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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.↗

NASA NTRS1984

A vectorized solution for incompressible flow

An algorithm is developed to obtain solutions to the unsteady Reynolds-averaged incompressible Navier-Stokes equations in general curvilinear coordinates on a vector processor. The governing equations are in nonconservative form with the velocity and pressure as dependent variables. Two momentum equations and the Poisson equation for pressure form a set of three governing equations for three flow field unknowns: u, v, and p. The governing equations and boundary conditions are expressed in terms of boundary-conforming curvilinear coordinates, and a checkerboard SOR iteration is used to solve the governing equations. Several possible sequences for a checkerboard SOR iteration are investigated for finding the best overall convergence rate. The efficiency and capability of the present algorithm was assessed using the example of an 18 percent thick NACA 66(3)018 airfoil at zero degree angle of attack for chord Reynolds number range 1000-40,000.

Patel, N. R.↗

NASA NTRS1993

A Fractional-Step Method Of Computing Incompressible Flow

Method of computing time-dependent flow of incompressible, viscous fluid involves numerical solution of Navier-Stokes equations on two- or three-dimensional computational grid based on generalized curvilinear coordinates. Equations of method derived in primitive-variable formulation. Dependent variables are pressure at center of each cell of computational grid and volume fluxes across faces of each cell. Volume fluxes replace Cartesian components of velocity; these fluxes correspond to contravariant components of velocity multiplied by volume of computational cell, in staggered grid. Choice of dependent variables enables simple extension of previously developed staggered-grid approach to generalized curvilinear coordinates and facilitates enforcement of conservation of mass.

Kwak, Dochan↗

DOE OSTI2020

The Athena++ Adaptive Mesh Refinement Framework: Design and Magnetohydrodynamic Solvers

The design and implementation of a new framework for adaptive mesh refinement calculations are described. It is intended primarily for applications in astrophysical fluid dynamics, but its flexible and modular design enables its use for a wide variety of physics. The framework works with both uniform and nonuniform grids in Cartesian and curvilinear coordinate systems. It adopts a dynamic execution model based on a simple design called a "task list" that improves parallel performance by overlapping communication and computation, simplifies the inclusion of a diverse range of physics, and even enables multiphysics models involving different physics in different regions of the calculation. We describe physics modules implemented in this framework for both nonrelativistic and relativistic magnetohydrodynamics (MHD). These modules adopt mature and robust algorithms originally developed for the Athena MHD code and incorporate new extensions: support for curvilinear coordinates, higher-order time integrators, more realistic physics such as a general equation of state, and diffusion terms that can be integrated with super-time-stepping algorithms. The modules show excellent performance and scaling, with well over 80% parallel efficiency on over half a million threads. The source code has been made publicly available.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1992

A non-linearly stable implicit finite element algorithm for hypersonic aerodynamics

A generalized curvilinear coordinate Taylor weak statement implicit finite element algorithm is developed for the two-dimensional and axisymmetric compressible Navier-Stokes equations for ideal and reacting gases. For accurate hypersonic simulation, air is modeled as a mixture of five perfect gases, i.e., molecular and atomic oxygen and nitrogen as well as nitric oxide. The associated pressure is then determined via Newton solution of the classical chemical equilibrium equation system. The directional semidiscretization is achieved using an optimal metric data Galerkin finite element weak statement, on a developed 'companion conservation law system', permitting classical test and trial space definitions. Utilizing an implicit Runge-Kutta scheme, the terminal algorithm is then nonlinearly stable, and second-order accurate in space and time on arbitrary curvilinear coordinates. Subsequently, a matrix tensor product factorization procedure permits an efficient numerical linear algebra handling for large Courant numbers. For ideal- and real-gas hypersonic flows, the algorithm generates essentially nonoscillatory numerical solutions in the presence of strong detached shocks and boundary layer-inviscid flow interactions.

