Search NASA⌕ Search

SEARCH · Search NASA

Results for “boundary value problems”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,207 records · Page 67
All sourcesNASA NTRSDOE OSTI
NASA NTRS1974

A finite-element analysis for steady and oscillatory subsonic flow around complex configurations

The problem of potential subsonic flow around complex configurations is considered. The solution is given of an integral equation relating the values of the potential on the surface of the body to the values of the normal derivative, which is known from the boundary conditions. The surface of the body is divided into small (hyperboloidal quadrilateral) surface elements, which are described in terms of the Cartesian components of the four corner points. The values of the potential (and its normal derivative) within each element is assumed to be constant and equal to its value at the centroid of the element. The coefficients of the equation are given by source and doublet integrals over the surface elements. Closed form evaluations of the integrals are presented. The results obtained with the above formulation are compared with existing analytical and experimental results.

Chen, L. T.↗
NASA NTRS1981

Efficient computation of the stability of three-dimensional compressible boundary layers

Methods for the computer analysis of the stability of three-dimensional compressible boundary layers are discussed and the user-oriented Compressible Stability Analysis (COSAL) computer code is described. The COSAL code uses a matrix finite-difference method for local eigenvalue solution when a good guess for the eigenvalue is available and is significantly more computationally efficient than the commonly used initial-value approach. The local eigenvalue search procedure also results in eigenfunctions and, at little extra work, group velocities. A globally convergent eigenvalue procedure is also developed which may be used when no guess for the eigenvalue is available. The global problem is formulated in such a way that no unstable spurious modes appear so that the method is suitable for use in a black-box stability code. Sample stability calculations are presented for the boundary layer profiles of an LFC swept wing.

Malik, M. R.↗
DOE OSTI2017

Solar Models in Light of New High Metallicity Measurements from Solar Wind Data

We study the impact of new metallicity measurements, from solar wind data, on the solar model. The “solar modeling problem” refers to the persisting discrepancy between helioseismological observations and predictions of solar models computed implementing state-of-the-art photospheric abundances. We critically reassess the problem, in particular considering the new set of abundances of von Steiger and Zurbuchen, determined through the in situ collection of solar wind samples from polar coronal holes. This new set of abundances indicates a solar metallicity Z{sub ⊙}⩾0.0196±0.0014, significantly higher than the currently established value. The new values hint at an abundance of volatile elements (i.e., C, N, O, Ne) close to previous results of Grevesse and Sauval, whereas the abundance of refractory elements (i.e., Mg, Si, S, Fe) is considerably increased. Using the Linear Solar Model formalism, we determine the variation of helioseismological observables in response to the changes in elemental abundances, in order to explore the consistency of these new measurements with constraints from helioseismology. We find that for observables that are particularly sensitive to the abundance of volatile elements, in particular the radius of the convective zone boundary (CZB) and the sound speed around the radius of CZB, improved agreement over previous models is obtained. Conversely, the high abundance of refractories correlates with a higher core temperature, resulting in an overproduction of neutrinos and a huge increase in the surface helium abundance. We conclude that the “solar modeling problem” remains unsolved.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

