Theory of weakly nonlinear self sustained detonations

We propose a theory of weakly nonlinear multi-dimensional self sustained detonations based on asymptotic analysis of the reactive compressible Navier-Stokes equations. We show that these equations can be reduced to a model consisting of a forced, unsteady, small disturbance, transonic equation and a rate equation for the heat release. In one spatial dimension, the model simplifies to a forced Burgers equation. Through analysis, numerical calculations and comparison with the reactive Euler equations, the model is demonstrated to capture such essential dynamical characteristics of detonations as the steady-state structure, the linear stability spectrum, the period-doubling sequence of bifurcations and chaos in one-dimensional detonations and cellular structures in multi- dimensional detonations

Using the generalized interpolation material point method for fluid-solid interactions induced by surface tension

This thesis is devoted to the development of new, Generalized Interpolation Material Point Method (GIMP)-based algorithms for handling surface tension and contact (wetting) in fluid-solid interaction (FSI) problems at small scales. In these problems, surface tension becomes so dominant that its influence on both fluids and solids must be considered. Since analytical solutions for most engineering problems are usually unavailable, numerical methods are needed to describe and predict complicated time-dependent states in the solid and fluid involved due to surface tension effects. Traditional computational methods for handling fluid-solid interactions may not be effective due to their weakness in solving large-deformation problems and the complicated coupling of two different types of computational frameworks: one for solid, and the other for fluid. On the contrary, GIMP, a mesh-free algorithm for solid mechanics problems, is numerically effective in handling problems involving large deformations and fracture. Here we extend the capability of GIMP to handle fluid dynamics problems with surface tension, and to develop a new contact algorithm to deal with the wetting boundary conditions that include the modeling of contact angle and slip near the triple points where the three phases -- fluid, solid, and vapor -- meet. The error of the new GIMP algorithm for FSI problems at small scales, as verified by various benchmark problems, generally falls within the 5% range. In this thesis, we have successfully extended the capability of GIMP for handling FSI problems under surface tension in a one-solver numerical framework, a unique and innovative approach.Chapter 1. Introduction -- Chapter 2. Using the generalized interpolation material point method for fluid dynamics at low reynolds numbers -- Chapter 3. On the modeling of surface tension and its applications by the generalized interpolation material point method -- Chapter 4. Using the generalized interpolation material point method for fluid-solid interactions induced by surface tension -- Chapter 5. Conclusions

Autoignition in nonpremixed flow

The objective of this investigation has been to improve understanding of autoignition processes in nonpremixed flow fields of the types encountered in Diesel-engine ignition, through theoretical analyses that employ asymptotic methods of applied mathematics. The work was intended to develop formulas and equations that can be used in activities of applied research, such as code development, aimed at providing tools useful for the design of Diesel engines. The formulas may also be used directly for ignition estimates.Characteristic time scales were identified for these ignition problems. Their relative magnitudes were employed to define different regimes of ignition and to obtain simplified partial differential equations that describe ignition in these regimes. Effects of turbulence on ignition were addressed. Special attention was devoted to unsteady mixing layers, involving both variable strain and variable pressure, for which ignition-time formulas were derived. In addition, ignition analyses were completed for variable-volume chambers with arbitrary initial spatial variations of temperature and composition, to determine pressure histories produced by ignition-front propagation. These studies were based on one-step, Arrhenius approximations for the chemical kinetics and were restricted to ignition stages that precede ordinary flame propagation. Additional work considered triple-flame propagation that can odcur in mixing layers after ignition, with this same chemical-kinetic description, and asymptotic analysis of n-heptane ignition on the basis of a four-step, semi-empirical model for the chemical kinetics. In this latter study, the region of negative effective overall activation energy, between 800 K and 1100 K, was identified as exhibiting unusual ignition dynamics, and the asymptotic ignition-time formulas were shown to give good agreement with predictions of numerical integrations. This research has helped to strengthen the foundations of ignition theory for nonuniform media. It provided simplified descriptions of ignition processes that can be employed in studies of Diesel combustion that are oriented more towards development than are the present investigations. The asymptotic methods employed in this work thus appear capable of providing quite useful results

On the nonlinear growth of single three-dimensional disturbances in boundary layers

Experiments indicate the importance of three-dimensional action during transition, while high-Reynolds-number-flow theory indicates a multi-structured type of analysis. In line with this, the three-dimensional nonlinear unsteady triple-deck problem is addressed here, for slower transition. High-amplitude/high-frequency properties show enhanced disturbance growth occurring downstream for single nonlinear oblique waves inclined at angles greater than tan−1 √2 (≈54.7°) to the free stream, in certain interesting special cases. The three-dimensional response there is very ‘spiky’ and possibly random, with sideband instabilities present. A second nonlinear stage, and then an Euler stage, are entered further downstream, although faster transition can go straight into these more nonlinear stages. More general cases are also considered. Sideband effects, sublayer bursting and secondary instabilities are discussed, along with the relation to experimental observations

Numerical and approximate solution of the high Reynolds number small separation problem

Several possible methods of solving the small separation problem at high Reynolds number are investigated. In addition to using analytical methods, there are several numerical approaches which are used. High Reynolds number laminar two dimensional problems are used for simplicity. A brief discussion is given of the finite difference methods since these methods are discussed in detail. Most of the emphasis is placed on developing an approximate integral method. As a model problem the supersonic compression ramp problem is chosen since several numerical solutions along with experimental data are available. The techniques discussed are modified and applied to other similar type wall geometries

Backward variational approach on time scales with an action depending on the free endpoints

We establish necessary optimality conditions for variational problems with an action depending on the free endpoints. New transversality conditions are also obtained. The results are formulated and proved using the recent and general theory of time scales via the backward nabla differential operator.Comment: Submitted 17-Oct-2010; revised 18-Dec-2010; accepted 4-Jan-2011; for publication in Zeitschrift fuer Naturforschung

A numerical study of the alpha model for two-dimensional magnetohydrodynamic turbulent flows

We explore some consequences of the ``alpha model,'' also called the ``Lagrangian-averaged'' model, for two-dimensional incompressible magnetohydrodynamic (MHD) turbulence. This model is an extension of the smoothing procedure in fluid dynamics which filters velocity fields locally while leaving their associated vorticities unsmoothed, and has proved useful for high Reynolds number turbulence computations. We consider several known effects (selective decay, dynamic alignment, inverse cascades, and the probability distribution functions of fluctuating turbulent quantities) in magnetofluid turbulence and compare the results of numerical solutions of the primitive MHD equations with their alpha-model counterparts' performance for the same flows, in regimes where available resolution is adequate to explore both. The hope is to justify the use of the alpha model in regimes that lie outside currently available resolution, as will be the case in particular in three-dimensional geometry or for magnetic Prandtl numbers differing significantly from unity. We focus our investigation, using direct numerical simulations with a standard and fully parallelized pseudo-spectral method and periodic boundary conditions in two space dimensions, on the role that such a modeling of the small scales using the Lagrangian-averaged framework plays in the large-scale dynamics of MHD turbulence. Several flows are examined, and for all of them one can conclude that the statistical properties of the large-scale spectra are recovered, whereas small-scale detailed phase information (such as e.g. the location of structures) is lost.Comment: 22 pages, 20 figure