The Effect of Volume Expansion on the Propagation of

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Combust. Sci. and Tecb., 1999. Vol. 146, pp. 85·103 Reprints available directly from the publisher Photocopying permitted by license only

The Effect of Volume Expansion on the Propagation of Wrinkled Laminar Premixed Flame E.H. CHUNG and SEJIN KWON' Department of Aerospace Engineering, KAIST 373-1 Kusong-dong, Yusung-ku, Taejon 305-701, Korea (Received November 25, 1998; Revised March 17, 1999) Past studies using G-equation successfully described the effect of flame stretch on the laminar flame propagation. In those studies, flames were regarded as a passive interface that did not influence the flow field, The experimental evidences, however, suggested that flow field was significantly modified by the flames as the burned gas expanded at the flame, A method using G-equation and Biot-Savart law to approximate induced velocity field is described to estimate the effect of volume expansion. Present method was applied to initially wrinkled and planar flames propagating in an imposed velocity field and the average flame speed was evaluated from the ratio of simulated flame surface area and projected area of unburned stream channel. It was found that the initial wrinkling of flame could not sustain itself without velocity disturbance but decayed into planar flame. The rate of decay of the wrinkles increased as the volume expansion ratio increased, The asymptotic change in the average burning speed occurred only in a disturbed velocity field. The average burning speed was always affected by the volume expansion that directly influenced the velocity field. With relatively small expansion ratio of 3, the average flame speed increased 10%. The comparison of the relative significance of volume expansion and flame stretch suggested that the effect of volume expansion was no less important than that of flame stretch and warranted that both of the effects should be taken into account in the simulation of flame propagation.

1. INTRODUCTION Turbulence enhances burning velocities of premixed flame and the rate of reaction per unit volume of combustion chambers. The interaction of turbulence and flame has been exploited in devices such as internal combustion engines, gas turbine combustors and industrial furnaces to achieve higher efficiency. In premixed hydrocarbon flames, the flame thickness is smaller than the scale of the turbulence and turbulent flame can be treated as wrinkled laminar flame. The

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E.H. CHUNG and SEJIN KWON

effect of turbulence is to increase the flame surface area while the rate of reactant consumption per unit flame area is relatively invariant when the distortion is modest (Turns, 1996). The governing equations of turbulent combustion involve non-linearity due to convection and chemical reaction. Direct solution of the equations has been bcyond the capacity of currently available resources and alternative methods (Sivashinsky, 1979; Kerstein et al., 1988; Ashurst and Sivashinsky, 1991) using laminar flamelet assumption were proposed to deal with this problem. These methods regard flame as a wave that propagates at a laminar burning velocity normal to the flame surface. At the same time, the surface is convected like a material surface by the flow field. Reaction is confined in infinitely thin flame and influences the flame propagation through laminar burning velocity (Su) only. The prerequisites for the successful laminar flame model are efficient algorithms for flow field simulation and interface motion. Ghoniem et al. (1982) used random vortex method for velocity simulation and SUC (Simple Line Interface Calculations; Noh and Woodward, 1976) for flame propagation. Kwon et al. (1992) used stochastic time series technique to simulate isotropic turbulence and SUC for flame motion. Some authors (Sethian, 1984; Osher and Sethian, 1988; Kwon and Faeth, 1992) recognized that an equation of Hamilton-Jacobi type (G-equation) could replace conventional ad hoc flame propagation algorithms. The G-equation can be derived from the Lagrangian description of flame surface motion. Aldredge (1992) studied steady flame propagation in a periodic shear flow and considered the effect of flame stretch using G-equation. Zhu and Ronney (1994) calculated the flame motion in a Taylor-Couette flow and showed that simulation accurately predicted the measurement of liquid flame propagation. Sung et al. (1996) modified G-equation using its gradient and obtained analytical solution of laminar flame propagating in a periodic velocity field. A drawback of a laminar flamelet model is that major effects of reaction must be evaluated separately, because it does not calculate reaction. Two of the known consequences of reaction are volume expansion and flame stretch. However, studies described above considered only flame stretch neglecting the effect of volume expansion. In the present study, a method to include the effect of volume expansion is described. The method was applied to simple flame propagation problems and the effect of volume expansion was compared with that of flame stretch.

