Application of the method of integral relations to real-gas flows past pointed bodies.
Frozen flow and nonequilibrium flow of real gas past pointed bodies, solving partial differential equation by integral relations method
NUMERICAL INTEGRATION↗
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Application of the method of integral relations to real-gas flows past pointed bodies.
Frozen flow and nonequilibrium flow of real gas past pointed bodies, solving partial differential equation by integral relations method
NUMERICAL INTEGRATION↗
One-way nesting for a primitive equation ocean model
Prognostic numerical models for atmospheric and oceanic circulations require initial fields, boundary conditions, and forcing functions in addition to a consistent set of partial differential equations, including a state relation and equations expressing conservation of mass, momentum, and energy. Depending on the horizontal domain to be modeled, the horizontal boundary conditions are either physically obvious or extremely difficult to specify consistently. If the entire atmosphere is modeled, periodic horizontal boundary conditions are appropriate. On the other hand, the physical horizontal boundaries on the entire ocean are solid walls. Obviously, the normal velocity at a solid wall is zero while the specification of the tangential velocity depends on the mathematical treatment of the horizontal viscous terms. Limitations imposed by computer capacity and cost, as well as research interests, have led to the use of limited area models to study flows in the atmosphere and ocean. The limited area models do not have physical horizontal boundaries, merely numerical ones. Correctly determining these open boundary conditions for limited-area numerical models has both intrigued and frustrated numerical modelers for decades. One common approach is to use the closed or solid wall boundary conditions for a limited-area model. The argument given for this approach is that the boundary conditions affect flow near the walls but that none of these effects are propagated into the interior. Therefore, one chooses a big enough domain that the central region of interest is not corrupted by the boundary flow. Research in progress to model the North Atlantic circulation vividly illustrates the pitfalls of this approach. Two model runs are compared: (1) the southern boundary at 20S between latitudes 0 and 40W is artificially closed; and (2) the same boundary is specified as open with an inward transport of 15 Sv (determined from a global model with the same physics) uniformly spread across the boundary. A comparison of both runs is presented.
Blake, D. W.↗
Comparison of finite-difference schemes for analysis of shells of revolution
Several finite difference schemes are applied to the stress and free vibration analysis of homogeneous isotropic and layered orthotropic shells of revolution. The study is based on a form of the Sanders-Budiansky first-approximation linear shell theory modified such that the effects of shear deformation and rotary inertia are included. A Fourier approach is used in which all the shell stress resultants and displacements are expanded in a Fourier series in the circumferential direction, and the governing equations reduce to ordinary differential equations in the meridional direction. While primary attention is given to finite difference schemes used in conjunction with first order differential equation formulation, comparison is made with finite difference schemes used with other formulations. These finite difference discretization models are compared with respect to simplicity of application, convergence characteristics, and computational efficiency. Numerical studies are presented for the effects of variations in shell geometry and lamination parameters on the accuracy and convergence of the solutions obtained by the different finite difference schemes. On the basis of the present study it is shown that the mixed finite difference scheme based on the first order differential equation formulation and two interlacing grids for the different fundamental unknowns combines a number of advantages over other finite difference schemes previously reported in the literature.
Noor, A. K.↗
Numerical determination of the fundamental eigenvalue for the Laplace operator on a spherical domain
Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (i.e., the membrane eignevalue problem for the vibration equation) on the unit spherical surface are developed. Two types of spherical surface domains are considered: the interior of a spherical triangle, and the exterior of a great circle arc extending for less than pi radians (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is employed. The fundamental eigenvalue is approximated by iteration utilizing the power method and point successive overrelaxation. Some numerical results are given and compared, in certain special cases, with analytical solutions to the eigenvalue problem. The significance of the numerical eigenvalue results is discussed in terms of the singularities in the solution of three-dimensional boundary-value problems near a polyhedral corner of the domain.
Walden, H.↗
Numerical calculation of flow fields about rectangular wings of finite thickness in supersonic flow
The calculation of the outer inviscid flow about a rectangular wing moving at supersonic speeds is reported. The inviscid equations of motion governing the flow generated by the wing form a set of hyperbolic differential equations. The flow field about the rectangular wing is separated into three regions consisting of the forebody, the afterbody, and the wing wake. Solutions for the forebody are obtained using conical flow techniques while the afterbody and the wing wake regions are treated as initial value problems. The numerical solutions are compared in the two dimensional regions with known exact solutions.
Vogel, J. M.↗
Convergence of step-by-step methods for non-linear integro-differential equations.
The theory of consistent step-by-step methods for solving Volterra integral equations is extended to nonsingular Volterra integro-differential equations. It is shown that standard step-by-step algorithms for these more general equations are convergent. Several numerical examples are included.
Mocarsky, W. L.↗
Effect of the nonuniform dc field on carrier waves in negative-differential-mobility semiconductors.
The growth of a carrier wave propagating through a negative-differential-mobility semiconducting sample is discussed under nonuniform dc bias conditions. A simple analytical expression for the over-all numerical gain is given in terms of current density and of the input and output carrier-wave velocities only. Applications to n-type GaAs are discussed.
Bini, M.↗
Development of a nonlinear unsteady transonic flow theory
A nonlinear, unsteady, small-disturbance theory capable of predicting inviscid transonic flows about aerodynamic configurations undergoing both rigid body and elastic oscillations was developed. The theory is based on the concept of dividing the flow into steady and unsteady components and then solving, by method of local linearization, the coupled differential equation for unsteady surface pressure distribution. The equations, valid at all frequencies, were derived for two-dimensional flows, numerical results, were obtained for two classses of airfoils and two types of oscillatory motions.
