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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.

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At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

Wall functions for the kappa-epsilon turbulence model in generalized nonorthogonal curvilinear coordinates

A k-epsilon turbulence model suitable for compressible flow, including the new wall function formulation, has been incorporated into an existing compressible Reynolds-averaged Navier-Stokes code, F3D. The low Reynolds number k-epsilon model of Chien (1982) was added for comparison with the present method. A number of features were added to the F3D code including improved far-field boundary conditions and viscous terms in the streamwise direction. A series of computations of increasing complexity was run to test the effectiveness of the new formulation. Flow over a flat plate was computed by using both orthogonal and nonorthogonal grids, and the friction coefficients and velocity profiles compared with a semi-empirical equation. Flow over a body of revolution at zero angle of attack was then computed to test the method's ability to handle flow over a curved surface. Friction coefficients and velocity profiles were compared to test data. All models gave good results on a relatively fine grid, but only the wall function formulation was effective with coarser grids. Finally, in order to demonstrate the method's ability to handle complex flow fields, separated flow over a prolate spheroid at angle of attack was computed, and results were compared to test data. The results were also compared to a k-epsilon model by Kim and Patel (1991), in which one equation model patched in at the wall was employed. Both models gave reasonable solutions, but improvement is required for accurate prediction of friction coefficients in the separated regions.

Sondak, D. L.↗
NASA NTRS1992

Multigrid acceleration of a fractional-step solver in generalized curvilinear coordinate systems

The efficiency of the INS3D-fractional step (FS) code has been significantly enhanced by accelerating the Poisson solver with a multigrid (MG) method. This is apparently the first implementation of the MG method for solving the 3D discrete Poisson-like equation obtained in the derivation of FS solution methods for the incompressible Navier-Stokes equations in generalized nonorthogonal coordinate systems using a staggered arrangement of the variables. The MG solver is insensitive to the geometry, even in cases of highly nonorthogonal clustered meshes, as well as to the free parameters of MG methods.

Rosenfeld, Moshe↗
NASA NTRS1976

On differential transformations between Cartesian and curvilinear (geodetic) coordinates

Differential transformations are developed between Cartesian and curvilinear orthogonal coordinates. Only matrix algebra is used for the presentation of the basic concepts. After defining the reference systems used the rotation (R), metric (H), and Jacobian (J) matrices of the transformations between cartesian and curvilinear coordinate systems are introduced. A value of R as a function of H and J is presented. Likewise an analytical expression for J(-1) as a function of H(-2) and R is obtained. Emphasis is placed on showing that differential equations are equivalent to conventional similarity transformations. Scaling methods are discussed along with ellipsoidal coordinates. Differential transformations between elipsoidal and geodetic coordinates are established.

Soler, T.↗
NASA NTRS1979

Numerical method for predicting flow characteristics and performance of nonaxisymmetric nozzles, theory

The theoretical foundation and formulation of a numerical method for predicting the viscous flowfield in and about isolated three dimensional nozzles of geometrically complex configuration are presented. High Reynolds number turbulent flows are of primary interest for any combination of subsonic, transonic, and supersonic flow conditions inside or outside the nozzle. An alternating-direction implicit (ADI) numerical technique is employed to integrate the unsteady Navier-Stokes equations until an asymptotic steady-state solution is reached. Boundary conditions are computed with an implicit technique compatible with the ADI technique employed at interior points of the flow region. The equations are formulated and solved in a boundary-conforming curvilinear coordinate system. The curvilinear coordinate system and computational grid is generated numerically as the solution to an elliptic boundary value problem. A method is developed that automatically adjusts the elliptic system so that the interior grid spacing is controlled directly by the a priori selection of the grid spacing on the boundaries of the flow region.

Thomas, P. D.↗
NASA NTRS1978

Numerical solution of flow problems using body-fitted coordinate systems

The technique of boundary-fitted coordinate systems is based on a method of automatic numerical generation of a general curvilinear coordinate system having a coordinate line coincident with each boundary of a general multi-connected region containing any number of arbitrarily shaped bodies. Once the curvilinear coordinate system is generated, any partial differential system of interest can be solved on this coordinate system by transforming the equations and solving the resulting system in finite difference approximation on the rectangular transformed plane. This method of automatic body-fitted curvilinear coordinate generation is used to construct finite-difference solutions of the full, time dependent Navier-Stokes equations for the unsteady viscous flow about arbitrary two-dimensional airfoils, or any other two-dimensional bodies. Finally, initial results for three-dimensional applications are also presented.

