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NASA NTRS2015

Advances in Distance-Based Hole Cuts on Overset Grids

An automatic and efficient method to determine appropriate hole cuts based on distances to the wall and donor stencil maps for overset grids is presented. A new robust procedure is developed to create a closed surface triangulation representation of each geometric component for accurate determination of the minimum hole. Hole boundaries are then displaced away from the tight grid-spacing regions near solid walls to allow grid overlap to occur away from the walls where cell sizes from neighboring grids are more comparable. The placement of hole boundaries is efficiently determined using a mid-distance rule and Cartesian maps of potential valid donor stencils with minimal user input. Application of this procedure typically results in a spatially-variable offset of the hole boundaries from the minimum hole with only a small number of orphan points remaining. Test cases on complex configurations are presented to demonstrate the new scheme.

Advances↗

NASA NTRS1986

Numerical simulation of backward step and jet exhaust flows

Preparatory to a continuing study of efficient, high resolution flow structure capture on patched mesh systems, turbulent flowfield solutions for a tactical missile base flow problem are computed on two first level composite grids of differing topologies. One mesh is of the wraparound character and the other is a step mesh on the Cartesian topology. A simplified Baldwin-Lomax algebraic eddy viscosity model is applied in the wake region. Results for the two meshes are compared with each other and with other computations reported in the literature. The totality of results suggest both mesh topology and resolution are important factors in the accuracy of predictive results as well as turbulence modeling. Related to interests in high altitude laminar flows bluff bodies such as the AOTV, the paper also presents results from a solution from laminar flow over a backward facing step and makes comparison with the available classical experiment of Chapman, et al.

Lombard, C. K.↗

NASA NTRS2014

Summary and Statistical Analysis of the First AIAA Sonic Boom Prediction Workshop

A summary is provided for the First AIAA Sonic Boom Workshop held 11 January 2014 in conjunction with AIAA SciTech 2014. Near-field pressure signatures extracted from computational fluid dynamics solutions are gathered from nineteen participants representing three countries for the two required cases, an axisymmetric body and simple delta wing body. Structured multiblock, unstructured mixed-element, unstructured tetrahedral, overset, and Cartesian cut-cell methods are used by the participants. Participants provided signatures computed on participant generated and solution adapted grids. Signatures are also provided for a series of uniformly refined workshop provided grids. These submissions are propagated to the ground and loudness measures are computed. This allows the grid convergence of a loudness measure and a validation metric (dfference norm between computed and wind tunnel measured near-field signatures) to be studied for the first time. Statistical analysis is also presented for these measures. An optional configuration includes fuselage, wing, tail, flow-through nacelles, and blade sting. This full configuration exhibits more variation in eleven submissions than the sixty submissions provided for each required case. Recommendations are provided for potential improvements to the analysis methods and a possible subsequent workshop.

Park, Michael A.↗

NASA NTRS2008

The Adaptively-Refined MHD Solver (ARMS)

ARMS is a robust code for solving the single-fluid, fully compressible equations of magnetohydrodynamics. In this presentation I will describe how to access and run ARMS and how to use HelioSpace to analyze the results. The main features of the code are the following: 1. Equations are solved in either Cartesian or Spherical coordinates. For the latter, either a linear or a Log(r) coordinate can be used. 2. The latest flux-corrected-transport schemes developed at NRZ, are used to advance the convective equations. All waves and shocks are accurately captured. 3. A fully adaptive grid can be used. The code incorporates the PARAMESH technology for block-adaptation. Arbitrary adaptation criteria can be specified. 4. The code maintains a divergence-free magnetic field to machine accuracy throughout all refinement and de-refinement. 5. A uniform resistivity and thermal conduction can be included. Solution is either explicit or implicit. 6. Arbitrary boundary and initial conditions can be used in either Cartesian or spherical coordinates. 7. An accompanying visualization package, HelioSpace, is available for viewing and analyzing the native ARMS output, (without need for interpolating onto a structured grid).

Antichos, S. K.↗

NASA NTRS1992

An alternative to unstructured grids for computing gas dynamic flows around arbitrarily complex two-dimensional bodies

In this paper we describe an approach for dealing with arbitrary complex, two dimensional geometries, the so-called cartesian boundary method. Conceptually, the cartesian boundary method is quite simple. Solid bodies blank out areas of a background, cartesian mesh, and the resultant cut cells are singled out for special attention. However, there are several obstacles that must be overcome in order to achieve a practical scheme. We present a general strategy that overcomes these obstacles, together with some details of our successful conversion of an adaptive mesh algorithm from a body-fitted code to a cartesian boundary code.

Quirk, James J.↗

NASA NTRS1989

Algorithm For Hypersonic Flow In Chemical Equilibrium

Implicit, finite-difference, shock-capturing algorithm calculates inviscid, hypersonic flows in chemical equilibrium. Implicit formulation chosen because overcomes limitation on mathematical stability encountered in explicit formulations. For dynamical portion of problem, Euler equations written in conservation-law form in Cartesian coordinate system for two-dimensional or axisymmetric flow. For chemical portion of problem, equilibrium state of gas at each point in computational grid determined by minimizing local Gibbs free energy, subject to local conservation of molecules, atoms, ions, and total enthalpy. Major advantage: resulting algorithm naturally stable and captures strong shocks without help of artificial-dissipation terms to damp out spurious numerical oscillations.

