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NASA NTRS1992

Grid generation and its application to separated flows

A method for the computation of recirculating flows on a nonorthogonal boundary-fitted coordinate system is presented. The algebraic grid generation technique is used to generate grids in a flow field of arbitrary shape. Computations are performed by using the set of governing equations transformed for use on a generalized curvilinear coordinate system. Turbulent flow computations were made by using the k-epsilon model for most of the flow cases. Three pressure correction algorithms - SIMPLE, SIMPLEC, and PISO - were incorporated for use on a nonorthogonal grid. Comparisons were made on the basis of numerical stability and accuracy. The present method was found to be efficient for computations of recirculating flows.

Maruszewski, J. P.↗
NASA NTRS2010

Ocean Turbulence, III: New GISS Vertical Mixing Scheme

We have found a new way to express the solutions of the RSM (Reynolds Stress Model) equations that allows us to present the turbulent diffusivities for heat, salt and momentum in a way that is considerably simpler and thus easier to implement than in previous work. The RSM provides the dimensionless mixing efficiencies Gamma-alpha (alpha stands for heat, salt and momentum). However, to compute the diffusivities, one needs additional information, specifically, the dissipation Epsilon. Since a dynamic equation for the latter that includes the physical processes relevant to the ocean is still not available, one must resort to different sources of information outside the RSM to obtain a complete Mixing Scheme usable in OGCMs. As for the RSM results, we show that the Gamma-alpha s are functions of both Ri and Rq (Richardson number and density ratio representing double diffusion, DD); the Gamma-alpha are different for heat, salt and momentum; in the case of heat, the traditional value Gamma-h = 0.2 is valid only in the presence of strong shear (when DD is inoperative) while when shear subsides, NATRE data show that Gamma-h can be three times as large, a result that we reproduce. The salt Gamma-s is given in terms of Gamma-h. The momentum Gamma-m has thus far been guessed with different prescriptions while the RSM provides a well defined expression for Gamma-m(Ri,R-rho). Having tested Gamma-h, we then test the momentum Gamma-m by showing that the turbulent Prandtl number Gamma-m/Gamma-h vs. Ri reproduces the available data quite well. As for the dissipation epsilon, we use different representations, one for the mixed layer (ML), one for the thermocline and one for the ocean;s bottom. For the ML, we adopt a procedure analogous to the one successfully used in PB (planetary boundary layer) studies; for the thermocline, we employ an expression for the variable epsilon/N(exp 2) from studies of the internal gravity waves spectra which includes a latitude dependence; for the ocean bottom, we adopt the enhanced bottom diffusivity expression used by previous authors but with a state of the art internal tidal energy formulation and replace the fixed Gamma-alpha = 0.2 with the RSM result that brings into the problem the Ri, R-rho dependence of the Gamma-alpha; the unresolved bottom drag, which has thus far been either ignored or modeled with heuristic relations, is modeled using a formalism we previously developed and tested in PBL studies. We carried out several tests without an OGCM. Prandtl and flux Richardson numbers vs. Ri. The RSM model reproduces both types of data satisfactorily. DD and Mixing efficiency Gamma-h(Ri,Rq). The RSM model reproduces well the NATRE data. Bimodal epsilon-distribution. NATRE data show that epsilon (Ri < 1) approximately equals 10epsilon(Ri > 1), which our model reproduces. Heat to salt flux ratio. In the Ri much greater than 1 regime, the RSM predictions reproduce the data satisfactorily. NATRE mass diffusivity. The z-profile of the mass diffusivity reproduces well the measurements at NATRE. The local form of the mixing scheme is algebraic with one cubic equation to solve.

