(PHD, 2011)

Abstract:

In this work, several low-order models are derived to describe and simulate fluid-structure interaction problems with rigid bodies at a modest computational cost. The models are based on the inviscid flow assumption such that potential theory can be used with, in some cases, point vortices in the flow. Three general areas of application are considered. First, a thin airfoil undergoing small-scale unsteady motions in the presence of a freestream flow is investigated. The low-order model that is developed has only one ordinary differential equation for the fluid dynamic variables. This model is used to briefly investigate vortex-induced flutter in the attached-flow regime and control of a free-flying airfoil using synthetic jet actuators. Second, the vortex-induced vibrations of an arbitrary bluff body in the presence of vortices, with or without a freestream flow, are considered. Several examples of the canonical mass-spring-damper system for a circular cylinder and a flat plate are given to demonstrate the use of the vortex-based model for these applications. Finally, the two-body problem in a potential flow is addressed. A relatively simple solution specific to the doubly connected domain is determined and its resulting force and moment are coupled to the rigid bodies to investigate the mutual interactions between the two bodies. Aspects of drafting behind a forced body, the role of the fluid in elastic collision, and flapping flight are discussed in this context. Although a few specific examples and applications are given for each chapter, the main purpose of the thesis is to present low-order potential flow methods that are applicable to a variety of situations.

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(PHD, 2007)

Abstract:

The approximate dynamics of many physical phenomena, including turbulence, can be represented by dissipative systems of ordinary differential equations. One often turns to numerical integration to solve them. There is an incompatibility, however, between the answers it can produce (i.e., specific solution trajectories) and the questions one might wish to ask (e.g., what behavior would be typical in the laboratory?) To determine its outcome, numerical integration requires more detailed initial conditions than a laboratory could normally provide. In place of initial conditions, experiments stipulate how tests should be carried out: only under statistically stationary conditions, for example, or only during asymptotic approach to a final state. Stipulations such as these, rather than initial conditions, are what determine outcomes in the laboratory.

This theoretical study examines whether the points of view can be reconciled: What is the relationship between one’s statistical stipulations for how an experiment should be carried out–stationarity or asymptotic approach–and the expected results? How might those results be determined without invoking initial conditions explicitly?

To answer these questions, stationarity and asymptotic approach conditions are analyzed in detail. Each condition is treated as a statistical constraint on the system–a restriction on the probability density of states that might be occupied when measurements take place. For stationarity, this reasoning leads to a singular, invariant probability density which is already familiar from dynamical systems theory. For asymptotic approach, it leads to a new, more regular probability density field. A conjecture regarding what appears to be a limit relationship between the two densities is presented.

By making use of the new probability densities, one can derive output statistics directly, avoiding the need to create or manipulate initial data, and thereby avoiding the conceptual incompatibility mentioned above. This approach also provides a clean way to derive reduced-order models, complete with local and global error estimates, as well as a way to compare existing reduced-order models objectively.

The new approach is explored in the context of five separate test problems: a trivial one-dimensional linear system, a damped unforced linear oscillator in two dimensions, the isothermal Rayleigh-Plesset equation, Lorenz’s equations, and the Stokes limit of Burgers’ equation in one space dimension. In each case, various output statistics are deduced without recourse to initial conditions. Further, reduced-order models are constructed for asymptotic approach of the damped unforced linear oscillator, the isothermal Rayleigh-Plesset system, and Lorenz’s equations, and for stationarity of Lorenz’s equations.

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(PHD, 2007)

Abstract:

Vortex-induced vibrations have been studied experimentally with emphasis on damping and Reynolds number effects. Our system was an elastically-mounted rigid circular cylinder, free to oscillate only transverse to the flow direction, with very low inherent damping. We were able to prescribe the mass, damping, and elasticity of the system over a wide range of values, with the damping controlled by a custom-made variable magnetic eddy-current damping system.

Special emphasis is put on a nontraditional parameter formulation. The advantages of this formulation are explained, and an important new parameter, effective stiffness, is introduced. Using this new formulation, the amplitude and frequency responses are only a function of damping, Reynolds number, and effective stiffness. We show the effects that damping and Reynolds number each have on the amplitude and frequency response profiles and make the interesting observation that changes in damping or Reynolds number have similar effects.

The maximum amplitudes of our systems are studied in detail. We theoretically show that they should be functions of both damping and Reynolds number. This allows us to create constant-Reynolds-number curves of maximum amplitude over a large range of damping values, which we call a “generalized” Griffin plot. We also define maximum amplitudes in the case of zero damping as limiting amplitudes, and show that they are only a function of Reynolds number. We experimentally determine our limiting amplitude dependence on Reynolds number over the range 200 < Reynolds number < 5050.

Discontinuities in the amplitude response profile are also investigated. The discontinuity between the initial branch and the large-amplitude, upper branch is studied in two ways. First, the time-averaged behavior is examined to understand what controls the discontinuity and look for damping and Reynolds number effects. Second, we track the cycle-by-cycle transient response through this discontinuous amplitude change, induced either by changes in the tunnel velocity or system damping. Finally, we also find a new discontinuity hysteresis region between the lower branch and the desynchronized region, which appears to be a low Reynolds number effect and is only seen in systems with Reynolds number < 1000.

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(PHD, 2005)

Abstract:

Several contributions to the three-dimensional vortex element method for incompressible flows are presented. We introduce redistribution schemes based on the hexagonal lattice in two dimensions, and the face-centered cubic lattice in three dimensions. Interpolation properties are studied in the frequency domain and are used to build high-order schemes that are more compact and isotropic than equivalent cubic schemes. We investigate the reconnection of vortex rings at small Reynolds numbers for a variety of configurations. In particular, we trace their dissipative nature to the formation of secondary structures.

A method for flows with moving boundaries is implemented. The contributions of rotating or deforming boundaries to the Biot-Savart law are derived in terms of surface integrals. They are implemented for rigid boundaries in a fast multipole algorithm. Near-wall vorticity is discretized with attached panels. The shape function and Biot-Savart contributions of these elements account for the presence of the boundary and its curvature. A conservative strength exchange scheme was designed to compute the viscous flux from these panels to free elements.

The flow past a spinning sphere is studied for a Reynolds number of 300 and a wall velocity that is equal to half the free-stream velocity. Three directions of the angular velocity are considered. Good agreement with previous numerical and experimental measurements of the force coefficients is observed. Topological features such as the separation and critical points are investigated and compared amongst the configurations.

