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The effects of aerodynamic forces on trajectories at orbital speeds are discussed in terms of atmospheric models. The assumptions for the model are spherical symmetry, nonrotating, and an exponential atmosphere. The equations of flight, and the performance in extra-atmospheric flight are discussed along with the return to the atmosphere, and the entry. Solutions of the exact equations using directly matched asymptotic expansions are presented.
A central crack, symmetrically growing at a constant speed in a two dimensional ductile material subject to uniform tension at infinity, is investigated using the integral transform methods. The crack is assumed to be the Dugdale crack, and the finite stress condition at the crack tip is satisfied during the propagation of the crack. Exact expressions of solution are obtained for the finite stress condition at the crack tip, the crack shape, the crack opening displacement, and the energy release rate. All those expressions are written as the product of explicit dimensional quantities and a nondimensional dynamic correction function. The expressions reduce to the associated static results when the crack speed tends to zero, and the nondimensional dynamic correction functions were calculated for various values of the parameter involved.
The development of an analytical technique used in the solution of nonlinear velocity-sensitive combustion instability problems is presented. The Galerkin method was used and proved successful. The pressure wave forms exhibit a strong second harmonic distortion and a variety of behaviors are possible depending on the nature of the combustion process and the parametric values involved. A one dimensional model provides insight into the problem by allowing a comparison of Galerkin solutions with more exact finite difference computations.
The wave tilt of a transverse electric (TE) electromagnetic wave over a linearly inhomogeneous lossy layer overlying a homogeneous half-space is studied. Two approaches are used: an exact formulation using solutions of Airy's equation and an approximate numerical solution using a large number of homogeneous layers with a linearly increasing dielectric constant. The numerical results of both solutions are practically identical as long as the thickness of the layers in the approximate model are somewhat smaller than a quarter-wave length.
The development of a technique to be used in the solution of nonlinear velocity-sensitive combustion instability problems is described. The orthogonal collocation method was investigated. It found that the results are heavily dependent on the location of the collocation points and characteristics of the equations, so the method was rejected as unreliabile. The Galerkin method, which has proved to be very successful in analysis of the pressure sensitive combustion instability was found to work very well. It was found that the pressure wave forms exhibit a strong second harmonic distortion and a variety of behaviors are possible depending on the nature of the combustion process and the parametric values involved. A one-dimensional model provides further insight into the problem by allowing a comparison of Galerkin solutions with more exact finite-difference computations.
Higher order mode propagation in a nonuniform circular duct without mean flow was investigated. An approximate wave equation is derived on the assumptions that the duct cross section varies slowly and that mode conversion is negligible. Exact closed form solutions are obtained for a particular class of converging-diverging circular duct which referred to as 'circular cosh duct.' Numerical results are presented in terms of the transmission loss for the various duct shapes and frequencies. The results are applicable to multimodal propagation, single mode propagation, and sound radiation from certain types of contoured inlet ducts, or of sound propagation in a converging-diverging duct of somewhat different shape from a cosh duct.
This paper presents an analytical investigation of higher order mode propagation in a nonuniform circular duct without mean flow. An approximate wave equation is derived on the assumptions that the duct cross section varies slowly and that mode conversion is negligible. Exact closed form solutions are obtained for a particular class of converging-diverging circular duct which is here referred to as 'circular cosh duct'. Numerical results are presentd in terms of the transmission loss for the various duct shapes and frequencies. The results are applicable to studies of multimodal propagation as well as single mode propagation. The results are also applicable to studies of sound radiation from certain types of contoured inlet ducts, or of sound propagation in a converging-diverging duct of somewhat different shape from a cosh duct.
The behavior of an isolated, stationary, multicomponent gas bubble in a glassmelt containing several dissolved gases is considered. The relevant mass-transport equations are formulated and calculations are performed for the case of two diffusing gases using a quasi-stationary model and a numerical solution of the exact mass-transfer equations. The results obtained from these two approaches are compared. The factors which govern the dissolution or growth of a bubble are thermodynamic and kinetic in origin. The tendency of a bubble to grow or shrink at long times is controlled by departure from overall equilibrium, whereas the short-time bubble dynamics may be dominated by kinetic effects. As a result of the existence of these dual influences, maxima and/or minima occur in the functional dependence of the bubble radius on time.
Computational procedures that would be useful in finding the Fresnel field from a knowledge of the Jacobi-Bessel expansion of the far field are considered. The range of validity of the Fresnel approximation is carefully examined by comparing it with the exact closed form solution for the uniform circular aperture. Also investigated numerically, and in great detail, is the range of validity (over theta) of the Fresnel small angle (FSA) approximation. For moderate sized apertures as small as 10 wavelengths, it is found that the FSA approximation is very accurate to angles as wide as four or more sidelobes (as seen in the far zone). A very efficient computational method is shown to exist for the radiation integral in the form of a single series expansion that is analytically continuous and convergent for a wide range of observation points in three-dimensional space.
A method is presented for computing the unsteady three-dimensional transonic flowfield about twin wings of arbitrary planform and section geometry including the effects of inviscid/viscous interaction. Dominant effects of the shock-boundary-layer interaction are accounted for by a simple empirically defined model, while viscous regions adjacent to the wing surface and along the trailing wake are described by a set of integral equations appropriate for two-dimensional turbulent flows. The validity of the proposed method is verified by comparison with steady three-dimensional experimental data and a solution of more exact equations for a typical transport wing configuration.
