Recent results on stability of planar detonations. (English) Zbl 1366.76033
Colombini, Ferruccio (ed.) et al., Shocks, singularities and oscillations in nonlinear optics and fluid mechanics. Papers based on the workshop, Rome, Italy, September 2015. Cham: Springer (ISBN 978-3-319-52041-4/hbk; 978-3-319-52042-1/ebook). Springer INdAM Series 17, 273-308 (2017).
Summary: We describe recent analytical and numerical results on stability and behavior of viscous and inviscid detonation waves obtained by dynamical systems/Evans function techniques like those used to study shock and reaction diffusion waves. In the first part, we give a broad description of viscous and inviscid results for 1D perturbations; in the second, we focus on inviscid high-frequency stability in multi-D and associated questions in turning point theory/WKB expansion.
For the entire collection see [Zbl 1371.76002].
For the entire collection see [Zbl 1371.76002].
MSC:
| 76E17 | Interfacial stability and instability in hydrodynamic stability |
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 35Q35 | PDEs in connection with fluid mechanics |
Software:
STABLABReferences:
| [1] | M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55 For sale by the Superintendent of Documents (U.S. Government Printing Office, Washington DC, 1964), xiv+1046pp. · Zbl 0171.38503 |
| [2] | J. Alexander, R. Gardner, C. Jones, A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410, 167–212 (1990) · Zbl 0705.35070 |
| [3] | S. Alinhac, Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels (French) [Existence of rarefaction waves for multidimensional hyperbolic quasilinear systems]. Commun. Partial Differ. Equ. 14 (2), 173–230 (1989) · Zbl 0692.35063 · doi:10.1080/03605308908820595 |
| [4] | B. Barker, J. Humpherys, K. Zumbrun, STABLAB: a MATLAB-based numerical library for Evans function computation (2015). Available at: http://impact.byu.edu/stablab/ |
| [5] | B. Barker, J. Humpherys, G. Lyng, K. Zumbrun, Viscous hyperstabilization of detonation waves in one space dimension. SIAM J. Appl. Math. 75 (3), 885–906 (2015) · Zbl 1320.35271 · doi:10.1137/140980223 |
| [6] | B. Barker, K. Zumbrun, A numerical investigation of stability of ZND detonations for Majda’s model, preprint. arxiv:1011.1561 |
| [7] | B. Barker K. Zumbrun, Numerical stability of ZND detonations, in preparation |
| [8] | G. K. Batchelor. An Introduction to Fluid Dynamics, paperback edn. Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1999) |
| [9] | M. Beck, B. Sandstede, K. Zumbrun, Nonlinear stability of time-periodic viscous shocks. Arch. Ration. Mech. Anal. 196 (3), 1011–1076 (2010) · Zbl 1221.35054 · doi:10.1007/s00205-009-0274-1 |
| [10] | A. Bourlioux, A. Majda, V. Roytburd, Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Math. 51, 303–343 (1991) · Zbl 0731.76076 · doi:10.1137/0151016 |
| [11] | L.Q. Brin, Numerical testing of the stability of viscous shock waves. Math. Comput. 70 (235), 1071–1088 (2001) · Zbl 0980.65092 · doi:10.1090/S0025-5718-00-01237-0 |
| [12] | J. Buckmaster, The contribution of asymptotics to combustion. Phys. D 20 (1), 91–108 (1986) · Zbl 0604.76056 · doi:10.1016/0167-2789(86)90098-9 |
| [13] | J. Buckmaster, J. Neves, One-dimensional detonation stability: the spectrum for infinite activation energy. Phys. Fluids 31 (12), 3572–3576 (1988) · Zbl 0656.76061 · doi:10.1063/1.866874 |
| [14] | D.L. Chapman, VI. On the rate of explosion in gases. Philos. Mag. Ser. 5 47 (284), 90–104 (1899). doi: 10.1080/14786449908621243 · doi:10.1080/14786449908621243 |