Finally, preliminary results for flapping motions are presented. Simple rigid geometries are used to model a fish swimming in a free-stream and a flapping plate.

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(PHD, 2004)

Abstract:

For the applications of high Reynolds number flows, the vortex method presents the advantage of being free from numerically dissipative truncation error. In practice, however, many vortex methods introduce some numerical dissipation in mesh-based spatial adaption stages, or schemes such as vortex particle splitting. The need for spatial adaption in vortex methods arises from the Lagrangian framework, which results in an increase of discretization error over time. Presently, a vortex method is devised that incorporates radial basis function (RBF) interpolation to provide spatial adaption in a fully mesh-less formulation. Numerical experiments show that there is a potential for higher accuracy in comparison with the standard remeshing techniques. The rate of convergence of the new spatial adaption method is exponential, however convection error limits the vortex method to second order convergence. Avenues for future research involve decreasing convection error, for example by means of deformable basis functions. Nevertheless, the RBF-based spatial adaption scheme has various advantages, in addition to a demonstrated higher accuracy and the obvious benefit of not requiring a regular arrangement of particles or mesh. One presently demonstrated advantage is automatic core size control for the core spreading viscous method, without the need for vortex particle splitting.

Three applications have been successfully treated with the presently developed vortex method. The relaxation of monopoles under non-linear perturbations has been computed, resulting in noticeable improvements compared to previously published results. The existence of a quasi-steady state consisting of a rotating tripole has been corroborated, for the case of large amplitude perturbations. The second application consists of the early adaptation of two co-rotating vortices at high Reynolds number, characterized by elliptical deformation of the cores, as well as small scale deformation in the weak areas of vorticity. This is considered to pose a severe test on the present method, or indeed any method. Comparison with results using spectral methods demonstrate in practice the potential for high accuracy computations using a mesh-less method, and in addition show that the naturally adaptive vortex method can result in considerably reduced problem sizes. Finally, for the calculation of non-symmetric Burgers vortices, a correction to the core spreading method for out-of-plane strain was developed. The results establish the capability of the vortex method for the computation of vortices under three-dimensional strain.

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(PHD, 2002)

Abstract:

Vortex methods have become useful tools for the computation of incompressible fluid flow. In the present work, a vortex particle method for the simulation of unsteady two-dimensional compressible flow is developed and applied to several problems. The method is the first Langrangian simulation method for the full compressible Navier-Stokes equations. By decomposing the velocity into irrotational and solenoidal parts, and using particles that are able to change volume and that carry vorticity, dilation, enthalpy, entropy, and density, the equations of motion are satisfied. A general deterministic treatment of spatial derivatives in particle methods is developed by extending the method of particle strength exchange through the construction of higher-order-accurate, non-dissipative kernels for use in approximating arbitrary differential operators. The application of this technique to wave propagation problems is thoroughly explored. A one-sided operator is developed for approximating derivatives near the periphery of particle coverage; the operator is used to enforce a non-reflecting boundary condition for the absorption of acoustic waves at this periphery. Remeshing of the particles and the smooth interpolation of their strengths are addressed, and a criterion for the frequency of remeshing is developed on the principle axes of the rate-of-strain tensor. The fast multipole method for the fast summation of the velocity field is adapted for use with compressible particles. The new vortex method is applied to co-rotating and leapfrogging vortices in compressible flow, with the acoustic field computed using a two-dimensional Kirchoff surface, and the results agree will with those of previous work or analytical prediction. The method is also applied to the baroclinic generation of vorticity, and to the steepening of waves in the one-dimensional Burgers’ equation, with favorable results in both cases.

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(PHD, 2000)

Abstract: A computational method is presented to follow the evolution of regularized three-dimensional (3D) vortex sheets through an otherwise irrotational, inviscid, constant-density fluid. The sheet surface is represented by a triangulated mesh with interpolating functions locally defined inside each triangle. C^1 continuity is maintained between triangles via combinations of cubic Bézier triangular interpolants. The self-induced sheet motion generally results in a highly deformed surface which is adaptively refined as needed to capture regions of increasing curvature and to avoid severe Lagrangian deformation. Automatic mesh refinement is implemented with an advancing front technique. Sheet motion is regularized by adding a length scale cut-off to the BiotSavart kernel. Velocity evaluation takes less time than the standard O(N^2) scaling, due to utilization of multi-pole expansions of the kernel. Zero, singly, and doubly periodic vortex sheets are simulated, modeling vortex rings, vortex/jet combinations and standard shear layers. Comparisons with previous two-dimensional (2D) results are favorable and 3D simulations are presented. The perturbed 3D planar shear layer is simulated and compared with a similar experiment revealing qualitatively similar results and agreement on the mechanism by which streamwise vorticity is created. We find the ratio of spanwise to streamwise vorticity to vary between 7 and 9 during early stages of roll-up. A new technique for estimating the curvature singularity time of true vortex sheets (i.e., non-regularized motion) is presented. The motion and singularity time of a planar, doubly periodic sheet, evolving from a 3D normal mode perturbation, is found to reduce to that of a well known singly periodic (and only two-dimensional) problem, an unexpected extension of Moore’s [38] non-linear analysis for 2D vortex sheets.

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(PHD, 2000)

Abstract:

This research focuses on the incompressible scalar advection-diffusion equation. After applying a Gaussian filter, an infinite series expansion is found for the advection term to obtain a closed equation. Only the first two terms in this expansion are retained yielding the tensor-diffusivity subgrid model. This model can be interpreted as a tensor-diffusivity term which is proportional to the rate-of-strain tensor of the large-scale filtered velocity field. Due to the negative diffusion in the stretching directions, care needs to be taken in the choice of a numerical method. The scalar field is decomposed in a collection of anisotropic or axisymmetric Gaussian particles. Equations of motion for the location and the shape/size of the particles are derived using an expansion in Hermite polynomials. A novel, accurate remeshing scheme was found resulting in explicit expressions for the amplitudes of the new set of particles. A stagnation flow was used for illustrative purposes and validation. Using a 2D time-dependent velocity field yielding chaotic advection, both axisymmetric and anisotropic particles yield good agreement with filtered direct numerical simulations and compare favorably with the Smagorinsky subgrid model. Computational efficiency makes axisymmetric particles the preferred choice. A literature study using a 3D stationary one-parameter chaotic velocity field was used to validate model and particle-method in 3D. For highly chaotic fields good agreement was obtained with this study. Computations have been performed for 3D forced isotropic periodic turbulence to study scalar mixing. Comparisons with literature are made. It was shown that when the unfiltered velocity field is known, the most accurate results are obtained by moving particles using this field. It was concluded that a good subgrid model modifies the equation of motion to get a good approximation to the unfiltered velocity field.

