Flame front propagation IV: Random Noise and Pole-Dynamics in Unstable Front Propagation II
📝 Abstract
The current paper is a corrected version of our previous paper arXiv:adap-org/9608001. Similarly to previous version we investigate the problem of flame propagation. This problem is studied as an example of unstable fronts that wrinkle on many scales. The analytic tool of pole expansion in the complex plane is employed to address the interaction of the unstable growth process with random initial conditions and perturbations. We argue that the effect of random noise is immense and that it can never be neglected in sufficiently large systems. We present simulations that lead to scaling laws for the velocity and acceleration of the front as a function of the system size and the level of noise, and analytic arguments that explain these results in terms of the noisy pole dynamics.This version corrects some very critical errors made in arXiv:adap-org/9608001 and makes more detailed description of excess number of poles in system, number of poles that appear in the system in unit of time, life time of pole. It allows us to understand more correctly dependence of the system parameters on noise than in arXiv:adap-org/9608001
💡 Analysis
The current paper is a corrected version of our previous paper arXiv:adap-org/9608001. Similarly to previous version we investigate the problem of flame propagation. This problem is studied as an example of unstable fronts that wrinkle on many scales. The analytic tool of pole expansion in the complex plane is employed to address the interaction of the unstable growth process with random initial conditions and perturbations. We argue that the effect of random noise is immense and that it can never be neglected in sufficiently large systems. We present simulations that lead to scaling laws for the velocity and acceleration of the front as a function of the system size and the level of noise, and analytic arguments that explain these results in terms of the noisy pole dynamics.This version corrects some very critical errors made in arXiv:adap-org/9608001 and makes more detailed description of excess number of poles in system, number of poles that appear in the system in unit of time, life time of pole. It allows us to understand more correctly dependence of the system parameters on noise than in arXiv:adap-org/9608001
📄 Content
arXiv:1106.0558v4 [nlin.CD] 9 Dec 2012 UDC 536.46 Random Noise and Pole-Dynamics in Unstable Front Propagation II Oleg Kupervasser, Zeev Olami Department of Chemical Physics The Weizmann Institute of Science Rehovot 76100, Israel November 18, 2018 Abstract The current paper is a corrected version of our previous paper (Olami et al., PRE 55 (3),(1997)). Similarly to previous version we investigate the problem of flame propagation. This problem is studied as an example of unstable fronts that wrinkle on many scales. The analytic tool of pole expansion in the complex plane is em- ployed to address the interaction of the unstable growth process with random initial conditions and perturbations. We argue that the effect of random noise is immense and that it can never be neglected in sufficiently large systems. We present simu- lations that lead to scaling laws for the velocity and acceleration of the front as a function of the system size and the level of noise, and analytic arguments that ex- plain these results in terms of the noisy pole dynamics.This version corrects some very critical errors made in (Olami et al., PRE 55 (3),(1997)) and makes more detailed description of excess number of poles in system , number of poles that appear in the system in unit of time, life time of pole. It allows us to understand more correctly dependence of the system parameters on noise than in (Olami et al., PRE 55 (3),(1997)) Keywords: flame propagation, pole dynamics, unstable front, random noise, self- acceleration 1 Introduction This article considers the very interesting problem of describing the nonlinear stage of development of hydrodynamic instability of the flame. This problem can be considered for 1D (channel propagation) , 2D (cylindrical case) , 3D (spherical case). The direct numerical simulations can be made based on of the NavierStokes equations including chemical kinetics in the form of the Arrhenius law [1]. For 2D and 3D cases we can 1 see experimentally observable effects [2], [3] of the self-acceleration of the front of a divergent flame, the formation of cellular structure, and other effects. Much more simple equation for flame front propagation can be obtained [4], [5] . It is Michelson-Sivashinsky approximation model. The Michelson-Sivashinsky model assumes very serious limitations, such as the smallness of the coefficient of gas ex- pansion and, consequently, the potential flow in the combustion products and fresh mixture, weak nonlinearity, the assumptions of the stabilizing effect of the curvature of the flame front and a linear dependence on the curvature of the normal speed, and others. The calculations with a noise term was performed in both one-dimensional and two-dimensional formulations of the problem in our previous papers [6], [7], [8], [9], [10], [11] and recently by Karlin and Sivashinsky [12] , [13] for 1D, 2D and 3D cases. Interest of this model stems, firstly, from the fact that despite the serious limita- tions, this model can qualitatively describe the scenario of hydrodynamic instability and, in particular, the self-acceleration of the front of a divergent flame, the formation of a cellular structure, and other effects. Secondary, this nonlinear model has exact solutions which can be constructed on the basis of pole expansions [14], [15], [16], [17]. This pole expansion method got development in the following papers. There was investigated relationships between pole solutions and partial decomposition in Fourier series [18], between the gas flow field and pole solutions [19]. The future development can be found in our previous papers and the correspondent references inside of [6], [7], [8], [9], [10], [11]. It must be mentioned that the simplest 1D case of the flame front propagation is very important. It was the main reason for creation this papers considering in detail the 1D case. Such investigation of this case allows us to understand qualitatively and quantitatively the pole dynamics. This understanding is a basis for the consideration more complex 2D and 3D cases. The our paper [8] clearly demonstrate this fact. The cellular structure, acceleration exponent for 2D case was found on the basis of 1D results. The future development of our results is gotten in papers [20], [21], [22]. In the very interesting paper [20] the noise term was considered in the poles-like form. The numerical results for 1D case are in a good consistence with the theoretical results of this paper. This paper is update version of our previous version [6]. In this version we correct some error. For example, we choose incorrect spectrum width of white noise with con- stant amplitude during changing size of channel L. As result we didn’t obtain in [6] saturation of the frame front velocity during increasing L in contradiction with this pa- per. The second important error connected to number of regimes defined on the “phase diagram” of the system as a function of L and f. Indeed not three but four such regimes exist. The noi
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