| [15] | P. Clavin, L. He, Stability and nonlinear dynamics of one-dimensional overdriven detonations in gases. J. Fluid Mech. 306, 306–353 (1996) · Zbl 0857.76040 · doi:10.1017/S0022112096001334 |
| [16] | E.A. Coddington, M. Levinson, Theory of Ordinary Differential Equations (McGraw–Hill Book Company, Inc., New York, 1955) · Zbl 0064.33002 |
| [17] | R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves (Springer–Verlag, New York, 1976), xvi+464pp. · Zbl 0365.76001 |
| [18] | W. Döring, Über Detonationsvorgang in Gasen [On the detonation process in gases]. Ann. Phys. 43, 421–436 (1943). doi: 10.1002/andp.19434350605 · doi:10.1002/andp.19434350605 |
| [19] | J.J. Erpenbeck, Stability of steady-state equilibrium detonations. Phys. Fluids 5, 604–614 (1962) · doi:10.1063/1.1706664 |
| [20] | J.J. Erpenbeck, Stability of step shocks. Phys. Fluids 5 (10), 1181–1187 (1962) · Zbl 0111.38403 · doi:10.1063/1.1706503 |
| [21] | J.J. Erpenbeck, Stability of idealized one-reaction detonations. Phys. Fluids 7, 684 (1964) · Zbl 0123.42901 · doi:10.1063/1.1711269 |
| [22] | J.J. Erpenbeck, Stability of Detonations for Disturbances of Small Transverse Wave-Length Los Alamos, LA-3306 (1965), 136pp. https://www.google.com/search?tbm=bks&hl=en&q=Stability+of+detonation+for+disturbances+of+small+transverse+wavelength+Report+of+los+Alamos+LA3306+1965 |
| [23] | J.J. Erpenbeck, Detonation stability for disturbances of small transverse wave length. Phys. Fluids 9, 1293–1306 (1966) · Zbl 0144.47204 · doi:10.1063/1.1761844 |
| [24] | L.M. Faria, A.R. Kasimov, R.R. Rosales, Study of a model equation in detonation theory. SIAM J. Appl. Math. 74 (2), 547–570 (2014) · Zbl 1304.35419 · doi:10.1137/130938232 |
| [25] | W. Fickett, W.C. Davis, Detonation (University of California Press, Berkeley, 1979); reissued as Detonation: Theory and Experiment (Dover Press, Mineola, New York 2000). ISBN:0-486-41456-6 |
| [26] | W. Fickett, W. Wood, Flow calculations for pulsating one-dimensional detonations. Phys. Fluids 9, 903–916 (1966) · doi:10.1063/1.1761791 |
| [27] | R.A. Gardner, K. Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles. Commun. Pure Appl. Math. 51 (7), 797–855 (1998) · Zbl 0933.35136 · doi:10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1 |
| [28] | I. Gasser, P. Szmolyan, A geometric singular perturbation analysis of detonation and deflagration waves. SIAM J. Math. Anal. 24, 968–986 (1993) · Zbl 0783.76099 · doi:10.1137/0524058 |
| [29] | O. Gues, G. Métivier, M. Williams, K. Zumbrun, Existence and stability of multidimensional shock fronts in the vanishing viscosity limit. Arch. Ration. Mech. Anal. 175 (2), 151–244 (2005) · Zbl 1072.35122 · doi:10.1007/s00205-004-0342-5 |
| [30] | O. Gues, G. Métivier, M. Williams, K. Zumbrun, Existence and stability of noncharacteristic boundary layers for the compressible Navier-Stokes and MHD equations. Arch. Ration. Mech. Anal. 197 (1), 1–87 (2010) · Zbl 1217.35136 · doi:10.1007/s00205-009-0277-y |
| [31] | M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. Math. Ann. 342 (1), 177–243 (2008) · Zbl 1151.35063 · doi:10.1007/s00208-008-0230-7 |
| [32] | J. Hendricks, J. Humpherys, G. Lyng, K. Zumbrun, Stability of viscous weak detonation waves for Majda’s model. J. Dyn. Differ. Equ. 27 (2), 237–260 (2015) · Zbl 1326.35199 · doi:10.1007/s10884-015-9440-3 |
| [33] | J. Humpherys, G. Lyng, K. Zumbrun, Multidimensional spectral stability of large-amplitude Navier-Stokes shocks, preprint. arxiv:1603.03955 · Zbl 1386.35326 |