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(PHD, 1998)

Abstract:

Bluff body flows controlled in various manners are simulated with a high-resolution, gridless vortex method. Two-dimensional, unsteady, viscous simulations are utilized to illuminate the physical phenomenon underpinning certain flows of this class. Flows past a rotationally oscillating circular cylinder and flows past an elastically mounted circular cylinder are studied, providing a variety of new insights about these systems. A computational method facilitating longtime, high-resolution vortex simulations is developed whose grid-free nature enables future extension to complex geometries.

The significant fluid forces experienced by bluff bodies are of much practical concern and are induced by flowfields that are often complex. The studies in this thesis aim to contribute to the understanding of the relation between wake development and forces and how to exploit this relationship to achieve flow control. A circular cylinder undergoing rotational oscillation is known to experience a significant deviation in forces from unforced flow. Computations from Re=150-15000 verify past experimental observation of significant drag reduction for certain forcing parameters. These simulations also illuminate the mechanism which renders this control effective - a forced boundary layer instability triggering premature shedding of multipole vortex structures.

New insights were also provided by studies of flow over a model of an elastically mounted cylinder. A two-dimensional cylinder modeled as a damped oscillator can serve as an approximation to three-dimensional situations such as a cable under tension. Simulations clarified the behavior of such a two-dimensional system and, contrary to a line of classical thinking, revealed an unexpected adaptivity in wake evolution. New scaling is also suggested which better classifies these systems under certain conditions.

Vortex methods are well-suited for incompressible bluff body flow in many ways. However, the handling of viscous diffusion causes complications for such simulations. A relatively unexplored approach, the core expansion method, is studied, extended, and implemented in this work in order to balance accuracy with preservation of the gridless foundation of vortex methods. This viscous technique is found to enable long-time calculations that are prohibitive with other techniques while preserving a high level of accuracy.
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(PHD, 1997)

Abstract:

In this thesis, a new method for the design of unsplit numerical schemes for hyperbolic systems of conservation laws with source terms is developed. Appropriate curves in space-time are introduced, along which the conservation equations decouple to the characteristic equations of the corresponding one-dimensional homogeneous system. The local geometry of these curves depends on the source terms and the spatial derivatives of the solution vector. Numerical integration of the characteristic equations is performed on these curves.

In the first chapter, a scalar conservation law with a stiff, nonlinear source term is studied using the proposed unsplit scheme. Various tests are made, and the results are compared with the ones obtained by conventional schemes. The effect of the stiffness of the source term is also examined.

In the second chapter, the scheme is extended to the one-dimensional, unsteady Euler equations for compressible, chemically-reacting flows. A numerical study of unstable detonations is performed. Detonations in the regime of low overdrive factors are also studied. The numerical simulations verify that the dynamics of the flow-field exhibit chaotic behavior in this regime.

The third chapter deals with the development and implementation of the unsplit scheme, for the two-dimensional, reactive Euler equations. In systems with more than two independent variables there are one-parameter families of curves, forming manifolds in space-time, along which the one-dimensional characteristic equations hold. The local geometry of these manifolds and their position relative to the classical characteristic rays are studied. These manifolds might be space-like or time-like, depending on the local flow gradients and the source terms.

In the fourth chapter a numerical study of two-dimensional detonations in performed. These flows are intrinsically unstable and produce very complicated patterns, such as cellular structures and vortex sheets. The proposed scheme appears to be capable of capturing many of the the important details of the flow-fields. Unlike traditional schemes, no explicit artificial-viscosity mechanisms need to be used with the proposed scheme.

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(PHD, 1995)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

To understand the basic contribution of vortex motion in the transport and mixing of passive fluid, we study a system of N discrete vortices. With variation of N and […] (a vorticity distribution parameter), we are able to experiment with a range of vortex dynamics sufficient to capture many of the features of two-dimensional turbulence in their elementary form - such as vortex merging (inverse cascade of energy), filamentation (enstrophy cascade), etc. With this model the mixing of the fluid is numerically studied via stretch statistics and the spatial distribution of a non-diffusive scalar interface. The spectrum of spatial distribution of scalars as a result of the stirring motion of the N vortices is particularly important in view of the recent (as well as historical) interest in the characterization of the scalar distribution in turbulence. We also examine the velocity field statistics and the Lagrangian motion of fluid particles. It is also instructive to look at the kinematic causes behind the types of statistics that are obtained for the velocity structure functions. A ‘building block’ approach to understanding these effects in turbulence may lie in building up from a collection of discrete vortices, as done in this thesis, to adding vortices of different scales and the three-dimensional effects. It is in the context of these wider issues that we study the N-vortex problem.

In the final part of this thesis we investigate the two-dimensional mixing produced by large scale vortical structures during the evolution of a spatially developing mixing layer. Although the advent of three-dimensionality and fully developed turbulence are essential features of mixing layers, it is still dominated by the large scale two-dimensional structures and its effect on the mixing is illustrated here.

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(PHD, 1994)

Abstract:

