zbMATH — the first resource for mathematics

Sharp asymptotic growth laws of turbulent flame speeds in cellular flows by inviscid Hamilton-Jacobi models. (English) Zbl 1366.35018
Summary: We study the large time asymptotic speeds (turbulent flame speeds \(s_T\)) of the simplified Hamilton-Jacobi (HJ) models arising in turbulent combustion. One HJ model is G-equation describing the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are HJ equations with convex (\(L^1\) type) but non-coercive Hamiltonians. The other is the quadratically nonlinear (\(L^2\) type) inviscid HJ model of Majda-Souganidis derived from the Kolmogorov-Petrovsky-Piskunov reactive fronts. Motivated by a question posed by P. F. Embid et al. [Phys. Fluids 7, No. 8, 2052–2060 (1995; Zbl 1039.80504)], we compare the turbulent flame speeds \(s_T\)’s from these inviscid HJ models in two-dimensional cellular flows or a periodic array of steady vortices via sharp asymptotic estimates in the regime of large amplitude. The estimates are obtained by analyzing the action minimizing trajectories in the Lagrangian representation of solutions (Lax formula and its extension) in combination with delicate gradient bound of viscosity solutions to the associated cell problem of homogenization. Though the inviscid turbulent flame speeds share the same leading order asymptotics, their difference due to nonlinearities is identified as a subtle double logarithm in the large flow amplitude from the sharp growth laws. The turbulent flame speeds differ much more significantly in the corresponding viscous HJ models.

35F21 Hamilton-Jacobi equations
35Q35 PDEs in connection with fluid mechanics
80A25 Combustion
70H20 Hamilton-Jacobi equations in mechanics
76F99 Turbulence
Full Text: DOI
[1] Abel, M.; Cencini, M.; Vergni, D.; Vulpiani, A., Front speed enhancement in cellular flows, Chaos, 12, 481-488, (2002) · Zbl 1080.80501
[2] Audoly, B.; Berestycki, H.; Pomeau, Y., Réaction diffusion en écoulement stationnaire rapide, C. R. Acad. Sci. Paris, 328, IIb, 255-262, (2000) · Zbl 0992.76097
[3] Berestycki, H.; Hamel, F., Front propagation in periodic excitable media, Comm. Pure Appl. Math., 60, 949-1032, (2002) · Zbl 1024.37054
[4] Cardaliaguet, P.; Nolen, J.; Souganidis, P. E., Homogenization and enhancement for the G-equation, Arch. Ration. Mech. Anal., 199, 2, 527-561, (2011) · Zbl 1294.35002
[5] P. Cardaliaguet, P.E. Souganidis, Homogenization and enhancement the G-equation in random environments, Comm. Pure Appl. Math., in press. · Zbl 1284.60126
[6] Childress, S.; Soward, A. M., Scalar transport and alpha-effect for a family of catʼs-eye flows, J. Fluid Mech., 205, 99-133, (1989) · Zbl 0675.76091
[7] Clavin, P.; Williams, F., Theory of premixed-flame propagation in large-scale turbulence, J. Fluid Mech., 90, 598-604, (1979) · Zbl 0434.76052
[8] Constantin, P.; Kiselev, A.; Oberman, A.; Ryzhik, L., Bulk burning rate in passive-reactive diffusion, Arch. Ration. Mech. Anal., 154, 53-91, (2000) · Zbl 0979.76093
[9] Fannjiang, A.; Papanicolaou, G., Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54, 333-408, (1992) · Zbl 0796.76084
[10] Embid, P.; Majda, A.; Souganidis, P., Comparison of turbulent flame speeds from complete averaging and the G-equation, Phys. Fluids, 7, 8, 2052-2060, (1995) · Zbl 1039.80504
[11] Evans, L. C., Partial differential equations, Graduate Studies in Mathematics, (1998), AMS Providence, RI
[12] Lax, P. D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 11, (1973), SIAM Philadelphia · Zbl 0268.35062
[13] Majda, A.; Souganidis, P., Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7, 1-30, (1994) · Zbl 0839.76093
[14] Majda, A.; Souganidis, P., Flame fronts in a turbulent combustion model with fractal velocity fields, Comm. Pure Appl. Math., 51, 1337-1348, (1998) · Zbl 0939.35097
[15] Nolen, J.; Novikov, A., Homogenization of the G-equation with incompressible random drift in two dimensions, Commun. Math. Sci., 9, 2, 561-582, (2011) · Zbl 1241.35021
[16] Nolen, J.; Xin, J., Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows, Ann. Inst. Henri Poincaré, Anal. Non Lineaire, 26, 3, 815-839, (May-June 2009)
[17] Nolen, J.; Xin, J.; Yu, Y., Bounds on front speeds for inviscid and viscous G-equations, Methods Appl. Anal., 16, 4, 507-520, (2009) · Zbl 1256.35090
[18] Novikov, A.; Ryzhik, L., Boundary layers and KPP fronts in a cellular flow, Arch. Ration. Mech. Anal., 184, 1, 23-48, (2007) · Zbl 1109.76064
[19] A. Oberman, Ph.D thesis, University of Chicago, 2001.
[20] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, Applied Mathematical Sciences, vol. 153, (2003), Springer New York · Zbl 1026.76001
[21] Peters, N., Turbulent combustion, (2000), Cambridge University Press Cambridge · Zbl 0955.76002
[22] Ronney, P., Some open issues in premixed turbulent combustion, (Buckmaster, J. D.; Takeno, T., Modeling in Combustion Science, Lecture Notes in Physics, vol. 449, (1995), Springer-Verlag Berlin), 3-22
[23] Sivashinsky, G., Cascade-renormalization theory of turbulent flame speed, Combust. Sci. Techol., 62, 77-96, (1988)
[24] Sivashinsky, G., Renormalization concept of turbulent flame speed, (Lecture Notes in Physics, vol. 351, (1989))
[25] Williams, F., Turbulent combustion, (Buckmaster, J., The Mathematics of Combustion, (1985), SIAM Philadelphia), 97-131
[26] Xin, J., Front propagation in heterogeneous media, SIAM Rev., 42, 2, 161-230, (June 2000)
[27] Xin, J., An introduction to fronts in random media, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 5, (2009), Springer · Zbl 1188.35003
[28] Xin, J.; Yu, Y., Periodic homogenization of inviscid G-equation for incompressible flows, Commun. Math. Sci., 8, 4, 1067-1078, (2010) · Zbl 1372.76085
[29] Yakhot, V., Propagation velocity of premixed turbulent flames, Combust. Sci. Techol., 60, 191-241, (1988)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.