Stahara, S. S.↗
Spline approximations for nonlinear hereditary control systems
A sline-based approximation scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The approximating framework reduces the original control problem to a sequence of optimization problems governed by ordinary differential equations. Convergence proofs, which appeal directly to dissipative-type estimates for the underlying nonlinear operator, are given and numerical findings are summarized.
Daniel, P. L.↗
Assessing the quality of curvilinear coordinate meshes by decomposing the Jacobian matrix
An algebraic decomposition of the Jacobian matrix which relates physical and computational variables is presented. This invertible decomposition parameterizes the mesh by the physically intuitive qualities of cell orientation, cell orthogonality, cell volume, and cell aspect ratio. The decomposition can be used to analyze numerically generated curvilinear coordinate meshes and to assess the contribution of the mesh to the truncation error for any specific differential operator and algorithm. This is worked out in detail for Laplace's equation in nonconservative and conservative forms. The analysis is applied to the solution of the full potential code TAIR, showing grid plots, carpet plots, and truncation error for a NACA 0012 airfoil.
Kerlick, G. D.↗
Cubic spline approximation techniques for parameter estimation in distributed systems
Approximation schemes employing cubic splines in the context of a linear semigroup framework are developed for both parabolic and hyperbolic second-order partial differential equation parameter estimation problems. Convergence results are established for problems with linear and nonlinear systems, and a summary of numerical experiments with the techniques proposed is given.
Banks, H. T.↗
Second-order explicit finite-difference methods for transient-flow analysis
Three second-order accurate numerical methods - MacCormack's method, Lambda scheme and Gabutti scheme - are introduced to solve the quasi-linear, hyperbolic partial differential equations describing transient flows in closed conduits. The details of these methods and the treatment of boundary conditions are presented and the results computed by using these methods for a typical piping system are compared. It is shown that for the same accuracy, second-order methods require considerably lesser number of computational nodes and computer time as compared to those required by the first-order methods.
Chaudhry, M. H.↗
A numerical study of the effects of a large sandbar upon sea breeze development
Two-dimensional numerical simulations of sea breeze development over a large sandbar on the North Sea coast of Germany are reported. The numerical model used in these experiments contains a detailed treatment of soil moisture, which allows evaluation of the effects of differential surface characteristics on the airflow pattern. Results of the simulations indicate that the contrast between the moist sandbar and adjacent dry land, the tidal inundation of the sandbar, and the westward penetration of the Baltic sea breeze play important roles in the development of mesoscale airflow patterns in the sandbar region.
Kessler, R. C.↗
Concurrent processing in nonlinear structural stability
A concurrent processing algorithm is developed for materially nonlinear stability analysis of imperfect columns with biaxial partial rotational end restraints. The algorithm for solving the governing nonlinear ordinary differential equations is implemented on a multiprocessor computer called the 'Finite Element Machine', developed at the NASA Langley Research Center. Numerical results are obtained on up to nine concurrent processors. A substantial computational gain is achieved in using the parallel processing approach.
Darbhamulla, S. P.↗
The origin of rigidly rotating magnetic field patterns on the sun
Using analytical calculations and numerical simulations, it is shown that a meridional component of magnetic-flux transport will offset the shearing effect of differential rotation and give rise to rigidly rotating patterns of large-scale magnetic field. The nonaxisymmetric field attains a striped polarity pattern which rotates rigidly like a barber pole while its individual small-scale flux elements rotate at the differential rate of the latitudes they are crossing. On the sun, the meridional transport is provided by supergranular diffusion possibly assisted by a small poleward flow. New sources of flux retard this process and exclude the rigid rotation from the sunspot belts until well into the declining phase of the sunspot cycle. This mechanism accounts for a number of heretofore unexplained phenomena including the tendency for coronal holes to rotate rigidly during the declining phase of the sunspot cycle.
Sheeley, N. R., Jr.↗
A numerical simulation of the microburst phenomenon - Comparison of a discrete and a parameterized microphysical representation
Two versions of a microburst model that differ in how they represent microphysical processes are described and comparison of the numerical solutions is made. This study focuses on how the versions differ in representing the evaporation of raindrops and the differential sedimentation of drops that develops because of their fallspeeds. The two versions are then compared to demonstrate how predicted microburst characteristics are influenced by the choice of microphysical scheme.
Coen, Janice L.↗
Accurate computation and continuation of homoclinic and heteroclinic orbits for singular perturbation problems
In earlier papers, Doedel and the authors have developed a numerical method and derived error estimates for the computation of branches of heteroclinic orbits for a system of autonomous ordinary differential equations in R(exp n). The idea of the method is to reduce a boundary value problem on the real line to a boundary value problem on a finite interval by using a local (linear or higher order) approximation of the stable and unstable manifolds. A practical limitation for the computation of homoclinic and heteroclinic orbits has been the difficulty in obtaining starting orbits. Typically these were obtained from a closed form solution or via a homotopy from a known solution. Here we consider extensions of our algorithm which allow us to obtain starting orbits on the continuation branch in a more systematic way as well as make the continuation algorithm more flexible. In applications, we use the continuation software package AUTO in combination with some initial value software. The examples considered include computation of homoclinic orbits in a singular perturbation problem and in a turbulent fluid boundary layer in the wall region problem.
Vaughan, William W.↗