Iannelli, G. S.↗

DOE OSTI2025

Directional finite difference method for directly solving 3D gyrokinetic field equations with enhanced accuracy

The gyrokinetic (GK) field equation is a three-dimensional (3D) elliptic equation, but it is often simplified to a set of two-dimensional (2D) equations by assuming that the field does not vary along a specific direction. However, this simplification can introduce inevitable 0th-order numerical errors, as nonlinear mode coupling in toroidal geometry can produce undesirable harmonic modes that violate the assumption. In this work, we propose a novel directional finite difference method (FDM) with a local coordinate transformation to better resolve the target field of interest. The directional FDM can accurately solve 3D GK field equations without simplifications, which can overcome the limitations of conventional methods. The accuracy and efficiency of different FDMs are analyzed in great detail for a variety of geometries, from simple 2D Cartesian coordinates to realistic 3D curvilinear coordinates. The 0th-order numerical errors of simplified 2D GK equations were found to be more problematic for low-harmonic modes and low aspect ratio geometries such as spherical tokamaks. On the other hand, the directional 3D FDM can accurately resolve a much wider range of harmonic modes aligned to the direction of interest, including the low-harmonic modes. In conclusion, we demonstrate that the directional 3D FDM is a highly effective algorithm for solving the 3D GK field equations, achieving accuracy improvements of 10 to 100 times or more, particularly for low-harmonic modes in spherical tokamaks.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1996

A Study of the Development of Steady and Periodic Unsteady Turbulent Wakes Through Curved Channels at Positive, Zero, and Negative Streamwise Pressure Gradients, Part 1

The turbomachinery wake flow development is largely influenced by streamline curvature and streamwise pressure gradient. The objective of this investigation is to study the development of the wake under the influence of streamline curvature and streamwise pressure gradient. The experimental investigation is carried out in two phases. The first phase involves the study of the wake behind a stationary circular cylinder (steady wake) in curved channels at positive, zero, and negative streamwise pressure gradients. The mean velocity and Reynolds stress components are measured using a X-hot-film probe. The measured quantities obtained in probe coordinates are transformed to a curvilinear coordinate system along the wake centerline and are presented in similarity coordinates. The results of the steady wakes suggest strong asymmetry in velocity and Reynolds stress components. However, the velocity defect profiles in similarity coordinates are almost symmetrical and follow the same distribution as the zero pressure gradient straight wake. The results of Reynolds stress distributions show higher values on the inner side of the wake than the outer side. Other quantities, including the decay of maximum velocity defect, growth of wake width, and wake integral parameters, are also presented for the three different pressure gradient cases of steady wake. The decay rate of velocity defect is fastest for the negative streamwise pressure gradient case and slowest for the positive pressure gradient case. Conversely, the growth of the wake width is fastest for the positive streamwise pressure gradient case and slowest for the negative streamwise pressure gradient. The second phase studies the development of periodic unsteady wakes generated by the circular cylinders of the rotating wake generator in a curved channel at zero streamwise pressure gradient. Instantaneous velocity components of the periodic unsteady wakes, measured with a stationary X-hot-film probe, are analyzed by the phase averaging techniques. The temporal distribution of velocity and Reynolds stress components obtained in a stationary frame of reference are transformed to a spatial distribution in a relative frame of reference. Profiles of phase-averaged velocity and Reynolds stress distributions in the relative frame of reference and similarity coordinates are presented. The velocity defect and Reynolds stress distributions agree with the results of the wake development behind a stationary cylinder in the curved channel at zero streamwise pressure gradient. The phase-averaged third-order correlations, presented in the relative frame of reference and similarity coordinates, show pronounced asymmetric features.

