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NASA NTRS1993

Modeling void growth and movement with phase change in thermal energy storage canisters

A scheme was developed to model the thermal hydrodynamic behavior of thermal energy storage salts. The model included buoyancy, surface tension, viscosity, phases change with density difference, and void growth and movement. The energy, momentum, and continuity equations were solved using a finite volume formulation. The momentum equation was divided into two pieces. The void growth and void movement are modeled between the two pieces of the momentum equations. Results showed this scheme was able to predict the behavior of thermal energy storage salts.

Darling, Douglas↗

NASA NTRS1993

Modeling void growth and movement with phase change in thermal energy storage canisters

Darling, Douglas↗

DOE OSTI2021

Numerical and Analytical Modeling of the Effect of Cracks on the Self-Inductance of a COTS YJ-41003-TC Toroid

COTS inductors and transformers often contain partial cracks whose effect on inductance, a key performance parameter, have not been carefully studied. In this report, the effects of both partial and complete cracks on the self-inductance of a 100 turn square cross section COTS YJ-41003-TC toroid comprised of J Material was comprehensively investigated using both analytically derived closed form expressions and 3D computational techniques employing commercial codes. Both partial (half-penny) and complete (air gap) cracks of 10 and 25 μm were investigated. The crack is defined as the physical distance between two faces of the toroid's magnetic core, such that the surface normal of either face is along the Φ-direction, in alignment with the B-field. For the purposes of validation, two different approaches were incorporated for both the analytical and numerical models. The two analytical methods are comprised of a first principles approach based on the physics of electromagnetics, as well as linear circuit theory. The former directly utilizes the integral form of Maxwell's equations while the latter exploits the interchangeable relationship between electric and magnetic circuits. Validation within the computational scheme is realized through a code-to-code comparison between commercial solvers, COMSOL Multiphysics and CST, with the former employing the Finite Element Method (FEM) and the latter the Finite Difference Time Domain (FDTD) technique. Sound agreement between all four methods (ie., two analytical and two numerical) is observed, with results indicating that only a perturbation in self-inductance occurs for the half-penny cracks, while a substantial reduction takes place for the case of complete cracks. It is important to note that even though a static μ r is applied, representing the linear region of the BH curve (based on manufacturer specifications), the complete crack results still place a lower conservative bound on the inductance. This follows from the fact that even in the case of a half-penny crack, if the magnetic core portion of the crack approaches saturation, the crack begins to behave like an air gap, or complete crack. When an air gap is introduced into a magnetic core, a substantial reduction in inductance can occur due to the significant difference in permeabilities between the two mediums (ie., μ core >> μ air ). The once intact bulk magnetic core of the toroid essentially begins to behave like an air core.

36 MATERIALS SCIENCE↗

NASA NTRS1985

Use of a hyperbolic grid generation scheme in simulating supersonic viscous flow about three-dimensional winged configuration

The present paper describes a numerical mesh generation technique to be used with an implicit finite difference method for simulating visous supersonic flow about low-aspect-ratio wing body configurations using a single grid strategy. The computational domain is segmented into multiple regions, with borders located in supersonic areas to avoid the otherwise costly interfacing procedure between adjacent segments. The numerical procedure is applied to calculate the turbulent flow around the shuttle orbiter and a canard projectile at supersonic free stream Mach number.

Rizk, Y. M.↗

DOE OSTI2022

Progress on Demonstration of a MOOSE-Based Coupled Capability for Hot Channel Factors in Fast Reactors

