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NASA NTRS1994

New Developments in the Method of Space-Time Conservation Element and Solution Element-Applications to Two-Dimensional Time-Marching Problems

A new numerical discretization method for solving conservation laws is being developed. This new approach differs substantially in both concept and methodology from the well-established methods, i.e., finite difference, finite volume, finite element, and spectral methods. It is motivated by several important physical/numerical considerations and designed to avoid several key limitations of the above traditional methods. As a result of the above considerations, a set of key principles for the design of numerical schemes was put forth in a previous report. These principles were used to construct several numerical schemes that model a 1-D time-dependent convection-diffusion equation. These schemes were then extended to solve the time-dependent Euler and Navier-Stokes equations of a perfect gas. It was shown that the above schemes compared favorably with the traditional schemes in simplicity, generality, and accuracy. In this report, the 2-D versions of the above schemes, except the Navier-Stokes solver, are constructed using the same set of design principles. Their constructions are simplified greatly by the use of a nontraditional space-time mesh. Its use results in the simplest stencil possible, i.e., a tetrahedron in a 3-D space-time with a vertex at the upper time level and other three at the lower time level. Because of the similarity in their design, each of the present 2-D solvers virtually shares with its 1-D counterpart the same fundamental characteristics. Moreover, it is shown that the present Euler solver is capable of generating highly accurate solutions for a famous 2-D shock reflection problem. Specifically, both the incident and the reflected shocks can be resolved by a single data point without the presence of numerical oscillations near the discontinuity.

Chang, Sin-Chung↗

NASA NTRS2001

The Numerical Analysis of a Turbulent Compressible Jet

A numerical method to simulate high Reynolds number jet flows was formulated and applied to gain a better understanding of the flow physics. Large-eddy simulation was chosen as the most promising approach to model the turbulent structures due to its compromise between accuracy and computational expense. The filtered Navier-Stokes equations were developed including a total energy form of the energy equation. Subgrid scale models for the momentum and energy equations were adapted from compressible forms of Smagorinsky's original model. The effect of using disparate temporal and spatial accuracy in a numerical scheme was discovered through one-dimensional model problems and a new uniformly fourth-order accurate numerical method was developed. Results from two- and three-dimensional validation exercises show that the code accurately reproduces both viscous and inviscid flows. Numerous axisymmetric jet simulations were performed to investigate the effect of grid resolution, numerical scheme, exit boundary conditions and subgrid scale modeling on the solution and the results were used to guide the three-dimensional calculations. Three-dimensional calculations of a Mach 1.4 jet showed that this LES simulation accurately captures the physics of the turbulent flow. The agreement with experimental data was relatively good and is much better than results in the current literature. Turbulent intensities indicate that the turbulent structures at this level of modeling are not isotropic and this information could lend itself to the development of improved subgrid scale models for LES and turbulence models for RANS simulations. A two point correlation technique was used to quantify the turbulent structures. Two point space correlations were used to obtain a measure of the integral length scale, which proved to be approximately 1/2 D(sub j). Two point space-time correlations were used to obtain the convection velocity for the turbulent structures. This velocity ranged from 0.57 to 0.71 U(sub j).

DeBonis, James R.↗

DOE OSTI2024

Verification of electromagnetic simulation capabilities in global gyrokinetic particle-in-cell code GTS

Recently, the numerical scheme presented by Mishchenko et al. enabled explicit gyrokinetic simulations of low-frequency electromagnetic instabilities in tokamaks at experimentally relevant values of plasma β⁠. This scheme resolved the long-standing cancellation problem that previously hindered gyrokinetic particle-in-cell code simulations of magnetohydrodynamic phenomena with inherently small parallel electric fields. Moreover, the scheme did not employ approximations that eliminate critical tearing-type instabilities. Here, we report on the implementation of this numerical scheme in the global gyrokinetic particle-in-cell code GTS. This implementation allows for a more complete and accurate picture of interaction between small scale turbulence and MHD modes in tokamaks. Additionally, we present a comprehensive set of verification simulations of numerous electromagnetic instabilities relevant to present-day tokamaks. These simulations encompass the kinetic ballooning mode, the internal kink mode, the tearing mode, the micro-tearing mode, and the toroidal Alfven eigenmode destabilized by energetic ions, which are all instrumental in understanding tokamak physics. We will also showcase the preliminary nonlinear simulations of kinetic ballooning instabilities and (2,1) island formation due to tearing mode instability. These simulations validate the accuracy of the scheme implementation and pave the way for studying how these instabilities affect plasma confinement and performance.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2003

