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Aerospace Engineering

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#### Solution of viscous flow problems by using the boundary element method

A scheme based on the Boundary Element Method (BEM) for solving the problem of steady flow of an incompressible viscous fluid is presented in this thesis. The problem is governed by both Navier-Stokes (N-S) equations and the continuity equation. The fundamental solution of the two-dimensional N-S is derived, and the partial differential equations are converted to an integral equation;The computer code is flexible enough to handle a variety of boundary and domain elements with different degrees of interpolation polynomial. Boundary and domain integrals over corresponding elements are evaluated analytically. The Newton Raphson iteration scheme accompanied by a relaxation factor is used to solve the nonlinear equations. The code includes a post processor that calculates the velocity components at any point inside the domain;The scheme has been applied to three test problems. The first concerns Couette flow, which has been used as a test case for testing the rate of convergence and accuracy. The second and the third concern the driven cavity and the flow in a stepped channel, respectively;In the integral equation formulation, the primary unknowns are tractions on the domain boundary and velocities in the interior. Because the shear stress, drag, and lift can be simply computed from the values of tractions along the boundary, such a formulation is markedly superior to either the finite-difference or the finite-element formulation. In customary pressure-velocity or streamfunction-vorticity formulations, employed in the finite-difference or finite-element methods, calculation of stress, drag, and lift involves extensive postprocessing.

#### Numerical solution of inverse problems in nondestructive evaluation using the boundary element method and multivariate adaptive regression splines

Flaw identification is an important inverse problem that underlies techniques for nondestructive evaluation (NDE). In this study, a known steady state thermal field is used to identify multiple flaws in a material. The problem is to determine locations and sizes of the multiple flaws if the number of flaws and the temperature at certain probe locations on the boundary are known. The boundary element method (BEM) is used as a computational tool in this task;Earlier work in this area has dealt with the case of a single flaw, while we address the case of multiple flaws. The identification of the multiple flaws is difficult because it is impossible to identify the disturbances caused by each individual fLaw; As a result, the iterative methods, used in the single flaw identification, typically fail to converge unless approximate locations of the multiple flaws are known;In our method, the characterization of flaws is performed in two stages. First, the specimen probe data is compared with a set of known cases of probe data (training set) to predict the approximate locations and sizes of the multiple flaws. Second, the final prediction of flaws is determined using a nonlinear optimization method;To prepare the training set, we need only the information of a single flaw of fixed size at various locations. The superposition principle and a special scaling are used to create the multiple flaws information. This procedure is developed as an extension of the theory of potential flows in fluid mechanics. The distinguishing feature of this technique is that only a small training set is stored in the memory;In this study, the final characterizations are made by two different methods. One of them is an iteration method, which minimizes an error functional. The other is called the multivariate adaptive regression splines (MARS). Various test cases yielded excellent solutions. The tolerance of both methods to experimental errors is also discussed. It is found that the iterative method performs better than MARS.

#### Unsteady incompressible viscous flow past stationary, pitching or oscillating airfoil leading edges

The objective of this study is to obtain a better understanding of the fundamental mechanisms governing the unsteady flow past airfoil leading edges. The approach taken is to study the leading edge in isolation and to use the semi-infinite parabola as a model for the leading-edges of conventional airfoils. Numerical solution methods were developed and implemented for the two-dimensional, unsteady, incompressible Navier-Stokes and boundary-layer equations for arbitrary motion of the parabola. Navier-Stokes solutions of the impulsively-started flow past a stationary parabola and of the flow past a pitching parabola compare well with the corresponding computational results for the NACA0012 airfoil. Navier-Stokes solutions for the pitching leading edge were obtained for chord Reynolds numbers up to half-a-million. The sequence of events leading to the unsteady breakaway of the boundary layer, in both the impulsive and pitchup cases, was qualitatively similar for the range of Reynolds numbers considered;We show using Navier-Stokes simulations that small perturbations in the flow field can lead to the formation of eddies in the boundary layer before flow reversal occurs in the base flow. The cases considered here are impulsive changes in the angle of attack, smooth but rapid variations in the angle of attack and introduction of small-amplitude inviscid vortices in the freestream. This type of eddy creation prior to base-flow reversal is a feature of the high-frequency Rayleigh instability. A study of the Reynolds-number scaling of the wavelength of these instabilities yielded a value reasonably close to that predicted by theory. A linear stability analysis of the boundary layer over the parabola was carried out to map the neutral curve for the Rayleigh instability. Preliminary indications are that the disturbances that lead to the eddies in the Navier-Stokes simulations are being initiated within the linearly unstable region bounded by the neutral curve. The linear stability analysis showed that the Rayleigh instability occurs a little after boundary layer velocity profiles become inflectional but much before flow reversal sets in.

