Frankel equation for turbulent flames in the presence of a hydrodynamic instability

Abstract

An extension of the Frankel equation, coupled to a treatment of reconnections, is used to describe numerically turbulent flames submitted to the hydrodynamic Darrieus-Landau instability. The role played by this instability on the fractal properties of the front is evaluated. ͓S1063-651X͑97͒03506-X͔ PACS number͑s͒: 47.70.Fw, 82.40.Py It has been customary to describe turbulent premixed flames as interfaces propagating normally with a given laminar velocity, and submitted to an imposed turbulent flow field ͓1͔. However, it is well known that there is a retroaction of the flame front, which induces a new velocity field, which in turn will also affect the flame wrinkling.In the case of laminar flames, this retroaction effect leads to the hydrodynamic Darrieus-Landau instability. A nonlinear equation, the Michelson-Sivashinsky equation ͓2͔, has been introduced to describe this instability in the limit where the burnt gases are irrotational and where the flame is close to a plane flame. Despite these simplifying hypotheses, this equation had a lot of success in describing correctly most of the nonlinear features observed in laminar flames ͓3-6͔. Extensions of this equation have also been successful in describing spherically expanding flames ͓7,8͔. This Michelson-Sivashinsky equation has been generalized to obtain a coordinate-free equation describing a closed front, the Frankel equation ͓9͔ ͑an alternative way to incorporate heat release into a lagrangian method has been proposed in ͓10͔͒. A numerical solution of the Frankel equation can be found in ͓11͔, and in ͓12͔ an interesting study of the behavior of laminar flames with a very large radius. We shall be concerned here with the problem of turbulent flame fronts submitted to the Darrieus-Landau instability; a variant of the Frankel equation, coupled to a treatment of reconnections occurring on the front, will be used for this purpose.Let us first recall the usual form of the Frankel equation. Let us note by n the local normal vector at a given point of the flame, which is assumed to have the topology of a closed contour. n will be chosen to point in the direction of flame propagation. The evolution of the front is completely described by specifying the normal velocity of the flame at each point of the contour. This normal velocity is the sum of two terms: a first term u 1 (1Ϫ) represents the flame velocity: it is given by the laminar velocity u 1 corrected by a term proportional to the curvature ͑ being a constant usually called the Markstein length͒. The values u 1 ϭ1 and ϭ0.1 will be used in the simulations presented in this paper. A second term corresponding to the normal velocity caused by hydrodynamics has to be added to the flame velocity, and the Frankel equation ͓9͔ is obtained,where the integration is taken over the contour, and ␣ is a parameter controlling the importance of the hydrodynamic instability. In the case of a circle and ϭ0, the normal velocity V n becomes simply 1, as has been shown in the original article of Frankel ͓9͔. If we consider the g...

Assessment of the Evolution Equation Modelling approach for three-dimensional expanding wrinkled premixed flames

“…Since then, many other attempts and techniques to improve this equation or to propose other kinds of EEM equations can be found in the literature : e.g. higher order expansions in α for MS type equations [28,29,30], second order in time equations for transients or acoustics [31,32] non perturbative approaches [33,34], asymptotic expansion based on flame aspect ratio [35], 3D planar equations [36,37,38], equations dealing with non connected or non stellate front topology [39,40,41]. In the context of 3D expanding flames [13,42,43], many of the proposed equations can be seen as different extensions of Michelson-Sivashinsky equation.…”

Turbulent Flow Produced by Piston Motion in a Spark-ignition Engine

“…One of the popular models of turbulent burning treats a flame front as a passively propagating surface, e.g. see [3,[35][36][37][38][39]. Formally, this model corresponds to "burning" with zero thermal expansion, which does not influence the turbulent flow.…”

Flames with Realistic Thermal Expansion in a Time-Dependent Turbulent Flow

“…However, though the Taylor hypothesis has been used in many works [1][2][3][4][5][6][7][8][9][10], it has not been proven theoretically or checked experimentally. A more general turbulent model [5] takes into account temporal pulsations of the flow like…”

“…According to the Taylor hypothesis, temporal pulsations of the turbulent flow are negligible in comparison with time variations caused by flame propagation and can be ignored. In this case, a two-dimensional turbulent flow can be described in the laboratory reference frame by the stationary isotropic model [5] 1 Institute of Physics, Umeå University, S-90187 Umeå, Sweden. 2 Nuclear Safety Institute, Russian Academy of Sciences, Moscow 113191; slava.akkerman@physics.umu.se.…”

“…Moreover, our knowledge of turbulence in real laboratory and industrial applications is quite limited. The so-called Taylor hypothesis of "stationary" turbulence is one of the standard simplifying assumptions used to describe a turbulent flow [1][2][3][4][5][6]. According to the Taylor hypothesis, temporal pulsations of the turbulent flow are negligible in comparison with time variations caused by flame propagation and can be ignored.…”