Implementation of a pressure correction term in the CTF sub-channel code

CTF [1] is a modernized and improved version of the legacy subchannel code, COBRA-TF [2], that is being jointly developed and maintained by Oak Ridge National Laboratory (ORNL) and North Carolina State University. The code was adopted by ORNL for use in the Consortium for Advanced Simulation of Light Water Reactors (CASL) in 2012 for aiding in addressing CASL challenge problems. Since that time, activities related to CTF have included implementing software quality assurance measures, implementing new features and models, performing validation and verification testing, establishing and supporting a CTF User Group, and developing the code for use in coupled applications [3, 4, 5]. CTF is applicable to single- and two-phase flows in light water reactor geometries at normal and accident operating conditions. The two-phase flow model consists of the six-equation two-phase flow model [6] augmented by equations for the droplet field and also includes appropriate source terms. This work focuses on the numerical stability of the six-equation two-phase flow model and is part of a recent eort to improve CTF capabilities for simulating flows in boiling nuclear reactors (BWRs). As it is of common knowledge among the thermodynamic community, the six-equation two-phase flow model is not strongly hyperbolic, meaning some of its eigenvalues are either not real or not distinct [7]. In a Cauchy sense and for an initial-value problem, strong hyperbolicity was found to be equivalent to well-posedness which ensures uniqueness of the solution in the space-time used. This concept is also of importance when designing high-order numerical methods and also when implementing boundary conditions that rely on real eigenvalues such as Riemann solvers. A tremendous amount of research is available in the literature on how to hyperbolize the six-equation two-phase flow model by adding ad hoc terms to the system of equations often referred to as virtual mass or interfacial pressure correction term. From a mathematical prospective, the main idea driving this approach is to add terms to the phasic momentum equation containing first-order partial derivatives of the unknowns to modify the Jacobian matrix and thus canceling the imaginary part of the eigenvalues.For this work it was chosen to implement a interfacial pressure correction (IPC) term on the same model as what was done for the CATHARE system code [8], that allows to recover a two-pressure formulation. In the following, the six-equation two-phase flow model is recalled along with the IPC term and its influence on the eigenvalues. Then, numerical results and convergence studies are presented for a water-faucet problem [9] at a pressure of 10 bar. Finally, conclusions and future work are addressed.

Delchini, Marco↗
DOE OSTI2015

Nonlinear galactic dynamos and the magnetic pitch angle

Pitch angles p of the large-scale magnetic fields B-bar of spiral galaxies have previously been inferred from observations to be systematically larger in magnitude than predicted by standard mean-field dynamo theory. This discrepancy is more pronounced if dynamo growth has saturated, which is reasonable to assume given that such fields are generally inferred to be close to energy equipartition with the interstellar turbulence. This “pitch angle problem” is explored using local numerical mean-field dynamo solutions as well as asymptotic analytical solutions. It is first shown that solutions in the saturated or kinematic regimes depend on only five dynamo parameters, two of which are tightly constrained by observations of galaxy rotation curves. The remaining three-dimensional (dimensionless) parameter space can be constrained to some extent using theoretical arguments. Predicted values of |p| can be as large as ∼40{sup ∘}, which is similar to the largest values inferred from observations, but only for a small and non-standard region of parameter space. We argue, based on independent evidence, that such non-standard parameter values are plausible. However, these values are located toward the boundary of the allowed parameter space, suggesting that additional physical effects may need to be incorporated. We therefore suggest possible directions for extending the basic model considered.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1975

A finite element analysis of the exact nonlinear formulation of a lifting surface in steady incompressible flow, with the evaluation of the correct wake geometry

The problem of steady incompressible flow for lifting surfaces is considered. An integral equation is solved relating the values of the potential discontinuity on the lifting surface and its wake to the values of the normal derivative of the potential which are known from the boundary conditions. The lifting surface and the wake are divided into small quadrilateral surface elements. The values of the potential discontinuity and the normal derivative of the potential are assumed to be constant within each lifting surface element and equal to their values at the centroids of the lifting surface elements. This yields a set of linear algebraic equations. An iteration procedure is used to obtain the wake geometry: the velocities at the corner points of the wake elements are calculated and the wake streamlines are aligned to be parallel to the velocity vector. The procedure is repeated until convergence is attained.

Suciu, E. O.↗
NASA NTRS1976

Influence of specimen boundary on the dynamic stress intensity factor

The problem to be considered is the sudden appearance of a flaw or crack in a strip of material of finite height subjected to tensile loading. Stress waves are generated within the strip and are reflected from boundary to boundary. Of interest is the maximum value of the dynamic stress intensity factor at a given instance of time as the strip height to crack length ratio is varied.

Chen, E. P.↗
NASA NTRS1978

The Incorporation of Truncated Fourier Series into Finite Difference Approximations of Structural Stability Equations

A new trigonometric approach to the finite difference calculus was applied to the problem of beam buckling as represented by virtual work and equilibrium equations. The trigonometric functions were varied by adjusting a wavelength parameter in the approximating Fourier series. Values of the critical force obtained from the modified approach for beams with a variety of boundary conditions were compared to results using the conventional finite difference method. The trigonometric approach produced significantly more accurate approximations for the critical force than the conventional approach for a relatively wide range in values of the wavelength parameter; and the optimizing value of the wavelength parameter corresponded to the half-wavelength of the buckled mode shape. It was found from a modal analysis that the most accurate solutions are obtained when the approximating function closely represents the actual displacement function and matches the actual boundary conditions.