WRINKLED PREMIXED FLAME

87

2. THEORETICAL METHOD


2.1 Analysis of G-Equation Consider a scalar function C(P, t) and let the level surface, C(P" t) = 0, represent the flame surface. Differentiating the level surface equation with respect to time, an equation of motion is obtained for the flame surface.

dC) = DC + ,[,7, . VC = 0 dt G=o Dt di

(1)

In Eq. (I), the flame propagation velocity, dP,ldt , is related to the laminar burning velocity, S", by

S u _- (dP, _ .~) u

dt

(2)

where S" is the velocity of the unburned gas at the flame front and "'1 = -VCI IVCI is a unit normal vector of the flame surface. Eliminating 7 d1 , I dt from Eqs. (I) and (2), the well known G-equation is obtained.

DC +v. VC = S IVCI Dt U

(3)

S)\7GI in the right hand side of Eq. (3) represents wave propagation that moves the flame surface in accordance with Huygens' principle. Convection term represents passive interface advection by the imposed flow field. The Lagrangian description of flame surface motion (Ghoniem et al., 1982) by the two velocities is 7

d1 , S-/rt = v + ,,"1

(4)

where ·v is advection velocity due to the imposed velocity field and S" ...., is self-propagation of flame. Multiplying -\7G to Eq. (4) and using the definition of "'1 , we obtain d17 ,

-_._. VC +v· VC

dt

= S" IVCI

(5)

Therefore, we can conclude that G-equation is simply Eulerian transformation of originally Lagrangian description of flame motion. Also, we can expect G-equation is more amenable to numerical analysis than its Lagrangian counterpart.

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E.H. CHUNG and SEJIN KWON

2.2 The Induced Velocity Field Figure I illustrates a stationary planar flame to show the velocity adjustment in the burned region by the volume expansion of the burned gas. Continuity across the flame surface area requires


(6)

The adjusted velocity field can be modeled by distributing volume source elements with source strength of (v-I)Su along the flame. v is the density ratio and expressed in terms of temperature ratio for ideal gases. (J"

Tb

_

-=-='1 pIJ 1',.,

(7)

Figure 2 is a schematic of velocity induced by a volume source segment on the flame surface. Although following derivation is based on two-dimensional flame propagation, analogous approach can be used for the analysis of three-dimensional flame. Velocity potential induced by infinitesimal length ds of the segment is (8)

By integrating Eq. (8) over the flame segment length, following velocity potential is obtained.

tJ.'

>'

0'

velocity potential

t. j

x- and y- directional grid indices

p

density

p, n

parellel and normal to a line source segment

v

density ratio

e

directional angle

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Ghoniem, A. F.. Chorin, A. J., and Oppenheim, A. K. (1982). Numerical Modeling of Turbulent Flow in a Combustion Tunnel, Phil. TrailS. R. Soc. Lond. A-304. 303-325. Kerstein. A.. Ashurst, W. T. and Williams, F. A. (1988). Field Equation for Interface Propagation in an Unsteady Homogeneous Flow Field, Phys. Rev. A., 37. 2728. Kwon, S. and Faeth. G. M. (1992). Stochastic Simulationof Free Turbulent Premixed Flames in Isotropic Turbulence, Twenty-fourth Symposium (International) all Combustion, Combustion Institute, pp.444-451. Kwon, S.. Wu, M. S.. Driscoll, J. F. and Faeth, G. M. (1992). Flame Surface Properties of Premixed Flames in Isotropic Turbulence: Measurements and Simulations, Combust. Flame. 88, 221-238. Noh. W. T. and Woodward, P. (1976). Sl.JC, Simple Line-Interface Calculations. International Conference on Numerical Methods in Fluid Dynamics, 5th, A. I. Vooren and P. J. Zandenbergen. eds., Springer- Verlag, pp.330-339.

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Osher, S. and Sethian, J. (1988). Front Propagation with Curvature-Dependent Speed: Algorithm Based on Hamilton-Jacobi Formulations, J. Com. Phys., 79. 12. Sethian, J. (1984). Turbulent Combustion in Open and Closed Vessels, J. Com. Phys., 54, 425-456. Sivashinsky, G. I. (1979). Hydrodynamic theory of flame propagation in an enclosed volume, Acta. Astronaut;ca. 6, 631.

Sung, C. 1., Sun, C. J., and Law, C. K. (1996). Analytic Description of the Evolution of Two-dimensional Flame Surfaces, Combust. Flame. 107, 114-124.

Turns, S. R. (1996). An Introduction to Combustion, McGraw-Hill. Zhu. J., and Ronney, P. D. (1994). Simulation of Front Propagation at Large Nondimensional Flow Downloaded By: [Massachusetts Institute of Technology] At: 03:04 5 November 2008

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