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The formal derivation of the three-dimensional parabolic Navier-Stokes equations for subsonic turbulent flow is reviewed. A penalty finite element algorithm is established for numerical solution of the sixteen dependent variable system. Key numerical results are summarized documenting applications in various problem definitions.
Quasi-closed form solutions are derived for the finite time, minimum force rate-feedback threshold controller to bring a system with or without known external disturbances back into an 'allowable' state manifold in finite time. The disturbances are assumed to be expandable in terms of Fourier series. The quasi-closed form solutions replace the solution of the two-point boundary value problem and definite integral constraints with the solution of algebraic equations and the calculation of matrix exponentials. Examples demonstrate the threshold control technique and compare the quasi-closed form solutions with MACSYMA generated exact solutions (for small system order) and with the numerical solution of the two-point boundary value problem.
Two contrasting representations of land surface variability used in general circulation models (GCMs) are compared through an analysis of their corresponding surface energy balance equations. In one representation (the 'mixture' approach), different vegetation types are assumed to be homogeneously mixed over a grid square, so that the GCM atmosphere sees near-surface conditions pertaining to the mixture only. In the second representation (the 'mosaic' approach), different vegetation types are viewed as separate 'tiles' of a grid-square 'mosaic', and each tile interacts with the atmosphere independently. The mosaic approach is computationally simpler and in many ways more flexible than the mixture approach. Analytical solutions to the linearized energy balance equations and numerical solutions to the nonlinear equations both demonstrate that the mixture strategy, when applied to two coexisting vegetation types that differ only in canopy transpiration resistance, promotes both total turbulent flux and latent heat flux relative to the mosaic strategy. The effective differences between the strategies, however, are small over a wide range of conditions. In particular, the strategies are effectively equivalent when the transpiration resistances of the different vegetation types are of the same order of magnitude.
The turbine blades in the high-pressure fuel turbopump/alternate turbopump (HPFTP/AT) are subjected to hot gases rapidly flowing around them. This flow excites vibrations in the blades. Naturally, one has to worry about resonance, so a damping device was added to dissipate some energy from the system. The foundation is now laid for a very complex problem. The damper is in contact with the blade, so now there are contact stresses (both normal and tangential) to contend with. Since these stresses can be very high, it is not all that difficult to yield the material. Friction is another non-linearity and the blade is made out of a Nickel-based single-crystal superalloy that is orthotropic. A few approaches exist to solve such a problem and computer models, using contact elements, have been built with friction, plasticity, etc. These models are quite cumbersome and require many hours to solve just one load case and material orientation. A simpler approach is required. Ideally, the model should be simplified so the analysis can be conducted faster. When working with contact problems determining the contact patch and the stresses in the material are the main concerns. Closed-form solutions for non-conforming bodies, developed by Hertz, made out of isotropic materials are readily available. More involved solutions for 3-D cases using different materials are also available. The question is this: can Hertzian1 solutions be applied, or superimposed, to more complicated problems-like those involving anisotropic materials? That is the point of the investigation here. If these results agree with the more complicated computer models, then the analytical solutions can be used in lieu of the numerical solutions that take a very long time to process. As time goes on, the analytical solution will eventually have to include things like friction and plasticity. The models in this report use no contact elements and are essentially an applied load problem using Hertzian assumptions to determine the contact patch dimensions.
A three-state model is presented for the aeroassisted orbital plane change problem. A further model order reduction to a single state model is examined using singular perturbation theory. The optimal solution for this single state model compares favorably with the exact numerical solution using a four-state model; however, a separate boundary layer solution is required to satisfy the terminal constraint on altitude. This, in general, involves the solution of a two-point boundary value problem, but for a two-state model. An approximation is introduced to obtain an analytical control solution for lift and bank angle. Included are numerical simulation results of a guidance law derived from this analysis, along with comparison to earlier work by other researchers.
A weak-interaction solution algorithm is established for aerodynamic flow field prediction about an isolated airfoil. It requires numerical solution of differential equations governing potential flow, viscous and turbulent boundary layer flow, and the turbulent wake flow downstream of a trailing edge. The algorithm accounts for computed viscous displacement effects on the potential flow. These in turn alter the viscous flow through imposed pressure gradients. Closure for turbulence is accomplished using a second order model. Numerical evaluations assess factors affecting solution accuracy, convergence and stability for the combined potential, boundary layer, and parabolic Navier-Stokes equation systems as solved using a finite element algorithm.
Numerical analysis of chemical kinetics in reacting gas equations applied to rocket nozzle and plume flows
Numerical solution procedures evaluation for geometrically nonlinear structural analysis by direct stiffness method, noting capability of self correcting initial value formulation
A numerical solution to a theoretical model of vapor cavitation in a dynamically loaded journal bearing is developed, utilizing a multigrid iterative technique. The code is compared with a presently existing direct solution in terms of computational time and accuracy. The model is based on the Elrod algorithm, a control volume approach to the Reynolds equation which mimics the Jakobssen-Floberg and Olsson cavitation theory. Besides accounting for a moving cavitation boundary and conservation of mass at the boundary, it also conserves mass within the cavitated region via liquid striations. The mixed nature of the equations (elliptic in the full film zone and nonelliptic in the cavitated zone) coupled with the dynamic aspects of the problem create interesting difficulties for the present solution approach. Emphasis is placed on the methods found to eliminate solution instabilities. Excellent results are obtained for both accuracy and reduction of computational time.
Numerical schemes of the first and second order of approximation introduce numerical distortion when the wave propagation over a long distance is investigated. To alleviate this problem, the fourth-order leapfrog scheme is constructed. The standard leapfrog method is based on the truncated Taylor series expansion which depicts an error proportional to the second-order terms. In the proposed method the numerical solution is corrected for these terms. The space and time corrections work well in diminishing numerical dispersion and dissipation.
