Exact Equation for Curved Stationary Flames with Arbitrary Gas Expansion

Kazakov, Kirill A.

doi:10.1103/physrevlett.94.094501

Cited by 21 publications

(40 citation statements)

References 10 publications

(32 reference statements)

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“…It was mentioned there that even if one is interested only in the on-shell values of the vorticity component, the τ -dependence of the function M (x, t − τ ) cannot be neglected. Evidently, doing so amounts simply to rewriting the equation obtained by Kazakov [11,12] for steady flames in terms of the local current flow velocity relative to the front. It is not difficult to verify that in the case under consideration, this would change the coefficient of ν 2 in Eq.…”

Section: Equation (27) Thus Becomesmentioning

confidence: 99%

“…By virtue of the properties a), b) the complex quantity dω p /dz, where ω p = u p + iw p , is an analytical function of the complex variable z in the downstream region. In conjunction with the property c), analyticity of dω p /dz can be expressed in the form of the following dispersion relation [11,12] 1 + iĤ ω p…”

Section: Closed Description Of Non-stationary Flamesmentioning

confidence: 99%

“…As to the steady configurations, this question was settled out in the negative by [11,12]. More precisely, it was shown that the only piece of information about the gas flow downstream that is really necessary to derive an equation for the flame front position is the value of the vortex component of burnt gas along the front or, using the terminology used by [12], the on-shell value of this component.…”

Section: Introductionmentioning

confidence: 99%

“…However, it is the evolution of the flame front, its position and shape that usually constitute the main concern in practice. This limitation of the problem raises naturally the following dilemma [see 11,12]: On the one hand, deflagration is an essentially non-local and nonlinear process with all its complications mentioned above; on the other hand, this non-locality itself is determined by the flame front configuration and the gas velocity distribution along it, which play the role of boundary conditions for the flow equations and thus control the bulk flow. In such circumstances, is it really necessary to know explicitly the flow structure in the entire downstream region in order to describe the front evolution in a closed form, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Although the procedure is essentially the same as in the steady case, a subtle point is worth to be emphasized. According to [11,12], the derivative of the vortex component along the flame front is local, i.e., its value at a given point is a function of the on-shell fuel velocity, its derivatives, and the front shape at the same point. In view of this, one might expect the generalization to the unsteady case to be purely kinematic, namely, that it would amount to rewriting the steady equation in terms of the gas velocities relative to the local front velocity.…”

Section: Introductionmentioning

confidence: 99%

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On-shell description of unsteady flames

2008

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The problem of non-perturbative description of unsteady premixed flames with arbitrary gas expansion is solved in the two-dimensional case. Considering the flame as a surface of discontinuity with arbitrary local burning rate and gas velocity jumps given on it, we show that the front dynamics can be determined without having to solve the flow equations in the bulk. On the basis of the Thomson circulation theorem, an implicit integral representation of the gas velocity downstream is constructed. It is then simplified by a successive stripping of the potential contributions to obtain an explicit expression for the vortex component near the flame front. We prove that the unknown potential component is left bounded and divergence-free by this procedure, and hence can be eliminated using the dispersion relation for its on-shell value (i.e., the value along the flame front). The resulting system of integro-differential equations relates the on-shell fuel velocity and the front position. As limiting cases, these equations contain all theoretical results on flame dynamics established so far, including the linear equation describing the Darrieus-Landau instability of planar flames, and the nonlinear Sivashinsky-Clavin equation for flames with weak gas expansion.

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Section: Equation (27) Thus Becomesmentioning

confidence: 99%

Section: Closed Description Of Non-stationary Flamesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On-shell description of unsteady flames

2008

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Nonlinear evolution of 2D cellular lean hydrogen/air premixed flames with varying initial perturbations in the elevated pressure environment

Bai

et al. 2017

International Journal of Hydrogen Energy

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Analytical treatment of 2D steady flames anchored in high-velocity streams

Kazakov

2010

Physica D: Nonlinear Phenomena

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The problem of burning of high-velocity gas streams in channels is revisited. Previous treatments of this issue are found to be incomplete. It is shown that despite relative smallness of the transversal gas velocity, it plays crucial role in determining flame structure. In particular, it is necessary in formulating boundary conditions near the flame anchor, and for the proper account of the flame propagation law. Using the on-shell description of steady anchored flames, a consistent solution of the problem is given. Equations for the flame front position and gas-velocity at the front are obtained. It is demonstrated that they reduce to a second-order differential equation for the front position. Numerical solutions of the derived equations are found.PACS numbers: I. INTRODUCTIONThe problem of flame propagation in a flow of gaseous mixture with a fixed ignition point is important from both experimental and theoretical points of view. In fact, relative simplicity of practical realization and importance in applications made anchored flames one of the most popular topics in combustion science. Despite these circumstances theoretical description of the subject is far from being complete. It would not even be exaggeration to say that the very mechanism of formation of steady flame configurations is not fully understood. One of the most difficult problems here is the influence of anchoring system on the flame, in particular, the question of its locality. Analytical investigation of this arduous question is so complicated that it is usually not raised at all. Another closely related problem is the identification of mechanisms driving the development of flame disturbances. It is known for a long time that anchored flames may develop bulbous structures [1], their stability properties are affected by gravity diversely and deeper than in the case of freely propagating flames [2,3], which means that the stabilizing mechanisms in the two instances are quite different.These issues are especially nontrivial in the case of two-dimensional flames. Namely, reduction of dimensionality changes the long-range behavior of the Green functions, leading to enhanced interaction between distant parts of the flame. Specifically, pressure distribution is determined by the Green function of the Laplace operator, whose integral kernel grows logarithmically with distance in two dimensions, and therefore so does the response to a point source. This indicates that the regions with large velocity gradients, in particular, vicinity of the anchor, may have strong nonlocal impact on the global flame structure.Of particular interest are the high-velocity streams. Experiments indicate that when the incoming gas velocity significantly exceeds the normal flame burning speed, the flame front assumes highly elongated shape which often can be well approximated by straight lines (V-flames). Even if the front shape is not piecewise linear, simplifications admitted by the high-velocity limit make it accessible for theoretical investigation. An important example...

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Exact Equation for Curved Stationary Flames with Arbitrary Gas Expansion

Cited by 21 publications

References 10 publications

On-shell description of unsteady flames

On-shell description of unsteady flames

Nonlinear evolution of 2D cellular lean hydrogen/air premixed flames with varying initial perturbations in the elevated pressure environment

Analytical treatment of 2D steady flames anchored in high-velocity streams

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