Thompson, J. F.↗
NASA NTRS1991

An intrinsically n-dimensional generalized flux vector splitting implicit finite element Euler algorithm

A generalized flux-vector splitting implicit Galerkin finite-element algorithm for the Euler equations in curvilinear coordinates for ideal and reacting gases is derived. For an arbitrary equation of state, the curvilinear-coordinate flux vector is split in kinematic and kinetic components, and the associated jacobian matrix eigenvalues explicitly depend on the metric data. After directional semidiscretization, the terminal ordinary differential-equation system is solved via a nonlinearly stable implicit Runge-Kutta scheme in concert with an accurate tensor matrix product factorization. The results for selected two-dimensional supersonic and axisymmetric hypersonic flows validate the algorithm and verify its robustness for curvilinear-coordinate computations. The evolution towards a steady state is achieved for large Courant numbers without indication of numerical instabilities.

Iannelli, G. S.↗
NASA NTRS1982

Error induced by coordinate systems

It is pointed out that the choice of a curvilinear coordinate system can have a substantial effect on the error in the numerical solution of a partial differential equation. The truncation error is dependent not only on the higher order derivatives of the solution and the local grid spacing, but also on the rate-of-change of the grid spacing and on the departure of the grid from orthogonality. In connection with the present investigation, an analysis is conducted of the local truncation error in the approximation of first and second order derivatives on a curvilinear grid. Attention is given to a number of examples which illustrate the two fundamental sources of truncation error in the numerical solution of partial differential equations on curvilinear coordinate systems. The first is the grid spacing and changes in grid spacing which is measured by the first and second order derivatives of the functions defining the coordinate system. The second source is the higher order derivatives of the solution itself.

Mastin, C. W.↗
NASA NTRS1991

A General-Coordinate Formulation For Boundary-Layer Flow

Formulation for solution of equations of boundary-layer flow in general body-fitted curvilinear coordinates retains velocities in Cartesian coordinates. Increases stability of numerical simulations by avoiding coordinate source terms. In formulation, curvilinear coordinates do not have to be orthogonal, and much of software developed previously for use in numerical simulations of flow based on Navier-Stokes equations used.

Steger, Joseph L.↗
NASA NTRS1998

Numerical Investigation of Hot Gas Ingestion by STOVL Aircraft

This report compiles the various research activities conducted under the auspices of the NASA Grant NAG3-1026, "Numerical Investigation of Hot Gas Ingestion by STOVL Aircraft" during the period of April 1989 to April 1994. The effort involved the development of multigrid based algorithms and computer programs for the calculation of the flow and temperature fields generated by Short Take-off and Vertical Landing (STOVL) aircraft, while hovering in ground proximity. Of particular importance has been the interaction of the exhaust jets with the head wind which gives rise to the hot gas ingestion process. The objective of new STOVL designs to reduce the temperature of the gases ingested into the engine. The present work describes a solution algorithm for the multi-dimensional elliptic partial-differential equations governing fluid flow and heat transfer in general curvilinear coordinates. The solution algorithm is based on the multigrid technique which obtains rapid convergence of the iterative numerical procedure for the discrete equations. Initial efforts were concerned with the solution of the Cartesian form of the equations. This algorithm was applied to a simulated STOVL configuration in rectangular coordinates. In the next phase of the work, a computer code for general curvilinear coordinates was constructed. This was applied to model STOVL geometries on curvilinear grids. The code was also validated in model problems. In all these efforts, the standard k-Epsilon model was used.

Vanka, S. P.↗
NASA NTRS1991

Micro- and meso-scale simulations of magnetospheric processes related to the aurora and substorm morphology

The primary methodology during the grant period has been the use of micro or meso-scale simulations to address specific questions concerning magnetospheric processes related to the aurora and substorm morphology. This approach, while useful in providing some answers, has its limitations. Many of the problems relating to the magnetosphere are inherently global and kinetic. Effort during the last year of the grant period has increasingly focused on development of a global-scale hybrid code to model the entire, coupled magnetosheath - magnetosphere - ionosphere system. In particular, numerical procedures for curvilinear coordinate generation and exactly conservative differencing schemes for hybrid codes in curvilinear coordinates have been developed. The new computer algorithms and the massively parallel computer architectures now make this global code a feasible proposition. Support provided by this project has played an important role in laying the groundwork for the eventual development or a global-scale code to model and forecast magnetospheric weather.