Canuto, V. M.↗
NASA NTRS1974

A Computing Method for Sound Propagation Through a Nonuniform Jet Stream

Understanding the principles of jet noise propagation is an essential ingredient of systematic noise reduction research. High speed computer methods offer a unique potential for dealing with complex real life physical systems whereas analytical solutions are restricted to sophisticated idealized models. The classical formulation of sound propagation through a jet flow was found to be inadequate for computer solutions and a more suitable approach was needed. Previous investigations selected the phase and amplitude of the acoustic pressure as dependent variables requiring the solution of a system of nonlinear algebraic equations. The nonlinearities complicated both the analysis and the computation. A reformulation of the convective wave equation in terms of a new set of dependent variables is developed with a special emphasis on its suitability for numerical solutions on fast computers. The technique is very attractive because the resulting equations are linear in nonwaving variables. The computer solution to such a linear system of algebraic equations may be obtained by well-defined and direct means which are conservative of computer time and storage space. Typical examples are illustrated and computational results are compared with available numerical and experimental data.

Padula, S. L.↗
NASA NTRS2000

TOA Lightning Location Retrieval on Spherical and Oblate Spheroidal Earth Geometries

A simple linear algebraic solution is introduced for retrieving the location and time-of-occurrence of lightning ground strikes on a spherical Earth from a network of four or more time-of-arrival (TOA) sensors. Since the solution accounts for Earth curvature, it represents an extension to earlier planar model results described by Koshak et al. A test of the retrieval method is provided using computer-simulated data sets. The method is easy to comprehend and completely avoids reference to the mathematics of spherical hyperbolas such as discussed by Lewis. A quasi-analytic extension to the spherical Earth solution is provided for an oblate spheroidal Earth geometry, and the importance/relevance of oblate effects are discussed. Future application of these methods in support of the North American National Lightning Detection Network (NALDN) described by Cummins et al. is desirable, but additional theoretical investigations are required to incorporate magnetic bearing information into the present solution process.

Koshak, W. J.↗
NASA NTRS1992

Geometry modeling and multi-block grid generation for turbomachinery configurations

An interactive 3D grid generation code, Turbomachinery Interactive Grid genERation (TIGER), was developed for general turbomachinery configurations. TIGER features the automatic generation of multi-block structured grids around multiple blade rows for either internal, external, or internal-external turbomachinery flow fields. Utilization of the Bezier's curves achieves a smooth grid and better orthogonality. TIGER generates the algebraic grid automatically based on geometric information provided by its built-in pseudo-AI algorithm. However, due to the large variation of turbomachinery configurations, this initial grid may not always be as good as desired. TIGER therefore provides graphical user interactions during the process which allow the user to design, modify, as well as manipulate the grid, including the capability of elliptic surface grid generation.

Shih, Ming H.↗
NASA NTRS1988

Calculation of symmetric and asymmetric vortex seperation on cones and tangent ogives based on discrete vortex models

An inviscid discrete vortex model, with newly derived expressions for the tangential velocity imposed at the separation points, is used to investigate the symmetric and asymmetric vortex separation on cones and tangent ogives. The circumferential locations of separation are taken from experimental data. Based on a slender body theory, the resulting simultaneous nonlinear algebraic equations in a cross-flow plane are solved with Broyden's modified Newton-Raphson method. Total force coefficients are obtained through momentum principle with new expressions for nonconical flow. It is shown through the method of function deflation that multiple solutions exist at large enough angles of attack, even with symmetric separation points. These additional solutions are asymmetric in vortex separation and produce side force coefficients which agree well with data for cones and tangent ogives.

Chin, S.↗
NASA NTRS1977

Frequency domain compensation of a DYNGEN turbofan engine model

Following Rosenbrock's ideas regarding the advantages of dominance in linear multivariable control systems, a new graphical technique is used for the design of compensators that achieve dominance. The technique is illustrated with an application to the problem of designing compensators for a linear turbofan-engine model. The resulting design is put into perspective by examining it in the light of two other multivariable frequency-domain methods. One, MacFarlane's method of characteristic loci, is used to realize a final design for stability and low interaction. The other is a direct technique based upon the algebraic expansion of the determinant of the return difference in terms of it's elements. Results from simulations carried out on the NASA DYNGEN software are included.