Large-Eddy Simulation Code Developed for Propulsion Applications

A large-eddy simulation (LES) code was developed at the NASA Glenn Research Center to provide more accurate and detailed computational analyses of propulsion flow fields. The accuracy of current computational fluid dynamics (CFD) methods is limited primarily by their inability to properly account for the turbulent motion present in virtually all propulsion flows. Because the efficiency and performance of a propulsion system are highly dependent on the details of this turbulent motion, it is critical for CFD to accurately model it. The LES code promises to give new CFD simulations an advantage over older methods by directly computing the large turbulent eddies, to correctly predict their effect on a propulsion system. Turbulent motion is a random, unsteady process whose behavior is difficult to predict through computer simulations. Current methods are based on Reynolds-Averaged Navier- Stokes (RANS) analyses that rely on models to represent the effect of turbulence within a flow field. The quality of the results depends on the quality of the model and its applicability to the type of flow field being studied. LES promises to be more accurate because it drastically reduces the amount of modeling necessary. It is the logical step toward improving turbulent flow predictions. In LES, the large-scale dominant turbulent motion is computed directly, leaving only the less significant small turbulent scales to be modeled. As part of the prediction, the LES method generates detailed information on the turbulence itself, providing important information for other applications, such as aeroacoustics. The LES code developed at Glenn for propulsion flow fields is being used to both analyze propulsion system components and test improved LES algorithms (subgrid-scale models, filters, and numerical schemes). The code solves the compressible Favre-filtered Navier- Stokes equations using an explicit fourth-order accurate numerical scheme, it incorporates a compressible form of Smagorinsky s model for the subgrid-scale turbulence, and it uses generalized curvilinear coordinates to allow analysis of a wide range of geometries. The code runs in parallel on shared memory multiprocessor computers and is written in Fortran 90 with dynamic memory allocation. A sample result for a Mach-1.4 round jet is presented in the figure. Instantaneous Mach number contours in several cross-planes downstream of the nozzle exit are shown, illustrating how an LES captures the large unsteady three-dimensional turbulent structures present in the jet.

DeBonis, James R.↗

NASA NTRS1973

Study of hypervelocity meteoroid impact on orbital space stations

Structural damage resulting in hypervelocity impact of a meteorite on a spacecraft is discussed. Of particular interest is the backside spallation caused by such a collision. To treat this phenomenon two numerical schemes were developed in the course of this study to compute the elastic-plastic flow fracture of a solid. The numerical schemes are a five-point finite difference scheme and a four-node finite element scheme. The four-node finite element scheme proved to be less sensitive to the type of boundary conditions and loadings. Although further development work is needed to improve the program versatility (generalization of the network topology, secondary storage for large systems, improving of the coding to reduce the run time, etc.), the basic framework is provided for a utilitarian computer program which may be used in a wide variety of situations. Analytic results showing the program output are given for several test cases.

Leimbach, K. R.↗

NASA NTRS1981

A solution for two-dimensional Fredholm integral equations of the second kind with periodic, semiperiodic, or nonperiodic kernels

A numerical scheme for solving two dimensional Fredholm integral equations of the second kind is developed. The proof of the convergence of the numerical scheme is shown for three cases: the case of periodic kernels, the case of semiperiodic kernels, and the case of nonperiodic kernels. Applications to the incompressible, stationary Navier-Stokes problem are of primary interest.

Gabrielsen, R. E.↗

NASA NTRS1992

Boundary conditions for direct computation of aerodynamic sound generation

A numerical scheme suitable for the computation of both the near field acoustic sources and the far field sound produced by turbulent free shear flows utilizing the Navier-Stokes equations is presented. To produce stable numerical schemes in the presence of shear, damping terms must be added to the boundary conditions. The numerical technique and boundary conditions are found to give stable results for computations of spatially evolving mixing layers.