Hot channel factors (HCFs) are computed values that account for the impact on predicted peak fuel, cladding, and coolant temperatures due to uncertainties in the as-built reactor’s material properties and geometry as well as uncertainties due to modeling approximations. Reduction in computed HCF values via reduction or elimination of modeling approximations may translate to significant economic savings if the reactor power can be raised due to the extra temperature margin gained. While limited historical datasets exist for sodium-cooled fast reactors (SFRs), there are no available HCF data for lead-cooled fast reactors (LFRs) outside of work generated previously within NEAMS. The computation of HCFs involves insights from reactor physics, thermal fluids and heat conduction calculations to determine how the peak temperatures respond to various uncertainties in the design. Due to the significant advantages for multi-physics coupling offered by the MOOSE framework, Griffin (MOOSE-based reactor physics code), MOOSE Heat Conduction Module, and Cardinal (MOOSE-wrapped multi-physics application which includes the NekRS thermal fluids code) are being coupled together using the MOOSE MultiApp System to develop a highfidelity multi-physics modeling capability for HCF simulations. This high-fidelity coupling workflow may also be beneficial for other fast reactor applications in the future. In previous work, Griffin and NekRS were individually assessed to ensure the necessary capabilities were in place. This work describes initial efforts to couple the codes (including folding in the MOOSE Heat Conduction Module) and determining the workflow for the perturbed calculations which will leverage the Stochastic Tools Module (STM). To our knowledge, this is the first coupling of Griffin and NekRS as well as the first exploratory use of Stochastic Tools Module for Cardinal. In this report, the neutronics code Griffin, the heat conduction solver in MOOSE, and the MOOSE-wrapped application containing NekRS (Cardinal) are linked together to demonstrate the coupled capability. Griffin and Cardinal are linked dynamically by specifying shared libraries. Different coupling hierarchies are tested for selecting the most appropriate coupling strategy. A coupling scheme is selected based on the efficiency of calculation and ease of data communication. Multiple tests are performed to choose suitable mesh structure, model configurations, scheme setup and boundary conditions to avoid loss of energy due to data interpolation between different modules or weak imposition of fluxes in finite element codes. Computational experiments are performed to study the tolerance control of each type of iteration to avoid false convergence. The coupled capability is demonstrated in both single pin and 7-pin models based on LFR materials and geometry. The study finds that the use of too large a time step size in the heat conduction module can lead to temperature oscillation even though the heat conduction equation does not have a time-derivative kernel, but only the time-dependent boundary condition. A 7-pin model without duct region achieved good convergence in the coupled calculation while a 7 pin model with duct region experienced data communication issues which need to be resolved.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS1978

Finite elements and finite differences for transonic flow calculations

The paper reviews the chief finite difference and finite element techniques used for numerical solution of nonlinear mixed elliptic-hyperbolic equations governing transonic flow. The forms of the governing equations for unsteady two-dimensional transonic flow considered are the Euler equation, the full potential equation in both conservative and nonconservative form, the transonic small-disturbance equation in both conservative and nonconservative form, and the hodograph equations for the small-disturbance case and the full-potential case. Finite difference methods considered include time-dependent methods, relaxation methods, semidirect methods, and hybrid methods. Finite element methods include finite element Lax-Wendroff schemes, implicit Galerkin method, mixed variational principles, dual iterative procedures, optimal control methods and least squares.

Hafez, M. M.↗

NASA NTRS1982

Computer program for solving laminar, transitional, or turbulent compressible boundary-layer equations for two-dimensional and axisymmetric flow

A numerical algorithm and computer program are presented for solving the laminar, transitional, or turbulent two dimensional or axisymmetric compressible boundary-layer equations for perfect-gas flows. The governing equations are solved by an iterative three-point implicit finite-difference procedure. The software, program VGBLP, is a modification of the approach presented in NASA TR R-368 and NASA TM X-2458, respectively. The major modifications are: (1) replacement of the fourth-order Runge-Kutta integration technique with a finite-difference procedure for numerically solving the equations required to initiate the parabolic marching procedure; (2) introduction of the Blottner variable-grid scheme; (3) implementation of an iteration scheme allowing the coupled system of equations to be converged to a specified accuracy level; and (4) inclusion of an iteration scheme for variable-entropy calculations. These modifications to the approach presented in NASA TR R-368 and NASA TM X-2458 yield a software package with high computational efficiency and flexibility. Turbulence-closure options include either two-layer eddy-viscosity or mixing-length models. Eddy conductivity is modeled as a function of eddy viscosity through a static turbulent Prandtl number formulation. Several options are provided for specifying the static turbulent Prandtl number. The transitional boundary layer is treated through a streamwise intermittency function which modifies the turbulence-closure model. This model is based on the probability distribution of turbulent spots and ranges from zero to unity for laminar and turbulent flow, respectively. Several test cases are presented as guides for potential users of the software.

