A parallel adaptive method for simulating shock-induced combustion with detailed chemical kinetics in complex domains

doi:10.1016/j.compstruc.2008.11.007

Computers & Structures

Volume 87, Issues 11–12, June 2009, Pages 769-783

https://doi.org/10.1016/j.compstruc.2008.11.007 Get rights and content

Abstract

An adaptive finite volume approach is presented to accurately simulate shock-induced combustion phenomena in gases, particularly detonation waves. The method uses a Cartesian mesh that is dynamically adapted to embedded geometries and flow features by using regular refinement patches. The discretisation is a reliable linearised Riemann solver for thermally perfect gas mixtures; detailed kinetics are considered in an operator splitting approach. Besides easily reproducible ignition problems, the capabilities of the method and its parallel implementation are quantified and demonstrated for fully resolved triple point structure investigations of Chapman–Jouguet detonations in low-pressure hydrogen–oxygen–argon mixtures in two and three space dimensions.

Introduction

A detonation wave is a self-sustained, violent form of shock-induced combustion that is characterised by a subtle interplay between the processes of hydrodynamic shock propagation and chemical reaction. In here, we focus on gaseous detonations, especially in mixtures of hydrogen and oxygen, that are characteristic for accident scenarios. Innovative technical applications of gaseous detonations include pulsed detonation engines and ram accelerators.

While the classical Chapman–Jouguet (CJ) theory gives the average speed of propagation of a detonation with good accuracy, no theory is available for accurately predicting the detailed flow conditions in a multi-dimensional detonation wave. The theory after Zel’dovich, von Neumann and Döring (ZND) provides the one-dimensional solution of a steady planar detonation wave, but already early experiments showed that it can only be used for rough estimates in practically relevant cases since real detonations never remain planar. Instead, detonations exhibit instationary multi-dimensional sub-structures with dramatically enhanced flow and combustion conditions [1]. As experimental investigations of detonations and shock-induced combustion waves are already challenging for simple setups, numerical simulation can provide a means for understanding all aspects and consequences of detonation propagation in technically relevant configurations.

The greatest challenge in simulating shock-induced combustion is the inherent disparity of the scales involved. A typical detonation wave propagates roughly at speeds of 1000–2000 m s⁻¹ but the combustion is usually completed in less than 1 μs. In order to predict critical, hot sub-structures (triple points) at the detonation front correctly, it is mandatory to fully resolve the shock and the following combustion zone. Note that a straightforward multiscale approach, that ensures chemical reaction to the equilibrium state at all grid resolutions [2], eliminates such sub-structures from the computation and cannot be trusted in predicting combustion extinction and re-ignition (cf. Section 5.4). Similarly, the application of an operator splitting approach alone fails to capture the complex energy exchange between the chemical reactions and the shock wave of a self-sustained detonation (cf. Section 5.1). Predictive computations require sufficient resolution both in space and time around the leading shock and in the combustion zone. Since these resolution requirements are very localised, shock-induced combustion phenomena are very suitable for dynamic mesh adaptation.

While earlier efforts [3], [4] utilised unstructured adaptive meshes, the relative geometric simplicity of the configurations of interest to combustion researchers still favours computationally more efficient structured meshing approaches. Among the techniques considered so far are elaborate moving mesh methods [5], dynamically adaptive overlapping structured meshes [6], boundary-aligned structured meshes [7], and purely Eulerian Cartesian methods that either use octree-based cell-wise refinement [8], [9] or a patch-based approach with block-structured refinement grids [10], [11], [12], [13]. In here, we focus on the latter adaptation methodology, originally proposed for non-reactive gas dynamics [14], since it promises greatest performance on the current generation of super-scalar computers. While most previous publications are restricted to computationally inexpensive one-step reaction models and/or two-dimensional configurations, we consider in here multi-species thermally perfect gas mixtures with arbitrary chemical kinetics and demonstrate the applicability of the proposed methods for cases relevant to combustion research in all three space dimensions.

After a brief description of the governing equations in Section 2, we discuss all components of the employed finite volume method in Section 3. Of particular importance is the detailing of the linearised Riemann solver of Roe-type in Section 3.2 that has been employed as a hydrodynamic upwind scheme in all computations in Section 5. In particular, we describe all adjustments found necessary to make this low numerical diffusion scheme robust and reliable in practical relevant computations. In Section 3.3 we sketch our approach to considering complex, possibly moving, objects on a Cartesian mesh. Section 4 describes the block-structured mesh adaptation method, refinement indicators, and our parallelisation approach based on rigorous domain decomposition of the mesh hierarchy. Several numerical examples for detailed hydrogen–oxygen chemistry are discussed in Section 5. Special emphasis is put on quantifying the savings in wall time from mesh adaptation and parallelisation. The first two configurations, an ignition problem in one space dimension [15] and a steady shock-induced combustion problem with cylindrical symmetry [16], are excellent, easily reproducible verification cases. In Section 5.3, typical adaptive simulations in two and three space dimensions are presented that aim at fully resolving and understanding triple point formation in CJ detonation waves in the idealised, perfectly regular case. Finally, we briefly describe large-scale massively parallel computations of fully resolved transient detonation structures as a CJ detonation propagates through smooth pipe bends. These computations demonstrate that it is nowadays feasible to resolve all relevant sub-scale features in two-dimensional detonation waves in certain practically relevant applications.