Colonius, Tim↗

NASA NTRS1985

A Factored Implicit Scheme for Numerical Weather Prediction with Small Factorization Error

Numerical results show that, for large time steps, the factorization error can be significant, even for the slowly propagating Rossby modes. A new scheme is formulated based on a more accurate factorization of the equations. By grouping separately the terms of the equations which give rise to the fast and slow motion, the equations are factored more accurately. The fast-slow factorization eliminated the factorization error. If each of the fast and slow factors are factored again according to spatial components, the resulting scheme only involves the solution of one dimensional linear systems, and computational efficient. It is shown that the factorization error for the slow made component is negligible for this new scheme.

Augenbaum, J. M.↗

NASA NTRS1986

Aerothermal modeling program, phase 2

The main objective of the NASA sponsored Aerothermal Modeling Program, Phase 2--Element A, is to develop an improved numerical scheme for predicting combustor flow fields. This effort consists of the following three technical tasks. Task 1 involves the selection and evaluation of various candidate numerical techniques. Task 2 involves an in-depth evaluation of the selected numerical schemes. Task 3 involves the convection-diffusion scheme and the direct solver that will be incorporated in the NASA 3-D elliptic code (COM3S).

Karki, K. C.↗

NASA NTRSDate not supplied

Numerical Investigation of Combustor-Turbine Interactions with a Two-Stage High-Pressure Turbine Including Cooling Airflows

Combustor–turbine interactions (CTIs) are investigated by performing three-dimensional unsteady simulations using a realistic combustor and high-pressure turbine (HPT) configurations from the Energy Efficient Engine (E3) program. To understand CTIs, we compare the numerically predicted flow fields from single-component simulations (Step 1: the combustor + the first-stage stator of turbine; Step 2: the two-stage HPT imposing the time-averaged flow solution from Step 1 as the inflow boundary condition) and a fully coupled combustor–turbine simulation (Step 3) at the sea-level takeoff (SLTO) condition. In addition to three previous simulations where the cooling airflows inside the HPT had been neglected for all Step 1, Step 2, and Step 3, two new simulations of Step 2 and Step 3 take into account the cooling airflows using the source team approach. In this approach, to mimic the cooling airflows, we impose the source term at a specific area of a cooling airflow hole at the solid surfaces without making a mesh of each hole. The objective of this study is twofold. One is to investigate the effect of the presence of the cooling airflows on the aerodynamics of the combustor and HPT as well as the HPT performance. The second is to perform a detailed comparison among the calculated flowfields by two different numerical schemes, the central-difference with the standard Jameson–Schmidt–Turkel (CD-JST) scheme and the AUSM scheme. There is a noticeable difference in the hot-streak distributions at the first-stage stator. In addition, depending on the choice of the numerical scheme and the presence of the cooling airflows, an occurrence of shock waves appearing at the first-stage stator is greatly influenced. Thus, this has a noticeable impact on the HPT performance. It is shown that the estimated turbine efficiencies from Step 3 are about 7 % less than the ones from Step 2.

Combustor-turbine interaction↗

NASA NTRS2008

Problems Associated with Grid Convergence of Functionals

The current use of functionals to evaluate order-of-convergence of a numerical scheme can lead to incorrect values. The problem comes about because of interplay between the errors from the evaluation of the functional, e.g., quadrature error, and from the numerical scheme discretization. Alternative procedures for deducing the order-property of a scheme are presented. The problem is studied within the context of the inviscid supersonic flow over a blunt body; however, the problem and solutions presented are not unique to this example.

Salas, Manuel D.↗

NASA NTRS2009

On Problems Associated with Grid Convergence of Functionals

The current use of functionals to evaluate order-of-convergence of a numerical scheme can lead to incorrect values. The problem comes about because of interplay between the errors from the evaluation of the functional, e.g., quadrature error, and from the numerical scheme discretization. Alternative procedures for deducing the order property of a scheme are presented. The problems are studied within the context of the inviscid supersonic flow over a blunt body; however, the problems and solutions presented are not unique to this example.

