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Turbulent Flame Speeds of G-equation Models in UnsteadyCellular Flows

Published online by Cambridge University Press:  12 June 2013

Y-Y Liu ,
J. Xin  and
Y. Yu
Y-Y Liu*
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan 70101, Taiwan
J. Xin
Affiliation:
Department of Mathematics, University of California, Irvine, CA 92697, USA
Y. Yu
Affiliation:
Department of Mathematics, University of California, Irvine, CA 92697, USA
*
⋆ Corresponding author. E-mail: yuyul@ncku.edu.tw
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Abstract

We perform a computationl study of front speeds of G-equation models in time dependentcellular flows. The G-equations arise in premixed turbulent combustion, and areHamilton-Jacobi type level set partial differential equations (PDEs). The curvature-strainG-equations are also non-convex with degenerate diffusion. The computation is based onmonotone finite difference discretization and weighted essentially nonoscillatory (WENO)methods. We found that the large time front speeds lock into the frequency of timeperiodic cellular flows in curvature-strain G-equations similar to what occurs in thebasic inviscid G-equation. However, such frequency locking phenomenon disappears inviscous G-equation, and in the inviscid G-equation if time periodic oscillation of thecellular flow is replaced by time stochastic oscillation.

Keywords

G-equationsfront speed computationcellular flowsfrequency locking
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 3: Front Propagation , 2013 , pp. 198 - 205
Copyright
© EDP Sciences, 2013

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References

Abel, M., Cencini, M., Vergni, D., Vulpiani, A.. Front Speed Enhancement in Cellular Flows. Chaos 12 (2002), 481-488. CrossRefGoogle ScholarPubMed
Cencini, M., Torcini, A., Vergni, D., Vulpiani, A.. Thin front propagation in steady and unsteady cellular flows. Phys. Fluids 15 (2003), 679-688. CrossRefGoogle Scholar
Cardaliaguet, P., Nolen, J., Souganidis, P. E.. Homogenization and Enhancement for the G-Equation. Arch. Rational Mech. Analysis 199 (2011), 527-561. CrossRefGoogle Scholar
Denet, B.. Possible Role of Temporal Correlations in the Bending of Turbulent Flame Velocity. Combust. Theory Model. 3 (1999), 585-589. CrossRefGoogle Scholar
Khouider, B., Bourlioux, A., Majda, A.. Parametrizing Turbulent Flame Speed-Part I: Unsteady Shears, Flame Residence Time and bending. Combust. Theory Model. 5 (2001), 295-318. CrossRefGoogle Scholar
Nolen, J., Xin, J.. Reaction-Diffusion Front Speeds in Spatially-Temporally Periodic Shear Flows. SIAM J. Multiscale Modeling and Simulation 1 (2003), 554-570. CrossRefGoogle Scholar
Nolen, J., Xin, J.. Asymptotic Spreading of KPP Reactive Fronts in Incompressible Space-Time Random Flows. Ann Inst. H. Poincaré, Analyse Non Lineaire 26 (2009), 815-839. CrossRefGoogle Scholar
P. Cardaliaguet, P. E. Souganidis. Homogenization and Enhancement of the G-equation in Random Environments. Comm. Pure Appl. Math, to appear.
Liu, Y.-Y., Xin, J., Yu, Y.. Asymptotics for turbulent flame speeds of the viscous G-equation enhanced by cellular and shear flows. Arch. Rational Mech. Analysis 202 (2011), 461-492. CrossRefGoogle Scholar
Liu, Y.-Y., Xin, J., Yu, Y.. A Numerical Study of Turbulent Flame Speeds of Curvature and Strain G-equations in Cellular Flows. Physica D 243 (2013), 20-31. CrossRefGoogle Scholar
Nolen, J., Novikov, A.. Homogenization of the G-equation with incompressible random drift in two dimensions. Comm. Math Sci. 9 (2011), 561-582. CrossRefGoogle Scholar
Nolen, J., Xin, J.. Computing reactive front speeds in random flows by variational principle. Physica D 237 (2008), 3172-3177. CrossRefGoogle Scholar
A. Oberman. Ph.D. thesis, University of Chicago, Chicago, IL, 2001.
S. Osher, R. Fedkiw. Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag, New York, NY 2002.
N. Peters. Turbulent Combustion. Cambridge University Press, 2000.
Ronny, P.D.. Some Open Issues in Premixed Turbulent Combustion. Lecture Notes in Physics 449 (1995), 3-22. Google Scholar
F. Williams, Turbulent Combustion. The Mathematics of Combustion (J. Buckmaster, ed.), SIAM, Philadelphia (1985) 97-131.
Xin, J., Front Propagation in Heterogeneous Media. SIAM Review 42 (2000), 161-230. CrossRefGoogle Scholar
J. Xin. An Introduction to Fronts in Random Media. Surveys and Tutorials in the Applied Mathematical Sciences 5, Springer, 2009.
Xin, J., Yu, Y.. Periodic Homogenization of Inviscid G-equation for Incompressible Flows. Comm. Math Sci. 8 (2010), 1067-1078. CrossRefGoogle Scholar