This thesis is divided in two parts: in Part I, using a time-periodic perturbation of a two-dimensional steady separation bubble on a plane no-slip boundary to generate chaotic particle trajectories in a localized region of an unbounded boundary-layer flow, we study the impact of various geometrical structures that arise naturally in chaotic advection fields on the transport of a passive scalar from a local “hot spot” on the no-slip boundary. We consider here the full advection-diffusion problem, though attention is restricted to the case of small scalar diffusion, or large Peclet number. In this regime, a certain one-dimensional unstable manifold is shown to be the dominant organizing structure in the distribution of the passive scalar. In general, it is found that the chaotic structures in the flow strongly influence the scalar distribution while, in contrast, the flux of passive scalar from the localized active no-slip surface is, to dominant order, independent of the overlying chaotic advection. Increasing the intensity of the chaotic advection by perturbing the velocity field further away from integrability results in more non-uniform scalar distributions, unlike the case in bounded flows where the chaotic advection leads to rapid homogenization of diffusive tracer. In the region of chaotic particle motion the scalar distribution attains an asymptotic state which is time-periodic, with the period same as that of the time-dependent advection field. Some of these results are understood by using the shadowing property from dynamical systems theory. The shadowing property allows us to relate the advection-diffusion solution at large Peclet numbers to a fictitious zero-diffusivity or frozen-field solution - the so-called stirring solution - corresponding to infinitely large Peclet number. The zero-diffusivity solution is an unphysical quantity, but it is found to be a powerful heuristic tool in understanding the role of small scalar diffusion. A novel feature in this problem is that the chaotic advection field is adjacent to a no-slip boundary. The interaction between the necessarily non-hyperbolic particle dynamics in a thin near-wall region with the strongly hyperbolic dynamics in the overlying chaotic advection field is found to have important consequences on the scalar distribution: that this is indeed the case is shown using shadowing. Comparisons are made throughout with the flux and the distributions of the passive scalar for the advection-diffusion problem corresponding to the steady, unperturbed, integrable advection field. In Part II, the transport of a passive scalar from a no-slip boundary into a two-dimensional steady boundary-layer flow is studied in the vicinity of a laminar separation point, where the dividing streamline - which is also a one-dimensional unstable manifold - is assumed to be normal to the boundary locally near the separation point. The novelty of the ensuing convection-diffusion process derives from the convective transport normal to the active boundary resulting from convection along the dividing streamline, and because of which the standard thermal boundary-layer approximations become invalid near the separation point. Using only the topology of the laminar, incompressible separated flow, a local solution of the Navier-Stokes equations is constructed in the form of a Taylor-series expansion from the separation point. The representation is universal, without regard to the outer inviscid flow and it is used in obtaining an asymptotically exact solution for the steady scalar distribution near the separation point at large Peclet number, using matched asymptotic expansions. The method demonstrates the application of local solutions of the Navier-Stokes equations in seeking asymptotic solutions to convection-diffusion problems. Verification of the asymptotic result is obtained from numerical computations based on the Wiener bundle solution - which is particularly well-suited to the large-Peclet-number transport problem.

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(PHD, 1994)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

Numerical simulations are presented for viscous incompressible homogeneous turbulent flows with periodic boundary conditions. Our numerical method is based on the spectral Fourier method. Rogallo’s code is modified and extended to trace fluid particles and simulate the evolution of material line elements.

The first part of the thesis is about modifying and applying the code to simulate a passive vector field convected and stretched by the so-called ABC flows in the presence of viscosity. The correlation of the geometry of the physical structures of the passive vector with the external straining is investigated. It is observed that most amplifications either occur in the neighborhoods of local unstable manifolds of the stagnation points of the ABC flows, if they exist, especially those with only one positive eigenvalue, or they are confined within the chaotic regions of the ABC flows if there is no stagnation point. Tube-like structures in all simulations are observed.

The second part of the thesis is an investigation of the power-law energy decay of turbulence. Two decay exponents, 1.24 and 1.54, are measured from simulations. A new similarity form for the double and triple velocity autocorrelation functions using the Taylor microscale as the scaling, consistent with the Karman-Howarth equation and a power-law, energy decay, is proposed and compared with numerical results. The proposed similarity form seems applicable at small to intermediate Reynolds number. For flows with very large Reynolds number, an expansion form of energy spectrum is proposed instead. Two lengthscales are used to express the energy spectrum in the energy-containing range and in the dissipation range of wave numbers. The uniform expansion is obtained by matching spectra in the inertial subrange to the famous Kolmogorov’s […] spectrum.

The third part of the thesis is a presentation of the Lagrangian data collected by tracking fluid particles in decaying turbulent flows. The mean growth rates of the magnitudes of material line elements, that of the vorticity due to nonlinear forces, and the mean principal rates of strain tensors are found to be proportional to the square root of the mean enstrophy. The proportional coefficients remain constant during the decay. The mean angles between material line elements and the principal directions of the strain tensors corresponding to the most stretching and the intermediate principal rates are about the same which is probably caused by the averaging process and by the possible switch of principal directions. The evolution of these angles under the action of one Burger’s vortex is examined and the results support the conjecture. Following fluid particles which suffer substantial stretching in their history, we, through use of flow visualization tools, observe the evolution of nearby vorticity structures. It is observed that vortex sheets are created first by the nonlinear stretching which gradually become tubes at later times by diffusion.

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(PHD, 1993)

Abstract:

In the first part of this dissertation a two-dimensional unsteady separated flow past a semi-infinite plate with transverse motion is considered. The flow is assumed incompressible and at high Reynolds number. The rolling-up of the separated shear-layer is modelled by a point vortex whose time dependent circulation is predicted by an unsteady Kutta condition. A power-law starting flow is assumed along with a power-law for the transverse motion. The effects of the motion of the plate on the starting vortex circulation and trajectory are presented. A suitable vortex shedding mechanism is introduced and a class of flows involving several vortices is presented. Subsequently, a control strategy able to maintain constant circulation when a vortex is present is derived. An exact solution for the non-linear controller is then obtained. Dynamical system analysis is used to explore the performance of the controlled system. Finally, the control strategy is applied to a class of flows and the results are discussed.

In the second part of this dissertation the previous results are extended to the case of a two-dimensional unsteady separated flow past a plate of variable length. Again the rolling-up of the separated shear-layer is modelled by a vortex pair whose time dependent circulation is predicted by an unsteady Kutta condition. A power-law starting flow is assumed while the plate length is kept constant. The results of the simulations are presented and the model validated. A time-dependent scaling which unveils the universality of the phenomenon is discussed. The previous vortex shedding mechanism is implemented and a vortex merging scheme is tested in a class of flows involving several vortices and is shown to be highly accurate. Subsequently, a control strategy able to maintain constant circulation when a vortex pair is present is derived. An exact solution for the non-linear controller is obtained in the form of an ordinary differential equation. Dynamical system analysis is used to explore the performance of the controlled system and the existence of a controllability region is discussed. Finally, the control strategy is applied to two classes of flows and the results are presented.

