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NASA NTRS1978

Mixed boundary-value problems in mechanics

Definitions in the case of multiple series equations and multiple integral equations are examined. In considering the solution of a given mixed boundary value problem perhaps the simplest technique is the direct application of the method of complex potentials provided the problem admits such potentials and the domain and the boundary conditions are suitable for such an application. The direct application of complex potentials is described with the aid of examples, taking into account a problem in potential theory, the case of periodic cuts, and an elasticity problem for a nonhomogeneous plane. The reduction to singular integral equations is discussed along with the numerical solution of singular integral equations of the first kind, integral equations with generalized Cauchy kernels, and singular integral equations of the second kind.

Erdogan, F.↗

NASA NTRS1978

Critical study of higher order numerical methods for solving the boundary-layer equations

A fourth order box method is presented for calculating numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations. The method, which is the natural extension of the second order box scheme to fourth order, was demonstrated with application to the incompressible, laminar and turbulent, boundary layer equations. The efficiency of the present method is compared with two point and three point higher order methods, namely, the Keller box scheme with Richardson extrapolation, the method of deferred corrections, a three point spline method, and a modified finite element method. For equivalent accuracy, numerical results show the present method to be more efficient than higher order methods for both laminar and turbulent flows.

Wornom, S. F.↗

NASA NTRS1980

Exospheric perturbations by radiation pressure. II - Solution for orbits in the ecliptic plane

A previous study (Chamberlain, 1979) gave solutions for the mean time rates of change of orbital elements of satellite atoms in an exosphere influenced by solar radiation pressure; each element was assumed to behave independently. In the present paper, the instantaneous rates of changes for three elements (e, Omega, and phi = omega + Omega) are integrated simultaneously for the case of the inclination i = 0. The results confirm the validity of using mean rates when the orbits are tighly bound to the planet, and serve as examples to be reproduced by the complicated numerical solutions required for arbitrary inclination. Strongly bound hydrogen atoms perturbed in earth orbit by radiation pressure do not seem a likely cause of the geotail extending in the anti-sun direction. Instead, radiation pressure will cause those particles' orbits to form a broad fan-shaped tail and to deteriorate into the earth's atmosphere.

Chamberlain, J. W.↗

DOE OSTI2022

A Lagrangian dual method for two-stage robust optimization with binary uncertainties

This report presents a new exact method to calculate worst-case parameter realizations in two-stage robust optimization problems with categorical or binary-valued uncertain data. Traditional exact algorithms for these problems, notably Benders decomposition and column-and-constraint generation, compute worst-case parameter realizations by solving mixed-integer bilinear optimization subproblems. However, their numerical solution can be computationally expensive not only due to their resulting large size after reformulating the bilinear terms, but also because decision-independent bounds on their variables are typically unknown. We propose an alternative Lagrangian dual method that circumvents these difficulties and is readily integrated in either algorithm. We specialize the method to problems where the binary parameters switch on or off constraints as these are commonly encountered in applications, and discuss extensions to problems that lack relatively complete recourse and to those with integer recourse. Numerical experiments provide evidence of significant computational improvements over existing methods.

42 ENGINEERING↗

NASA NTRS1991

The symmetric turbulent plane wake downstream of a sharp trailing edge

The analysis and modeling of the symmetric turbulent plane wake downstream of a sharp trailing edge is addressed. A compact description of the flow near the trailing edge is formulated using the results of a previous asymptotic analysis. The new description retains the two-layered structure identified in the previous work, and it clarifies the principal dynamics of the flow in the near-wake outer layer, away from the wake centerline. For zero-pressure-gradient flow, the near-wake outer layer is shown to be represented to leading order by the similarity solution that governs the outer region of the surface boundary layer. The leading perturbation in the outer layer due to the developing near-wake inner-layer flow is identified, and this is shown to be asymptotically smaller than undetermined higher-order terms associated with the surface boundary-layer flow. Results of the new near-wake analysis are used to formulate an algebraic eddy viscosity model for wake flow predictions at arbitrary distances from the trailing edge. The model is used in a numerical solution of the boundary layer equations, and computed velocity and Reynolds stress profiles are shown to compare well with experimental data.

