Pinch-off in forced and non-forced, buoyant laminar jet diffusion flames
Introduction
Jet diffusion flames subjected to a gravity field g have been object of study for many years. A series of papers published by Chamberlain and Thrun [2] and Chamberlain and Rose [3], [4] provided the first specific investigation of the buoyancy phenomenon. Chamberlain and his co-authors were the first to describe the onset of periodic flame oscillations due to buoyancy forces, measure its frequency and point out the possibility of creation of pockets of gas detaching from the tip of the flame, a phenomenon that later would be known as pinch-off. Apart from the seminal works by Chamberlain and Thrun [2] and Chamberlain and Rose [3], flame break-up in non-forced flames has also been observed by Ballantyne and Bray [15], Becker and Liang [16] and Grant and Jones [17].
Different arguments have been used to explain the emergence of these oscillations. The first attempt by Kimura [5] and later the paper by Peters and Buckmaster [6] and Lingens et al. [7] pointed at a Kelvin–Helmoltz instability initiated by the large density gradients found at the flame. This line of thinking was complemented some years later when Jiang and Luo [8] proposed an absolute instability initiated by the buoyancy force as the responsible of flame flickering. According to their work, the hot gas inside the flame is pushed upwards by buoyancy while the cold outer fluid moves downwards, creating a high-vorticity region that interacts with the flame and makes it flicker. The transition from shear to buoyancy dominance probably takes place at some intermediate value of the Froude number, similarly to what happen in low density non-reacting jets [9].
Several experimental works have reported flickering frequencies that remain independent of the Reynolds number and of the fuel considered, at least while the flow remains laminar [10], [11], but change considerably with the Froude number , where Uj and a are the maximum jet velocity and the pipe radius respectively and g is the acceleration of gravity. According to this work, as we move towards microgravity conditions, the instabilities disappear and the flame attains a steady state. Alternatively, and contrary to what could be expected, extremely small Froude numbers lead also to steady flames [12]. In this case, the flame is so short that the destabilizing structures created by buoyancy do not affect the region where the heat is released but rather the plume downstream.
The relationship between the unsteady velocity field and the unsteady reaction field in a turbulent flame is still nowdays a challenging problem. In this regard, the instabilities observed in buoyant jet diffusion flames offer a simplified frame to study that interaction in low Reynolds number configurations. This research line was followed by Strawa and Cantwell [13], [14], who axially forced a jet flame at different frequencies to produce a periodic flow suitable for conditional sampling. They found that the structure of the flow field could be controlled by the influence that the pulsation exerts in the mixing and the combustion process. Also, if the flame is forced at a frequency close to the natural flickering frequency, the flame breaks up and a pocket of gas is detached from the tip of the flame, in a process highly repeatable from cycle to cycle.
Further insight on the interaction between unsteadiness and chemical kinetic was obtained by the experimental studies by Shaddix et al. [18] and Smyth et al. [19] or the computational works by Sánchez-Sanz et al. [20], Dworkin et al. [21], Mohammed et al. [22] and others. All those studies have reported higher concentrations of pollutants in unsteady flames than in their steady counterparts. Specifically, Shaddix et al. [18] used tomographic reconstruction and laser-induced incandescence to measure the soot volume fraction in both steady and forced flames, finding, in the latter, a fourfold increase in the time-averaged soot volume. Smyth et al. [19] went a little farther and acoustically forced a methane flame at a frequency close to the flickering frequency. At that excitation frequency, flame pinch-off occurs and seven times more soot was found in their laser-induced fluorescence measurements. Computational works [21], [22], [20] deepen in that conclusion and offer more details about the formation of carbon monoxide and soot in unsteady flames.
Motivated by the influence of pinch-off on the formation of pollutants, this work tries to elucidate the conditions for which the surface of the flame breaks in forced and non-forced flames. To this end, we have carried out a computational study that uses an infinitely fast chemistry model that allows the parametric study of the problem for fuels with unity and non-unity Lewis number.
Section snippets
Formulation
Consider the combustion problem, sketched in Fig. 1, of a fuel jet issuing from a pipe of radius a into a quiescent air atmosphere at temperature T0. The fuel emerges with a Poiseuille velocity profile modulated with an oscillatory component of amplitude A and dimensional frequency w′. The chemical reaction between the fuel and the oxidizer is assumed to be infinitely fast, with an overall stoichiometryin which a mass s of oxygen is consummed, generating a thermal energy q and a
Details of the solver
The set of equations given above in (16), (17), (18), (19) can be written in general form aswhere the solution stands for Z, H and . In the above equation, represents the position vector in the spatial domain Σ with smooth boundary ∂Σ. Γc and Γc⧹∂Σ are the parts of the boundary in which Dirichlet and Neumann conditions are applied, respectively. The coefficient Dc changes for every variable, adapting (21)
Flame pinch-off
Flame pinch-off is defined to occur whenare satisfied simultaneously at the axis of the jet r = 0. Conditions (22) are simply the mathematical representation of a break of the stoichiometric surface that generates two combustion regions. The first one constitutes the main flame and remains attached to the fuel injector. The second region is formed by a pocket of hot gas enclosed by a flame that travels downstream with velocity , and disappears when the
Conclusions
The paper analyzes numerically the conditions under which flame pinch-off is observed in forced and non-forced, buoyant laminar jet diffusion flames. Intrinsic buoyancy instabilities lead to flame oscillations even when the flow inlet conditions are steady. Its oscillation frequency ωl,0 has been measured previously and is only a function of the Froude number, being almost independent of the fuel considered and of the Reynolds number of the flow [10]. Under certain conditions, these oscillation
Acknowledgements
This work was supported by the Spanish MICINN under Projects # ENE2008-06515-C04 (MSS, EFT) and # MTM2010-18079 (JC).
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