Hannah, S. R.↗
NASA NTRS1990

Inelastic transient dynamic analysis of three-dimensional problems by BEM

A general direct boundary element formulation and its numerical implementation for solving transient dynamic problems of three-dimensional isotropic homogeneous or piecewise homogeneous solids involving material nonlinearities are presented. The algorithm produces accurate results for static nonlinear problems by using large time steps. When a large value of yield stress is selected, the incremental inelastic transient algorithm produces results identical to those obtained by elastodynamic analysis.

Ahmad, S.↗
NASA NTRS1949

On the Theory of the Laval Nozzle

In the present paper, the motion of a gas in a plane-parallel Laval nozzle in the neighborhood of the transition from subsonic to supersonic velocities is studied. In a recently published paper, F. I. Frankl, applying the holograph method of Chaplygin, undertook a detailed investigation of the character of the flow near the line of transition from subsonic to supersonic velocities. From the results of Tricomi's investigation on the theory of differential equations of the mixed elliptic-hyperbolic type, Frankl introduced as one of the independent variables in place of the modulus of the velocity, a certain specially chosen function of this modulus. He thereby succeeded in explaining the character of the flow at the point of intersection of the transition line and the axis of symmetry (center of the nozzle) and in studying the behavior of the stream function in the neighborhood of this point by separating out the principal term having, together with its derivatives, the maximum value as compared with the corresponding corrections. This principal term is represented in Frankl's paper in the form of a linear combination of two hypergeometric functions. In order to find this linear combination, it is necessary to solve a number of boundary problems, which results in a complex analysis. In the investigation of the flow with which this paper is concerned, a second method is applied. This method is based on the transformation of the equations of motion to a form that may be called canonical for the system of differential equations of the mixed elliptic-hyperbolic type to which the system of equations of the motion of an ideal compressible fluid refers. By studying the behavior of the integrals of this system in the neighborhood of the parabolic line, the principal term of the solution is easily separated out in the form of a polynomial of the third degree. As a result, the computation of the transitional part of the nozzle is considerably simplified.

Falkovich, S. V.↗
DOE OSTI2017

Implementation of a Pressure Correction Term in the CTF Subchannel Code

CTF is a modernized and improved version of the legacy subchannel code COBRA-TF, being jointly developed and maintained by Oak Ridge National Laboratory (ORNL) and North Carolina State University. The code was adopted by ORNL for use in the Consortium for Advanced Simulation of Light Water Reactors (CASL) in 2012 to help address CASL challenge problems. Since that time, activities related to CTF have included implementing software quality assurance measures, implementing new features and models, performing validation and verification testing, establishing and supporting a CTF user group, and developing the code for use in coupled applications. CTF is applicable to single- and two-phase flows in light water reactor geometries at normal and accident operating conditions. The two-phase flow model consists of the six-equation two-phase flow model augmented by equations for the droplet field, and it also includes appropriate source terms. This work focuses on the numerical stability of the six-equation two-phase flow model and is part of a recent effort to improve CTF capabilities for simulating flows in boiling water reactors (BWRs). The six-equation two-phase flow model is not strongly hyperbolic, meaning that some of its eigenvalues are either not real or not distinct. In a Cauchy sense, and for an initial-value problem, strong hyperbolicity was found to be equivalent to well-posedness, which ensures the uniqueness of the solution in the space-time used. This concept is also of importance when designing high-order numerical methods and also when implementing boundary conditions that rely on real eigenvalues such as Riemann solvers. A tremendous amount of research is available in the literature on how to hyperbolize the six-equation two-phase flow model by adding ad hoc terms to the system of equations. Such terms are often referred to as virtual mass or interfacial pressure correction terms. From a mathematical perspective, this approach adds terms to the phasic momentum equation containing first-order partial derivatives of the unknowns to modify the Jacobian matrix, thus the imaginary part of the eigenvalues. For this work, an interfacial pressure correction (IPC) term on the same model as what was used for the CATHARE system code, is implemented in CTF. This allows for recovery of a two-pressure formulation. In the following, the six-equation two-phase flow model is recalled, along with the IPC term and its influence on the eigenvalues. Then, numerical results and convergence studies are presented for a water-faucet problem at a pressure of 10 bar. Finally, conclusions and future work are addressed.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2020

