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The direct analytical solution to the vibration of non-uniform beams with and without discontinuities and with various boundary conditions is presented. Results are compared to results from the exact solution for certain cases where the exact solution has been obtained. It is shown that the direct solution converges to the exact solution, in fact, with 'indefinite accuracy' just as Hamilton stated that it would.
A fourth-order accurate numerical solution algorithm is derived using finite element interpolation theory for the non-linear parabolic equations governing turbulent boundary layer flow including a two-equation turbulence closure model. The results of carefully controlled numerical experiments firmly quantize for the first time performance differences between finite element and finite difference solution methodology for this type of equation. The developed algorithm takes advantage of the apparent semi-analytical formulational procedure, in establishment of a single, retarded-evaluation Jacobian matrix iterative solution algorithm. Numerical results document performance of solution economy features in terms of computer requirements and solution accuracy. The developed algorithm should find wide application in aerodynamics analysis.
A previous global solution for the earth's gravitational field and station positions employed a combination of 21 weeks of Doppler tracking data from seven satellites, a complete set of 5-deg mean gravity anomalies, and geometric constraints from the worldwide satellite triangulation network on 17 of the stations. To this is now added geoid heights over the major part of the oceans obtained from GEOS 3 altimeter observations. The solution has a form based on the simple density-layer method and is equivalent to a solution in spherical harmonics to the tenth degree and order. The rms difference of 5-deg mean geoid heights of the new solution with the GEM 10 solution is 5.6 m.
Laminar and turbulent flow-field solutions with coupled carbon-phenolic mass injection are presented for the forebody of a probe entering a nominal Jupiter atmosphere. Solutions are obtained for a 35-degree hyperboloid and for a 45-degree spherically blunted cone using a time-dependent, finite-difference method. The radiative heating rates for the coupled laminar flow are significantly reduced as compared to the corresponding no-blowing case; however, for the coupled turbulent flow, it is found that the surface radiative heating rates are substantially increased and often exceed the corresponding no-blowing values. Turbulence is found to have no effect on the surface radiative heating rates for the no-blowing solutions. The present results are compared with the other available solutions, and some additional solutions are presented.
Laminar and turbulent flowfield solutions with coupled carbon-phenolic mass injection are presented for the forebody of a probe entering a nominal Jupiter atmosphere. Solutions are obtained for a 35-deg hyperboloid and for a 45-deg spherically blunted cone using a time-dependent finite-difference method. The radiative heating rates for the coupled laminar flow are significantly reduced as compared to the corresponding no-blowing case; however, for the coupled turbulent flow, it is found that the surface radiative heating rates are substantially increased and often exceed the corresponding no-blowing values. Turbulence is found to have no effect on the surface radiative heating rates for the no-blowing solutions. The present results are compared with the other available solutions and some additional solutions are presented.
Methods which yield closed form performability solutions for continuous valued variables are developed. The models are similar to those employed in performance modeling (i.e., Markovian queueing models) but are extended so as to account for variations in structure due to faults. In particular, the modeling of a degradable buffer/multiprocessor system is considered whose performance Y is the (normalized) average throughput rate realized during a bounded interval of time. To avoid known difficulties associated with exact transient solutions, an approximate decomposition of the model is employed permitting certain submodels to be solved in equilibrium. These solutions are then incorporated in a model with fewer transient states and by solving the latter, a closed form solution of the system's performability is obtained. In conclusion, some applications of this solution are discussed and illustrated, including an example of design optimization.
A new approach to the solution of matrix equations resulting from integral equations is presented and applied to the solution of two-dimensional Neumann problems describing the inviscid, incompressible flow past an airfoil. The problem is reformulated in terms of a preselected set of mode functions giving an equivalent matrix equation to be solved for the mode-function expansion coefficients. Because of the inherent smoothness of the original problem, the coefficient problem can be solved approximately without significantly affecting the accuracy of the final solution. Very promising two-dimensional results are obtained and the extension of the method to three-dimensional problems is investigated. On the basis of these results it is shown that the computing time for the matrix solution for a large three-dimensional panel method calculation could be reduced by an order of magnitude compared with that required for a direct solution.
Heat transfer in cavity flow is numerically analyzed by a new numerical method called the finite-analytic method. The basic idea of the finite-analytic method is the incorporation of local analytic solutions in the numerical solutions of linear or nonlinear partial differential equations. In the present investigation, the local analytic solutions for temperature, stream function, and vorticity distributions are derived. When the local analytic solution is evaluated at a given nodal point, it gives an algebraic relationship between a nodal value in a subregion and its neighboring nodal points. A system of algebraic equations is solved to provide the numerical solution of the problem. The finite-analytic method is used to solve heat transfer in the cavity flow at high Reynolds number (1000) for Prandtl numbers of 0.1, 1, and 10.
The fundamental nature of the boundary-layer effect in fiber-reinforced composite laminates is formulated in terms of the theory of anisotropic elasticity. The basic structure of the boundary-layer field solution is obtained by using Lekhnitskii's stress potentials (1963). The boundary-layer stress field is found to be singular at composite laminate edges, and the exact order or strength of the boundary layer stress singularity is determined using an eigenfunction expansion method. A complete solution to the boundary-layer problem is then derived, and the convergence and accuracy of the solution are analyzed, comparing results with existing approximate numerical solutions. The solution method is demonstrated for a symmetric graphite-epoxy composite.
