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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 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

On the Connection Between One-and Two-Equation Models of Turbulence

A formalism will be presented that allows the transformation of two-equation eddy viscosity turbulence models into one-equation models. The transformation is based on an assumption that is widely accepted over a large range of boundary layer flows and that has been shown to actually improve predictions when incorporated into two-equation models of turbulence. Based on that assumption, a new one-equation turbulence model will be derived. The new model will be tested in great detail against a previously introduced one-equation model and against its parent two-equation model.

Menter, F. R.↗
NASA NTRS1994

Fluid/Structure Interaction Studies of Aircraft Using High Fidelity Equations on Parallel Computers

Abstract Aeroelasticity which involves strong coupling of fluids, structures and controls is an important element in designing an aircraft. Computational aeroelasticity using low fidelity methods such as the linear aerodynamic flow equations coupled with the modal structural equations are well advanced. Though these low fidelity approaches are computationally less intensive, they are not adequate for the analysis of modern aircraft such as High Speed Civil Transport (HSCT) and Advanced Subsonic Transport (AST) which can experience complex flow/structure interactions. HSCT can experience vortex induced aeroelastic oscillations whereas AST can experience transonic buffet associated structural oscillations. Both aircraft may experience a dip in the flutter speed at the transonic regime. For accurate aeroelastic computations at these complex fluid/structure interaction situations, high fidelity equations such as the Navier-Stokes for fluids and the finite-elements for structures are needed. Computations using these high fidelity equations require large computational resources both in memory and speed. Current conventional super computers have reached their limitations both in memory and speed. As a result, parallel computers have evolved to overcome the limitations of conventional computers. This paper will address the transition that is taking place in computational aeroelasticity from conventional computers to parallel computers. The paper will address special techniques needed to take advantage of the architecture of new parallel computers. Results will be illustrated from computations made on iPSC/860 and IBM SP2 computer by using ENSAERO code that directly couples the Euler/Navier-Stokes flow equations with high resolution finite-element structural equations.

Guruswamy, Guru↗
NASA NTRS1995

On the Dynamics of Some Discretizations of Convection-Diffusion Equations

Numerical discretizations of differential equations which model physical processes can possess dynamics quite different from that of the equations themselves. Recently the emphasis has been on the the dynamics of numerical discretizations for Ordinary Differential Equations (ODEs). For Partial Differential Equations (PDEs) using a method of lines approach the situation is more complex. First, the spatial discretisation may introduce dynamics not present in the original equations; second, the solution of the resulting system of ODEs is open to the modified dynamics of the ODE solver used. These two effects may interact in a complex manner. In this talk we present some results of our recent work on the dynamics of discretizations of convection-diffusion equations, including those produced using Total Variation Diminishing (TVD) schemes and adaptive grid techniques. A more general overview of the area may be found on our accompanying poster presentation.

Sweby, Peter K.↗
NASA NTRS2001

A Highly Accurate Technique for the Treatment of Flow Equations at the Polar Axis in Cylindrical Coordinates using Series Expansions

Numerical methods for solving the flow equations in cylindrical or spherical coordinates should be able to capture the behavior of the exact solution near the regions where the particular form of the governing equations is singular. In this work we focus on the treatment of these numerical singularities for finite-differences methods by reinterpreting the regularity conditions developed in the context of pseudo-spectral methods. A generally applicable numerical method for treating the singularities present at the polar axis, when nonaxisymmetric flows are solved in cylindrical, coordinates using highly accurate finite differences schemes (e.g., Pade schemes) on non-staggered grids, is presented. Governing equations for the flow at the polar axis are derived using series expansions near r=0. The only information needed to calculate the coefficients in these equations are the values of the flow variables and their radial derivatives at the previous iteration (or time) level. These derivatives, which are multi-valued at the polar axis, are calculated without dropping the accuracy of the numerical method using a mapping of the flow domain from (0,R)*(0,2pi) to (-R,R)*(0,pi), where R is the radius of the computational domain. This allows the radial derivatives to be evaluated using high-order differencing schemes (e.g., compact schemes) at points located on the polar axis. The proposed technique is illustrated by results from simulations of laminar-forced jets and turbulent compressible jets using large eddy simulation (LES) methods. In term of the general robustness of the numerical method and smoothness of the solution close to the polar axis, the present results compare very favorably to similar calculations in which the equations are solved in Cartesian coordinates at the polar axis, or in which the singularity is removed by employing a staggered mesh in the radial direction without a mesh point at r=0, following the method proposed recently by Mohseni and Colonius (1). Extension of the method described here for incompressible flows or for any other set of equations that are solved on a non-staggered mesh in cylindrical or spherical coordinates with finite-differences schemes of various level of accuracy is immediate.