An analysis of the spatio-temporal resolution of the immersed boundary method with direct forcing

The immersed boundary method (IBM) with direct forcing is very popular in the simulation of rigid particulate flows. In the IBM, an interaction force is introduced at the interface between fluid and particle in order to approximate the no-slip boundary condition. The interaction force is calculated through dividing the velocity difference (or error) between fluid and particle at the interface by the time step. Here, a dynamic equation for the velocity difference is derived. Additionally, analyses on the dynamic equation provide a few new findings: (i) The interaction force is the solution of a least-squares error problem, with the direct implication that the Lagrangian marker distribution has no effect on the large scale flow structure once the distribution of Lagrangian markers become saturated along the interface (i.e., each marker remains properly correlated with all its neighbors); (ii) The Lagrangian volume-weight is a relaxation factor to control how fast the velocity error decays to the ideal value of zero; (iii) The optimal choice of the Lagrangian volume-weight is the largest value permissible by a stability condition. A comprehensive convergence analysis with regard to the spatial and temporal resolution is presented for the velocity error and also for the shear-stress and surface pressure. In three simple canonical problems, it is analytically and numerically shown that the IBM results converge to the theoretical solutions obtained with precise imposition of no-slip and no-penetration boundary conditions. It is observed that it is not necessary to match the Lagrangian marker volume-weight to that of the local Eulerian cell volume and in fact this matching leads to lower than optimal computational efficiency. However, it is found that extremely high Eulerian grid resolution and small time step have to be used to obtain high precision simulation results. Especially, the time step should be inversely proportional to the particle Reynolds number for low Reynolds number flows. For high frequency oscillation problems, the grid size needs to be reduced by a factor of the square root of the frequency, and the time step to be reduced by a factor of the frequency. The theoretical findings here can be used to alleviate the technical difficulties in simulating non-spherical particles by not requiring the Lagrangian marker distribution to match the Eulerian grids and also in the implementation of IBM on non-uniform Eulerian grids. The present work also provides simple practical guidance on the choice of temporal and spatial resolution so as to control the simulation error a priori.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2016

Tangle-Free Mesh Motion for Ablation Simulations

Problems involving mesh motion-which should not be mistakenly associated with moving mesh methods, a class of adaptive mesh redistribution techniques-are of critical importance in numerical simulations of the thermal response of melting and ablative materials. Ablation is the process by which material vaporizes or otherwise erodes due to strong heating. Accurate modeling of such materials is of the utmost importance in design of passive thermal protection systems ("heatshields") for spacecraft, the layer of the vehicle that ensures survival of crew and craft during re-entry. In an explicit mesh motion approach, a complete thermal solve is first performed. Afterwards, the thermal response is used to determine surface recession rates. These values are then used to generate boundary conditions for an a posteriori correction designed to update the location of the mesh nodes. Most often, linear elastic or biharmonic equations are used to model this material response, traditionally in a finite element framework so that complex geometries can be simulated. A simple scheme for moving the boundary nodes involves receding along the surface normals. However, for all but the simplest problem geometries, evolution in time following such a scheme will eventually bring the mesh to intersect and "tangle" with itself, inducing failure. This presentation demonstrates a comprehensive and sophisticated scheme that analyzes the local geometry of each node with help from user-provided clues to eliminate the tangle and enable simulations on a wide-class of difficult problem geometries. The method developed is demonstrated for linear elastic equations but is general enough that it may be adapted to other modeling equations. The presentation will explicate the inner workings of the tangle-free mesh motion algorithm for both two and three-dimensional meshes. It will show abstract examples of the method's success, including a verification problem that demonstrates its accuracy and correctness. The focus of the presentation will be on the algorithm; specifics on how the techniques may be used in spacecraft design will be not discussed.

