Journal of Combustion

Volume 2018 (2018), Article ID 3635797, 8 pages

https://doi.org/10.1155/2018/3635797

## Numerical Study of Detonation Wave Propagation in the Variable Cross-Section Channel Using Unstructured Computational Grids

^{1}Institute for Computer Aided Design, Russian Academy of Sciences, Moscow, Russia^{2}Moscow Institute of Physics and Technology, Dolgoprudny, Russia

Correspondence should be addressed to Pavel Utkin; ur.liam@ktu_levap

Received 29 January 2018; Accepted 20 February 2018; Published 22 April 2018

Academic Editor: Richard Saurel

Copyright © 2018 Alexander Lopato and Pavel Utkin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The work is dedicated to the numerical study of detonation wave initiation and propagation in the variable cross-section axisymmetric channel filled with the model hydrogen-air mixture. The channel models the large-scale device for the utilization of worn-out tires. Mathematical model is based on two-dimensional axisymmetric Euler equations supplemented by global chemical kinetics model. The finite volume computational algorithm of the second approximation order for the calculation of two-dimensional flows with detonation waves on fully unstructured grids with triangular cells is developed. Three geometrical configurations of the channel are investigated, each with its own degree of the divergence of the conical part of the channel from the point of view of the pressure from the detonation wave on the end wall of the channel. The problem in consideration relates to the problem of waste recycling in the devices based on the detonation combustion of the fuel.

#### 1. Introduction

Mathematical modeling of two-dimensional flows with detonation waves (DW) as a result of solving of the Euler equations for the inviscid gas supplemented by the chemical reaction kinetics model originates in the late 1970s [1, 2]. Since then with the development of the computational methods for the solution of gas dynamics problems, the improvement of the kinetic schemes, and the growth of available computational resources, the continuous clarification of qualitative and quantitative characteristics of the process has occurred (for example, obtaining the three-dimensional spin mode of propagation, the thin structure of cellular detonation, and the detonation limits [3]). One can agree with [4] in which the current understanding of the mechanisms of DW propagation is put into direct dependence on the maximum computer performance achieved at the moment. At the same time, the growth of computational powers and as a result the possibility of carrying out the calculations with more and more detailed spatial-temporal resolution revealed a number of unusual effects of the detonation modeling that have not had full explanation yet. The effects include, for example, the possible detonation decay for the kinetics parameters close to the hydrocarbon fuels in two-dimensional calculations of the long time propagation of DW in the plane channel [5]. Thus, despite almost forty-year history of computational works in the field of detonation, a number of fundamental questions of DW mathematical modeling remain.

Numerical research of problems of DW initiation and propagation in technical systems and facilities involves the necessity of consideration of computational domains of a complex shape. This fact leads to the set of questions related to the construction of computational grids in such domains, construction of monotone in some sense schemes of high approximation order on the selected class of grids, and also infrastructure problems with large amounts of poorly structured data including visualization issues. Despite all the achievements in the field of multiprocessor computing, the numerical studies of problems of physical and chemical hydrodynamics with a volume exceeding 10^{8} computational cells are at the limit of the capabilities of the researchers due to the factors listed above. Thus, the development of computational technologies for the modeling of high speed flows with chemical reactions in areas of complex shape remains the actual issue.

In [5, 6] for carrying out three-dimensional computations of the DW initiation and propagation in complex-shaped domains, the block-structured grids and the classical first approximation order scheme of S. K. Godunov [6] or its modification based on some interpolation schemes for the approximation order increase [5] are used. At the same time, in relation to gas dynamics problems of chemically inert media the apparatus of high approximation order schemes on completely unstructured computational grids including shock wave problems has been developed; see the most known paper [7]. The analysis of publications about the usage of such approaches to the study of high speed flows with chemical reactions gives a few works [8–10]. In [9, 10], the finite element approach is applied. The most probable reason is the fact that although the integration of the gas dynamics equations is a key element of the computational algorithm for the simulation of high speed flows with chemical reactions, the presence of strongly nonlinear sources in the right-hand sides of the equations significantly reduces the possibilities of applying many numerical methods that successfully deal with the gas dynamics problems of inert media.

#### 2. Statement of the Problem

The axisymmetric channel of a variable cross-section filled with quiescent model hydrogen-air mixture under normal conditions (see Figure 1) that simulates the facility for utilization of used automobile tires [11, 12] is considered. The geometry of the problem corresponds to that considered in [12]. The channel consists of a short narrow segment for detonation initiation (preliminary chamber), a conical part for DW passing into a segment of a larger diameter (detonation chamber), and a working chamber. The cases of different expansion angles of the conical part corresponding to the values cm, 30 cm, and 50 cm are investigated. For the detonation initiation, the pressure 40 atm and temperature 1500 K are set in a region with the length of 4 cm in the preliminary chamber (colored gray in Figure 1). The boundary conditions of impermeability are set on all boundaries of the computational area. The pressure is recorded by the sensors and . The objectives are the description of the mechanism of the development of detonation process in the channel of such geometry, the comparative analysis of the pressure curves at the sensors under variation of the geometrical parameter , comparison of the obtained results with the data from [12], and research of the applicability of the developed computational algorithm of second approximation order on completely unstructured computational grids for solving large-scale practical tasks. Note that the problem in consideration relates to the fundamental problem of the investigation of mechanism of DW passing from a narrow channel into the wide, which was discussed in theoretical and experimental works of many authors; see, for example, [13, 14]. The theory of this phenomenon which would predict the realized regime depending on the geometrical characteristics of the channel, mixture properties, and degree of DW overdriving has not been built.