Schafer, R. M.↗
NASA NTRS1995

Elliptic surface grid generation on minimal and parametrized surfaces

An elliptic grid generation method, which generates boundary conforming grids in a two dimensional physical space, is presented. The method is based on the composition of an algebraic and elliptic transformation. The composite mapping obeys the Poisson grid generation system with control functions specified by the algebraic transformation. It is shown that the grid generation on a minimal surface in a three dimensional space is equivalent to the grid generation in a two dimensional domain in physical space. A second elliptic grid generation method, which generates boundary conforming grids on smooth surfaces, is presented. Concerning surface modeling, it is shown that bicubic Hermit interpolation is an excellent method to generate a smooth surface crossing a discrete set of control points.

Spekreijse, S. P.↗
NASA NTRS2009

A Module Language for Typing by Contracts

Assume-guarantee reasoning is a popular and expressive paradigm for modular and compositional specification of programs. It is becoming a fundamental concept in some computer-aided design tools for embedded system design. In this paper, we elaborate foundations for contract-based embedded system design by proposing a general-purpose module language based on a Boolean algebra allowing to define contracts. In this framework, contracts are used to negotiate the correctness of assumptions made on the definition of a component at the point where it is used and provides guarantees to its environment. We illustrate this presentation with the specification of a simplified 4-stroke engine model.

Glouche, Yann↗
NASA NTRS2016

Introduction to COFFE: The Next-Generation HPCMP CREATE-AV CFD Solver

HPCMP CREATE-AV Conservative Field Finite Element (COFFE) is a modular, extensible, robust numerical solver for the Navier-Stokes equations that invokes modularity and extensibility from its first principles. COFFE implores a flexible, class-based hierarchy that provides a modular approach consisting of discretization, physics, parallelization, and linear algebra components. These components are developed with modern software engineering principles to ensure ease of uptake from a user's or developer's perspective. The Streamwise Upwind/Petrov-Galerkin (SU/PG) method is utilized to discretize the compressible Reynolds-Averaged Navier-Stokes (RANS) equations tightly coupled with a variety of turbulence models. The mathematics and the philosophy of the methodology that makes up COFFE are presented.

Glasby, Ryan S.↗
NASA NTRS1986

Equations for Annular-Heat-Transfer Coefficients

Tables of coefficients converted to algebraic expressions. Plot of Equation for Nusselt number agrees closely with points from tabulated data. Equation for Nusselt number and those for coefficients A and B obtained by regression analysis of data. Other plots also show close agreement for radius of 0.1 and 0.2. In equation form, coefficients incorporated into mathematical models more readily than as tabular data. Equations simplify design and analysis of heat exchangers.

Yao, B.↗
NASA NTRS1995

Estimation of attitude sensor timetag biases

This paper presents an extended Kalman filter for estimating attitude sensor timing errors. Spacecraft attitude is determined by finding the mean rotation from a set of reference vectors in inertial space to the corresponding observed vectors in the body frame. Any timing errors in the observations can lead to attitude errors if either the spacecraft is rotating or the reference vectors themselves vary with time. The state vector here consists of the attitude quaternion, timetag biases, and, optionally, gyro drift rate biases. The filter models the timetags as random walk processes: their expectation values propagate as constants and white noise contributes to their covariance. Thus, this filter is applicable to cases where the true timing errors are constant or slowly varying. The observability of the state vector is studied first through an examination of the algebraic observability condition and then through several examples with simulated star tracker timing errors. The examples use both simulated and actual flight data from the Extreme Ultraviolet Explorer (EUVE). The flight data come from times when EUVE had a constant rotation rate, while the simulated data feature large angle attitude maneuvers. The tests include cases with timetag errors on one or two sensors, both constant and time-varying, and with and without gyro bias errors. Due to EUVE's sensor geometry, the observability of the state vector is severely limited when the spacecraft rotation rate is constant. In the absence of attitude maneuvers, the state elements are highly correlated, and the state estimate is unreliable. The estimates are particularly sensitive to filter mistuning in this case. The EUVE geometry, though, is a degenerate case having coplanar sensors and rotation vector. Observability is much improved and the filter performs well when the rate is either varying or noncoplanar with the sensors, as during a slew. Even with bad geometry and constant rates, if gyro biases are independently known, the timetag error for a single sensor can be accurately estimated as long as its boresight is not too close to the spacecraft rotation axis.