Bogucz, E. A.↗

DOE OSTI2024

Simplified Universal Equations for Ionic Conductivity and Transference Number

Nernst-Einstein equation can provide a reasonable estimate of the ionic conductivity of dilute solutions. For concentrated solutions, alternate methods such as Green–Kubo relations and Einstein relations are more suitable to account for ion-ion interactions. Such computations can be expensive for multicomponent systems. Simplified mathematical expressions like the Nernst-Einstein equation do not exist for concentrated multicomponent mixtures. Newman’s treatment of multicomponent concentrated solutions yields a conductivity relation in terms of species concentration and Onsager phenomenological coefficients. However, the estimation of these phenomenological coefficients is not straightforward. Here, mathematical formulations that relate the phenomenological coefficients with the friction coefficients are developed, leading to simplified, ready-to-use expressions of conductivity and transference numbers that can be used for a wide range of ionic mixtures. This approach involves spectral decomposition of the matrix of Onsager phenomenological coefficients. The general analytical expressions for conductivity and transference number are simplified for binary electrolytes, and numerical solutions are provided for ternary and quaternary mixtures with ion dissociation.

Electrochemistry↗

NASA NTRS1977

A critical study of higher-order numerical methods for solving the boundary-layer equations

A fourth-order box method is presented for calculating numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations. The method is the natural extension of the second-order Keller Box Scheme to fourth order and is demonstrated with application to the incompressible, laminar and turbulent boundary-layer equations. The efficiency of the present method is compared with other two-point and three-point higher-order methods; namely, the Keller Box Scheme with Richardson extrapolation, the method of deferred corrections, and the three-point spline methods. For equivalent accuracy, numerical results show the present method to be more efficient than the other higher-order methods for both laminar and turbulent flows.

Wornom, S. F.↗

DOE OSTI2022

Spatial resolution of different discretizations over long-time for the Dirac equation with small potentials

In this report we compare the long-time error bounds and spatial resolution of finite difference methods with different spatial discretizations for the Dirac equation with small electromagnetic potentials characterized by $\varepsilon \in (0, 1]$ a dimensionless parameter. We begin with the simple and widely used finite difference time domain (FDTD) methods, and establish rigorous error bounds of them, which are valid up to the time at $O(1/\varepsilon)$. In the error estimates, we pay particular attention to how the errors depend explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $\varepsilon$. Based on the results, in order to obtain "correct" numerical solutions up to the time at $O(1/\varepsilon)$, the $\varepsilon$-scalability (or meshing strategy requirement) of the FDTD methods should be taken as $h = O(\varepsilon^{1/2})$ and $\tau = O(\varepsilon^{1/2})$. To improve the spatial resolution capacity, we apply the Fourier spectral method to discretize the Dirac equation in space. Error bounds of the resulting finite difference Fourier pseudospectral (FDFP) methods show that they exhibit uniform spatial errors in the long-time regime, which are optimal in space as suggested by the Shannon's sampling theorem. Extensive numerical results are reported to confirm the error bounds and demonstrate that they are sharp.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS2017

Numerical Modeling of Saturated Boiling in a Heated Tube

This paper describes a mathematical formulation and numerical solution of boiling in a heated tube. The mathematical formulation involves a discretization of the tube into a flow network consisting of fluid nodes and branches and a thermal network consisting of solid nodes and conductors. In the fluid network, the mass, momentum and energy conservation equations are solved and in the thermal network, the energy conservation equation of solids is solved. A pressure-based, finite-volume formulation has been used to solve the equations in the fluid network. The system of equations is solved by a hybrid numerical scheme which solves the mass and momentum conservation equations by a simultaneous Newton-Raphson method and the energy conservation equation by a successive substitution method. The fluid network and thermal network are coupled through heat transfer between the solid and fluid nodes which is computed by Chen's correlation of saturated boiling heat transfer. The computer model is developed using the Generalized Fluid System Simulation Program and the numerical predictions are compared with test data.

