A DNS study of the impact of gravity on spherically expanding laminar premixed flames
Introduction
The laminar unstretched burning velocity is a characteristic property of a premixed flame. It is used to validate chemical reaction mechanisms [1], [2], [3] and it is an ingredient of many combustion models for the simulation of turbulent premixed flames like the G-equation model [4], [5], [6]. Recently, it has been widely used to assess the safety hazards of hydrofluorocarbon refrigerants [7] that are mildly flammable. To measure laminar burning velocities, spherically expanding flames in a closed vessel are commonly studied [8], [9] since, in the limit of sufficiently large radii, the flame structure of a spherically expanding flame approaches the structure of an unstretched laminar premixed flame. However, very slow flames such as hydrofluorocarbon refrigerant flames are affected by buoyancy, which makes the determination of the laminar burning velocity challenging. The objective of this work is to understand if and how laminar burning velocities of unstretched premixed flames may be extracted from flames that are significantly affected by buoyancy.
The propagation speed of an unstretched laminar premixed flame with respect to the burned mixture, is referred to as the laminar burning velocity . The flame speed of spherical flames with respect to the burned gases deviates from due to the effect of flame stretch. Karlovitz et al. [10] suggested that the flame speed is proportional to stretch. This was confirmed in theoretical studies by Clavin and Williams [11] and Matalon and Matkowsky [12], who assumed a global one-step reaction with large activation energy. In this model, the flame sheet is located in the reaction zone, which is assumed to be small compared to the preheat zone. However, if considering different iso-surfaces within the flame or assuming a two-step chemical mechanism, Giannakopoulos et al. [13] and Clavin and Graa-Otero [14] showed that the flame speed does not only depend on stretch, but is proportional to strain rate and curvature. As will be shown for the case of a perfectly spherical flame, strain rate and curvature are proportional to stretch and can be expressed in terms of stretch, so the flame speed again yields a linear dependence on stretch. This relationship is a well established concept that enables the determination of . It has been the subject of extensive studies and it is in good agreement with experimental results of various groups [15], [16].
For slowly burning flames, the flame expansion is no longer spherical as the flame motion is significantly affected by buoyancy. Due to the density difference between the burned and unburned gas, these flames develop an upward motion and the initially spherical flame deforms and attains a mushroom-like shape. Choi et al. [17] experimentally investigated the dynamics of C2H2F4/CH4 flames with different O2 and N2 compositions as oxidizer. Mixtures with high C2H2F4 and low O2 concentrations are most significantly affected by buoyancy due to the slow flame propagation. Similar results were obtained by Zingale and Dursi [18] who conducted simulations of flame bubbles in type Ia Supernovae. They performed three simulations with different fuel densities on a two-dimensional cartesian grid to modulate the influence of gravity on the flame evolution. The thermonuclear burning process is described by a one-step mechanism. Similar to Choi et al. [17], an initially spherical flame expansion is seen and, eventually, buoyancy effects set in and the flames obtain a mushroom-like shape as the upward motion induces a toroidal vortex.
The Richardson number Ri defined asis a measure of the relative importance of buoyancy [19]. In Eq. (1), ρu and ρb represent the density in the unburned and burned gas, g is the gravitational acceleration, R is the flame radius, whose definition is discussed later, and refers to the speed of the flame front relative to the flame origin. If the Richardson number is sufficiently small, the effects of gravity on the flame expansion process are negligible, and if it is sufficiently large, the flame expansion is slow enough to allow the gravitational force to affect the flow.
The upward motion of buoyant flames induces a complex flow, so in contrast to ideally spherical flames, different points of the flame front propagate with a different absolute speed, making the extraction of difficult. Pfahl et al. [20] and Choi et al. [17] conducted experiments of buoyant flames and suggested determining from the speed of the most outer horizontal flame front position as the effect of buoyancy is minimal for this point on the flame front. However, it will be shown below that the induced velocity field substantially modifies the dynamics of this point and determining from the most outer horizontal flame front position introduces significant errors.
In this work, the flame dynamics of rich hydrogen/air flames diluted with nitrogen and a lean methane/air flame are analyzed by means of Direct Numerical Simulations (DNS). The detailed data provided by DNS are used to analyze the impact of buoyancy on flame curvature, strain rate, stretch, and flame speed. Additionally, a hydrogen/air flame DNS with unity Lewis number for all species is computed to assess Lewis number effects on the flame evolution. The paper is structured as follows: first, the configuration of the simulations and the numerical methods are discussed. Second, the concept of flame speed and the flame speed of spherical flames without the impact of buoyancy is recapped to be used as a reference with respect to the buoyant flames of this study. Third, the dynamics and motion of buoyant flames are discussed and an analysis of the flame speed and its dependence on stretch, strain rate, and curvature is presented. Finally, different techniques to determine from buoyant flames are discussed.
Section snippets
Configuration
All simulations have been conducted in a cylindrical closed vessel. Figure 1 shows the simulation domain, in which a small flame kernel expands and moves upwards due to gravity. Figure 1(a) shows a temperature iso-surface of the full flame that is enclosed in the cylindrical vessel. However, the actual simulation domain is shown in Fig. 1(b) and is marked by the dotted lines. Since cylindrical coordinates are employed and due to the axisymmetric flame configuration, only a two-dimensional plane
Governing equations and numerical methods
The flow is modeled by the reacting Navier–Stokes equations in the low-Mach limit [22]. The species diffusion velocity appearing in the species and temperature equations is modeled with the Curtiss–Hirschfelder approximation [23], [24] and the species diffusivities Di are determined from the thermal conductivity λ, the density ρ, and the specific heat capacity cp as by imposing spatially homogeneous Lewis numbers. The Lewis numbers were taken from the burned gas region of
Spherically expanding flames
For large radii, the flame speed of a spherical flame that is unaffected by buoyancy is proportional to stretch [15], [16]. Flame stretch, strain rate and curvature of a spherical flame are coupled and cannot change independently. These relations are well established in literature, but do not hold for buoyant flames as will be shown later. Thus, this section recaps the dependency of flame speed on stretch, curvature, and strain rate to contrast the findings of buoyant flames with the ones of
Temporal evolution of buoyant flames
Figure 5 shows the temporal evolution of all cases where the flame is affected by buoyancy. The change in time is expressed by the Richardson number, which monotonically increases in time, and the spatial coordinates are normalized by an equivalent radius Req.-V of the flame volume. The equivalent radius Req.-V is obtained from a sphere that has the same volume as the flame volume VfIn each sub-figure, a region of size 4Req.-V × 5Req.-V is shown. The Richardson number is defined
Conclusion
Direct Numerical Simulations of a lean methane/air flame and rich hydrogen/air flames diluted with molecular nitrogen whose motion is affected by buoyancy have been performed. The flames feature different laminar burning velocities to assess the competition of the gravitational force and flame propagation. For all flames, the temporal evolution and the rising velocity of the buoyant flames is found to be determined by the Richardson number unless the flame propagation is dominated by non-unity
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The authors from RWTH Aachen University gratefully acknowledge the generous support of the National Institute of Standards and Technology, U. S. Department of Commerce, (Grant No. 70NANB17H276). Computational resources have been provided by the Gauss Centre for Supercomputing e.V. on the GCS Supercomputer SuperMuc at Leibniz Supercomputing Centre in Munich.
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