model has a maximum number of adjustable parameters for the given set of discrete velocities. Generalized hydrodynamics characterizes dispersion, dissipation (hyper-viscosities), anisotropy, and lack of Galilean invariance of the model, and can be applied to select the values of the adjustable parameters which optimize the properties of the model. The proposed generalized hydrodynamic analysis also provides some insights into stability and proper initial conditions for LBE simulations. The stability properties of some 2D LBE models are analyzed and compared with each other in the parameter space of the mean streaming velocity and the viscous relaxation time. The procedure described in this work can be applied to analyze other LBE models. As examples, LBE models with various interpolation schemes are analyzed. Numerical results on shear flow with an initially discontinuous velocity profile (shock) with or without a constant streaming velocity are shown to demonstrate the dispersion effects in the LBE model; the results compare favorably with our theoretical analysis. We also show that whereas linear analysis of the LBE evolution operator is equivalent to Chapman-Enskog analysis in the long wave-length limit (wave vector k = 0), it can also provide results for large values of k. Such results are important for the stability and other hydrodynamic properties of the LBE method and cannot be obtained through Chapman-Enskog analysis.

Lallemand, Pierre↗