Nikolaos Nikiforakis

Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations

To understand the nonlinear dynamical behaviour of a one-dimensional pulsating detonation, results obtained from numerical simulations of the Euler equations with simple one-step Arrhenius kinetics are analysed using basic nonlinear dynamics and chaos theory. To illustrate the transition pattern from a simple harmonic limit-cycle to a more complex irregular oscillation, a bifurcation diagram is constructed from the computational results. Evidence suggests that the route to higher instability modes may follow closely the Feigenbaum scenario of a period-doubling cascade observed in many generic nonlinear systems. Analysis of the one-dimensional pulsating detonation shows that the Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be in reasonable agreement with the universal value of d = 4.669. Using the concept of the largest Lyapunov exponent, the existence of chaos in a one-dimensional unsteady detonation is demonstrated.

Numerical investigation of the instability for one-dimensional Chapman–Jouguet detonations with chain-branching kinetics

The dynamics of one-dimensional Chapman–Jouguet detonations driven by chain-branching kinetics is studied using numerical simulations. The chemical kinetic model is based on a two-step reaction mechanism, consisting of a thermally neutral induction step followed by a main reaction layer, both governed by Arrhenius kinetics. Results are in agreement with previous studies that detonations become unstable when the induction zone dominates over the main reaction layer. To study the nonlinear dynamics, a bifurcation diagram is constructed from the computational results. Similar to previous results obtained with a single-step Arrhenius rate law, it is shown that the route to higher instability follows the Feigenbaum route of a period-doubling cascade. The corresponding Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be close to the universal value of 4.669. The present parametric analysis determines quantitatively the relevant non-dimensional parameter χ, defined as the activation energy for the induction process ϵ I multiplied by the ratio of the induction length Δ I to the reaction length Δ R . The reaction length Δ R is estimated by the inverse of the maximum thermicity (1/ max) multiplied by the Chapman–Jouguet particle velocity u CJ . An attempt is made to provide a physical explanation of this stability parameter from the coherence concept. A series of computations is carried out to obtain the neutral stability curve for one-dimensional detonation waves over a wide range of chemical parameters for the model. These results are compared with those obtained from numerical simulations using detailed chemistry for some common gaseous combustible mixtures.

A Three-Dimensional, Adaptive, Godunov-Type Model for Global Atmospheric Flows

Abstract In this paper a Godunov-type methodology is applied to three-dimensional global atmospheric modeling. Numerical issues are addressed regarding the formulation of the tracer advection problem, the application of dimensional splitting, and the implementation of a Godunov-type scheme, based on the WAF approach, on spherical geometries. Particular attention is paid to addressing the problems that arise because of the convergence of the grid lines toward the Poles. A three-dimensional model is then built on the sphere that is based on a uniform longitude–latitude–height grid. This provides the framework within which an adaptive mesh refinement (AMR) algorithm is applied, to enhance the efficiency and accuracy with which results are obtained. These methods are not commonly used in the area of atmospheric modeling, but AMR in particular is commonly used with great success in other areas of computational fluid dynamics. The model is initially validated using a series of idealized case studies that have e...

Richtmyer-Meshkov instability induced by the interaction of a shock wave with a rectangular block of SF6

In this article the interaction of a shock wave with a rectangular block of sulphur hexafluoride (SF6), occupying part of the test section of a shock tube, is studied by experimental and numerical means. The difference between the ratios of the specific heats of the two gases (air and SF6) gives rise to numerical problems (generation of spurious waves at their interface). This necessitated the development of a multifluid algorithm (augmented Navier-Stokes formulation). The governing equations are based on a thermodynamically consistent and fully conservative formulation. A Riemann-problem-based scheme (the weighted average flux method) is used to integrate the hyperbolic part of the system. To this end, a new approximate Riemann problem solver has been formulated to account for the variable ratio of specific heats. The resulting algorithm was implemented in an adaptive mesh refinement code, which allowed high-resolution simulations to be performed on desktop computers. The evolution of the flow is well ca...

Well-balanced compressible cut-cell simulation of atmospheric flow

Cut-cell meshes present an attractive alternative to terrain-following coordinates for the representation of topography within atmospheric flow simulations, particularly in regions of steep topographic gradients. In this paper, we present an explicit two-dimensional method for the numerical solution on such meshes of atmospheric flow equations including gravitational sources. This method is fully conservative and allows for time steps determined by the regular grid spacing, avoiding potential stability issues due to arbitrarily small boundary cells. We believe that the scheme is unique in that it is developed within a dimensionally split framework, in which each coordinate direction in the flow is solved independently at each time step. Other notable features of the scheme are: (i) its conceptual and practical simplicity, (ii) its flexibility with regard to the one-dimensional flux approximation scheme employed, and (iii) the well-balancing of the gravitational sources allowing for stable simulation of near-hydrostatic flows. The presented method is applied to a selection of test problems including buoyant bubble rise interacting with geometry and lee-wave generation due to topography.