Most fluid flows are turbulent rather than laminar and the reason for this was studied. One of the earliest explanations was that laminar flow is unstable, and the linear instability theory was first developed to explore this possibility. A series of early papers by Rayleigh produced many notable results concerning the instability of inviscid flows, such as the discovery of inflectional instability. Viscosity was commonly thought to act only to stabilize the flow, and flows with convex velocity profiles appeared to be stable. The investigations that led to a viscous theory of boundary layer instability was reported. The earliest application of linear stability theory to transition prediction calculated the amplitude ratio of the most amplified frequency as a function of Reynolds number for a Blasius boundary layer, and found that this quantity had values between five and nine at the observed Ret. The experiment of Schubauer and Skramstad (1947) completely reversed the prevailing option and fully vindicated the Gottingen proponents of the theory. This experiment demonstrated the existence of instability waves in a boundary layer, their connection with transition, and the quantitative description of their behavior by the theory of Tollmien and Schlichting. It is generally accepted that flow parameters such as pressure gradient, suction and heat transfer qualitatively affect transition in the manner predicted by the linear theory, and in particular that a flow predicted to be stable by the theory should remain laminar. The linear theory, in the form of the e9, or N-factor is today in routine use in engineering studies of laminar flow. The stability theory to boundary layers with pressure gradients and suction was applied. The only large body of numerical results for exact boundary layer solutions before the advent of the computer age by calculating the stability characteristics of the Falkner-Skan family of velocity profiles are given. When the digital computer reached a stage of development which permit the direct solution of the primary differential equations, numerical results were obtained from the linear theory during the next 10 years for many different boundary layer flows: three dimensional boundary layers; free convention boundary layers; compressible boundary layers; boundary layers on compliant walls; a recomputation of Falkner-Skan flows; unsteady boundary layers; and heated wall boundary layers.
A study of electromagnetic satellite de-spin system for orbiting satellites was conducted. Exact closed form solutions for the interacting magnetic flux have been formulated both for the case of the current loop and the bar magnet. The magnet system has been extended to the case of a U shaped magnet. Magnet design details are outlined. Computer simulation of the interacting flux density profile has been attempted. Data for the de-spin times has been extended to higher rpms.
A numerical algorithm is presented which is used to solve the unsteady, fully three-dimensional, incompressible Navier-Stokes equations in vorticity-velocity variables. A discussion of the discrete approximation scheme is presented as well as the solution method used to solve the resulting algebraic set of difference equations. Second order spatial and temporal accuracy is verified through solution comparisons with exact results obtained for steady three-dimensional stagnation point flow and unsteady axisymmetric vortex spin-up. In addition, results are presented for the problem of unsteady bubble-type vortex breakdown with emphasis on internal bubble dynamics and structure.
A review of numerical methods for one-dimensional reaction-diffusion equations arising in combustion theory is presented. The methods reviewed include explicit, implicit, quasi-linearization, time linearization, operator-splitting, random walk and finite-element techniques and methods of lines. Adaptive and nonadaptive procedures are also reviewed. These techniques are applied first to solve two model problems which have exact traveling wave solutions with which the numerical results can be compared. This comparison is performed in terms of both the wave profile and computed wave speed. It is shown that the computed wave speed is not a good indicator of the accuracy of a particular method. A fourth-order time-linearized, Hermitian compact operator technique is found to be the most accurate method for a variety of time and space sizes.
Using ideas from kinetic theory, the Navier-Stokes equations are modified in such a way that they can be cast as a set of first order hyperbolic equations. This is achieved by incorporating time dependent terms into the definition of the stress tensor and the heat flux vectors. The boundary conditions are then determined from the theory of characteristics. Because the resulting equations reduce to the traditional Navier-Stokes equations when the steady state is reached, the present approach provides a straightforward scheme for the determination of inflow and outflow boundary conditions. The method is validated by comparing its predictions with exact and numerical solutions of the steady Navier-Stokes equations.
This paper describes a study in which structural optimization techniques are used to minimize the n/rev vertical hub shears in forward flight, subject to aeroelastic stability constraints and frequency placement constraints. A special technique is used to build a sequence of approximate, inexpensive to solve optimization problems, the solutions of which converge to the solution of the exact, expensive to solve optimization problem. Blade configurations with both straight and swept tips, and single- and double-cell cross sections are analyzed. The results show that the approach used in this study is very efficient, and produces improved designs with a very small number of blade aeroelastic analyses.
A finite difference technique is used to solve the transonic small disturbance flow equation making use of shock capturing to treat wave discontinuities. Thus the nonlinear effects of thickness and angle of attack are considered. Such an approach is made feasible by the development of a new code called CAP-TSD (Computational Aeroelasticity Program - Transonic Small Disturbance), and is based on a fully implicit approximate factorization (AF) finite difference method to solve the time dependent transonic small disturbance equation. The application of the CAP-TSD code to the calculation of low to moderate supersonic steady and unsteady flows is presented. In particular, comparisons with exact linear theory solutions are made for steady and unsteady cases to evaluate shock capturing and other features of the current method. In addition, steady solutions obtained from an Euler code are used to evaluate the small disturbance aspects of the code. Steady and unsteady pressure comparisons are made with measurements for an F-15 wing model and for the RAE tailplane model.
Numerical flux functions for solving the Euler equations using exact and/or approximate solutions of the Riemann problem of gasdynamics are discussed. Under certain restrictive conditions, schemes using these flux functions produce systems of equations which can exhibit a single degree of freedom. In some instances, the solutions represented by this degree of freedom are unstable to perturbations. This local instability can seriously degrade the temporal convergence of numerical schemes. This point is demonstrated by numerical example.