Constantinescu, George S.↗
NASA NTRS2002

Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

In this paper we review the existing and develop new continuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L(exp 2) stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.

Yan, Jue↗
NASA NTRS2010

Forces Associated with Nonlinear Nonholonomic Constraint Equations

A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by nonholonomic equations that are inherently nonlinear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known nonholonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as nonholonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor to form the former, larger set first and subsequently perform matrix multiplications.

Roithmayr, Carlos M.↗
NASA NTRS2010

An Attempt to Derive the epsilon Equation from a Two-Point Closure

The goal of this paper is to derive the equation for the turbulence dissipation rate epsilon for a shear-driven flow. In 1961, Davydov used a one-point closure model to derive the epsilon equation from first principles but the final result contained undetermined terms and thus lacked predictive power. Both in 1987 and in 2001, attempts were made to derive the epsilon equation from first principles using a two-point closure, but their methods relied on a phenomenological assumption. The standard practice has thus been to employ a heuristic form of the equation that contains three empirical ingredients: two constants, c(sub 1 epsilon), and c(sub 2 epsilon), and a diffusion term D(sub epsilon) In this work, a two-point closure is employed, yielding the following results: 1) the empirical constants get replaced by c(sub 1), c(sub 2), which are now functions of Kappa and epsilon; 2) c(sub 1) and c(sub 2) are not independent because a general relation between the two that are valid for any Kappa and epsilon are derived; 3) c(sub 1), c(sub 2) become constant with values close to the empirical values c(sub 1 epsilon), c(sub epsilon 2), (i.e., homogenous flows); and 4) the empirical form of the diffusion term D(sub epsilon) is no longer needed because it gets substituted by the Kappa-epsilon dependence of c(sub 1), c(sub 2), which plays the role of the diffusion, together with the diffusion of the turbulent kinetic energy D(sub Kappa), which now enters the new equation (i.e., inhomogeneous flows). Thus, the three empirical ingredients c(sub 1 epsilon), c(sub epsilon 2), D (sub epsilon)are replaced by a single function c(sub 1)(Kappa, epsilon ) or c(sub 2)(Kappa, epsilon ), plus a D(sub Kappa)term. Three tests of the new equation for epsilon are presented: one concerning channel flow and two concerning the shear-driven planetary boundary layer (PBL).

Canuto, V. M.↗
NASA NTRS2018

Output Measurement Equations for Flexible Aircraft Flight Dynamics

A summary of output measurement equations for onboard sensors used in flight testing of flexible aircraft is presented. These equations include the effects of structural flexibility and are considerably more complex than the standard equations for rigid-body aircraft. The output equations discussed include accelerations from linear accelerometers, strains, angular rates, angular accelerations, Euler angles, true airspeed, and air flow angles. The output equations are derived in full form and then simplified in some cases. Linearized output equations, suitable for state-space or transfer function models, are also developed. Example flight test data from the X-56A subscale aeroelastic demonstrator is discussed, for reference.

Grauer, Jared A.↗
NASA NTRS2022

Transition Analysis for the CRM-NLF Wind Tunnel Configuration using Transport Equation Models and Linear Stability Correlations