Majumdar, Alok↗

NASA NTRS1991

Faster Algorithm For Computation Of Incompressible Flow

Improved algorithm yields faster numerical solutions of Navier-Stokes equations of steady or unsteady three-dimensional flow of incompressible fluid. In artificial-compressibility method, unsteady flow treated as incompressible in advancing from one time step to next, but at each time step (or in steady state), fluid treated as having variable compressibility enabling propagation of flow field, and subiterations performed in increments of pseudotime until effects of compressibility subside. Directly couples pressure and velocity fields at same time step and converts elliptic incompressible Navier-Stokes equations to hyperbolic form more amenable to numerical integration.

Rogers, S. E.↗

NASA NTRS1990

Resolution of the 1D regularized Burgers equation using a spatial wavelet approximation

The Burgers equation with a small viscosity term, initial and periodic boundary conditions is resolved using a spatial approximation constructed from an orthonormal basis of wavelets. The algorithm is directly derived from the notions of multiresolution analysis and tree algorithms. Before the numerical algorithm is described these notions are first recalled. The method uses extensively the localization properties of the wavelets in the physical and Fourier spaces. Moreover, the authors take advantage of the fact that the involved linear operators have constant coefficients. Finally, the algorithm can be considered as a time marching version of the tree algorithm. The most important point is that an adaptive version of the algorithm exists: it allows one to reduce in a significant way the number of degrees of freedom required for a good computation of the solution. Numerical results and description of the different elements of the algorithm are provided in combination with different mathematical comments on the method and some comparison with more classical numerical algorithms.

Liandrat, J.↗

NASA NTRS1990

A computational procedure for multibody systems including flexible beam dynamics

A computational procedure suitable for the solution of equations of motions for flexible multibody systems has been developed. A fully nonlinear continuum approach capable of accounting for both finite rotations and large deformations has been used to model a flexible beam component. The beam kinematics are referred directly to an inertial reference frame such that the degrees of freedom embody both the rigid and flexible deformation motions. As such, the beam inertia expression is identical to that of rigid body dynamics. The nonlinear coupling between gross body motion and elastic deformation is contained in the internal force expression. Numerical solution procedures for the integration of spatial kinematic systems can be directily applied to the generalized coordinates of both the rigid and flexible components. An accurate computation of the internal force term which is invariant to rigid motions is incorporated into the general solution procedure.

Downer, J. D.↗

NASA NTRS1999

A User's Manual for ROTTILT Solver: Tiltrotor Fountain Flow Field Prediction

A CFD solver has been developed to provide the time averaged details of the fountain flow typical for tiltrotor aircraft in hover. This Navier-Stokes solver, designated as ROTTILT, assumes the 3-D fountain flowfield to be steady and incompressible. The theoretical background is described in this manual. In order to enable the rotor trim solution in the presence of tiltrotor aircraft components such as wing, nacelle, and fuselage, the solver is coupled with a set of trim routines which are highly efficient in CPU and suitable for CFD analysis. The Cartesian grid technique utilized provides the user with a unique capability for insertion or elimination of any components of the bodies considered for a given tiltrotor aircraft configuration. The flowfield associated with either a semi or full-span configuration can be computed through user options in the ROTTILT input file. Full details associated with the numerical solution implemented in ROTTILT and assumptions are presented. A description of input surface mesh topology is provided in the appendices along with a listing of all preprocessor programs. Input variable definitions and default values are provided for the V22 aircraft. Limited predicted results using the coupled ROTTILT/WOPWOP program for the V22 in hover are made and compared with measurement. To visualize the V22 aircraft and predictions, a preprocessor graphics program GNU-PLOT3D was used. This program is described and example graphic results presented.

