Sharp asymptotic growth laws of turbulent flame speeds in cellular flows by inviscid Hamilton–Jacobi models
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 6, p. 1049-1068

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 Embid, Majda and Souganidis (1995) [10], we compare the turbulent flame speeds s T ʼs 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.

DOI : https://doi.org/10.1016/j.anihpc.2012.11.004
Classification:  70H20,  76M50,  76M45,  76N20
Keywords: Inviscid Hamilton–Jacobi equations, Cellular flows, Sharp front speed asymptotics
@article{AIHPC_2013__30_6_1049_0,
     author = {Xin, Jack and Yu, Yifeng},
     title = {Sharp asymptotic growth laws of turbulent flame speeds in cellular flows by inviscid Hamilton--Jacobi models},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {30},
     number = {6},
     year = {2013},
     pages = {1049-1068},
     doi = {10.1016/j.anihpc.2012.11.004},
     zbl = {06286079},
     mrnumber = {3132416},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2013__30_6_1049_0}
}
Xin, Jack; Yu, Yifeng. Sharp asymptotic growth laws of turbulent flame speeds in cellular flows by inviscid Hamilton–Jacobi models. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 6, pp. 1049-1068. doi : 10.1016/j.anihpc.2012.11.004. http://www.numdam.org/item/AIHPC_2013__30_6_1049_0/

[1] M. Abel, M. Cencini, D. Vergni, A. Vulpiani, Front speed enhancement in cellular flows, Chaos 12 (2002), 481-488 | MR 1907659 | Zbl 1080.80501

[2] B. Audoly, H. Berestycki, Y. Pomeau, Réaction diffusion en écoulement stationnaire rapide, C. R. Acad. Sci. Paris 328 no. IIb (2000), 255-262

[3] H. Berestycki, F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math. 60 (2002), 949-1032 | MR 1900178 | Zbl 1024.37054

[4] P. Cardaliaguet, J. Nolen, P.E. Souganidis, Homogenization and enhancement for the G-equation, Arch. Ration. Mech. Anal. 199 no. 2 (2011), 527-561 | MR 2763033 | Zbl 1294.35002

[5] P. Cardaliaguet, P.E. Souganidis, Homogenization and enhancement the G-equation in random environments, Comm. Pure Appl. Math., in press. | MR 3084699

[6] S. Childress, A.M. Soward, Scalar transport and alpha-effect for a family of catʼs-eye flows, J. Fluid Mech. 205 (1989), 99-133 | MR 1014361 | Zbl 0675.76091

[7] P. Clavin, F. Williams, Theory of premixed-flame propagation in large-scale turbulence, J. Fluid Mech. 90 (1979), 598-604 | Zbl 0434.76052

[8] P. Constantin, A. Kiselev, A. Oberman, L. Ryzhik, Bulk burning rate in passive–reactive diffusion, Arch. Ration. Mech. Anal. 154 (2000), 53-91 | MR 1778121 | Zbl 0979.76093

[9] A. Fannjiang, G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math. 54 (1992), 333-408 | MR 1265233 | Zbl 0796.76084

[10] P. Embid, A. Majda, P. Souganidis, Comparison of turbulent flame speeds from complete averaging and the G-equation, Phys. Fluids 7 no. 8 (1995), 2052-2060 | MR 1346082 | Zbl 1039.80504

[11] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, AMS, Providence, RI (1998) | MR 1625845

[12] P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-NSF Regional Conference Series in Applied Mathematics vol. 11, SIAM, Philadelphia (1973) | MR 350216

[13] A. Majda, P. Souganidis, Large scale front dynamics for turbulent reaction–diffusion equations with separated velocity scales, Nonlinearity 7 (1994), 1-30 | MR 1260130 | Zbl 0839.76093

[14] A. Majda, P. Souganidis, Flame fronts in a turbulent combustion model with fractal velocity fields, Comm. Pure Appl. Math. 51 (1998), 1337-1348 | MR 1639220 | Zbl 0939.35097

[15] J. Nolen, A. Novikov, Homogenization of the G-equation with incompressible random drift in two dimensions, Commun. Math. Sci. 9 no. 2 (2011), 561-582 | MR 2815685 | Zbl 1241.35021

[16] J. Nolen, J. Xin, Asymptotic spreading of KPP reactive fronts in incompressible space–time random flows, Ann. Inst. Henri Poincaré, Anal. Non Lineaire 26 no. 3 (May–June 2009), 815-839 | Numdam | MR 2526403

[17] J. Nolen, J. Xin, Y. Yu, Bounds on front speeds for inviscid and viscous G-equations, Methods Appl. Anal. 16 no. 4 (2009), 507-520 | MR 2734499 | Zbl 1256.35090

[18] A. Novikov, L. Ryzhik, Boundary layers and KPP fronts in a cellular flow, Arch. Ration. Mech. Anal. 184 no. 1 (2007), 23-48 | MR 2289862 | Zbl 1109.76064

[19] A. Oberman, Ph.D thesis, University of Chicago, 2001.

[20] S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences vol. 153, Springer, New York (2003) | MR 1939127 | Zbl 1026.76001

[21] N. Peters, Turbulent Combustion, Cambridge University Press, Cambridge (2000) | MR 1792350 | Zbl 0955.76002

[22] P. Ronney, Some open issues in premixed turbulent combustion, J.D. Buckmaster, T. Takeno (ed.), Modeling in Combustion Science, Lecture Notes in Physics vol. 449, Springer-Verlag, Berlin (1995), 3-22

[23] G. Sivashinsky, Cascade-renormalization theory of turbulent flame speed, Combust. Sci. Techol. 62 (1988), 77-96

[24] G. Sivashinsky, Renormalization concept of turbulent flame speed, Lecture Notes in Physics vol. 351 (1989) | MR 1037592

[25] F. Williams, Turbulent combustion, J. Buckmaster (ed.), The Mathematics of Combustion, SIAM, Philadelphia (1985), 97-131 | MR 806548

[26] J. Xin, Front propagation in heterogeneous media, SIAM Rev. 42 no. 2 (June 2000), 161-230 | MR 1778352

[27] J. Xin, An Introduction to Fronts in Random Media, Surveys and Tutorials in the Applied Mathematical Sciences vol. 5, Springer (2009) | MR 2527020 | Zbl 1188.35003

[28] J. Xin, Y. Yu, Periodic homogenization of inviscid G-equation for incompressible flows, Commun. Math. Sci. 8 no. 4 (2010), 1067-1078 | MR 2744920 | Zbl 05843226

[29] V. Yakhot, Propagation velocity of premixed turbulent flames, Combust. Sci. Techol. 60 (1988), 191-241