Transition models based on auxiliary transport equations augmenting the Reynolds-averaged Navier-Stokes (RANS) framework rely upon transition correlations that were derived from a limited number of low-speed experiments. Furthermore, these models often account for only a subset of the relevant transition mechanisms and/or cannot accurately predict the sensitivity of those mechanisms to the changes in significant flow parameters. A preceding investigation had targeted the assessment of the transport-equation-based transition models in NASA's OVERFLOW 2.3b solver, namely, the amplification factor transport (AFT-2017b) equation model coupled with the Spalart-Allmaras RANS model and the Langtry-Menter transition models (LM2009 without crossflow effects and LM2015 including the modeling of crossflow transition) implemented with Menter’s shear-stress transport equation (SST2003) RANS model. Comparisons with recent measurements at transonic freestream conditions on the Common Research Model with Natural Laminar Flow (CRM-NLF) reinforced our earlier finding that all three of the above models significantly underpredict the reported extent of the laminar flow region over the entire span of the wing, regardless of the dominant instability mechanism(s) underlying the onset of the transition process. The underprediction of the laminar flow extent was attributed to the failure of the above models in accounting for the stabilizing effect of compressibility on the amplification of Tollmien-Schlichting instabilities. Based on previous linear stability studies related to compressibility effects, the present work proposes modifications to the two classes of transition models that reduce to the original form of each model at low subsonic speeds and do not require any nonlocal flow information or additional transport equation(s). The modifications are shown to significantly improve the predicted laminar extent of the flow and compare well against the data from the CRM-NLF experiment. Additionally, a previous assessment of transition prediction based on the dual, nonparallel N -factor method in conjunction with linear parabolized stability equations (PSE) is extended to additional angles of attack to provide the first comprehensive assessment of transition models based on nonparallel disturbance amplification over the CRM-NLF. In general, the transition criterion based on the dual, nonparallel N-factor method with N TS = N CF = 6 is reasonably successful at correlating with the measured transition fronts at R eMAC = 15 million for all angles of attack investigated herein and provides additional validation of the improved predictions from the compressibility-corrected transition models.

CFD modeling↗
NASA NTRSDate not supplied

Evaluation of the Accuracy of the Load Prediction Equations of Low-Load Balance Calibration Data

Accuracy and reliability of the load prediction equations of a low-load calibration data set of a force balance were investigated. First, independent load prediction equations were generated from the data of a full-load and low-load machine calibration. Then, the low- load data set was processed as a set of precision check loads for the load prediction equations that were obtained from the full-load and low-load calibrations. Finally, the load prediction equations of the two calibrations were applied to four manual check load data sets that were recorded between 2009 and 2022. The residuals of the predicted calibration and check loads were compared in all cases. No systematic improvement of the load prediction accuracy was observed when the load prediction equations of the low-load calibration were applied to the check load data sets. Therefore, it is recommended to apply a full-load calibration to a six-component balance even if the balance is not used across its entire design envelope. This approach has the advantage that the resulting load prediction equations are less likely to be applied outside of the calibration load ranges. In addition, it is expected that the numerical estimates of the gage sensitivities are more reliable assuming that the uncertainties of the applied loads and measured outputs of the calibration data are more or less constant across the load range of the balance.

strain-gage balance↗
NASA NTRSDate not supplied

Evaluation of the Accuracy of the Load Prediction Equations of Low-Load Balance Calibration Data

Accuracy and reliability of the load prediction equations of a low-load calibration data set of a force balance were investigated. First, independent load prediction equations were generated from the data of a full-load and low-load machine calibration. Then, the low-load data set was processed as a set of precision check loads for the load prediction equations that were obtained from the full-load and low-load calibrations. Finally, the load prediction equations of the two calibrations were applied to four manual check load data sets that were recorded between 2009 and 2022. The residuals of the predicted calibration and check loads were compared in all cases. No systematic improvement of the load prediction accuracy was observed when the load prediction equations of the low-load calibration were applied to the check load data sets. Therefore, it is recommended to apply a full-load calibration to a six-component balance even if the balance is not used across its entire design envelope. This approach has the advantage that the resulting load prediction equations are less likely to be applied outside of the calibration load ranges. In addition, it is expected that the numerical estimates of the bridge sensitivities are more reliable assuming that the uncertainties of the applied loads and measured outputs of the calibration data are more or less constant across the load range of the balance.

strain-gage balance↗
NASA NTRSDate not supplied

Construction of an Exact Pressure-Equilibrium Scheme for the Five-Equation Two-Phase Flow Model With Thermal Relaxation

Numerical simulation of compressible multiphase flows based on the four-equation (homogeneous relaxation) model is known to suffer from two fundamental difficulties with (a) wave propagation and (b) pressure equilibrium preservation. First, the mixture sound speed exhibits non-monotonic dependency with respect to the volume fraction, which leads to robustness issues in the resolution of shocks and acoustic wave propagation across two-phase regions. This difficulty can be mitigated by solving Allaire’s five-equation model augmented with infinitely fast phasic temperature equilibrium, from which solutions of the four-equation model can be recovered. However, when temperature is non-uniform, this augmented five-equation formulation still fails to preserve pressure equilibrium across material interfaces. In this work, we propose a fully conservative numerical scheme that exactly preserves pressure equilibrium at the discrete level for the augmented five-equation model, for arbitrary initial distributions of temperature and volume fraction. Combined with the monotonic sound speed property of the five-equation formulation, the proposed pressure-equilibrium preserving scheme significantly improves robustness in the presence of strong multiphase interactions, including shock–interface interactions and advection of material interfaces.