Salas, Manuael D.↗

NASA NTRS1998

Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows

A higher order accurate numerical procedure has been developed for solving incompressible Navier-Stokes equations for 2D or 3D fluid flow problems. It is based on low-storage Runge-Kutta schemes for temporal discretization and fourth and sixth order compact finite-difference schemes for spatial discretization. The particular difficulty of satisfying the divergence-free velocity field required in incompressible fluid flow is resolved by solving a Poisson equation for pressure. It is demonstrated that for consistent global accuracy, it is necessary to employ the same order of accuracy in the discretization of the Poisson equation. Special care is also required to achieve the formal temporal accuracy of the Runge-Kutta schemes. The accuracy of the present procedure is demonstrated by application to several pertinent benchmark problems.

Wilson, Robert V.↗

NASA NTRS2016

Computational Flow Field in Energy Efficient Engine (EEE)

In this paper, preliminary results for the recently-updated Open National Combustion Code (Open NCC) as applied to the EEE are presented. The comparison between two different numerical schemes, the standard Jameson-Schmidt-Turkel (JST) scheme and the advection upstream splitting method (AUSM), is performed for the cold flow and the reacting flow calculations using the RANS. In the cold flow calculation, the AUSM scheme predicts a much stronger reverse flow in the central recirculation zone. In the reacting flow calculation, we test two cases: gaseous fuel injection and liquid spray injection. In the gaseous fuel injection case, the overall flame structures of the two schemes are similar to one another, in the sense that the flame is attached to the main nozzle, but is detached from the pilot nozzle. However, in the exit temperature profile, the AUSM scheme shows a more uniform profile than that of the JST scheme, which is close to the experimental data. In the liquid spray injection case, we expect different flame structures in this scenario. We will give a brief discussion on how two numerical schemes predict the flame structures inside the EEE using different ways to introduce the fuel injection.

CFD↗

DOE OSTI2023

A Probabilistic Scheme for Semilinear Nonlocal Diffusion Equations with Volume Constraints

This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial integro-differential equation (PIDE), in which the integro-differential operator consists of both local convection-diffusion and nonlocal diffusion operators. Here, our numerical scheme is based on the direct approximation of the nonlinear Feynman–Kac formula that establishes a link between nonlinear PIDEs and stochastic differential equations. The exploitation of the Feynman–Kac representation avoids solving dense linear systems arising from nonlocal operators. Compared with existing stochastic approaches, our method can achieve first-order convergence after balancing the temporal and spatial discretization errors, which is a significant improvement of existing probabilistic/stochastic methods for nonlocal diffusion problems. Error analysis of our numerical scheme is established. The effectiveness of our approach is shown in two numerical examples. The first example considers a three-dimensional nonlocal diffusion equation to numerically verify the error analysis results. The second example presents a physics problem motivated by the study of heat transport in magnetically confined fusion plasmas.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Image-Driven Hybrid Structural Analysis Based on Continuum Point Cloud Method with Boundary Capturing Technique

Conventional approaches for the structural health monitoring of infrastructures often rely on physical sensors or targets attached to structural members, which require considerable preparation, maintenance, and operational effort, including continuous on-site adjustments. This paper presents an image-driven hybrid structural analysis technique that combines digital image processing (DIP) and regression analysis with a continuum point cloud method (CPCM) built on a particle-based strong formulation. Polynomial regressions capture the boundary shape change due to the structural loading and precisely identify the edge and corner coordinates of the deformed structure. The captured edge profiles are transformed into essential boundary conditions. This allows the construction of a strongly formulated boundary value problem (BVP), classified as the Dirichlet problem. Capturing boundary conditions from the digital image is novel, although a similar approach was applied to the point cloud data. It was shown that the CPCM is more efficient in this hybrid simulation framework than the weak-form-based numerical schemes. Unlike the finite element method (FEM), it can avoid aligning boundary nodes with regression points. A three-point bending test of a rubber beam was simulated to validate the developed technique. The simulation results were benchmarked against numerical results by ANSYS and various relevant numerical schemes. The technique can effectively solve the Dirichlet-type BVP, yielding accurate deformation, stress, and strain values across the entire problem domain when employing a linear strain model and increasing the number of CPCM nodes. In addition, comparative analysis with conventional displacement tracking techniques verifies the developed technique’s robustness. The proposed technique effectively circumvents the inherent limitations of traditional monitoring methods resulting from the reliance on physical gauges or target markers so that a robust and non-contact solution for remote structural health monitoring in real-scale infrastructures can be provided, even in unfavorable experimental environments.