Palmer, Grant↗

NASA NTRS1983

A method for solving the transonic full-potential equation for general configurations

A method is developed for solving the full-potential equation for two-dimensional and axisymmetric flow which retains the grid and boundary condition simplicity of the transonic small-disturbance codes. The method is based on a finite-volume formulation of the mass conservation equation in a Cartesian coordinate system, and is an extension of the method of Purvis and Burkhalter (1979). This finite-volume approach, combined with the simple boundary treatment, is shown to result in a highly robust method applicable to a wide range of geometries and flow conditions. The accuracy of the method is demonstrated for general geometries in two-dimensional and axisymmetric flows. The use of this method results in significant gains in convergence rate over the vertical-line over-relaxation scheme by incorporating an AF2-type algorithm (Ballhaus et al., 1978). It is suggested that the simplicity of this method shold allow a relatively easy extension to complex geometries in three-dimensional flows, and complex two-dimensional configurations such as multielement airfoils should be amenable to this method.

Wedan, B.↗

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 NTRS1987

Inviscid transonic wing design using inverse methods in curvilinear coordinates

An inverse wing design method has been developed around an existing transonic wing analysis code. The original analysis code, TAWFIVE, has as its core the numerical potential flow solver, FLO30, developed by Jameson and Caughey. Features of the analysis code include a finite-volume formulation; wing and fuselage fitted, curvilinear grid mesh; and a viscous boundary layer correction that also accounts for viscous wake thickness and curvature. The development of the inverse methods as an extension of previous methods existing for design in Cartesian coordinates is presented. Results are shown for inviscid wing design cases in super-critical flow regimes. The test cases selected also demonstrate the versatility of the design method in designing an entire wing or discontinuous sections of a wing.

Gally, Thomas A.↗

NASA NTRS1989

Development of direct-inverse 3-D methods for applied transonic aerodynamic wing design and analysis

An inverse wing design method was developed around an existing transonic wing analysis code. The original analysis code, TAWFIVE, has as its core the numerical potential flow solver, FLO30, developed by Jameson and Caughey. Features of the analysis code include a finite-volume formulation; wing and fuselage fitted, curvilinear grid mesh; and a viscous boundary layer correction that also accounts for viscous wake thickness and curvature. The development of the inverse methods as an extension of previous methods existing for design in Cartesian coordinates is presented. Results are shown for inviscid wing design cases in super-critical flow regimes. The test cases selected also demonstrate the versatility of the design method in designing an entire wing or discontinuous sections of a wing.

Carlson, Leland A.↗

NASA NTRS1982

General curvilinear coordinate systems

The basic ideas of the construction and use of numerically-generated boundary-fitted coordinate systems for the numerical solution of partial differential equations are discussed. With such coordinate systems, all computation can be done on a fixed square grid in the rectangular transformed region regardless of the shape or movement of the physical boundaries. A number of different types of configurations for the transformed region and the basic transformation relations from a cartesian system to a general curvilinear system are given. The material of this paper is applicable to all types of coordinate system generation.

Thompson, J. P.↗

NASA NTRS1994

Investigation of Advanced Counterrotation Blade Configuration Concepts for High Speed Turboprop Systems. Task 8: Cooling Flow/heat Transfer Analysis

The focus of this task was to validate the ADPAC code for heat transfer calculations. To accomplish this goal, the ADPAC code was modified to allow for a Cartesian coordinate system capability and to add boundary conditions to handle spanwise periodicity and transpiration boundaries. The primary validation case was the film cooled C3X vane. The cooling hole modeling included both a porous region and grid in each discrete hold. Predictions for these models as well as smooth wall compared well with the experimental data.

Hall, Edward J.↗

NASA NTRS1995

Numerical Study of Boundary Layer Interaction with Shocks: Method Improvement and Test Computation

The objective is the development of a high-order and high-resolution method for the direct numerical simulation of shock turbulent-boundary-layer interaction. Details concerning the spatial discretization of the convective terms can be found in Adams and Shariff (1995). The computer code based on this method as introduced in Adams (1994) was formulated in Cartesian coordinates and thus has been limited to simple rectangular domains. For more general two-dimensional geometries, as a compression corner, an extension to generalized coordinates is necessary. To keep the requirements or limitations for grid generation low, the extended formulation should allow for non-orthogonal grids. Still, for simplicity and cost efficiency, periodicity can be assumed in one cross-flow direction. For easy vectorization, the compact-ENO coupling algorithm as used in Adams (1994) treated whole planes normal to the derivative direction with the ENO scheme whenever at least one point of this plane satisfied the detection criterion. This is apparently too restrictive for more general geometries and more complex shock patterns. Here we introduce a localized compact-ENO coupling algorithm, which is efficient as long as the overall number of grid points treated by the ENO scheme is small compared to the total number of grid points. Validation and test computations with the final code are performed to assess the efficiency and suitability of the computer code for the problems of interest. We define a set of parameters where a direct numerical simulation of a turbulent boundary layer along a compression corner with reasonably fine resolution is affordable.