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(PHD, 1993)

Abstract:

In the first part of this thesis, a method for computing one-dimensional, unsteady compressible flows, with and without chemical reactions, is presented. This work has focused on accurate computation of the discontinuous waves that arise in such flows. The main feature of the method is the use of an adaptive Lagrangian grid. This allows the computation of discontinuous waves and their interactions with the accuracy of front-tracking algorithms. This is done without the use of additional grid points representing shocks, in contrast to conventional, front-tracking schemes. The Lagrangian character of the present scheme also allows contact discontinuities to be captured easily. The algorithm avoids interpolation across discontinuities in a natural and efficient way. The method has been used on a variety of reacting and non-reacting flows in order to test its ability to compute complicated wave interactions accurately and in a robust way. In the second part of this thesis, a new approach is presented for computing multidimensional flows of an inviscid gas. The goal is to use the knowledge of the one-dimensional, characteristic problem for gas dynamics to compute genuinely multidimensional flows in a mathematically consistent way. A family of spacetime manifolds is found on which an equivalent 1-D problem holds. These manifolds are referred to as Riemann Invariant Manifolds. Their geometry depends on the local, spatial gradients of the flow, and they provide locally a convenient system of coordinate surfaces for spacetime. In the case of zero entropy gradients, functions analogous to the Riemann invariants of 1-D gas dynamics can be introduced. These generalized Riemann Invariants are constant on the Riemann Invariant Manifolds. The equations of motion are integrable on these manifolds, and the problem of computing the solution becomes that of determining the geometry of these manifolds locally in spacetime. The geometry of these manifolds is examined, and in particular, their relation to the characteristic surfaces. It turns out that they can be space-like or time-like, depending on the flow gradients. An important parameter is introduced, which plays the role of a Mach number for the wave fronts that these manifolds represent. Finally, the issue of determining the solution at points in spacetime, using information that propagates along space-like surfaces is discussed. The question of whether it is possible to use information outside the domain of dependence of a point in spacetime to determine the solution is discussed in relation to the existence and uniqueness theorems, which introduce the concept of domain of dependence. This theory can be viewed as an extension of the method of characteristics to multidimensional, unsteady flows. There are many ways of using the theory to develop practical, numerical schemes. It is shown how it is possible to correct a conventional, second-order Godunov scheme for multidimensional effects, using this theory. A family of second-order, conservative Godunov schemes is derived, using the theory of Riemann Invariant Manifolds, for the case of two-dimensional flow. The extension to three dimensions is straightforward. One of these schemes is used to compute two standard test cases and a two-dimensional, inviscid, shear layer.

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(Engineer, 1993)

Abstract:

Finite-rate chemistry of hydrogen-air combustion is to be investigated numerically in a one-dimensional constant pressure SCRAMJET combustor and two-dimensional nozzles. Detailed reaction mechanisms and temperature dependent thermodynamics are to be used in the models. The aspects of interest include the combustion characteristics at different fuel-air ratios, pressures and initial temperatures in the combustor. Methods for enhancing the combustion rate in the combustor is to be studied also. The effect of expansion rate on the hydrogen-air reactions is the prime focus of the nozzle calculation. The results from different inlet conditions and wall geometries are to be analyzed.

A computer model for a one-dimensional (channel-flow) combustor is constructed based on the chemical kinetics subroutine library CHEMKIN. Subsequent calculations show that the initial temperature is the most important parameter in the combustor. It is further discovered that certain reaction steps are responsible for the initial delay exhibited in all hydrogen-air combustion processes. Low temperature behavior is studied extensively and augmentation methods are developed. The introduction of a small percentage of the hydrogen radical into the initial mixture is found to be the most effective in reducing the reaction delay. The combustor pressure enters the overall reaction process in a linear manner. The calculations over five combustor pressures show that the initial delay in hydrogen-air reaction and the following period of explosion are proportional to the combustor pressure raised to certain powers.

The nozzle model is two-dimensional, steady and inviscid with no conductivity and diffusivity. Two schemes are developed to handle the boundary conditions. One is based on pure numerical interpolation/extrapolation methods while the other imposes analytical supersonic characteristic equations. The former scheme is found to be more efficient while the latter is more accurate. In analysing the response of the combustion product to an expansion, it is found that the formation of water is favoured by an expansion. A closer examination reveals that the behavior can be attributed to the abundance of free radicals in the nozzle inlet composition. Freezing is not clearly observed except for the NO_{x} species.

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(PHD, 1993)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

Numerical simulations are presented for viscous incompressible flows with and without solid wall boundaries. Our numerical method is based on vortex methods. The classical inviscid scheme is enhanced to account for viscous effects via the method of particle strength exchange. The method is extended to account for the enforcement of the no-slip boundary condition as well by appropriately modifying the strength of the particles. Computations are possible for extended times by periodically remeshing the vorticity field.

The particles are advanced using the Blot-Savart law for the evaluation of the velocity. Computations are made using up to … vortex particles by efficiently implementing the method of multipole expansions for vector computer architectures to obtain an … algorithm.

The method is used to simulate the inviscid evolution of an elliptical vortex in an unbounded fluid as well as unsteady separated flows around circular cylinders for a wide range of Reynolds numbers (40 - 9500). Direct comparisons are made of the results of the present method with those from a variety of theoretical, computational and experimental studies. The results exhibit the robustness and validity of the present method and allow to gain physical insight as to vorticity formation and its relation to the forces experienced by the body.
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(PHD, 1993)

Abstract:

We use global stable and unstable manifolds of invariant hyperbolic sets as templates for studying the dynamics within classes of homoclinic and heteroclinic chaotic tangles, focusing on transport, stretching, and mixing within these tangles. These templates are exploited in the context of lobes in phase space mapping within invariant lobe structures formed out of the intersecting global stable and unstable manifolds. Our interest lies in: (a) extending the templates and their applications to fundamentally larger classes of dynamical systems, (b) expanding the description of dynamics offered by the templates, and (c) applying the templates to the study of various nonlinear physical phenomena, such as stirring and mixing under chaotically advecting fluids and molecular dissociation under external electromagnetic forcing. These and other nonlinear physical phenomena are intimately connected to the underlying chaotic dynamics, and describing these processes encourages study of finite-time, or transient, phenomena as well as asymptotics, the former being much more virgin territory from a dynamical systems perspective. Under the rubric of themes (a)-(c) we offer five studies.