Sedlak, J.↗
NASA NTRS1977

A finite element algorithm for sound propagation in axisymmetric ducts containing compressible mean flow

The described investigation is concerned with he development of a finite element scheme which can be used in a study of the acoustics of aircraft-engine ducts. In the absence of suitable variational principles for acoustic fluctuations within an aircraft fan engine, an acoustic analysis must proceed directly from the differential equations which describe compressible flow. The derived equations cannot be solved algebraically. The numerical technique used for solving them makes use of a linear rectangular element of a type considered by Zienkiewicz (1971). Attention is given to aspects of element derivation, the global matrix assembly, the solution of the matrix equation, questions of acoustic attenuation, and illustrations of the potential of the current model in duct optimization.

Abrahamson, A. L.↗
NASA NTRS1986

Compliance matrices for cracked bodies

An algorithm is developed to construct the compliance matrix for a cracked solid in the integral-equation formulation of two-dimensional linear-elastic fracture mechanics. The integral equation is reduced to a system of algebraic equations for unknown values of the dislocation-density function at discrete points on the interval from -1 to 1, using the numerical procedure described by Gerasoulis (1982). Sample numerical results are presented, and it is suggested that the algorithm is especially useful in cases where iterative solutions are required; e.g., models of fiber-reinforced concrete, rocks, or ceramics where microcracking, fiber bridging, and other nonlinear effects are treated as nonlinear springs along the crack surfaces (Ballarini et al., 1984).

Ballarini, R.↗
NASA NTRS1976

New convergence criteria for the vortex-lattice models of the leading-edge separation

The convergence criterion for the vortex-lattice technique which deals with delta wings exhibiting significant leading-edge separation was studied. It was shown that one can predict pressure distributions without irregularities which agree fairly well with experimental data (which show some irregularities of their own) by replacing the system of discrete vortex lines with a single concentrated core. This core has a circulation equal to the algebraic sum of the circulations around the discrete lines and is located at the centroid of these lines. Moreover, there is a requirement that the position and strength of the core must converge as the number of elements increases. Because the calculation of the position and strength of the core is much less involved than the calculation of the loads, this approach has the additional desirable feature of requiring less computational time.

Kandil, O. A.↗
NASA NTRS1984

Stability of finite difference models containing two boundaries or interfaces

The stability of finite difference models of hyperbolic initial boundary value problems is connected with the propagation and reflection of parasitic waves. Wave propagation ideas are applied to models containing two boundaires or interfaces, where repeated reflection of trapped wave packets is a potential new source of instability. Various known instability phenomena are accounted for in a unified way. Results show: (1) dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface; (2) algebraic GKS instabilities can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present; and (3) GKS-stability and P-stability can be established in certain problems by showing that all numerical reflection coefficients have modulus less than 1.

Trefethen, L. N.↗
NASA NTRS1985

Stability of finite-difference models containing two boundaries or interfaces

The stability of finite difference models of hyperbolic initial boundary value problems is connected with the propagation and reflection of parasitic waves. Wave propagation ideas are applied to models containing two boundaries or interfaces, where repeated reflection of trapped wave packets is a potential new source of instability. Various known instability phenomena are accounted for in a unified way. Results show: (1) dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface; (2) algebraic GKS instabilities can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present; and (3) GKS-stability and P-stability can be established in certain problems by showing that all numerical reflection coefficients have modulus less than 1.

Trefethen, L. N.↗
NASA NTRS1989

A skin friction law for compressible turbulent flow

An algebraic skin friction law is derived for adiabatic, compressible, equilibrium, turbulent boundary layer flow. An outer solution in terms of the Clauser defect stream function is matched to an inner empirical expression composed of compressible laws of the wall and wake. The modified Crocco temperature-velocity relationship and the Clauser eddy viscousity model are used in the outer solution. The skin friction law pertains for all pressure gradients in the incompressible through supersonic range and for small pressure gradients in the hypersonic range. Excellent comparisons with experiment are obtained in the appropriate parameter ranges. The application to numerical computation is discussed.

Barnwell, Richard W.↗