ESG↗
DOE OSTI2025

Constrained or unconstrained? Neural-network-based equation discovery from data

Throughout many fields, practitioners often rely on differential equations to model systems. Yet, for many applications, the theoretical derivation of such equations and/or the accurate resolution of their solutions may be intractable. Instead, recently developed methods, including those based on parameter estimation, operator subset selection, and neural networks, allow for the data-driven discovery of both ordinary and partial differential equations (PDEs), on a spectrum of interpretability. The success of these strategies is often contingent upon the correct identification of representative equations from noisy observations of state variables and, as importantly and intertwined with that, the mathematical strategies utilized to enforce those equations. Specifically, the latter has been commonly addressed via unconstrained optimization strategies. Representing the PDE as a neural network, we propose to discover the PDE (or the associated operator) by solving a constrained optimization problem and using an intermediate state representation similar to a physics-informed neural network (PINN). The objective function of this constrained optimization problem promotes matching the data, while the constraints require that the discovered PDE is satisfied at a number of spatial collocation points. We present a penalty method and a widely used trust-region barrier method to solve this constrained optimization problem, and we compare these methods on numerical examples. Our results on several example problems demonstrate that the latter constrained method outperforms the penalty method, particularly for higher noise levels or fewer collocation points. This work motivates further exploration into using sophisticated constrained optimization methods in scientific machine learning, as opposed to their commonly used, penalty-method or unconstrained counterparts. For both of these methods, we solve these discovered neural network PDEs with classical methods, such as finite difference methods, as opposed to PINNs-type methods relying on automatic differentiation. Here, we briefly highlight how simultaneously fitting the data while discovering the PDE improves the robustness to noise and other small, yet crucial, implementation details.

Data-driven discovery↗
DOE OSTI2025

10-th order of accuracy for numerical solution of 3-D elasticity equations for heterogeneous materials on unfitted Cartesian meshes

We have developed the Optimal Local Truncation Error Method (OLTEM) with 10-th order of accuracy on unfitted Cartesian meshes for a system of 3-D elasticity equations with smooth irregular interfaces. 5 x 5 x 5 = 125-point stencils (similar to those for quadratic finite elements) for elastic heterogeneous materials are used for OLTEM. There are no unknowns at the interface points between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal 10-th order of accuracy for OLTEM on unfitted Cartesian meshes, i.e., the increase by 7 orders in accuracy compared to quadratic finite elements on conformal meshes. A new post-processing procedure provides the 9-th order of accuracy for stresses in the 3-D case. Similar to basic computations it uses OLTEM with the 125-point stencils, the interface conditions and the elasticity equations. It was shown that the use of the elasticity equations for post-processing improves the accuracy of 0.1% stresses by 6 orders compared to post-processing without the use of PDEs. At an accuracy of for stresses, OLTEM with the new post-processing procedure reduces the number of degrees of freedom by 360 - 8000 times compared to quadratic finite elements with similar stencils. OLTEM with the 125-point stencils yields even more accurate results than high-order finite elements with much wider stencils. OLTEM provides accurate numerical results for compressible and nearly incompressible materials.

elasticity equations↗
DOE OSTI2024

A second-order-in-time, explicit approach addressing the redundancy in the low-Mach, variable-density Navier-Stokes equations

A novel algorithm for explicit temporal discretization of the variable-density, low-Mach Navier-Stokes equations is presented here in this study. Recognizing there is a redundancy between the mass conservation equation, the equation of state, and the transport equation(s) for the scalar(s) which characterize the thermochemical state, and that it destabilizes explicit methods, we demonstrate how to analytically eliminate the redundancy and propose an iterative scheme to solve the resulting transformed scalar equations. The method obtains second-order accuracy in time regardless of the number of iterations, so one can terminate this subproblem once stability is achieved. Hence, flows with larger density ratios can be simulated while still retaining the efficiency, low cost, and parallelizability of an explicit scheme. The temporal discretization algorithm is used within a pseudospectral direct numerical simulation which extends the method of Kim, Moin, and Moser for incompressible flow to the variable-density, low-Mach setting, where we demonstrate stability for density ratios up to ~25.7.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

