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NASA NTRS1985

Formulation and application of Russell's method

It is shown that the numerical technique of Russell's momentum approach can be derived by using Hamilton's principle and Vance's numerical scheme. It results in a set of first order differnce equations for solving the angular velocities. The numerical examples show that the method is reliable. The algorithm is modified next to perform the analysis of N-body systems with closed loop topology. To increase the formulation flexibility, the equations of motion are represented by using Cartesian coordinates and Lagrange multipliers. The algorithm consists of two parts, Vance's scheme and an unconstrained minimization. The Vance's scheme is used to find the angular velocities, and the unconstrained minimization is applied to provide the correct angular displacements. The proposed scheme is further extended to find the design sensitivity of an N-body system with closed loop configuration, and to carry out the design optimization as well. The numerical example of a small-scaled mechanical system is presented to verify the proposed formulation.

Hou, J. W.↗

NASA NTRS1990

A three-dimensional space marching algorithm for the solution of the Euler equations on unstructured grids

A three-dimensional space marching algorithm using an unstructured discretization is proposed. The governing equations and the spatial discretization of the domain and the interpolation from one arbitrary unstructured mesh to another are discussed, and the grid generation technique is described. Solutions for several different geometries are presented, and the results are compared with existing numerical techniques and experimental results.

Mcgrory, W. D.↗

NASA NTRS1990

A numerical method for prediction of compressible turbulent flows with closure models

A new computer code to solve the time averaged Navier-Stokes equations is developed. Many of the state-of-the-art numerical techniques and algorithms have been tested and implemented in the program in order to achieve a better numerical accuracy and code efficiency. Various turbulence models are tested for a wide range of flows. The initial focus has been on two-equation eddy-viscosity models, which are the most advanced available in current compressible flow codes. The long term goal will be to test Reynolds-Stress models and to explore their performance in the high Mach number range. Although testing and improvement of turbulence models for supersonic and hypersonic flows is the primary objective of this research, part of the effort has been devoted to analyzing the vortex breakdown phenomena using new computer programs. Some preliminary results on the breakdown of a vortex flow in a tube are reported. Present calculations are restricted to two dimensional flow geometry.

Huang, P. G.↗

NASA NTRS1982

Calculation of three-dimensional, inviscid supersonic, steady flows

A numerical technique is described for the calculation of three dimensional, inviscid, supersonic, steady flows over wing-body configurations. A high degree of accuracy without increasing the number of computational nodes is obtained by means of a powerful conformal mapping technique. Results are presented for some simple body configurations and for a more complex arrow wing airframe. The numerical results show good agreement with experimental measurements.

Moretti, G.↗

NASA NTRS1982

On the modelling and simulation of the dynamics and control of large flexible orbiting systems

This paper attempts to review the steps involved in the development of mathematical models that can be used to simulate the in-orbit dynamics of large flexible systems. The use of graph theoretic techniques can often be used to reduce the computational effort involved for calculating the eigenvalues of large ordered systems. Computer generated graphical techniques may provide additional insight into the understanding of elastic modal shape functions of complex systems. Finally the numerical techniques commonly used to develop shape and attitude control laws will be briefly reviewed.

Bainum, P. M.↗

NASA NTRS1985

A numerical method for approximating antenna surfaces defined by discrete surface points

A simple numerical method for the quadratic approximation of a discretely defined reflector surface is described. The numerical method was applied to interpolate the surface normal of a parabolic reflector surface from a grid of nine closest surface points to the point of incidence. After computing the surface normals, the geometrical optics and the aperture integration method using the discrete Fast Fourier Transform (FFT) were applied to compute the radiaton patterns for a symmetric and an offset antenna configurations. The computed patterns are compared to that of the analytic case and to the patterns generated from another numerical technique using the spline function approximation. In the paper, examples of computations are given. The accuracy of the numerical method is discussed.

Lee, R. Q.↗

NASA NTRS1990

Thermal radiation characteristics of nonisothermal cylindrical enclosures using a numerical ray tracing technique

Analysis of energy emitted from simple or complex cavity designs can lead to intricate solutions due to nonuniform radiosity and irradiation within a cavity. A numerical ray tracing technique was applied to simulate radiation propagating within and from various cavity designs. To obtain the energy balance relationships between isothermal and nonisothermal cavity surfaces and space, the computer code NEVADA was utilized for its statistical technique applied to numerical ray tracing. The analysis method was validated by comparing results with known theoretical and limiting solutions, and the electrical resistance network method. In general, for nonisothermal cavities the performance (apparent emissivity) is a function of cylinder length-to-diameter ratio, surface emissivity, and cylinder surface temperatures. The extent of nonisothermal conditions in a cylindrical cavity significantly affects the overall cavity performance. Results are presented over a wide range of parametric variables for use as a possible design reference.

