Theoretical and numerical structure for unstable two-dimensional detonations

Bourlioux, Anne; Majda, Andrew J.

doi:10.1016/0010-2180(92)90084-3

Cited by 147 publications

(116 citation statements)

References 10 publications

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“…This applies to cases with a high degree of overdrive factor or moderate activation energy; or when seeking only a reasonable estimation of the stability boundary using numerical simulations (Grismer & Powers 1996;Bourlioux et al 1991;Sharpe & Falle 1999;Hwang et al 2000). For sufficiently regular detonation cells as for the case with high degree of overdrive, Bourlioux & Majda (1992) simulated detonation cells using a similar grid resolution of about 20 points/l 1/2 and agreement was obtained between computed detonation cell sizes and predictions from stability theories. The later work by Choi, Ma & Yang (2008) also performed a parametric study and proposed a general rule for simulating moderately unstable detonation that the detonation cell structure can be properly resolved if a minimum of five grid points are included in the heat-release zone of the corresponding steady ZND detonation wave.…”

Section: Resultsmentioning

confidence: 65%

“…For moderate heat release values, increases in detonation overdrive f cannot guarantee stability of the steady detonation wave to transverse linear disturbances, which can be excited to originate the detonation cell formation (Short & Stewart 1998). Numerical simulations also demonstrate that with sufficient resolution, regular detonation cells can still emerge at very high overdrive (Bourlioux & Majda 1992). In order to confirm whether a neutral stability boundary can be obtained, the overdrive f is further increased to 2.71 and 3.07 by using a wedge angle of 33 and 36 • , respectively.…”

Section: Resultsmentioning

confidence: 99%

“…The reaction is controlled by the activation energy E a and pre-exponential factor k. The latter is chosen to define the spatial and temporal scales. In this study, the dimensionless parameters are fixed with the values q = 50, E a = 50 and γ = 1.20 often used in previous studies of both oblique and normal onedimensional pulsating detonations (Bourlioux & Majda 1992;Grismer & Powers 1996;Short & Stewart 1998;Papalexandris 2000;Verreault et al 2013). The governing equations are discretized on Cartesian orthogonal uniform grids and solved with the MUSCL-Hancock scheme with Strang's splitting.…”

Section: Methodsmentioning

confidence: 99%

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Numerical study on unstable surfaces of oblique detonations

2014

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In this study, the onset of cellular structure on oblique detonation surfaces is investigated numerically using a one-step irreversible Arrhenius reaction kinetic model. Two types of oblique detonations are observed from the simulations. One is weakly unstable characterized by the existence of a planar surface, and the other is strongly unstable characterized by the immediate formation of the cellular structure. It is found that a high degree of overdrive suppresses the formation of cellular structures as confirmed by the results of many previous studies. However, the present investigation demonstrates that cellular structures also appear with degree of overdrive of 2.06 and 2.37, values much higher than ∼1.8 suggested previously in the literature for the critical value defining the instability boundary of oblique detonations. This contradiction could be explained by the use of differently shaped walls, a straight wall used in this study and a custom-designed curved wedge system so as to induce straight oblique detonations in previous studies. Another possible reason could be due to the low and possibly insufficient resolution used in previously published studies. Hence, simulations with different grid sizes are also performed to examine the effect of resolution on the numerical solutions. Using the present results, analysis also shows that although the characteristic lengths of unstable surfaces are different when the incident Mach number changes, these length scales are proportional to tangential velocities. Hence, the interior time determined by the overdrive degree is identified, and its limitation as the instability parameter is discussed.

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Section: Resultsmentioning

confidence: 65%

Section: Resultsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

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Numerical study on unstable surfaces of oblique detonations

2014

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“…This approach includes linear stability analyses [7][8][9], weakly non-linear theories [10,11] and direct numerical simulations [9,[12][13][14][15][16][17]. Due to the above empirical correlations with cell size, a major objective of theoretical or numerical studies is to predict and understand the dependence of cell size and regularity on the fuel mixture and initial conditions.…”

Section: Introductionmentioning

confidence: 99%

“…Much of this work employs a now standard idealized model as defined by Erpenbeck [18], which consists of an ideal gas (polytropic) equation of state, and a one-step chemistry model with Arrhenius kinetics. Numerical simulations using the idealized model have shown that the model appears to contain many of the features observed experimentally, including cells of varying regularity and size, complex transverse wave interactions, and the formation of unburnt pockets of fuel [9,[12][13][14][15][16][17][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear cellular dynamics of the idealized detonation model: Regular cells

Sharpe

Quirk

2007

Combustion Theory and Modelling

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High-resolution numerical simulations of cellular detonations are performed using a parallelized adaptive grid solver, in the case where the channel width is very wide. In particular, the nonlinear response of a weakly unstable ZND detonation to two-dimensional perturbations is studied in the context of the idealized one-step chemistry model. For random perturbations, cells appear with a characteristic size in good agreement with that corresponding to the maximum growth rate from a linear stability analysis. However, the cells then grow and equilibrate at a larger size. It is also shown that the linear analysis predicts well the ratio of cell lengths to cell widths of the fully developed cells. The evolutionary dynamics of the growth are nonetheless quite slow, in that the detonation needs to run of the order of 1000 reaction lengths before the final size and equilibrium state is reached. For sinusoidal perturbations, it is found that there is a large band of wavelengths/cell sizes which can propagate over very long distances (∼1000 reaction lengths). By perturbing the fully developed cells of each wavelength, it is found that smaller cells in this range are unstable to symmetry breaking, which again results in cellular growth to a larger final size. However, a range of larger cell sizes appear to be nonlinearly stable. As a result it is found that the final cell size of the model is non-unique, even for such a weakly unstable, regular cell case. Indeed, in the case studied, the equilibrium cell size varies by 100% with different initial conditions. Numerical dependencies of the cellular dynamics are also examined.

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Front capturing by level set method for the reactive Euler equations

Xiao

Wang

et al. 2021

Numerical Methods in Fluids

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In this article, we present a front capturing method for compressible reactive flows, where a shock tracking technique is applied with level set method. In stiff reaction problems, the difficulty arises when the time scale of the source term is significantly shorter than the time scale of the homogeneous conservation law. The spurious numerical phenomenon may occur due to the smeared out shock profiles. In order to overcome the difficulty, the detonation front is captured sharply by level set technique, and a modified finite volume scheme is developed at the computational cells cut by detonation front. The mass, momentum, and energy transitions are occurred on shock front for chemical reactive flows. In our method, the interface exchanges are calculated by considering a Riemann problem. Unlike the standard level set/ghost fluid method, our method can maintain the conservation on the shock discontinuity. Oneand two-dimensional numerical examples including stable and unstable detonation problems are illustrated to verify the good reliability and robustness of our method.

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Theoretical and numerical structure for unstable two-dimensional detonations

Cited by 147 publications

References 10 publications

Numerical study on unstable surfaces of oblique detonations

Numerical study on unstable surfaces of oblique detonations

Nonlinear cellular dynamics of the idealized detonation model: Regular cells

Front capturing by level set method for the reactive Euler equations

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