Schobeiri, M. T.↗

NASA NTRS1985

Implicit finite-difference methods for the Euler equations

The present paper is concerned with two-dimensional Euler equations and with schemes which are in use of the time of this writing. Most of the development presented carries over directly to three dimensions. The characteristics of the two-dimensional Euler equations in Cartesian coordinates are considered along with generalized curvilinear coordinate transformations, metric relations, invariants of the transformation, flux Jacobian matrices and eigensystems, numerical algorithms, flux split algorithms, implicit and explicit nonlinear control (smoothing), upwind differencing in supersonic regions, unsteady and steady-state computation, the diagonal form of implicit algorithm, metric differencing and invariants, boundary conditions, geometry and mesh generation, and sample solutions.

Pulliam, T. H.↗

NASA NTRS1993

Radiation heat transfer calculations using a control-angle, control-volume-based discrete ordinates method

A control-angle, control-volume-based discrete ordinates method (CA - CV DOM) is presented in this paper. A detailed formulation of the discretization equation is presented in two-dimensional Cartesian coordinate system. The procedure can be extended to curvilinear coordinate system with minor modifications. The step and modified-exponential schemes are used in this study. Present results converged to the grid independent solutions quickly and compared favorably against other published results for six test problems.

Chai, John C.↗

NASA NTRS1981

Numerical analysis of the scramjet inlet flow field using two-dimensional Navier-Stokes equations

A computer code has been developed to solve the full two-dimensional Navier-Stokes equations in a scramjet inlet. The analysis uses a numerical coordinate transformation which generates a set of boundary fitted curvilinear coordinates. The explicit finite-difference algorithm of MacCormack is used to solve the governing equations. A two-layer eddy viscosity model is used for the turbulent flow. The code can analyze both inviscid and viscous flows with multiple struts in the flow field. Detailed results are presented for two model problems and two scramjet inlets with one and two struts. The application of the two-dimensional analysis in the preliminary design of the actual scramjet inlet is briefly discussed.

Kumar, A.↗

NASA NTRS1982

Analysis of the Scramjet inlet flow field using two-dimensional Navier-Stokes equations

A computer code was developed to solve the full two dimensional Navier-Stokes equations in a scramjet inlet. The analysis uses a numerical coordinate transformation which generates a set of boundary-fitted curvilinear coordinates. The explicit finite difference algorithm of MacCormack is used to solve the governing equations. A two-layer eddy viscosity model is used for the turbulent flow. The code can analyze both inviscid and viscous flows with multiple struts in the flow field. Detailed results are presented for two model problems and two scramjet inlets with one and two struts. The application of the two dimensional analysis in the preliminary design of the actual scramjet inlet is briefly discussed.

Kumar, A.↗

NASA NTRS1981

Numerical analysis of the scramjet-inlet flow field by using two-dimensional Navier-Stokes equations

A computer code was developed to solve the full two dimensional Navier-Stokes equations in a supersonic combustion ramjet (scramjet) inlet. In order to be able to consider a general inlet geometry with embedded bodies, a numerical coordinate transformation is used which generates a set of boundary-fitted curvilinear coordinates. The explicit finite difference algorithm of MacCormack is used to solve the governing equations. An algebraic, two-layer eddy-viscosity model is used for the turbulent flow. The code can analyze both inviscid and viscous flows with no strut, one strut, or multiple struts in the flow field. The application of the two dimensional analysis in the preliminary parametric design studies of a scramjet inlet is discussed. Detailed results are presented for one model problem and for several actual scramjet-inlet configurations.

Kumar, A.↗

NASA NTRS1982

Three-dimensional inviscid analysis of the scramjet inlet flow field

A computer code has been developed to analyze the inviscid flow field in a supersonic combustion ramjet (scramjet) inlet. The code uses the three-dimensional Euler equations in full conservation form to describe the inlet flow. An algebraic numerical coordinate transformation is used to generate a set of boundary-fitted curvilinear coordinates. The governing equations are solved by a time-asymptotic, unsplit, two-step, finite-difference method. This method is highly efficient on the vector processing computers for which the current code is written. Detailed results are presented for two scramjet inlet configurations over a range of Mach numbers. The calculated results are compared with the available experimental and theoretical results.

Kumar, A.↗