#### The effect of rib attachment properties on structural acoustics

Sound Pressure measurements were taken of several 2.44m long thin beams which had 0.635cm wide t-ribs centered on them. The beams were excited by a chirp signal with the frequency range of 500Hz to 1500Hz. Two rib attachment methods were used on the beams. One group of beams were machined out of thick stock to the geometry of a rib welded to a beam to ensure that the rib and attachment were one continuous media. The other group of beams had ribs that were welded on. Fillets from the welding process were subjected to heat-treatment and machining to determine the effect of static stress in fillets. Farfield sound radiation from the beam and phase speeds of waves propagating through the beam were used to investigate the effects of the rib and its attachment properties on the beam response;The experimental results showed that the geometry and stress state of the attachment are the main parameters that alter wave propagation and sound radiation. Also, the maximum sound radiation from the rib was not centered directly over the rib, but rather ahead of the rib location. Furthermore, substantial phase speed increases were observed around the shaker location and the rib location;Attachment geometry and stress information were incorporated into a Euler-Bernoulli wave-based model. This wave-based model reproduced the static stress effect on the farfield sound radiation, but didn't reproduce the position of the sound radiation from the rib nor the experimental phase speed increases around the shaker or rib locations;An energy-based model was derived that included the geometry of the rib and attachment. The model used the Extended Hamilton's Principle. Cubic spline weighting functions were applied via Galerkin's method of weighted residuals. This energy-based model did reproduce both the position and the magnitude of the peak in the sound radiation field from the rib. It also reproduced the phase speed increases around the shaker and the rib locations.

#### Computation of turbulent flow about unconventional airfoil shapes

A new nonequilibrium turbulence closure model has been developed for computing wall bounded two-dimensional turbulent flows. This two-layer eddy viscosity model was motivated by the success of the Johnson-King model in separated flow regions. The influence of history effects are described by an ordinary differential equation developed from the turbulent kinetic energy equation. The performance of the present model has been evaluated by solving the flow around three airfoils using the Reynolds time-averaged Navier-Stokes equations. Excellent results were obtained for both attached and separated flows about the NACA 0012 airfoil, the RAE 2822 airfoil, and the Integrated Technology A 153W airfoil. Based on the comparison of the numerical solutions with the available experimental data, it is concluded that the new nonequilibrium turbulence model accurately captures the history effects of convection and diffusion on turbulence.

#### An interacting boundary layer method for unsteady compressible flows

A time accurate compressible interactive boundary layer procedure for airfoils using the quasi-simultaneous method of Veldman is developed. It couples the high frequency transonic small disturbance equation with the complete set of unsteady compressible boundary equations in Levy-Lees variable form, using a pseudo-time derivative of displacement thickness for enhanced stability. Included is a simple procedure for time accurately updating the viscous wake location. The basis of the interaction is an extension of the asymptotic matching condition of Davis for unsteady compressible interaction. This analysis identifies several possible unsteady transonic separation structures and highlights the importance of the pseudo-time derivative in stabilizing the interaction. The method is applied to oscillating airfoils experiencing light shock-induced stall. Comparisons are made with several standard turbulence models. Shock-induced oscillatory flow about the 18% circular arc airfoil is investigated with this method and found to be modeled quite accurately.

#### A numerical investigation of laminar airfoil stall

The details of an interacting boundary layer algorithm capable of calculating large scale laminar separation past airfoils at low speeds is given. Rationale behind various convergence acceleration methods is given. It is shown that linear based acceleration methods are limited to 50% savings in convergence rate. A nonlinear extrapolation method is proposed and tested on two simple model problems. Savings exceed the 50% limitation of the previous methods. Boundary layer results for laminar flow past symmetric airfoils at zero incidence are presented as a test of the methods. Leading edge marginal separation results at finite Reynolds numbers are presented. Richardson extrapolation of successive calculations is used to improve accuracy. Results for a zero thickness uncambered plate at angle of attack are presented.

