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EFFICIENT COMPUTATION OF COORDINATE-FREE MODELS OF FLAME FRONTS

Part of: Thermodynamics and heat transfer Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  29 April 2021

B. F. AKERS  and
D. M. AMBROSE Open the ORCID record for D. M. AMBROSE [Opens in a new window]
B. F. AKERS*
Affiliation:
Department of Mathematics and Statistics, Air Force Institute of Technology, WPAFB, OH45433, USA
D. M. AMBROSE
Affiliation:
Department of Mathematics, Drexel University, Philadelphia, PA19104, USA; dma68@drexel.edu.
*
Email: benjamin.akers@afit.edu
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Abstract

We present an efficient, accurate computational method for a coordinate-free model of flame front propagation of Frankel and Sivashinsky. This model allows for overturned flames fronts, in contrast to weakly nonlinear models such as the Kuramoto–Sivashinsky equation. The numerical procedure adapts the method of Hou, Lowengrub and Shelley, derived for vortex sheets, to this model. The result is a nonstiff, highly accurate solver which can handle fully nonlinear, overturned interfaces, with similar computational expense to methods for weakly nonlinear models. We apply this solver both to simulate overturned flame fronts and to compare the accuracy of Kuramoto–Sivashinsky and coordinate-free models in the appropriate limit.

Keywords

flame frontscoordinate-free modelsoverturned interfaces

MSC classification

Primary: 65M70: Spectral, collocation and related methods
Secondary: 80A22: Stefan problems, phase changes, etc.
Type
Research Article
Information
The ANZIAM Journal , Volume 63 , Issue 1 , January 2021 , pp. 58 - 69
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Copyright
© Australian Mathematical Society 2021

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References

Akers, B. F., Ambrose, D. M., Pond, K. and Wright, J. D., “Overturned internal capillary-gravity waves”, Eur. J. Mech. B Fluids 57 (2016) 143–151; doi:10.1016/j.euromechflu.2015.12.006.CrossRefGoogle Scholar
Akers, B. F., Ambrose, D. M. and Wright, J. D., “Gravity perturbed crapper waves”, Proc. Roy. Soc. A Math. Phys. Eng. Sci. 470 (2014) 20130526; doi:10.1098/rspa.2013.0526.Google Scholar
Ambrose, D. M., Hadadifard, F. and Wright, J. D., “Well-posedness and asymptotics of a coordinate- free model of flame fronts”, Preprint, 2020, arXiv:2010.00737v1.Google Scholar
Ascher, U. M., Ruuth, S. J. and Wetton, B. T. R., “Implicit-explicit methods for time-dependent partial differential equations”, SIAM J. Numer. Anal. 32 (1995) 797–823; doi:10.1137/0732037.CrossRefGoogle Scholar
Brauner, C.-M., Frankel, M. L., Hulshof, J., Lunardi, A. and Sivashinsky, G. I., “On the $\kappa$ - $\theta$ model of cellular flames: existence in the large and asymptotics”, Discrete Contin. Dyn. Syst. Ser. S 1 (2008) 27–39; doi:10.3934/dcdss.2008.1.27.Google Scholar
Frankel, M. L., Gordon, P. V. and Sivashinsky, G. I., “On disintegration of near-limit cellular flames”, Phys. Lett. A 310 (2003) 389–392; doi:10.1016/S0375-9601(03)00385-2.CrossRefGoogle Scholar
Frankel, M. L. and Sivashinsky, G. I., “On the nonlinear thermal diffusive theory of curved flames”, J. Phys. 48 (1987) 25–28; doi:10.1051/jphys:0198700480102500.CrossRefGoogle Scholar
Frankel, M. L. and Sivashinsky, G. I., “On the equation of a curved flame front”, Phys. D 30 (1988) 28–42; doi:10.1016/0167-2789(88)90096-6.CrossRefGoogle Scholar
Goto, M., Kuwana, K. and Yazaki, S., “A simple and fast numerical method for solving flame/smoldering evolution equations”, JSIAM Lett. 10 (2018) 49–52; doi:10.14495/jsiaml.10.49.CrossRefGoogle Scholar
Hou, T. Y., Lowengrub, J. S. and Shelley, M. J., “Removing the stiffness from interfacial flows with surface tension”, J. Comput. Phys. 114 (1994) 312–338; doi:10.1006/jcph.1994.1170.CrossRefGoogle Scholar
Hou, T. Y., Lowengrub, J. S. and Shelley, M. J., “The long-time motion of vortex sheets with surface tension”, Phys. Fluids 9 (1997) 1933–1954; doi:10.1063/1.869313.CrossRefGoogle Scholar
Kalogirou, A., Keaveny, E. E. and Papageorgiou, D. T., “An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation’’, Proc. Roy. Soc. A Math. Phys. Eng. Sci. 471 (2015) 20140932; doi:10.1098/rspa.2014.0932.Google ScholarPubMed
Kassam, A.-K. and Trefethen, L. N, “Fourth-order time-stepping for stiff PDEs”, SIAM J. Sci. Comput. 26 (2005) 1214–1233; doi:10.1137/S1064827502410633.CrossRefGoogle Scholar
Milewski, P. A. and Tabak, E. G., “A pseudospectral procedure for the solution of nonlinear wave equations with examples from free-surface flows”, SIAM J. Sci. Comput. 21 (1999) 1102–1114; doi:10.1137/S1064827597321532.CrossRefGoogle Scholar
Papageorgiou, D. T. and Smyrlis, Y. S.., “The route to chaos for the Kuramoto–Sivashinsky equation”, Theor. Comput. Fluid Dyn. 3 (1991) 15–42; doi:10.1007/BF00271514.Google Scholar
Smyrlis, Y. S. and Papageorgiou, D. T., “Predicting chaos for infinite dimensional dynamical systems: the Kuramoto–Sivashinsky equation, a case study”, Proc. Natl. Acad. Sci. 88 (1991) 11129–11132; doi:10.1073/pnas.88.24.11129.CrossRefGoogle ScholarPubMed