Accumulating sequence of ignitions from a propagating pulse

Abstract Some surprising effects are seen in studying numerically the evolution of a propagating pulse of pressure in a medium reacting via a one-step exothermic Arrhenius reaction. The length and amplitude of the pulse are taken to be large enough for steepening effects to be important and for enhanced reaction to lead to a substantial reduction in ignition time. The evolution proceeds through a repeated sequence of similar stages involving: shock-formation and growth; ignition behind the shock; and the generation of another propagating pressure pulse. Substantial unsteady behavior is seen to be engendered by the entropy released through shock formation. A number of unsteady reignitions are seen to culminate in a pressure-peak, substantially higher than the von Neumann spike of a Chapman-Jouget wave, during the formation of a transient overdriven detonation; this decays subsequently towards a Chapman-Jouget state. It is conjectured that this sort of evolution may well be generic to ignition via a range of pressure-pulses in state-sensitive systems. A saturation of, or relative reduction in, the reactions thermal sensitivity ultimately prevents the reignition process after shock-formation from happening quickly enough to continue its repetition. As such, the behavior should be strongly dependent on the nature of the chemical model and is likely to be modified significantly by changes in the chemical mechanism.

Quasi-Steady Structures in the Two-Dimensional Initiation of Detonations

Two regimes of reflected-shock-induced ignition of explosive gases are known to exist, referred to as strong and weak (or mild) ignition. Experimental studies have shown that the former is manifested by the early appearance of a plane shock wave of chemical activity near the back wall of the shock tube and the whole strong-ignition process is a nominally one-dimensional phenomenon. When small distinct regions of increased chemical activity exist near the wall from which the incident wave reflects, localized thermal runaway leads directly to detonations that are multidimensional in character; this is the situation in what is called mild ignition. Although both strong and weak modes have been studied experimentally (in the 1960s and 1970s) and visualized by means of streak-camera and stroboscopic-laser-schlieren techniques, up to now most studies which use the methods of computational fluid dynamics have concentrated on the one-dimensional case. Previous analytical/numerical studies have shown how the three-dimensional structure of detonations can appear as a consequence of the instability of an initially ideal planar Zeldovich-von Neumann-Döring detonation. The latter is subject to some small-amplitude disturbances, and subsequent events lead to the eventual appearance of triple points along the front. In this paper some of the transient phenomena that take place in the first few microseconds after incident shock reflection from the closed end of a shock-tube are examined by means of a number of numerical simulations. A small hot-spot is assumed to exist in one of the corners between the reflective end plate and the walls of the shock tube. Evolution of the flow is followed, from the time of incident shock reflection, through the genesis of curved reaction waves, on to the appearance of an ‘explosion within the explosion’ ending with the creation of a nearly plane detonation wave and its gradual contamination by triple-shock-wave features. The events portrayed in this way are recognizable stages in the story that the experimental studies revealed 25 years ago. The all-important region between reflective wall and reflected shock, within which intense chemical activity begins and which, because of its small geometric extent, is not well resolved in the schlieren photographs, is here replaced by high resolution images of the primitive variables of the flow. The wealth of data provided by these simulations is subsequently correlated to reveal the existence of quasi-steady structures in the form of reaction waves (specifically, weak detonations and fast flames or deflagrations), which previously have been shown to be part of the evolutionary processes only in strictly one-dimensional phenomena.

The mathematical structure of multiphase thermal models of flow in porous media

This paper is concerned with the formulation and numerical solution of equations for modelling multicomponent, two-phase, thermal fluid flow in porous media. The fluid model consists of individual chemical component (species) conservation equations, Darcys law for volumetric flow rates and an energy equation in terms of enthalpy. The model is closed with an equation of state and phase equilibrium conditions that determine the distribution of the chemical components into phases. It is shown that, in the absence of diffusive forces, the flow equations can be split into a system of hyperbolic conservation laws for the species and enthalpy and a parabolic equation for pressure. This decomposition forms the basis of a sequential formulation where the pressure equation is solved implicitly and then the component and enthalpy conservation laws are solved explicitly. A numerical method based on this sequential formulation is presented and used to demonstrate some typical flow behaviour that occurs during fluid injection into a reservoir.

Assessment of a High Resolution Centered Scheme for the Solution of Hydrodynamical Semiconductor Equations

Hydrodynamical models are suitable to describe carrier transport in submicron semiconductor devices. These models have the form of nonlinear systems of hyperbolic conservation laws with source terms, coupled with Poissons equation. In this article we examine the suitability of a high resolution centered numerical scheme for the solution of the hyperbolic part of these extended models, in one space dimension. Because of the lack of physically significant exact analytical solutions, the method is assessed against a benchmark for the system of compressible, unsteady Euler equations with source terms, which has an exact solution; the latter is shown to be nearly identical to the numerical one. The method is then used to solve the extended hydrodynamical model (EM) based on the maximum entropy closure recently introduced by Anile, Romano, and Russo, simulating a ballistic diode n+-n-n+, which models a metal oxide semiconductor field effect transistor (MOSFET) channel. Results are presented for the reduced- and full-equation EM formulation at steady state, for an initially discontinuous electron density at the junctions. Transient results show the evolution of highly nonlinear waves emanating from the neighborhood of the junctions.

Numerical studies of the evolution of detonations

Some studies of the evolution of the shock-induced initiation of detonations are presented here. The aim is to examine the way that the dynamics and the chemistry of the system interact to produce detonation waves. Our present approach for constructing algorithms suitable for this type of flows is outlined, after a brief statement of the governing equations. A chemically-active flow governed by the reactive Euler equations, which has an exact solution, is used as a validation exercise. Various (previously loosely-connected) pieces of numerical and analytical work are brought together in lieu of a review, to set the scene for the multidimensional studies that follow. Two-dimensional flow is induced by a temperature perturbation @DT of order @e (the dimensionless inverse activation energy), in an otherwise planar flowfield. The effect of small variations in @DT on the ignition pattern is investigated by means of a series of numerical integrations. Analysis of the results gives some preliminary indications regarding the role of quasisteady structures during weak ignition.