Section snippets

Governing equations

The appropriate model for detonation propagation in premixed gases with realistic chemistry are the Euler equations for multiple thermally perfect species with reactive source terms [17], [18]. In d-dimensional Cartesian coordinates these equations can be written as an inhomogeneous conservation law of the structure $\frac{\partial}{\partial t} q (x, t) + \sum_{n = 1}^{d} \frac{\partial}{\partial x_{n}} f_{n} (q (x, t)) = s (q (x, t)) x = (x_{1}, \dots, x_{d})^{T} \in R^{d}, t \in R_{0}^{+},$ where $q = q (x, t)$ denotes the vector of conserved quantities. The functions $f_{n} (q)$ , $n = 1, \dots, d$ are the hydrodynamic fluxes; $s (q)$

Numerical methods

We apply the time-operator splitting approach or method of fractional steps [20] to decouple hydrodynamic transport and chemical reaction numerically. This technique is most frequently used for time-dependent reactive flow computations. The homogeneous partial differential equation $H^{(Δ t)} : \frac{\partial q}{\partial t} + \sum_{n = 1}^{d} \frac{\partial}{\partial x_{n}} f_{n} (q) = 0, IC : Q^{κ} \overset{Δ t}{\Rightarrow} {\tilde{Q}}^{κ + 1}$ and the usually stiff ordinary differential equation $S^{(Δ t)} : \frac{\partial q}{\partial t} = s (q), IC : {\tilde{Q}}^{κ + 1} \overset{Δ t}{\Rightarrow} Q^{κ + 1}$ are integrated successively with the data $Q$ from the preceding step as initial condition

Adaptive mesh refinement

In order to supply the required temporal and spatial resolution efficiently, we employ the block-structured adaptive mesh refinement (SAMR) method after Berger and Colella [14], [35] which is tailored especially for hyperbolic conservation laws on logically rectangular FV grids (not necessarily Cartesian). We have implemented the SAMR method in a generic, dimension-independent object-oriented framework in C++. It is called AMROC (adaptive mesh refinement in object-oriented C++) and is free of

Numerical results

In the following sub-sections we present typical dynamically adaptive simulations of detonation waves and shock-induced combustion that have been obtained with the previously described numerical method. All computations use the hydrogen–oxygen reaction section of Westbrock’s larger hydrocarbon mechanism [43]. The employed subset consists of 34 elementary reactions and considers the nine species $H$ , $O$ , $OH$ , $H_{2}$ , $O_{2}$ , $H_{2} O$ , ${HO}_{2}$ , $H_{2} O_{2}$ and $Ar$ . In all simulations, the combustible mixture is perfectly

Conclusions

We have described a numerical method for simulating detonation and shock-induced combustion waves with detailed chemical reaction at very high accuracy. A splitting approach combines an explicit high-resolution upwind scheme based on a linearised Riemann solver for thermally perfect mixtures with a semi-implicit method for integrating the chemical reaction rates. In addition, temporal and spatial stiffness introduced by the reaction terms is approached by dynamically adapting the resolution

Acknowledgements

This work is sponsored by the Office of Advanced Scientific Computing Research; U.S. Department of Energy (DOE) and was performed at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725. The large-scale computations in Sections 5.3 Regular cellular detonation structure, 5.4 Detonation structure in smooth pipe bends have been carried out while the author was at the California Institute of Technology and was supported by the ASC program of

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Published by Elsevier Ltd.

A parallel adaptive method for simulating shock-induced combustion with detailed chemical kinetics in complex domains

Abstract

Introduction

Section snippets

Governing equations

Numerical methods

Adaptive mesh refinement

Numerical results

Conclusions

Acknowledgements

J Combust Flame

J Comput Phys

J Comput Phys

J Comput Phys

J Comput Phys

Combust Flame

Appl Numer Math

J Combust Flame

J Comput Phys

J Comput Phys

J Comput Phys

J Comput Phys

J Comput Phys

J Comput Phys

J Comput Phys

J Combust Flame

J Comput Phys

J Combust Flame

J Combust Flame

Numerical modeling of detonations

Adaptive unstructured finite element method for two-dimensional detonation simulations

Shock Waves

Nonlinear cellular dynamics of the idealized detonation model: regular cells

Combust Theory Model

Two-dimensional numerical simulations of multi-headed detonations in oxygen–aluminum mixtures using adaptive mesh refinement

Shock Waves

Local adaptive mesh refinement for shock hydrodynamics

J Comput Phys

Detonation