Baumeister, Joseph F.↗

NASA NTRS1995

Shock-Induced Separated Structures in Symmetric Corner Flows

Three-dimensional supersonic viscous laminar flows over symmetric corners are considered in this paper. The characteristic features of such configurations are discussed and an historical survey on the past research work is presented. A new contribution based on a numerical technique that solves the parabolized form of the Navier-Stokes equations is presented. Such a method makes it possible to obtain very detailed descriptions of the flowfield with relatively modest CPU time and memory storage requirements. The numerical approach is based on a space-marching technique, uses a finite volume discretization and an upwind flux-difference splitting scheme (developed for the steady flow equations) for the evaluation of the inviscid fluxes. Second order accuracy is reached following the guidelines of the ENO schemes. Different free-stream conditions and geometrical configurations are considered. Primary and secondary streamwise vortical structures embedded in the boundary layer and originated by the interaction of the latter with shock waves are detected and studied. Computed results are compared with experimental data taken from literature.

DAmbrosio, Domenic↗

NASA NTRS2015

Analytic Thermoelectric Couple Modeling: Variable Material Properties and Transient Operation

To gain a deeper understanding of the operation of a thermoelectric couple a set of analytic solutions have been derived for a variable material property couple and a transient couple. Using an analytic approach, as opposed to commonly used numerical techniques, results in a set of useful design guidelines. These guidelines can serve as useful starting conditions for further numerical studies, or can serve as design rules for lab built couples. The analytic modeling considers two cases and accounts for 1) material properties which vary with temperature and 2) transient operation of a couple. The variable material property case was handled by means of an asymptotic expansion, which allows for insight into the influence of temperature dependence on different material properties. The variable property work demonstrated the important fact that materials with identical average Figure of Merits can lead to different conversion efficiencies due to temperature dependence of the properties. The transient couple was investigated through a Greens function approach; several transient boundary conditions were investigated. The transient work introduces several new design considerations which are not captured by the classic steady state analysis. The work helps to assist in designing couples for optimal performance, and also helps assist in material selection.

Seebeck Effect↗

NASA NTRS1978

Numerical spatial marching techniques for estimating duct attenuation and source pressure profiles

A numerical method was developed that could predict the pressure distribution of a ducted source from far field pressure inputs. Using an initial value formulation, the two-dimensional homogeneous Helmholtz wave equation (no steady flow) was solved using explicit marching techniques. The Von Neumann method was used to develop relationships which describe how sound frequency and grid spacing effect numerical stability. At the present time, stability considerations limit the approach to high frequency sound. Sample calculations for both hard and soft wall ducts compare favorably to known boundary value solutions. In addition, assuming that reflections in the duct are small, this initial value approach was successfully used to determine the attenuation of a straight soft wall duct. Compared to conventional finite difference or finite element boundary value approaches, the numerical marching technique is orders of magnitude shorter in computation time and required computer storage and can be easily employed in problems involving high frequency sound.

Baumeister, K. J.↗

NASA NTRS1978

Numerical spatial marching techniques for estimating duct attenuation and source pressure profiles

A numerical method is developed that could predict the pressure distribution of a ducted source from far-field pressure inputs. Using an initial value formulation, the two-dimensional homogeneous Helmholtz wave equation (no steady flow) is solved using explicit marching techniques. The Von Neumann method is used to develop relationships which describe how sound frequency and grid spacing effect numerical stability. At the present time, stability considerations limit the approach to high frequency sound. Sample calculations for both hard and soft wall ducts compare favorably to known boundary value solutions. In addition, assuming that reflections in the duct are small, this initial value approach is successfully used to determine the attenuation of a straight soft wall duct. Compared to conventional finite difference or finite element boundary value approaches, the numerical marching technique is orders of magnitude shorter in computation time and required computer storage and can be easily employed in problems involving high frequency sound.

Baumeister, K. J.↗

NASA NTRS1994

Prediction of Undsteady Flows in Turbomachinery Using the Linearized Euler Equations on Deforming Grids

A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and blade motion is presented. The model accounts for blade loading, blade geometry, shock motion, and wake motion. Assuming that the unsteadiness in the flow is small relative to the nonlinear mean solution, the unsteady Euler equations can be linearized about the mean flow. This yields a set of linear variable coefficient equations that describe the small amplitude harmonic motion of the fluid. These linear equations are then discretized on a computational grid and solved using standard numerical techniques. For transonic flows, however, one must use a linear discretization which is a conservative linearization of the non-linear discretized Euler equations to ensure that shock impulse loads are accurately captured. Other important features of this analysis include a continuously deforming grid which eliminates extrapolation errors and hence, increases accuracy, and a new numerically exact, nonreflecting far-field boundary condition treatment based on an eigenanalysis of the discretized equations. Computational results are presented which demonstrate the computational accuracy and efficiency of the method and demonstrate the effectiveness of the deforming grid, far-field nonreflecting boundary conditions, and shock capturing techniques. A comparison of the present unsteady flow predictions to other numerical, semi-analytical, and experimental methods shows excellent agreement. In addition, the linearized Euler method presented requires one or two orders-of-magnitude less computational time than traditional time marching techniques making the present method a viable design tool for aeroelastic analyses.