Harris, J. E.↗

NASA NTRS1991

Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics

A prescription is given for conservatively integrating generalized hydromagnetic equations using flux-corrected transport (FCT) techniques. By placing the magnetic-field components at the interface locations of the finite-difference grid, the field is kept divergence-free to within machine roundoff error. The use of FCT techniques allows an integration scheme of high accuracy to be employed, while the numerical ripples associated with large dispersion errors are avoided. The method is particularly well suited for problems involving magnetohydrodynamic shocks and other discontinuities.

Devore, C. Richard↗

NASA NTRS1991

A rheologically layered three-dimensional model of the San Andreas fault in central and southern California

The effects of rheological parameters and the fault slip distribution on the horizontal and vertical deformation in the vicinity of the fault are investigated using 3D kinematic finite element models of the San Andreas fault in central and southern California. It is shown that fault models with different rheological stratification schemes and slip distributions predict characteristic deformation patterns. Models that do not include aseismic slip below the fault locking depth predict deformation patterns that are strongly dependent on time since the last earthquake, while models that incorporate the aseismic slip below the locking depth depend on time to a significantly lesser degree.

Williams, Charles A.↗

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↗

NASA NTRS1991

Discretized Newton-relaxation solution of high resolution flux-difference split schemes

A method is presented for the implicit solution of high resolution flux-difference split schemes for which it is impractical to obtain the Jacobian matrix analytically. The method is used frequently in numerical schemes for solving nonlinear systems of equations, but apparently has not been used to date for approximating the Jacobian of the Roe numerical flux vectors. The method consists of obtaining a discretized Jacobian by using simple finite-differences of the flux vectors, and then using this discretized Jacobian in a Newton-relaxation solution method. The method is applied to both compressible and incompressible flows. Numerical results are presented that demonstrate the method works for both inviscid and viscous flows.

Whitfield, David L.↗

DOE OSTI2017

A posteriori error estimation for multi-stage Runge–Kutta IMEX schemes

Implicit–Explicit (IMEX) schemes are widely used for time integration methods for approximating solutions to a large class of problems. In this work, we develop accurate a posteriori error estimates of a quantity-of-interest for approximations obtained from multi-stage IMEX schemes. This is done by first defining a finite element method that is nodally equivalent to an IMEX scheme, then using typical methods for adjoint-based error estimation. Furthermore, the use of a nodally equivalent finite element method allows a decomposition of the error into multiple components, each describing the effect of a different portion of the method on the total error in a quantity-of-interest.

97 MATHEMATICS AND COMPUTING↗

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↗

NASA NTRS1988

Numerical simulation of axisymmetric turbulent flow in combustors and diffusors

A method for predicting turbulent flow in combustors and diffusers is developed. The Navier-Stokes equations, incorporating a turbulence kappa-epsilon model equation, were solved in a nonorthogonal curvilinear coordinate system. The solution applied the finite volume method to discretize the differential equations and utilized the SIMPLE algorithm iteratively to solve the differenced equations. A zonal grid method, wherein the flow field was divided into several subsections, was developed. This approach permitted different computational schemes to be used in the various zones. In addition, grid generation was made a more simple task. However, treatment of the zonal boundaries required special handling. Boundary overlap and interpolating techniques were used and an adjustment of the flow variables was required to assure conservation of mass, momentum and energy fluxes. The numerical accuracy was assessed using different finite differencing methods, i.e., hybrid, quadratic upwind and skew upwind, to represent the convection terms. Flows in different geometries of combustors and diffusers were simulated and results compared with experimental data and good agreement was obtained.