- One of the canonical classes of dynamical systems in which these templates have
been exploited is defined by 2D time-periodic vector fields, where the analysis reduces
to a 2D Poincaré map. In this instance, one is well-equipped with basic elements of
dynamical systems theory associated with 2D maps, such as the Smale horseshoe map
paradigm, KAM-tori, hyperbolic fixed points and their global stable and unstable manifolds
that define the tangle boundaries, classical Melnikov theory, and so on. Our first
study performs a systematic extension of the dynamical system constructs associated
with 2D time-periodic vector fields to apply to 2D vector fields with more complicated
time dependences. In particular, we focus on 2D vector fields with quasiperiodic, or
multiple-frequency, time dependence. Any extension past the time-periodic case requires
the fundamental generalization from 2D maps to sequences of 2D nonautonomous
maps. To large extent the constructs associated with 2D Poincaré maps are found to
be robust under this generalization. For example, the Smale horseshoe map generalizes
to a traveling horseshoe map sequence, hyperbolic fixed points generalize to points that
live on invariant normally hyperbolic tori, and invariant 2D chaotic tangles generalize to
sequences of 2D chaotic tangles derived from an invariant tangle embedded in a higher-dimensional
phase space. It is within the setting of 2D lobes mapping within a sequence
of 2D lobe structures that one has a template for systematic study of the dynamics
generated by multiple-frequency vector fields. Dynamical systems tools with which to
study these systems include: (i) a generalized Melnikov theory that offers an approximate
analytical measure of stable and unstable manifold separation in the tangles, the
basis for a variety of analytical studies, and (ii) a double phase slice sampling method
that allows for numerical computation of precise 2D slices of the higher-dimensional invariant
chaotic tangles, the basis for numerical work. The Melnikov function defines
relative scaling functions which give an analytical measure of the relative importance of
each frequency on manifold separation. With the template and tools in hand, we study
multiple-frequency dynamics and compare with single-frequency dynamics. We recast
lobe dynamics under a hi-infinite sequence of nonautonomous maps in closed form by
exploiting underlying periodicity properties of the vector field, and present numerical
simulations of sequences of chaotic tangles and lobe dynamics within these tangles. In
contrast to lobes of equal area mapping within a fixed 2D lobe structure found under
single-frequency forcing, we find lobes of varying areas mapping within a sequence of
lobe structures that are distorting and breathing from one time sample to the next, affording
greater freedom in the nature of the dynamics. Our primary focus in this new
setting is on phase space transport (we consider stretching and mixing in other contexts
in later studies). The non-integrable motion in chaotic tangles allows for transport between
various regions of phase space, in particular, between regions corresponding to
qualitatively different types of motion, such as bounded and unbounded motion. This
transport is intimately connected to basic physical processes, such as the fluid mixing
and molecular dissociation processes. Transport theory refers to the enterprise where
one uses a combination of invariant manifold theory, Melnikov theory, numerical simulation
and/or approximate models such as Markov models, to partition phase space into
regions of qualitatively different behavior (such as bounded and unbounded motion),
establish complete and partial barriers between the regions, identify the turnstile lobes
that are the gateways for transport across partial barriers, and then study in the context
of lobe dynamics such phase space transport issues as flux and escape rates from a
particular region. The formal construction of a transport theory for multiple-frequency
vector fields is more involved than in the single-frequency case, as a consequence of more
complicated manifold geometry. This geometry is uncovered and explored, however, via
theorems and numerical studies based on Melnikov theory. We then partition phase
space and define turnstiles in the higher-dimensional autonomous setting, and from this
obtain the sequence of partitions and turnstiles in the 2D nonautonomous setting. A
main new feature of transport is its manifestation in the context of a sequence of time-dependent
regions, and we argue this is consistent with a Lagrangian viewpoint. We
then perform a detailed study of such transport properties as flux, lobe geometry, and
lobe content. In contrast to the single-frequency case, where a single flux suffices, in the
multiple-frequency case a variety of fluxes are allowed, such as different types of instantaneous,
finite-time average, and infinite-time average flux. We find for certain classes
of multiple-frequency forcing that infinite-time average flux is maximal in a particular
single-frequency limit, but that the spatial variation of lobe areas found in multiple-frequency
systems affords greater freedom to enhance or diminish finite-time transport
quantities. We illustrate our study with a quasiperiodically oscillating vortical flow that
gives rise to chaotic fluid trajectories and a quasiperiodically forced Duffing oscillator.
We explain how the analysis generalizes to vector fields with more complicated time
dependences than quasiperiodic.
- Besides the destruction of phase space barriers, allowing for phase space transport,
other essential features of the dynamics in chaotic tangles include greatly enhanced
stretching and mixing. Our second study returns to 2D time-periodic vector fields and
uses invariant manifolds as templates for a global study of stretching and mixing in
chaotic tangles. The analysis here thus complements the one of transport via invariant
manifolds, and can essentially be viewed as a generalization of the horseshoe map construction
to apply to entire material interfaces inside the tangles. Given the dominant
role of the unstable manifold in chaotic tangles, we study the stretching of a material interface
originating on a segment of the unstable manifold associated with a turnstile lobe.
We construct a symbolic dynamics formalism that describes the evolution of the entire
material curve, which is the basis for a global understanding of the stretch processes in
chaotic tangles, such as the topology of stretching, the mechanisms for good stretching,
and the statistics of stretching. A central interest will be in understanding the stretch
profile of the material interface, which is the graph of finite-time stretch experienced as
a function of location on the interface. In a near-integrable setting (meaning we add a
perturbation to the vector field of an originally integrable system) we argue how the perturbed
stretch profile can be understood in terms of a corresponding unperturbed stretch
profile approximately repeating itself on smaller and smaller scales, as described by the
symbolic dynamics. The basic interest is in how the non-uniformity in the unperturbed
stretch profile can approximately carry through to the non-uniformity in the perturbed
stretch profile, and this non-uniformity can play a basic role in mixing properties and
stretch statistics. After the stretch analysis we then add to the deterministic flows a small
stochastic component, corresponding for example to molecular diffusion (with small diffusion
coefficient D) in a fluid flow, and study the diffusion of passive scalars across
material interfaces inside the chaotic tangles. For sufficiently thin diffusion zones, the
diffusion of passive scalars across interfaces can be treated as a one-dimensional process,
and diffusion rates across interfaces are directly related to the stretch history of the interface.
Our understanding of stretching thus directly translates into an understanding
of mixing. However, a notable exception to the thin diffusion zone approximation occurs
when an interface folds on top of itself so that neighboring diffusion zones overlap. We
present an analysis which takes into account the overlap of neighboring diffusion zones,
capturing a saturation effect in the diffusion process relevant to efficiency of mixing. We
illustrate the stretching and mixing study in the context of two oscillating vortex pair
flows, one corresponding to an open heteroclinic tangle, the other to a closed homoclinic
tangle. Though we focus here on single-frequency systems, from the previous study the
extensions to multiple-frequency systems should be clear.
- We then study stretching from a different perspective, focusing on rates of
strain experienced by infinitesimal line elements as they evolve under near-integrable
chaotic flows associated with 2D time-periodic velocity fields. We introduce the notion
of irreversible rate of strain responsible for net stretch, study the role of hyperbolic fixed
points as engines for good irreversible straining, and observe the role of turnstiles as
mechanisms for enhancing straining efficiency via re-orientation of line elements and
transport of line elements to regions of superior straining.
- The remaining two studies can be viewed as applications of the material developed
in the previous studies, although both applications develop new theory and/or new
ideas as well. The first application studies the dynamics associated with a quasi-periodically
forced Morse oscillator as a classical model for molecular dissociation under
external quasiperiodic electromagnetic forcing. The forcing entails destruction of phase
space barriers, allowing escape from bounded to unbounded motion, and we study this
transition in the context of our quasiperiodic theory, comparing with single-frequency
forcing. New and interesting features of this application beyond the subject matter of
the previous quasiperiodic study includes that the relevant fixed point of the unforced
system is non-hyperbolic and at infinity, and the study of additional transport issues,
such as escape (implying dissociation) from a particular level set of the unforced Hamiltonian
system corresponding to a quantum state. We find for example that though
infinite-time average flux can be maximal in a single-frequency limit, escape from a level
set, or equivalently lobe penetration, can be maximal in the multiple-frequency case.
- The second application studies statistical relaxation of distributions of finite-time Lyapunov exponents associated with interfaces evolving within the chaotic tangles of 2D time-periodic vector fields. Whereas recent studies claim or give evidence that distributions of finite-time Lyapunov exponents are essentially Gaussian, our previous analysis of stretching via the symbolic dynamics construction shows the wide variety of stretch processes and stretch scales involved in the tangle, motivating our further study of stretch statistics. In particular, we focus on the high-stretch tails of finite-time Lyapunov exponents, which have relevance in incompressible flows to the mixing properties and multifractal characteristics of passive scalars and vectors in the limit of small spatial scales. Previous studies of stretch distributions consider a fixed number of points, thus lacking adequate resolution to study these tails. Instead, we use a dynamic point insertion scheme to maintain adequate interfacial covering, entailing extremely good resolution at high-stretch tails. These tails show a great range in behavior, varying from essentially Gaussian to nearly exponential, and these non-Gaussian deviations can have a significant effect on interfacial stretching, one that persists asymptotically. These non-Gaussian deviations can be associated with very small probabilities, thus indicating the need for highly-resolved numerical studies of stretch statistics. We explain the nearly exponential tail in a particular limiting regime corresponding to highly non-uniform stretch profiles, and explore how the full statistics might be captured by elementary models for the stretch processes.