Tadghighi, Hormoz↗

NASA NTRS1978

Application of higher-order numerical methods to the boundary-layer equations

A fourth-order method is presented for calculating numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations. The method is the natural extension of the second-order Keller Box Scheme to fourth order and is demonstrated with application to the incompressible, laminar and turbulent boundary-layer equations for both attached and separated flows. The efficiency of the present method is compared with other higher-order methods; namely, the Keller Box Scheme with Richardson extrapolation, the method of deferred corrections, the three-point spline methods, and a modified finite-element method. For equivalent accuracy, numerical results show the present method to be more efficient than the other higher-order methods for both laminar and turbulent flows.

Wornom, S. F.↗

NASA NTRS1985

Combined aerodynamic and structural dynamic problem emulating routines (CASPER): Theory and implementation

The Combined Aerodynamic and Structural Dynamic Problem Emulating Routines (CASPER) is a collection of data-base modification computer routines that can be used to simulate Navier-Stokes flow through realistic, time-varying internal flow fields. The Navier-Stokes equation used involves calculations in all three dimensions and retains all viscous terms. The only term neglected in the current implementation is gravitation. The solution approach is of an interative, time-marching nature. Calculations are based on Lagrangian aerodynamic elements (aeroelements). It is assumed that the relationships between a particular aeroelement and its five nearest neighbor aeroelements are sufficient to make a valid simulation of Navier-Stokes flow on a small scale and that the collection of all small-scale simulations makes a valid simulation of a large-scale flow. In keeping with these assumptions, it must be noted that CASPER produces an imitation or simulation of Navier-Stokes flow rather than a strict numerical solution of the Navier-Stokes equation. CASPER is written to operate under the Parallel, Asynchronous Executive (PAX), which is described in a separate report.

Jones, William H.↗

NASA NTRS1987

Current-driven magnetohydrodynamic thermal instabilities in sheared fields

Approximate analytic solutions are sought for the dispersion relation for the MHD stability of magnetized medium in current-driven filamentation modes such as those observed in the solar atmosphere. The magnetic field is assumed to have a self-consistent sheared equilibrium structure. The analysis is carried out in the small wavenumber regime, where shear length is similar to the mode wavelength. Instability is found to depend on the ratio between the thermal and magnetic diffusivities, i.e., the Prandtl number, which identifies the unstable transverse wavenumbers. The instability conditions are expressed in an algebraic equation amenable to numerical solution. Results are provided from use of the model to determine the maximum growth rate and typical scale lengths of instabilities in a precoronal atmosphere and the lower transition region.

Bodo, G.↗

NASA NTRS1969

Thermal convection at infinite Prandtl number

Numerical solutions are developed for the full scaled three dimensional thermal convection problem, at infinite Prandtl number, and for rigid boundaries. The procedure is designed to give an approximate account of horizontally homogeneous and isotropic flow situations. Results, which included a maximum of 36 horizontal wave number vectors and 20 vertical wave numbers, appear to adequately describe the flow fill up to R = 100,000; beyond this R the results appear to show horizontal wave number truncation error. This error seems to affect the boundary slope of the mean temperature field more than other mean quantities. Despite some numerical uncertainities, certain of the qualitative features of the flow fill are predicted with reasonable confidence.

Herring, J. R.↗

NASA NTRS1979

A computational fluid dynamic technique valid at the centerline for non-axisymmetric problems in cylindrical coordinates

A technique is described for the numerical solution of non-axisymmetric flow problems posed in cylindrical coordinates when the z-axis is included in the flowfield. The highlight of the technique is the manner in which the singularities at the centerline are handled. Specifically, the governing flowfield equations at r = 0 are put in a special form by applying L'Hospital's Rule. The required radial derivatives are evaluated using a one-sided, second-order accurate, first-difference. This leads to a smooth, convergent calculation of the flowfield at the centerline. This appears to be the first generally applicable numerical method for avoiding coordinate system singularities in the context of a finite-difference scheme, and could have application to many nonaxisymmetric flows. The technique is illustrated by specific results for the time-dependent flowfield inside an internal combustion engine.

Griffin, M. D.↗