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DOE OSTI2023

Exact solutions for the electromagnetic fields of a flying focus

The intensity peak of a “flying” focus travels at a programmable velocity over many Rayleigh ranges while maintaining a near-constant profile. Assessing the extent to which these features can enhance laser-based applications requires an accurate description of the electromagnetic fields. Here we present exact analytical solutions to Maxwell’s equations for the electromagnetic fields of a constant-velocity flying focus, generalized for arbitrary polarization and orbital angular momentum. Furthermore, the approach combines the complex source-point method, which transforms multipole solutions into beam-like solutions, with the Lorentz invariance of Maxwell’s equations. Propagating the fields backward in space reveals the space-time profile that an optical assembly must produce to realize these fields in the laboratory. Comparisons with simpler paraxial solutions provide conditions for their reliable use when modeling a flying focus.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1937

Stress Analysis of Beams with Shear Deformation of the Flanges

This report discusses the fundamental action of shear deformation of the flanges on the basis of simplifying assumptions. The theory is developed to the point of giving analytical solutions for simple cases of beams and of skin-stringer panels under axial load. Strain-gage tests on a tension panel and on a beam corresponding to these simple cases are described and the results are compared with analytical results. For wing beams, an approximate method of applying the theory is given. As an alternative, the construction of a mechanical analyzer is advocated.

Kuhn, Paul↗

NASA NTRS2002

Steady State Transportation Cooling in Porous Media Under Local, Non-Thermal Equilibrium Fluid Flow

An analytical solution to the steady-state fluid temperature for 1-D (one dimensional) transpiration cooling has been derived. Transpiration cooling has potential use in the aerospace industry for protection against high heating environments for re-entry vehicles. Literature for analytical treatments of transpiration cooling has been largely confined to the assumption of thermal equilibrium between the porous matrix and fluid. In the present analysis, the fundamental fluid and matrix equations are coupled through a volumetric heat transfer coefficient and investigated in non-thermal equilibrium. The effects of varying the thermal conductivity of the solid matrix and the heat transfer coefficient are investigated. The results are also compared to existing experimental data.

Rodriquez, Alvaro Che↗

DOE OSTI2022

1D Heat Loss Models Validation Experiment

Contains data from the model validation in the 1D Heat Loss Models to Predict the Aquifer Temperature Profile during Hot/Cold Water Injection Project. The data include two COMSOL models (2D axisymmetric benchmark model and 2D Vinsome model), one python code (1D Vinsome based FEM numerical simulation), one matlab main code (1D Newton analytical solution and all results comparison visualization), and output files generated from the above models.

1D↗

NASA NTRS1992

Closed form solutions of constrained trajectories - Application in optimal ascent of aerospace plane

The present consideration of the flight trajectory of hypersonic aerospace vehicles subject to a class of path constraints notes the constrained dynamics to constitute a natural two-timescale system, so that problems of trajectory optimization and guidance can be dramatically simplified by means of the asymptotic analytical solutions thus obtained. An illustrative application in ascent trajectory optimization for an aerospace vehicle is presented.

Lu, Ping↗

DOE OSTI2021

Upscaling Reactive Transport and Clogging in Shale Microcracks by Deep Learning

Fracture networks in shales exhibit multiscale features. A rock system may contain a few main fractures and thousands of microcracks, whose length and aperture are orders of magnitude smaller than the former. It is computationally prohibitive to resolve all the fractures explicitly for such multiscale fracture networks. One traditional approach is to model the small-scale features (e.g., microcracks in shales) as an effective medium. Although this fracture-matrix conceptualization significantly reduces the problem complexity, there are classes of physical processes that cannot be accurately upscaled by effective medium approximations, e.g., microcrack clogging during mineral reactions. Here, we employ deep learning in place of effective medium theory to upscale physical processes in small-scale features. Specifically, we consider reactive transport in a fracture-microcrack network where microcracks can be clogged by precipitation. A deep learning multiscale algorithm is developed, in which the microcracks are upscaled as a wall boundary condition of the main fractures. The wall boundary condition is constructed by recurrent neural networks, which take concentration histories as input and predict the solute transport from main fractures to microcracks. The deep learning multiscale algorithm is firstly employed in specific scenarios, then a general model is developed which can work under various conditions. The new approach is validated against fully resolved simulations and an analytical solution, providing a reliable and efficient solution for problems that cannot be upscaled by effective medium models.