- The remaining two studies can be viewed as applications of the material developed
in the previous studies, although both applications develop new theory and/or new
ideas as well. The first application studies the dynamics associated with a quasi-periodically
forced Morse oscillator as a classical model for molecular dissociation under
external quasiperiodic electromagnetic forcing. The forcing entails destruction of phase
space barriers, allowing escape from bounded to unbounded motion, and we study this
transition in the context of our quasiperiodic theory, comparing with single-frequency
forcing. New and interesting features of this application beyond the subject matter of
the previous quasiperiodic study includes that the relevant fixed point of the unforced
system is non-hyperbolic and at infinity, and the study of additional transport issues,
such as escape (implying dissociation) from a particular level set of the unforced Hamiltonian
system corresponding to a quantum state. We find for example that though
infinite-time average flux can be maximal in a single-frequency limit, escape from a level
set, or equivalently lobe penetration, can be maximal in the multiple-frequency case.

- We then study stretching from a different perspective, focusing on rates of
strain experienced by infinitesimal line elements as they evolve under near-integrable
chaotic flows associated with 2D time-periodic velocity fields. We introduce the notion
of irreversible rate of strain responsible for net stretch, study the role of hyperbolic fixed
points as engines for good irreversible straining, and observe the role of turnstiles as
mechanisms for enhancing straining efficiency via re-orientation of line elements and
transport of line elements to regions of superior straining.

- Besides the destruction of phase space barriers, allowing for phase space transport,
other essential features of the dynamics in chaotic tangles include greatly enhanced
stretching and mixing. Our second study returns to 2D time-periodic vector fields and
uses invariant manifolds as templates for a global study of stretching and mixing in
chaotic tangles. The analysis here thus complements the one of transport via invariant
manifolds, and can essentially be viewed as a generalization of the horseshoe map construction
to apply to entire material interfaces inside the tangles. Given the dominant
role of the unstable manifold in chaotic tangles, we study the stretching of a material interface
originating on a segment of the unstable manifold associated with a turnstile lobe.
We construct a symbolic dynamics formalism that describes the evolution of the entire
material curve, which is the basis for a global understanding of the stretch processes in
chaotic tangles, such as the topology of stretching, the mechanisms for good stretching,
and the statistics of stretching. A central interest will be in understanding the stretch
profile of the material interface, which is the graph of finite-time stretch experienced as
a function of location on the interface. In a near-integrable setting (meaning we add a
perturbation to the vector field of an originally integrable system) we argue how the perturbed
stretch profile can be understood in terms of a corresponding unperturbed stretch
profile approximately repeating itself on smaller and smaller scales, as described by the
symbolic dynamics. The basic interest is in how the non-uniformity in the unperturbed
stretch profile can approximately carry through to the non-uniformity in the perturbed
stretch profile, and this non-uniformity can play a basic role in mixing properties and
stretch statistics. After the stretch analysis we then add to the deterministic flows a small
stochastic component, corresponding for example to molecular diffusion (with small diffusion
coefficient D) in a fluid flow, and study the diffusion of passive scalars across
material interfaces inside the chaotic tangles. For sufficiently thin diffusion zones, the
diffusion of passive scalars across interfaces can be treated as a one-dimensional process,
and diffusion rates across interfaces are directly related to the stretch history of the interface.
Our understanding of stretching thus directly translates into an understanding
of mixing. However, a notable exception to the thin diffusion zone approximation occurs
when an interface folds on top of itself so that neighboring diffusion zones overlap. We
present an analysis which takes into account the overlap of neighboring diffusion zones,
capturing a saturation effect in the diffusion process relevant to efficiency of mixing. We
illustrate the stretching and mixing study in the context of two oscillating vortex pair
flows, one corresponding to an open heteroclinic tangle, the other to a closed homoclinic
tangle. Though we focus here on single-frequency systems, from the previous study the
extensions to multiple-frequency systems should be clear.