#### Effects of ballistic impact damage on thin and thick composites

Use of composite materials is growing rapidly because they offer weight reduction, durability and better fatigue life as compared to metals. Despite of their advantages, there are some problems associated with the use of composites, and impact resistance is one of the weak areas in composites. Low speed impact damage has been the focus of attention of researchers for quite some time but studies of high speed impact damage is still in its preliminary stages. In this study, effects of ballistic impact on thin and thick composite panels were investigated. Damage due to high speed impact in composites is a complex phenomenon and it is difficult to analytically model all the events taking place during the impact;An experimental approach was used to study the damage from high speed impact and the Finite Element analysis was performed to understand the damage sequence. Thin symmetric quasi-isotropic and cross-ply laminates consisting of up to 16 layers were studied using experimental vibration and Finite Element analysis. It was found that frequency response of damaged plates is dependent on the stacking sequence of the laminate. Results also show that the natural frequencies of the damaged plates decrease for few initial modes and increase for some of the higher modes. It was observed that cross-ply laminates exhibit very little effect on the natural frequencies due to damage;Thick symmetric quasi-isotropic and cross-ply laminates consisting of 56 plies were also studied. Results demonstrate that damage size is dependent more on the shape and size of the impactor, rather than the impact energy. Faster bullets cause less damage as compared to slower bullets. Damage was found to be dependent on stacking sequence of the laminate. Cross-ply laminates suffer more damage than the quasi-isotropic laminates. Damage is dependent on the thickness of the laminate and is more in thicker laminates. Damage is always more towards the exit side than the entry surface;A model of bullet penetration into a composite laminate is presented and the failure due to inter-laminar shear stresses was explained through this model. The model was verified using a quasi-static Finite Element analysis. It was demonstrated that outer-most ply of the laminate fails first and maximum inter-laminar shear stresses occur between two outer-most plies, causing delamination. It was also demonstrated that inter-laminar shear stresses increase progressively, as the number of effective plies in the laminate reduce due to failure and it is strongly dependent on the stacking sequence.

#### Modeling of ultrasonic scattering experiments with applications to system and transducer characterization

The voltage signal output by the receiver electronics, which represents the observable quantity in an ultrasonic scattering experiment, is written as a product, in the frequency domain, of two factors: the system efficiency and the scattering coefficient. The system efficiency represents the combined electrical properties of both the generator and receiver electronics and is a function of frequency only. The scattering coefficient represents the acoustic nature of the experiment (the radiation, propagation, scattering and reception of ultrasonic waves) and depends on the distributed field properties of the transducers involved and their locations and orientations, on the number and type of scattering obstacles and their locations and orientations, on the acoustic properties of the media through which the waves travel, and on the nature and shape of any interfaces through which the waves pass. Based on a generalized principle of electroacoustic reciprocity, formulae are developed for the evaluation of the scattering coefficient. The most general of these involve an integration over either the volume or the surface of the scattering obstacle. More specific formulae are also developed which express the scattering coefficient in terms of either the spherical wave transition matrix or the plane wave scattering amplitude of the obstacle;In order to demonstrate the use of the formulae developed, the calculation of the scattering coefficient is considered for two common ultrasonic scattering experiments. The first experiment involves the pulse-echo scattering from an infinite, flat elastic plate immersed in water. This arrangement is often used for the measurement of the velocity and attenuation of elastic waves, and also as a reference experiment for the determination of the system efficiency. The second experiment involves the pulse-echo scattering from an elastic sphere immersed in water. Particular attention is given to the specular reflection component of the scattering, which is demonstrated to be approximately equivalent to a point measurement of the pressure field radiated by the transducer. This approximation is subsequently used as the basis for obtaining experimental data for transducer characterization. The characterization itself is based on expanding in a set of basis functions, each weighted by an unknown coefficient, the normal velocity profile across the plane flush with the face of the probe. Values for the coefficients are obtained by determining the best fit between the experimental pressure data and the pressure calculated from the assumed velocity profile. Results are presented for two commercially manufactured immersion transducers, one planar (unfocused) and the other focused.

#### Design, analysis, and modeling of giant magnetostrictuve transducers

The increased use of giant magnetostrictive, Terfenol-D transducers in a wide variety of applications has led to a need for greater understanding of the materials performance. This dissertation attempts to add to the Terfenol-D transducer body of knowledge by providing an in-depth analysis and modeling of an experimental transducer. A description of the magnetostriction process related to Terfenol-D includes a discussion of material properties, production methods, and the effect of mechanical stress, magnetization, and temperature on the material performance. The understanding of the Terfenol-D material performance provides the basis for an analysis of the performance of a Terfenol-D transducer. Issues related to the design and utilization of the Terfenol-D material in the transducers are considered, including the magnetic circuit, application of mechanical prestress, and tuning of the mechanical resonance. Experimental results from two broadband, Tonpilz design transducers show the effects of operating conditions (prestress, magnetic bias, AC magnetization amplitude, and frequency) on performance. In an effort to understand and utlilize the rich performance space described by the experimental results a variety of models are considered. An overview of models applicable to Terfenol-D and Terfenol-D transducers is provided, including a discussion of modeling criteria. The Jiles-Atherton model of ferromagnetic hysteresis is employed to describe the quasi-static transducer performance. This model requires the estimation of only six physically-based parameters to accurately simulate performance. The model is shown to be robust with respect to model parameters over a range of mechanical prestress, magnetic biases, and AC magnetic field amplitudes, allowing predictive capability within these ranges. An additional model, based on electroacoustics theory, explains trends in the frequency domain and facilitates an analysis of efficiency based on impedance and admittance analysis. Results and discussion explain the importance of the resonance of the electromechanical system, as distinct from the mechanical resonance. Conclusions are drawn based on the experimental work, transducer analysis, and modeling approaches.