58 GEOSCIENCES↗

DOE OSTI2023

Planar turbulent wakes under pressure gradient: Integral and self-similarity analyses

By using a combination of integral and self-similarity analyses, the generalized analytical solutions for the mean transverse velocity and Reynolds shear stress are rigorously derived for the first time for the far field of planar turbulent wakes under arbitrary pressure gradients. Specifically, by assuming self-similarity for the mean axial velocity, the analytical formulation for the mean transverse velocity is obtained from the integral of the mean continuity equation, and the analytical formulation for the Reynolds shear stress is obtained from the integral of the momentum equation. The generalized analytical formulations for the mean transverse velocity and Reynolds shear stress consist of multiple components, each with its unique scale and physical mechanism. In the zero pressure gradient limit, the generalized formulations recover the single-scale equations reported by Wei, Liu, and Livescu. Furthermore, simpler approximate formulations for the mean transverse velocity and Reynolds shear stress are also obtained, and show excellent agreement with the experimental measurements. Here, the findings provide new insights into the properties of planar turbulent wakes under pressure gradients, filling some long-standing gaps in the existing literature.

42 ENGINEERING↗

DOE OSTI2020

Local Truncation Error-Informed Code Verification

The method of manufactured solutions (MMS) has become increasingly popular in conducting code verification studies on predictive codes, such as nuclear power system codes and computational fluid dynamic codes. The reason for the popularity of this approach is that it can be used when an analytical solution is not available. Using MMS, code developers are able to verify that their code is free of coding errors that impact the observed order of accuracy. While MMS is still an excellent tool for code verification, it does not identify coding errors that are of the same order as the numerical method. This paper presents a method that combines MMS with modified equation analysis (MEA), which calculates the local truncation error (LTE) to identify coding error up to and including the order of the numerical method. This method is referred to as modified equation analysis methd of manufactured solutions (MEAMMS). MEAMMS is then applied to a custom-built code, which solves the shallow water equations, to test the performance of the code verification method. MEAMMS is able to detect all coding errors that impact the implementation of the numerical scheme. To show how MEAMMS is different than MMS, they are both applied to the same first-order numerical method test problem with a first-order coding error. When there are first-order coding errors, only MEAMMS is able to identify them. Finally, this shows that MEAMMS is able to identify a larger set of coding errors while still being able to identify the coding errors MMS is able to identify.

42 ENGINEERING↗

NASA NTRS1977

A numerical study of the unsteady leading edge separation bubble on an oscillating airfoil

A preliminary study of the unsteady viscous flow in the region of an airfoil leading edge was performed, in which the interaction between the viscous and inviscid flow fields is neglected. The solution method uses the finite difference form of the governing equations throughout the separated flow field and incorporates a transition model based on the integral turbulence kinetic energy equation. The validity of the numerical procedure is verified by making comparisons with analytical solutions to several test problems, including unsteady flow over a plate oscillating in its own plane. The method was then applied to the problem of unsteady viscous flow over a NACA 0012 airfoil oscillating sinusoidally in pitch. The flow field characteristics were in qualitative agreement with experimental results. The bubble moved forward on the airfoil and decreased in size as incidence was increased. Viscous flow in the leading edge region was found to be quasi-steady, while bubble height varied inversely with Reynolds number.

Kreskovsky, J. P.↗

NASA NTRS1980

Implicit computations of unsteady transonic flow governed by the full-potential equation in conservation form

An alternating-direction implicit algorithm is presented for solving the conservative, full-potential equation for unsteady, transonic flow. A new development is the time-linearization of the density function. This linearization reduces the solution process from one of solving a system of two equations at each mesh point to one of solving a single equation. Two sample cases are computed. First, a one-dimensional traveling shock wave is computed and compared with the analytic solution. Second, a two-dimensional case is computed of a flow field that results from a thickening and subsequently thinning airfoil. The resulting flow field, which includes a traveling shock wave, is compared to the flow field obtained from the low-frequency, small-disturbance, transonic equation.