| [34] | J. Humpherys, K. Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems. Physica D 220 (2), 116–126 (2006) · Zbl 1101.65082 · doi:10.1016/j.physd.2006.07.003 |
| [35] | J. Humpherys, G. Lyng, K. Zumbrun, Spectral stability of ideal gas shock layers. Arch. Ration. Mech. Anal. 194 (3), 1029–1079 (2009) · Zbl 1422.76122 · doi:10.1007/s00205-008-0195-4 |
| [36] | J. Humpherys, O. Lafitte, K. Zumbrun, Stability of isentropic Navier-Stokes shocks in the high-Mach number limit. Commun. Math. Phys. 293 (1), 1–36 (2010) · Zbl 1195.35244 · doi:10.1007/s00220-009-0885-2 |
| [37] | J. Humpherys, K. Zumbrun, Efficient numerical stability analysis of detonation waves in ZND. Q. Appl. Math. 70 (4), 685–703 (2012) · Zbl 1390.76090 · doi:10.1090/S0033-569X-2012-01276-X |
| [38] | E. Jouguet, Sur la propagation des réactions chimiques dans les gaz [On the propagation of chemical reactions in gases]. J. Math. Pures Appl. 6 (1), 347–425 (1905) · JFM 36.0976.02 |
| [39] | E. Jouguet, Sur la propagation des réactions chimiques dans les gaz [On the propagation of chemical reactions in gases]. J. Math. Pures Appl. 6 (2), 5–85 (1906) · JFM 37.0816.01 |
| [40] | H.K. Jenssen, G. Lyng, M. Williams, Equivalence of low-frequency stability conditions for multidimensional detonations in three models of combustion. Indiana Univ. Math. J. 54, 1–64 (2005) · Zbl 1127.35046 · doi:10.1512/iumj.2005.54.2685 |
| [41] | A.R. Kasimov, D.S. Stewart, Spinning instability of gaseous detonations. J. Fluid Mech. 466, 179–203 (2002) · Zbl 1013.76034 · doi:10.1017/S0022112002001192 |
| [42] | T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1985) |
| [43] | H.O. Kreiss, Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23, 277–298 (1970) · Zbl 0188.41102 · doi:10.1002/cpa.3160230304 |
| [44] | O. Lafitte, M. Williams, K. Zumbrun, The Erpenbeck high frequency instability theorem for Zeldovitch-von Neumann-Döring detonations. Arch. Ration. Mech. Anal. 204 (1), 141–187 (2012) · Zbl 1427.76081 · doi:10.1007/s00205-011-0472-5 |
| [45] | O. Lafitte, M. Williams, K. Zumbrun, High-frequency stability of detonations and turning points at infinity. SIAM J. Math. Anal. 47 (3), 1800–1878 (2015) · Zbl 1321.80004 · doi:10.1137/140987547 |
| [46] | O. Lafitte, M. Williams, K. Zumbrun, Block-diagonalization of ODEs in the semiclassical limit and C {\(\omega\)} vs. C stationary phase. SIAM J. Math. Anal. 48 (3), 1773–1797 (2016) · Zbl 1347.34094 · doi:10.1137/15M103042X |
| [47] | P.D. Lax, Hyperbolic systems of conservation laws. II. Commun. Pure Appl. Math. 10, 537–566 (1957) · Zbl 0081.08803 · doi:10.1002/cpa.3160100406 |
| [48] | P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, vol. 11 (Society for Industrial and Applied Mathematics, Philadelphia, 1973), v+48pp. · Zbl 0268.35062 · doi:10.1137/1.9781611970562 |
| [49] | H.I. Lee, D.S. Stewart, Calculation of linear detonation instability: one-dimensional instability of plane detonation. J. Fluid Mech. 216, 103–132 (1990) · Zbl 0698.76120 · doi:10.1017/S0022112090000362 |
| [50] | J.H.S. Lee, The Detonation Phenomenon (Cambridge University Press, Cambridge/New York, 2008). ISBN-13:978-0521897235 |
| [51] | N. Levinson, The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111–126 (1948) · Zbl 0040.19402 · doi:10.1215/S0012-7094-48-01514-2 |
| [52] | T.-P. Liu, S.-H. Yu, Nonlinear stability of weak detonation waves for a combustion model. Commun. Math. Phys. 204 (3), 551–586 (1999) · Zbl 0976.76036 · doi:10.1007/s002200050657 |