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(PHD, 1991)

Abstract:

Incompressible, viscous flows in the spherical gap between a rotating inner-sphere and a stationary outer-shell, Spherical Couette Flows (SCF), are studied via direct numerical simulations. The investigation covers both “small-gap” and “large-gap” geometries, and is concerned primarily with the first occurrence of transition in those flows. Strong emphasis is put on the physical understanding of the basic flows and their transition mechanisms.

An alias-free spectral method, based on divergence-free vector expansions for the 3-D velocity field in spherical coordinates, is developed. The vector expansions are constructed with Chebyshev polynomials in the radial direction and Vector Spherical Harmonics for the two angular directions. Accuracy and spectral convergence of the resulting initial-value code are thoroughly tested. Three-dimensional transitional flows in both narrow-gaps and large-gaps as well as axisymmetric transitions in moderate-gaps are simulated.

For small-gap SCF’s, this study shows that the formation of Taylor-vortices at transition is a deterministic process and not the result of the instability of initial perturbations. The formation process involves the sub-critical appearance of a saddle-stagnation point within the meridional circulation cell in each hemisphere. A minimum length-scale ratio is shown necessary, and for a given inner-sphere radius, this leads to a theoretical prediction of the largest gap-width in which Taylor-vortices may form.

This investigation confirms that the first transition in large-gap SCF’s is caused by a 3-D instability of a linear nature. It is found that the process is characterized by very small growth-rates of the disturbance and by the absence of a “jump” in the friction torque. The supercritical flow is a complex-structured, laminar, time-periodic flow that exhibits traveling azimuthal-waves. The physical mechanism responsible for the large-gap transition is shown to be related to a shear instability of the “radial-azimuthal jet” that develops at the equator of the basic flow. A physical model is proposed in which that jet is viewed as a sequence of adjacent “fan-spreading quasi-2-D plane jets”. Predictions from the model are presented and verified from the computed unstable disturbance field. Extension of the model to the transition toward waviness in the Taylor-Couette flow, the Gortler-vortex flow and the Dean-vortex flow is proposed.
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(PHD, 1990)

Abstract:

This thesis is concerned with the applications of vortex methods to the problem of unsteady, separated flows in two and three dimensions, and can be divided into three parts. In the first part, an improved method for satisfying the boundary conditions on a flat plate is developed and applied to the two-dimensional separated flow problem. In this method, boundary layers on both side of the plate are represented by stacks of multiple vortex panels, the strength of which are determined by enforcing both the no-through flow and no-slip boundary conditions at the plate. Vortex shedding at the sharp edge of the plate is represented as the separation of the boundary vortex elements. Both forced and unforced flows are studied and comparisons to experiments are carried out. For the case without forcing, large discrepancy between calculations and experiments, which is also reported by other workers using a different vortex method or Navier-Stokes calculations, is observed. In the case with forcing, the discrepancy is reduced with lateral forcing at low amplitude; and eliminated, regardless of amplitude, with streamwise forcing (acceleration). In the second part, an improved three-dimensional vortex particle method is developed. In this method, vortex elements of vorticity that move with the local velocity and are stretched and rotated according to the local strain field, are used. To mimic the effects of vorticity cancellations, close pairs of opposite sign vortex elements are replaced by high order dipoles. The method is designed to handle complex high Reynolds number vortical flows and a non-linear viscosity model is included to treat small-scale effects in such flows. Applications to two problems involving strong interactions of vortex tubes are carried out and core deformation with complex internal strucures and induced axial flow within vortex tubes are observed. Qualitative comparison to experiments are encouraging. In the third part, the two-dimensional method developed in the first part is modified and extended to three dimensions. Here, solenoidal condition for vorticity is considered and closed vortex loops are used to represent the boundary layer vorticity and the vorticity at shedding. For the evolution of the vortex wake, the vortex particle method developed in the second part is used. Applications to the flow past a normal square plate is carried out and the early stages of the flow are studied.

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(PHD, 1989)

Abstract:

Contributions to vortex methods for the computation of incompressible unsteady flows are presented. Three methods are investigated, both theoretically and numerically.

The first method to be considered is the inviscid method of vortex filaments in three dimensions, and the following topics are presented: (a) review of the method of regularized vortex filaments and of convergence results for multiple-filament computations, (b) modeling of a vortex tube by a single filament convected with the regularized Biot-Savart velocity applied on the centerline: velocity of the thin filament vortex ring and dispersion relation of the rectilinear filament, and (c) development of a new regularization of the Biot-Savart law that reproduces the lowest mode dispersion relation of the rectilinear vortex tube in the range of large to medium wavelengths.

Next the method of vortex particles in three dimensions is investigated, and the following contributions are discussed: (a) review of the method of singular vortex particles: investigation of different evolution equations for the particle strength vector and weak solutions of the vorticity equation, (b) review of the method of regularized vortex particles and of convergence results, and introduction of a new algebraic smoothing with convergence properties as good as those of Gaussian smoothing, (c) development of a new viscous method in which viscous diffusion is taken into account by a scheme that redistributes the particle strength vectors, and application of the method to the computation of the fusion of two vortex rings at *Re* = 400, and (d) investigation of the particle method with respect to the conservation laws and derivation of new expressions for the evaluation of the quadratic diagnostics: energy, helicity and enstrophy.

The third method considered is the method of contour dynamics in two dimensions. The particular efforts presented are (a) review of the classical inviscid method and development of a new regularized version of the method, (b) development of a new vector particle version of the method, both singular and regularized: the method of *particles of vorticity gradient*, (c) development of a viscous version of the method of regularized particles and application of the method to computation of the reconnection of two vortex patches of same sign vorticity, and (d) investigation of the particle method with respect to the conservation laws and derivation of new expressions for the evaluation of linear and quadratic diagnostics.

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