Droba, Justin↗
NASA NTRS1989

Supercritical flow past a symmetrical bicircular arc airfoil

A numerical scheme is developed for computing steady supercritical flow about symmetrical airfoils, applying it to an ellipse for zero angle of attack. An algorithmic description of this new scheme is presented. Application to a symmetrical bicircular arc airfoil is also proposed. The flow field before the shock is region 1. For transonic flow, singularity can be avoided by integrating the resulting ordinary differential equations away from the body. Region 2 contains the shock which will be located by shock fitting techniques. The shock divides region 2 into supersonic and subsonic regions and there is no singularity problem in this case. The Method of Lines is used in this region and it is advantageous to integrate the resulting ordinary differential equation along the body for shock fitting. Coaxial coordinates have to be used for the bicircular arc airfoil so that boundary values on the airfoil body can be taken with one direction of the coaxial coordinates fixed. To avoid taking boundary values at + or - infinity in the coaxial co-ordinary system, approximate analytical representation of the flow field near the tips of the airfoil is proposed.

Holt, Maurice↗
DOE OSTI2016

Scalar excursions in large-eddy simulations

Here, the range of values of scalar fields in turbulent flows is bounded by their boundary values, for passive scalars, and by a combination of boundary values, reaction rates, phase changes, etc., for active scalars. The current investigation focuses on the local conservation of passive scalar concentration fields and the ability of the large-eddy simulation (LES) method to observe the boundedness of passive scalar concentrations. In practice, as a result of numerical artifacts, this fundamental constraint is often violated with scalars exhibiting unphysical excursions. The present study characterizes passive-scalar excursions in LES of a shear flow and examines methods for diagnosis and assesment of the problem. The analysis of scalar-excursion statistics provides support of the main hypothesis of the current study that unphysical scalar excursions in LES result from dispersive errors of the convection-term discretization where the subgrid-scale model (SGS) provides insufficient dissipation to produce a sufficiently smooth scalar field. In the LES runs three parameters are varied: the discretization of the convection terms, the SGS model, and grid resolution. Unphysical scalar excursions decrease as the order of accuracy of non-dissipative schemes is increased, but the improvement rate decreases with increasing order of accuracy. Two SGS models are examined, the stretched-vortex and a constant-coefficient Smagorinsky. Scalar excursions strongly depend on the SGS model. The excursions are significantly reduced when the characteristic SGS scale is set to double the grid spacing in runs with the stretched-vortex model. The maximum excursion and volume fraction of excursions outside boundary values show opposite trends with respect to resolution. The maximum unphysical excursions increase as resolution increases, whereas the volume fraction decreases. The reason for the increase in the maximum excursion is statistical and traceable to the number of grid points (sample size) which increases with resolution. In contrast, the volume fraction of unphysical excursions decreases with resolution because the SGS models explored perform better at higher grid resolution.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1974

NASTRAN variance analysis and plotting of HBDY elements

The NASTRAN Thermal Analyzer has been intended to do variance analysis and plot the thermal boundary elements. The objective of the variance analysis addition is to assess the sensitivity of temperature variances resulting from uncertainties inherent in input parameters for heat conduction analysis. The plotting capability provides the ability to check the geometry (location, size and orientation) of the boundary elements of a model in relation to the conduction elements. Variance analysis is the study of uncertainties of the computed results as a function of uncertainties of the input data. To study this problem using NASTRAN, a solution is made for both the expected values of all inputs, plus another solution for each uncertain variable. A variance analysis module subtracts the results to form derivatives, and then can determine the expected deviations of output quantities.

Harder, R. L.↗
NASA NTRS1985

Parameterization in Grid Generation

The distribution of grid points for calculating the solution of partial differential equations is chosen so as to include consideration of truncation error, stability, and the resolution of the solution near boundary layers and shocks. It is important to specify the distribution of points along a grid line. The problem of distributing points along a curve is considered. It is assumed that the curve is defined parametrically. The objective is to select a set of parameter values so that the corresponding points on the curves are properly distributed. The distribution is based on some intrinsic property of the curve such as arc length or curvature.

Mastin, C. W.↗