Yung, Chain Nan↗

NASA NTRS1981

Optimal time splitting for two- and three-dimensional Navier-Stokes equations with mixed derivatives

A new explicit, time splitting algorithm has been developed for finite difference modelling of the full two and three-dimensional time-dependent, compressible, viscous Navier-Stokes equations of fluid mechanics. The scheme is optimal in the sense that the split operators achieve their maximum allowable time step, i.e., the corresponding Courant number. The algorithm allows a conservation-form formulation. Stability is proven analytically and verified numerically. In proving stability it was shown that all nine matrix coefficients of the Navier-Stokes equations are simultaneously symmetrizable by a similarity transformation. Two such transformations and their resulting symmetric matrix coefficients are presented explicitly.

Abarbanel, S.↗

NASA NTRS2004

Frequency Domain Calculations Of Acoustic Propagation

Two complex geometry problems are solved using the linearized Euler equations. The impedance mismatch method1 is used to impose the solid surfaces without the need to use a body-fitted grid. The problem is solved in the frequency domain to avoid long run times. Although the harmonic assumption eliminates all time dependence, a pseudo-time term is added to allow conventional iterative methods to be employed. A Jameson type, Runge-Kutta scheme is used to advance the solution in pseudo time. The spatial operator is based on a seven-point, sixth-order finite difference. Constant coefficient, sixth-derivative artificial dissipation is used throughout the domain. A buffer zone technique employing a complex frequency to damp all waves near the boundaries is used to minimize reflections. The results show that the method is capable of capturing the salient features of the scattering, but an excessive number of grid points are required to resolve the phenomena in the vicinity of the solid bodies because the wavelength of the acoustics is relatively short compared with the size of the bodies. Smoothly transitioning into the immersed boundary condition alleviates the difficulties, but a fine mesh is still required.

Lockard, David P.↗

NASA NTRS1987

Goertler instability on an airfoil

An effective computational scheme was developed to study the growth/damping of Goertler vortices along walls of variable curvature. Computational experiments indicate that when the amplification rates for the u-, v-, and w-perturbations are the same, the finite difference approach to solve the initial value problem and the normal mode approach give identical results for the Blasius boundary layer on constant curvature concave walls. The growth of Goertler vortices was rapid in the concave regions and was followed by sharp damping in the convex region. However, multiple sets of counter-rotating vortices were formed and remained far downstream in the convex region. The current computational scheme can be easily extended to more realistic problems including variable pressure gradients and suction effects.

Kalburgi, Vijay↗

NASA NTRS2003

Nonlinear Aeroacoustics Computations by the Space-Time CE/SE Method

The Space-Time Conservation Element and Solution Element Method, or CE/SE Method for short, is a recently developed numerical method for conservation laws. Despite its second order accuracy in space and time, it possesses low dispersion errors and low dissipation. The method is robust enough to cover a wide range of compressible flows: from weak linear acoustic waves to strong discontinuous waves (shocks). An outstanding feature of the CE/SE scheme is its truly multi-dimensional, simple but effective non-reflecting boundary condition (NRBC), which is particularly valuable for computational aeroacoustics (CAA). In nature, the method may be categorized as a finite volume method, where the conservation element (CE) is equivalent to a finite control volume (or cell) and the solution element (SE) can be understood as the cell interface. However, due to its careful treatment of the surface fluxes and geometry, it is different from the existing schemes. Currently, the CE/SE scheme has been developed to a matured stage that a 3-D unstructured CE/SE Navier-Stokes solver is already available. However, in the present review paper, as a general introduction to the CE/SE method, only the 2-D unstructured Euler CE/SE solver is chosen and sketched in section 2. Then applications of the 2-D and 3-D CE/SE schemes to linear, and in particular, nonlinear aeroacoustics are depicted in sections 3, 4, and 5 to demonstrate its robustness and capability.

Loh, Ching Y.↗