Goorjian, P. M.↗

DOE OSTI2023

Expanded verification and validation studies of hypersonic aerodynamics with multiple physics-fidelity models

Hypersonic aerothermodynamics is an important domain of modern multiphysics simulation. The Multi-Fidelity Toolkit is a simulation tool being developed at Sandia National Laboratories to predict aerodynamic properties for compressible flows from a range of physics fidelities and computational speeds. These models include the Reynolds-averaged Navier–Stokes (RANS) equations, the Euler equations with momentum-energy integral technique (MEIT), and modified Newtonian aerodynamics with flat-plate boundary layer (MNA+FPBL) equations, and they can be invoked independently or coupled with hierarchical Kriging to interpolate between high-fidelity simulations using lower-fidelity data. However, as with any new simulation capability, verification and validation are necessary to gather credibility evidence. This work describes formal code- and solution-verification activities, as well as model validation with uncertainty considerations. Code verification activities on the MNA+FPBL model build on previous work by focusing on the viscous portion of the model. Viscous quantities of interest are compared against those from an analytical solution for flat-plate, inclined-plate, and cone geometries. The code verification methodology for the MEIT model is also presented. Test setup and results of code verification tests on the laminar and turbulent models within MEIT are shown. Solution-verification activities include grid-refinement studies on simulations that model the HIFiRE-1 wind tunnel experiments. These experiments are used for validation of all model fidelities. A thorough validation comparison with prediction error and uncertainty is also presented. Three additional HIFiRE-1 experimental runs are simulated in this study, and the solution verification and validation work examines the effects of the associated parameter changes on model performance. Finally, a study is presented that compares the computational costs and fidelities from each of the different models.

42 ENGINEERING↗

NASA NTRS1991

On the numerical solution of the dynamically loaded hydrodynamic lubrication of the point contact

A numerical transient solution of the hydrodynamically lubricated point contact problem is obtained using the ball-on-plane model. Results, which include the variation of the minimum film thickness and phase-lag with time as functions of excitation frequency, are compared with the analytic solution of the transient step bearing problem with the same dynamic loading function.

Lim, Sang G.↗

NASA NTRS1993

Trajectory optimization for the National aerospace plane

While continuing the application of the inverse dynamics approach in obtaining the optimal numerical solutions, the research during the past six months has been focused on the formulation and derivation of closed-form solutions for constrained hypersonic flight trajectories. Since it was found in the research of the first year that a dominant portion of the optimal ascent trajectory of the aerospace plane is constrained by dynamic pressure and heating constraints, the application of the analytical solutions significantly enhances the efficiency in trajectory optimization, provides a better insight to understanding of the trajectory and conceivably has great potential in guidance of the vehicle. Work of this period has been reported in four technical papers. Two of the papers were presented in the AIAA Guidance, Navigation, and Control Conference (Hilton Head, SC, August, 1992) and Fourth International Aerospace Planes Conference (Orlando, FL, December, 1992). The other two papers have been accepted for publication by Journal of Guidance, Control, and Dynamics, and will appear in 1993. This report briefly summarizes the work done in the past six months and work currently underway.

Lu, Ping↗

NASA NTRS1999

Steady Secondary Flows Generated by Periodic Compression and Expansion of an Ideal Gas in a Pulse Tube

This study establishes a consistent set of differential equations for use in describing the steady secondary flows generated by periodic compression and expansion of an ideal gas in pulse tubes. Also considered is heat transfer between the gas and the tube wall of finite thickness. A small-amplitude series expansion solution in the inverse Strouhal number is proposed for the two-dimensional axisymmetric mass, momentum and energy equations. The anelastic approach applies when shock and acoustic energies are small compared with the energy needed to compress and expand the gas. An analytic solution to the ordered series is obtained in the strong temperature limit where the zeroth-order temperature is constant. The solution shows steady velocities increase linearly for small Valensi number and can be of order I for large Valensi number. A conversion of steady work flow to heat flow occurs whenever temperature, velocity or phase angle gradients are present. Steady enthalpy flow is reduced by heat transfer and is scaled by the Prandtl times Valensi numbers. Particle velocities from a smoke-wire experiment were compared with predictions for the basic and orifice pulse tube configurations. The theory accurately predicted the observed steady streaming.