Adams, N. A.↗

NASA NTRS1984

Further development of XTRAN3S computer program

This report describes modifications and enhancements to XTRAN3S, a computer program for aerodynamic, static aeroelastic, and dynamic aeroelastic analysis of three-dimensional wings in the transonic speed regime. Modifications to the program include incorporation of a viscous boundary layer, modified coefficient and generalized force integration, direct input of airfoil surface coordinates and slopes, variable dimensions of computational arrays and a modified grid mapping transformation to improve computational stability for highly swept and tapered planforms. Results obtained with the modified program are included. Modifications discussed but not incorporated include a state matrix integration method, enhanced vectorization, and use of a cartesian physical mesh.

Borland, C. J.↗

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 NTRS1979

Explicit large time-step schemes for the shallow water equations

Modifications to explicit finite difference schemes for solving the shallow water equations for meteorological applications by increasing the time step for the fast gravity waves are analyzed. Terms associated with the gravity waves in the shallow water equations are treated on a coarser grid than those associated with the slow Rossby waves, which contain much more of the available energy and must be treated with higher accuracy, enabling a several-fold increase in time step without degrading the accuracy of the solution. The method is presented in Cartesian and spherical coordinates for a rotating earth, using generalized leapfrog, frozen coefficient, and Fourier filtering finite difference schemes. Computational results verify the numerical stability of the approach.

Turkel, E.↗

NASA NTRS1997

Exact Integrations of Polynomials and Symmetric Quadrature Formulas over Arbitrary Polyhedral Grids

This paper is concerned with two important elements in the high-order accurate spatial discretization of finite volume equations over arbitrary grids. One element is the integration of basis functions over arbitrary domains, which is used in expressing various spatial integrals in terms of discrete unknowns. The other consists of quadrature approximations to those integrals. Only polynomial basis functions applied to polyhedral and polygonal grids are treated here. Non-triangular polygonal faces are subdivided into a union of planar triangular facets, and the resulting triangulated polyhedron is subdivided into a union of tetrahedra. The straight line segment, triangle, and tetrahedron are thus the fundamental shapes that are the building blocks for all integrations and quadrature approximations. Integrals of products up to the fifth order are derived in a unified manner for the three fundamental shapes in terms of the position vectors of vertices. Results are given both in terms of tensor products and products of Cartesian coordinates. The exact polynomial integrals are used to obtain symmetric quadrature approximations of any degree of precision up to five for arbitrary integrals over the three fundamental domains. Using a coordinate-free formulation, simple and rational procedures are developed to derive virtually all quadrature formulas, including some previously unpublished. Four symmetry groups of quadrature points are introduced to derive Gauss formulas, while their limiting forms are used to derive Lobatto formulas. Representative Gauss and Lobatto formulas are tabulated. The relative efficiency of their application to polyhedral and polygonal grids is detailed. The extension to higher degrees of precision is discussed.

Liu, Yen↗

NASA NTRS2010

Adaptive Numerical Algorithms in Space Weather Modeling

Space weather describes the various processes in the Sun-Earth system that present danger to human health and technology. The goal of space weather forecasting is to provide an opportunity to mitigate these negative effects. Physics-based space weather modeling is characterized by disparate temporal and spatial scales as well as by different physics in different domains. A multi-physics system can be modeled by a software framework comprising of several components. Each component corresponds to a physics domain, and each component is represented by one or more numerical models. The publicly available Space Weather Modeling Framework (SWMF) can execute and couple together several components distributed over a parallel machine in a flexible and efficient manner. The framework also allows resolving disparate spatial and temporal scales with independent spatial and temporal discretizations in the various models. Several of the computationally most expensive domains of the framework are modeled by the Block-Adaptive Tree Solar wind Roe Upwind Scheme (BATS-R-US) code that can solve various forms of the magnetohydrodynamics (MHD) equations, including Hall, semi-relativistic, multi-species and multi-fluid MHD, anisotropic pressure, radiative transport and heat conduction. Modeling disparate scales within BATS-R-US is achieved by a block-adaptive mesh both in Cartesian and generalized coordinates. Most recently we have created a new core for BATS-R-US: the Block-Adaptive Tree Library (BATL) that provides a general toolkit for creating, load balancing and message passing in a 1, 2 or 3 dimensional block-adaptive grid. We describe the algorithms of BATL and demonstrate its efficiency and scaling properties for various problems. BATS-R-US uses several time-integration schemes to address multiple time-scales: explicit time stepping with fixed or local time steps, partially steady-state evolution, point-implicit, semi-implicit, explicit/implicit, and fully implicit numerical schemes. Depending on the application, we find that different time stepping methods are optimal. Several of the time integration schemes exploit the block-based granularity of the grid structure. The framework and the adaptive algorithms enable physics based space weather modeling and even forecasting.

Toth, Gabor↗