Swift, Daniel W.↗
NASA NTRS1980

Numerical solution of flow problems using body-fitted coordinate systems

The paper deals with numerically generated boundary-fitted coordinate systems. This procedure eliminates the shape of the boundaries as a complicating factor and allows the flow about arbitrary boundaries to be treated essentially as easily as that about simple boundaries. The technique of boundary-fitted coordinate systems is based on a method of automatic numerical generation of a general curvilinear coordinate system having a coordinate line coincident with each boundary of a general multiconnected region involving any number of arbitrarily shaped boundaries. Once the curvilinear coordinate system is generated, any partial differential system of interest may be solved on the coordinate system by transforming the equations and solving the resulting system in finite-difference approximation on the rectangular transformed plane. Attention is given to the types of boundary-fitted coordinate systems, coordinate system control, operation of the coordinate codes, solution of partial differential equations, application to free-surface flow, and other applications of interest.

Thompson, J. F.↗
NASA NTRS1983

A fully vectorized numerical solution of the incompressible Navier-Stokes equations

A vectorizable algorithm is presented for the implicit finite difference solution of the incompressible Navier-Stokes equations in general curvilinear coordinates. The unsteady Reynolds averaged Navier-Stokes equations solved are in two dimension and non-conservative primitive variable form. A two-layer algebraic eddy viscosity turbulence model is used to incorporate the effects of turbulence. Two momentum equations and a Poisson pressure equation, which is obtained by taking the divergence of the momentum equations and satisfying the continuity equation, are solved simultaneously at each time step. An elliptic grid generation approach is used to generate a boundary conforming coordinate system about an airfoil. The governing equations are expressed in terms of the curvilinear coordinates and are solved on a uniform rectangular computational domain. A checkerboard SOR, which can effectively utilize the computer architectural concept of vector processing, is used for iterative solution of the governing equations.

Patel, N.↗
NASA NTRS1974

Quaternion wave equations in curved space-time

The quaternion formulation of relativistic quantum theory is extended to include curvilinear coordinates and curved space-time in order to provide a framework for a unified quantum/gravity theory. Six basic quaternion fields are identified in curved space-time, the four-vector basis quaternions are identified, and the necessary covariant derivatives are obtained. Invariant field equations are derived, and a general invertable coordinate transformation is developed. The results yield a way of writing quaternion wave equations in curvilinear coordinates and curved space-time as well as a natural framework for solving the problem of second quantization for gravity.

Edmonds, J. D., Jr.↗
NASA NTRS1975

Use of numerically generated body-fitted coordinate systems for solution of the Navier-Stokes equations

A procedure for numerical solution of the time-dependent, two-dimensional incompressible Navier-Stokes equations that can treat the unsteady laminar flow about bodies of arbitrary shape, such as two-dimensional airfoils, multiple airfoils, and submerged hydrofoils, as naturally as it can deal with the flow about simple bodies. The solution is based on a method of automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multiconnected region containing any number of arbitrarily shaped bodies. The curvilinear coordinates are generated as the solution of two elliptical partial differential equations with Dirichlet boundary conditions, one coordinate being specified to be constant on each of the boundaries, and a distribution of the other being specified along the boundaries. The solution compares excellently with the Blasius boundary layer solution for the flow past a semiinfinite flat plate.

Thompson, J. F.↗
NASA NTRS1986

Triangular element for analysis of a stretched plate weakened by a part-through hole

A C0-type triangular element formulation in orthogonal curvilinear coordinates has been developed, based on assumptions of transverse inextensibility and layerwise constant shear angle, for analysis of a stretched homogeneous plate weakened by a part-through hole. The element stiffness matrix and consistent load vector have been derived using an assumed-quadratic-displacement (in the curvilinear coordinate plane) potential-energy approach. Numerical results obtained using the straight-sided triangular element version indicate the presence of transverse shear deformation in a stretched homogeneous plate weakened by a symmetrically located concentric hole.

Chaudhuri, R. A.↗