Compositional segregation of solid solution semiconducting alloys in the radial direction during unidirectional solidification was investigated by calculating the effect of a curved solid liquid interface on solute concentration at the interface on the solid. The formulation is similar to that given by Coriell, Boisvert, Rehm, and Sekerka except that a more realistic cylindrical coordinate system which is moving with the interface is used. Analytical results were obtained for very small and very large values of beta with beta = VR/D, where V is the velocity of solidification, R the radius of the specimen, and D the diffusivity of solute in the liquid. For both very small and very large beta, the solute concentration at the interface in the solid C(si) approaches C(o) (original solute concentration) i.e., the deviation is minimal. The maximum deviation of C(si) from C(o) occurs for some intermediate value of beta.
Functions for the representation of the thermodynamic properties of nonideal solutions were applied to the experimental data for several highly nonideal solutions. The test solutions were selected to cover both electrolyte behavior. The results imply that the functions are fully capable of representing the experimental data within their accuracy over the whole composition range and demonstrate that many nonideal solutions can be regarded as members of the defined class of nonideal solutions.
A class of nonideal solutions is defined by constructing a function to represent the composition dependence of thermodynamic properties for members of the class, and some properties of these solutions are studied. The constructed function has several useful features: (1) its parameters occur linearly; (2) it contains a logarithmic singularity in the dilute solution region and contains ideal solutions and regular solutions as special cases; and (3) it is applicable to N-ary systems and reduces to M-ary systems (M or = N) in a form-invariant manner.
The accuracy of the finite difference method in the solution of linear elasticity problems that involve either a stress discontinuity or a stress singularity is considered. Solutions to three elasticity problems are discussed in detail: a semi-infinite plane subjected to a uniform load over a portion of its boundary; a bimetallic plate under uniform tensile stress; and a long, midplane symmetric, fiber reinforced laminate subjected to uniform axial strain. Finite difference solutions to the three problems are compared with finite element solutions to corresponding problems. For the first problem a comparison with the exact solution is also made. The finite difference formulations for the three problems are based on second order finite difference formulas that provide for variable spacings in two perpendicular directions. Forward and backward difference formulas are used near boundaries where their use eliminates the need for fictitious grid points.
Nonunique laminar boundary layer equation solutions in direct problem calculations are identified for the case of accelerating flow. As a separation or reattachment point is approached, the multiple solutions approach each other and become identical. The computer code used to generate these results was developed for the solution of compressible, laminar or turbulent boundary layer, and free wake problems, in either direct or inverse mode. Similarity solutions in either a primitive variable or a stream function form are possible, and the resulting equations are solved by means of a modified Keller's Box scheme in which the energy equation and turbulence modeling equations are solved simultaneously with the continuity and momentum equations. Examples illustrating the nature of the solutions at the separation and reattachment points are presented.
A study on the fundamental mechanics of fiber-reinforced composite laminates with stress singularities is presented. Based on the theory of anisotropic elasticity and Lekhnitskii's complex-variable stress potentials, a system of coupled governing partial differential equations are established. An eigenfunction expansion method is introduced to determine the orders of stress singularities in composite laminates with various geometric configurations and material systems. Complete elasticity solutions are obtained for this class of singular composite laminate mechanics problems. Homogeneous solutions in eigenfunction series and particular solutions in polynomials are presented for several cases of interest. Three examples are given to illustrate the method of approach and the basic nature of the singular laminate elasticity solutions. The first problem is the well-known laminate free-edge stress problem, which has a rather weak stress singularity. The second problem is the important composite delamination problem, which has a strong crack-tip stress singularity. The third problem is the commonly encountered bonded composite joints, which has a complex solution structure with moderate orders of stress singularities.
The solution of the three dimensional flow field for a flow through nacelle was studied. Both inviscid and viscous inviscid interacting solutions were examined. Inviscid solutions were obtained with two different computational procedures for solving the three dimensional Euler equations. The first procedure employs an alternating direction implicit numerical algorithm, and required the development of a complete computational model for the nacelle problem. The second computational technique employs a fourth order Runge-Kutta numerical algorithm which was modified to fit the nacelle problem. Viscous effects on the flow field were evaluated with a viscous inviscid interacting computational model. This model was constructed by coupling the explicit Euler solution procedure with a flag entrainment boundary layer solution procedure in a global iteration scheme. The computational techniques were used to compute the flow field for a long duct turbofan engine nacelle at free stream Mach numbers of 0.80 and 0.94 and angles of attack of 0 and 4 deg.
In a low-g environment, buoyancy driven convection effects in solution crystal growth are greatly reduced and, thus, one can study diffusion mass transport which in 1-g is masked by convective phenomena. Crystals of triglycine sulfate (TGS) will be grown aboard the Spacelab 3 mission, using a specially developed Fluids Experiments System (FES). The objectives of the experiment are: (1) to develop a technique for solution crystal growth in a low-g environment, (2) to characterize the growth environment provided by an orbiting spacecraft and to determine the influence of the environment on the growth behavior, and (3) to determine how gravity in a low-gravity environment influences the properties of a resulting TGS crystal. Single crystals of TGS have been grown using conventional low-temperature solution crystal growth method and the growth process has been extensively characterized. Various physical properties of TGS solution have been measured. Also, a unique technique of growing solution growth crystals by extracting heat at a programmed rate from the crystal through a semi-insulating sting has been developed and tested in 1-g environment.
Steady solutions about a slender sharp edged delta wing in a supersonic freestream for moderate and high angles of attack are obtained numerically by time integration of the unsteady compressible three dimensional laminar Navier-Stokes equations. The main features of the flow, including primary and secondary separation, and vortex position and strength, are adequately simulated in the numerical solutions. Improved resolution of the computational grid in the leading edge region from a previous solution had considerable effect on the accuracy of the solutions. Good agreement between numerical solutions and experimental data was obtained for two cases. A local timestepping procedure is used to speed convergence by approximately a factor of two.