Clark, William S.↗

NASA NTRS1990

Prediction of cascade performance using an incompressible Navier-Stokes technique

A fully elliptic, control volume solution of the two-dimensional incompressible Navier-Stokes equations for the prediction of cascade performance over a wide incidence range is presented. The numerical technique is based on a new pressure substitution method. A Poisson equation is derived from the pressure weighted substitution of the full momentum equations into the continuity equation. The analysis of a double circular arc compressor cascade is presented, and the results are compared with the available experimental data at various incidence angles. Good agreement is obtained for the blade pressure distribution, boundary layer and wake profiles, skin friction coefficient, losses and outlet angles. Turbulence effects are simulated by the Low-Reynolds-Number version of the k-epsilon turbulence model.

Hobson, G. V.↗

NASA NTRS1990

General method for assessing the geographically correlated error for an orbiter with applications to Topex

An estimation based computational algorithm for evaluating geographically dependent orbit determination errors is presented. A derivation is included of the generalized geographically correlated error analysis algorithm and the algorithm is applied to a Topex simulation. Then a general numerical technique which provides the capability to analyze any type of mismodeled parameter with the geopotential as a special case is described. The numerical algorithm is compared with the analytic method of Rosborough (1986) and it is shown that both methods generate consistent geographically dependent orbit error profiles when the ephemeris error is dominated by the mismodeled geopotential.

Wolff, P. J.↗

NASA NTRS1993

A comparison of high resolution upwind solvers on 3-D inviscid hypersonic flows

A detailed comparison of numerical results obtained by solving the Euler equations for the inviscid flow over delta wings in reentry configurations is presented. The investigation involves a side-by-side comparison between independently developed upwind Euler solvers at VKI (M3D) and NASA Johnson Space Center (E3D) using identical grids. In both solvers the governing equations are integrated by means of time marching finite volume shock capturing methods, based on a cell centered upwind evaluation of the cell face fluxes and nonlinear limiters. High resolution schemes are obtained via MUSCL characteristic variable extrapolation to ensure total variation diminishing (TVD) properties and therefore monotonic discontinuity capturing. Further comparisons are made with several published results by other authors based on both upwind and central discretizations. The present results contribute in asserting the high resolution upwind TVD schemes as the most reliable numerical technique to handle the strong discontinuities typical of high speed flows. The performances of the two upwind solvers are satisfactory and the numerical results in good agreement. However, the important issue of reaching a grid converged solution for the present complex three-dimensional inviscid flow problems was not achieved.

Manna, M.↗

NASA NTRS1994

A mathematical model for simulating noise suppression of lined ejectors

A mathematical model containing the essential features embodied in the noise suppression of lined ejectors is presented. Although some simplification of the physics is necessary to render the model mathematically tractable, the current model is the most versatile and technologically advanced at the current time. A system of linearized equations and the boundary conditions governing the sound field are derived starting from the equations of fluid dynamics. A nonreflecting boundary condition is developed. In view of the complex nature of the equations, a parametric study requires the use of numerical techniques and modern computers. A finite element algorithm that solves the differential equations coupled with the boundary condition is then introduced. The numerical method results in a matrix equation with several hundred thousand degrees of freedom that is solved efficiently on a supercomputer. The model is validated by comparing results either with exact solutions or with approximate solutions from other works. In each case, excellent correlations are obtained. The usefulness of the model as an optimization tool and the importance of variable impedance liners as a mechanism for achieving broadband suppression within a lined ejector are demonstrated.

Watson, Willie R.↗

NASA NTRS1992

PNS predictions for supersonic/hypersonic flows over finned missile configurations

Finned missile design entails accurate and computationally fast numerical techniques for predicting viscous flows over complex lifting configurations at small to moderate angles of attack and over Mach 3 to 15; these flows are often characterized by strong embedded shocks, so that numerical algorithms are also required to capture embedded shocks. The recent real-gas Flux Vector Splitting technique is here extended to investigate the Mach 3 flow over a typical finned missile configuration with/without side fin deflections. Elliptic grid-generation techniques for Mach 15 flows are shown to be inadequate for Mach 3 flows over finned configurations and need to be modified. Fin-deflection studies indicate that even small amounts of missile fin deflection can substantially modify vehicle aerodynamics. This 3D parabolized Navier-Stokes scheme is also extended into an efficient embedded algorithm for studying small axially separated flow regions due to strong fin and control surface deflections.

Bhutta, Bilal A.↗

NASA NTRS1976

Effective thermal property improves phase change paint data

The phase-change coating technique presents itself as a valuable tool in determining the heat transfer rate over the surface of small complex wind tunnel models. A numerical technique is described which shows that an effective thermophysical property - the square root of the product of thermal conductivity, density, and specific heat - may significantly improve the accuracy of the phase-change coating technique with allowance for model inhomogeneity and temperature dependency in a transient environment. Results of the measured steady-state variation of the effective thermophysical property with temperature and the effect of surface heating rate on the effective thermophysical property are plotted for a representative homogeneous model material and for an extreme nonhomogeneous model material. The use of an effective thermophysical property to reduce phase-change paint data is recommended. The analysis also confirms that the apparatus described by Corwin and Kramer (1975) can be used to measure directly this effective thermophysical property for use in wind tunnel model heat-transfer measurements.

Drummond, J. P.↗