A novel conditional formulation of the Vlasov–Ampère equations: a conservative, positivity, asymptotic and Gauss law preserving scheme

We propose a novel reformulation of the Vlasov–Ampère equations for plasmas that reveals discrete symmetries that enables simultaneous conservation of mass, momentum and energy; preservation of Gauss’s law; positivity of the distribution function; and consistency with quasi-neutral asymptotics. The approach employs variable and coordinate transformations to yield a coupled system comprising a modified Vlasov equation and associated moment–field equations. The modified Vlasov equation advances a conditional distribution function that excludes mass, momentum and energy densities, which are instead evolved through moment equations enforcing the relevant symmetries, conservation laws and involution constraints. This reformulation aligns naturally with a recent slow-manifold reduction technique, which separates fast electron time scales and simplifies the treatment of the quasi-neutral limit within the reduced moment–field subsystem. Using this framework, we develop a numerical method for the reduced 1D1V subsystem that, for the first time in the literature, satisfies all key physical constraints while maintaining a quasi-neutral asymptotic behaviour. The advantages of the method are demonstrated on canonical electrostatic test problems, including the multiscale ion acoustic shock wave.

1D1V↗
DOE OSTI2025

Optical neural engine for solving scientific partial differential equations

Abstract Solving partial differential equations (PDEs) is the cornerstone of scientific research and development. Data-driven machine learning (ML) approaches are emerging to accelerate time-consuming and computation-intensive numerical simulations of PDEs. Although optical systems offer high-throughput and energy-efficient ML hardware, their demonstration for solving PDEs is limited. Here, we present an optical neural engine (ONE) architecture combining diffractive optical neural networks for Fourier space processing and optical crossbar structures for real space processing to solve time-dependent and time-independent PDEs in diverse disciplines, including Darcy flow equation, the magnetostatic Poisson’s equation in demagnetization, the Navier-Stokes equation in incompressible fluid, Maxwell’s equations in nanophotonic metasurfaces, and coupled PDEs in a multiphysics system. We numerically and experimentally demonstrate the capability of the ONE architecture, which not only leverages the advantages of high-performance dual-space processing for outperforming traditional PDE solvers and being comparable with state-of-the-art ML models but also can be implemented using optical computing hardware with unique features of low-energy and highly parallel constant-time processing irrespective of model scales and real-time reconfigurability for tackling multiple tasks with the same architecture. The demonstrated architecture offers a versatile and powerful platform for large-scale scientific and engineering computations.

Tang, Yingheng (ORCID:0009000153622546)↗
DOE OSTI2025

Tree tensor network hierarchical equations of motion based on time-dependent variational principle for efficient open quantum dynamics in structured thermal environments

In this work, we introduce an efficient method, TTN-HEOM, for exactly calculating the open quantum dynamics for driven quantum systems interacting with highly structured bosonic baths by combining the tree tensor network (TTN) decomposition scheme with the bexcitonic generalization of the numerically exact hierarchical equations of motion (HEOM). The method yields a series of quantum master equations for all core tensors in the TTN that efficiently and accurately capture the open quantum dynamics for non-Markovian environments to all orders in the system–bath interaction. These master equations are constructed based on the time-dependent Dirac–Frenkel variational principle, which isolates the optimal dynamics for the core tensors given the TTN ansatz. The dynamics converges to the HEOM when increasing the rank of the core tensors, a limit in which the TTN ansatz becomes exact. We introduce TENSO, tensor equations for non-Markovian structured open systems, as a general-purpose Python code to propagate the TTN-HEOM dynamics. We implement three general propagators for the coupled master equations: two fixed-rank methods that require a constant memory footprint during the dynamics and one adaptive-rank method with a variable memory footprint controlled by the target level of computational error. We exemplify the utility of these methods by simulating a two-level system coupled to a structured bath containing one Drude–Lorentz component and eight Brownian oscillators, which is beyond what can presently be computed using the standard HEOM. Our results show that the TTN-HEOM is capable of simulating both dephasing and relaxation dynamics of driven quantum systems interacting with structured baths, even those of chemical complexity, with an affordable computational cost.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