The fabrication and testing of a pair of robot fingers designed to grasp two types of Orbital Replacement Unit (ORU) interfaces, the H Handle type and the Micro Square type are discussed. A closed-form solution is given for the force inverse kinematics. A numerical solution using the Newton-Raphson Method for force forward kinematics is given. Mathematical expressions are derived to compute forces/torques applied to the finger. Suggestions are given for improving finger fabrication. The results of numerous experiments conducted to study the characteristics and feasibility of the fingers are given. The first part of the study was devoted to obtaining data on the forces applied by the finger to the interfaces under various translational and rotational misalignments; the second part was devoted to determining the maximum allowed capture angles that would insure successful mating; and the third part was devoted to the processing and interpretation of the forces/torque data.
Here, we introduce a finite volume scheme to solve a special case of isotropic 3-wave kinetic equations. We test our numerical solution against theoretical results concerning the long time behavior of the energy and observe that our solutions verify the energy cascade phenomenon. To our knowledge, this is the first numerical scheme that can capture the long time asymptotic behavior of solutions to those isotropic 3-wave kinetic equations, where the energy cascade can be observed. Our numerical energy cascade rates are in good agreement with previously obtained theoretical results. The finite volume scheme given here relies on a new identity, allowing one to reduce the number of terms needed in the collision operators.
Code verification plays an important role in establishing the credibility of computational simulations by assessing the correctness of the implementation of the underlying numerical methods. In computational electromagnetics, the numerical solution to integral equations incurs multiple interacting sources of numerical error, as well as other challenges, which render traditional code-verification approaches ineffective. In this paper, we provide approaches to separately measure the numerical errors arising from these different error sources for the method-of-moments implementation of the combined-field integral equation. Here, we demonstrate the effectiveness of these approaches for cases with and without coding errors.
A theoretical and experimental study of slender tubular columns for possible use in space structures is conducted in the presence of partial rotational end restraints. Explicit formulas are derived for computing the buckling load and the lowest natural frequency of perfectly straight uniform elastic members with rotational end restraints possessing linear moment-rotation characteristics. An exact solution in the form of a transcendental equation, and a numerical solution using second-order finite-differences are also presented. The presence of an initial imperfection is also incorporated into the numerical procedure. Vibration tests are conducted on an imperfect tubular steel member in the absence of an axial load. A damping concept consisting of a string-mass assembly is explored. Three passive damping configurations involving combinations of three lead shots were investigated. The three lead shot configurations provided considerably greater damping than the single lead shot.
While several results have pointed to the existence of exactly quasisymmetric fields on a surface (Garren & Boozer, Phys. Fluids B, vol. 3, 1991, pp. 2805–2821; 2822–2834; Plunk & Helander, J. Plasma Phys. , vol. 84, 2018, 905840205), we have obtained the first such solutions using a vacuum surface expansion formalism. We obtain a single nonlinear parabolic partial differential equation for a function $\eta$ such the field strength satisfies $B = B(\eta )$ . Closed-form solutions are obtained in cylindrical, slab and isodynamic geometries. Numerical solutions of the full nonlinear equations in general axisymmetric toroidal geometry are obtained, resulting in a class of quasihelical local vacuum equilibria near an axisymmetric surface. The analytic models provide additional insight into general features of the nonlinear solutions, such as localization of the surface perturbations on the inboard side. The local solutions thus obtained can be continued globally only for special initial surfaces.
Calculation procedures for non-reacting compressible two- and three-dimensional turbulent boundary layers were reviewed. Integral, transformation and correlation methods, as well as finite difference solutions of the complete boundary layer equations summarized. Alternative numerical solution procedures were examined, and both mean field and mean turbulence field closure models were considered. Physics and related calculation problems peculiar to compressible turbulent boundary layers are described. A catalog of available solution procedures of the finite difference, finite element, and method of weighted residuals genre is included. Influence of compressibility, low Reynolds number, wall blowing, and pressure gradient upon mean field closure constants are reported.
Numerical solutions for two-dimensional, time-dependent, separated flows around bodies are obtained, using a new version of the vortex method. This method provides an efficient representation of flows involving large regions of separation. The modifications incorporated in the new version, which improve its accuracy, versatility, and computing speed, are described. The computer cost is only of the order of the 3/2 power on N, instead of N-squared, for each step with N vortices. Arbitrary shapes can be treated; a conformal mapping is not required. Special attention is paid to the viscous character of the solution and to the accurate computation of the pressure distribution at the body surface. The vortex solution for the outer flow is coupled to an inner solution for the attached part of the boundary layer. Numerical results are presented for several bluff bodies exhibiting dependence on Reynolds number, for stationary airfoils under steady or transient conditions and for oscillating airfoils, including dynamic stalls. These results are compared with other available results, analytical or experimental, and demonstrate the enhanced reliability and accuracy of the improved method.
A linear unsteady potential flow analysis, which accounts for the effects of blade geometry and steady turning, is being developed to predict the aerodynamic response to blade vibrations in the fan, compressor or turbine stages of modern jet engines. In previous work numerical solutions were restricted to cascades of sharp-edged blades aligned with the mean flow. Under the present effort the solution procedure has been extended to treat blades with rounded or blunt edges. As part of this effort an analytical model-problem study has been conducted to clarify the behavior of first-order perturbation solutions in the vicinity of airfoil edges. Further, a numerical approximation using concepts from singular perturbation theory has been developed to resolve the unsteady boundary value problem for cascades of blunt leading-edged blades. Numerical results for NACA 0012 cascades, including detailed results in the vicinity of a blade leading edge, are presented and evaluated.