Chemistry↗

NASA NTRS1993

Numerical Simulation of a High Mach Number Jet Flow

The recent efforts to develop accurate numerical schemes for transition and turbulent flows are motivated, among other factors, by the need for accurate prediction of flow noise. The success of developing high speed civil transport plane (HSCT) is contingent upon our understanding and suppression of the jet exhaust noise. The radiated sound can be directly obtained by solving the full (time-dependent) compressible Navier-Stokes equations. However, this requires computational storage that is beyond currently available machines. This difficulty can be overcome by limiting the solution domain to the near field where the jet is nonlinear and then use acoustic analogy (e.g., Lighthill) to relate the far-field noise to the near-field sources. The later requires obtaining the time-dependent flow field. The other difficulty in aeroacoustics computations is that at high Reynolds numbers the turbulent flow has a large range of scales. Direct numerical simulations (DNS) cannot obtain all the scales of motion at high Reynolds number of technological interest. However, it is believed that the large scale structure is more efficient than the small-scale structure in radiating noise. Thus, one can model the small scales and calculate the acoustically active scales. The large scale structure in the noise-producing initial region of the jet can be viewed as a wavelike nature, the net radiated sound is the net cancellation after integration over space. As such, aeroacoustics computations are highly sensitive to errors in computing the sound sources. It is therefore essential to use a high-order numerical scheme to predict the flow field. The present paper presents the first step in a ongoing effort to predict jet noise. The emphasis here is in accurate prediction of the unsteady flow field. We solve the full time-dependent Navier-Stokes equations by a high order finite difference method. Time accurate spatial simulations of both plane and axisymmetric jet are presented. Jet Mach numbers of 1.5 and 2.1 are considered. Reynolds number in the simulations was about a million. Our numerical model is based on the 2-4 scheme by Gottlieb & Turkel. Bayliss et al. applied the 2-4 scheme in boundary layer computations. This scheme was also used by Ragab and Sheen to study the nonlinear development of supersonic instability waves in a mixing layer. In this study, we present two dimensional direct simulation results for both plane and axisymmetric jets. These results are compared with linear theory predictions. These computations were made for near nozzle exit region and velocity in spanwise/azimuthal direction was assumed to be zero.

Hayder, M. Ehtesham↗

DOE OSTI2024

A numerical Poisson solver with improved radial solutions for a self-consistent locally scaled self-interaction correction method

Abstract The universal applicability of density functional approximations is limited by self-interaction error made by these functionals. Recently, a novel one-electron self-interaction-correction (SIC) method that uses an iso-orbital indicator to apply the SIC at each point in space by scaling the exchange-correlation and Coulomb energy densities was proposed. The locally scaled SIC (LSIC) method is exact for the one-electron densities, and unlike the well-known Perdew–Zunger SIC (PZSIC) method recovers the uniform electron gas limit of the uncorrected density functional approximation, and reduces to PZSIC method as a special case when isoorbital indicator is set to the unity. Here, we present a numerical scheme that we have adopted to evaluate the Coulomb potential of the electron density scaled by the iso-orbital indicator required for the self-consistent LSIC calculations. After analyzing the behavior of the finite difference method (FDM) and the green function solution to the radial part of the Poisson equation, we adopt a hybrid approach that uses the FDM for the Coulomb potential due to the monopole and the GF for all higher-order terms. The performance of the resultant hybrid method is assessed using a variety of systems. The results show improved accuracy than earlier numerical schemes. We also find that, even with a generic set of radial grid parameters, accurate energy differences can be obtained using a numerical Coulomb solver in standard density functional studies.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