Lee, Jeffrey M.↗

DOE OSTI2021

Computational Evaluation of Thermal Response of Open-Cell Foam With Circular Pore (Computional Evaluation of Thermal Response of Open-Cell Foam with Circular Pore)

The evaluation of effective material properties in heterogeneous materials (e.g., composites or multicomponent structures) critically relevant to a wide spectrum of applications, including nuclear power, electronic packaging, flame retardants, hypersonics, and gas turbine power. The work described in this paper is centered around the numerical assessment of the thermal behavior of porous materials obtained from finite element thermal modeling and simulation. Here, two-dimensional, steady state analyses were performed on unit cells with centered, circular pores using a second order accurate Galerkin finite element method (FEM). The effective thermal conductivities of the porous systems were examined, encompassing a range of porosities from 4.9% to 60.1%. The geometries of the models were generated based on ordered circular pores for each modeled porosity level. The system response quantity (SRQ) under investigation was the dimensionless effective thermal conductivity across the unit cell. The dimensionless effective thermal conductivity was compared across all simulated cases, producing a trend between porosity and effective thermal conductivity. In the presented investigation, the method of manufactured solutions (MMS) was used to perform code verification, and the grid convergence index (GCI) was employed to estimate discretization uncertainty as solution verification. Code verification concluded an approximately second order accurate Galerkin FEM solver. It was found that the introduction of porosity to the unit cell material structure reduces effective thermal conductivity, as anticipated. Numerical results obtained in this study are compared to an analytical solution and to a sample of empirical data.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Aquifer Injection Modeling (AIM) Toolbox User Guide

The Aquifer Injection Modeling Toolbox (“AIM Toolbox”) software was developed by the Pacific Northwest National Laboratory (PNNL) for the U.S. Environmental Protection Agency’s (EPA) to provide a collection of analytical solutions suitable for evaluating the potential extent of the area impacted by subsurface injection operations. Subsurface injection operations are regulated under the EPA’s Underground Injection Control (UIC) program and are typically related to oil/gas development, waste disposal, or subsurface mining or storage. The analytical algorithms provided in the AIM Toolbox each have different approaches/assumptions/focus with respect to the nature and processes in subsurface and the nature of the injection operations. Collectively the set of analysis algorithms provides a broader evaluation of the area that can potentially be impacted by an injection operation. By providing estimates for the injectate plume extent, the software supports technical aspects of planning, evaluation, and overseeing injection activities. That is, the results help assess when further regulatory controls (e.g., monitoring, reporting) may be required and, in the case of disposal into an underground source of drinking water, the extent of the impacted area would require exemption from protection under the Safe Drinking Water Act. The AIM Toolbox software is available as a single-page web application, providing an interface to provide the necessary inputs, and both chart and map panes for visualizing the results. This User Guide describes the AIM Toolbox software, information required, user interactions, and the underlying basis of the calculations.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2023

Aquifer Injection Modeling (AIM) Toolbox User Guide

58 GEOSCIENCES↗

NASA NTRS1971

Heat addition to a subsonic boundary layer: A preliminary analytical study

A preliminary analytical study of the effects of heat addition to the subsonic boundary layer flow over a typical airfoil shape is presented. This phenomenon becomes of interest in the space shuttle mission since heat absorbed by the wing structure during re-entry will be rejected to the boundary layer during the subsequent low speed maneuvering and landing phase. A survey of existing literature and analytical solutions for both laminar and turbulent flow indicate that a heated surface generally destabilizes the boundary layer. Specifically, the boundary layer thickness is increased, the skin friction at the surface is decreased and the point of flow separation is moved forward. In addition, limited analytical results predict that the angle of attack at which a heated airfoil will stall is significantly less than the stall angle of an unheated wing. These effects could adversely affect the lift and drag, and thus the maneuvering capabilities of booster and orbiter shuttle vehicles.

Macha, J. M.↗