| [53] | G. Lyng, K. Zumbrun, One-dimensional stability of viscous strong detonation waves. Arch. Ration. Mech. Anal. 173 (2), 213–277 (2004) · Zbl 1067.76041 · doi:10.1007/s00205-004-0317-6 |
| [54] | G. Lyng, K. Zumbrun, A stability index for detonation waves in Majda’s model for reacting flow. Physica D 194, 1–29 (2004) · Zbl 1061.35018 · doi:10.1016/j.physd.2004.01.036 |
| [55] | G. Lyng, M. Raoofi, B. Texier, K. Zumbrun, Pointwise Green function bounds and stability of combustion waves. J. Differ. Equ. 233, 654–698 (2007) · Zbl 1116.80017 · doi:10.1016/j.jde.2006.10.006 |
| [56] | A. Martinez, An Introduction to Semiclassical and Microlocal Analysis (Springer, New York, 2002) · Zbl 0994.35003 · doi:10.1007/978-1-4757-4495-8 |
| [57] | G. Métivier, K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Am. Math. Soc. 175 (826), vi+107pp (2005) · Zbl 1074.35066 |
| [58] | A. Majda, A qualitative model for dynamic combustion. SIAM J. Appl. Math. 41, 70–91 (1981) · Zbl 0472.76075 · doi:10.1137/0141006 |
| [59] | A. Majda, The stability of multi-dimensional shock fronts – a new problem for linear hyperbolic equations. Mem. Am. Math. Soc. 275, 1–95 (1983) · Zbl 0506.76075 |
| [60] | A. Majda, R. Rosales, A theory for spontaneous Mach stem formation in reacting shock fronts. I. The basic perturbation analysis. SIAM J. Appl. Math. 43, 1310–1334 (1983) · Zbl 0544.76135 |
| [61] | C. Mascia, K. Zumbrun, Pointwise Green’s function bounds for shock profiles with degenerate viscosity. Arch. Ration. Mech. Anal. 169, 177–263 (2003) · Zbl 1035.35074 · doi:10.1007/s00205-003-0258-5 |
| [62] | G. Métivier, The block structure condition for symmetric hyperbolic problems. Bull. Lond. Math. Soc. 32, 689–702 (2000) · Zbl 1073.35525 · doi:10.1112/S0024609300007517 |
| [63] | G. Métivier, Stability of multidimensional shocks, in Advances in the Theory of Shock Waves. Progress in Nonlinear Differential Equations and Applications, vol. 47 (Birkhäuser Boston, Boston, 2001), pp. 25–103 · doi:10.1007/978-1-4612-0193-9_2 |
| [64] | T. Nguyen, Stability of multi-dimensional viscous shocks for symmetric systems with variable multiplicities. Duke Math. J. 150 (3), 577–614 (2009) · Zbl 1197.35045 · doi:10.1215/00127094-2009-060 |
| [65] | F.W.U. Olver, Asymptotics and Special Functions (Academic Press, New York, 1974) · Zbl 0303.41035 |
| [66] | R. L. Pego, M. I. Weinstein, Eigenvalues, and instabilities of solitary waves. Philos. Trans. R. Soc. Lond. Ser. A 340 (1656), 47–94 (1992) · Zbl 0776.35065 · doi:10.1098/rsta.1992.0055 |
| [67] | R. Pemantle, M.C. Wilson, Asymptotic expansions of oscillatory integrals with complex phase, in Algorithmic Probability and Combinatorics. Contemporary Mathematics, vol. 520 (American Mathematical Society, Providence, 2010), pp. 221–240 · Zbl 1207.41020 · doi:10.1090/conm/520/10261 |
| [68] | R. Plaza, K. Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles. J. Discret. Cont. Dyn. Sys. 10, 885–924 (2004) · Zbl 1058.35164 · doi:10.3934/dcds.2004.10.885 |
| [69] | J.M. Powers, S. Paolucci, Accurate spatial resolution estimates for reactive supersonic flow with detailed chemistry. AIAA J. 43 (5), 1088–1099 (2005) · doi:10.2514/1.11641 |
| [70] | C.M. Romick, T.D. Aslam, J.D. Powers, The Dynamics of Unsteady Detonation with Diffusion. 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando (2011), http://dx.doi.org/10.2514/6.2011-799 · doi:10.2514/6.2011-799 |
| [71] | C.M. Romick, T.D. Aslam, J.D. Powers, The effect of diffusion on the dynamics of unsteady detonations. J. Fluid Mech. 699, 453–464 (2012) · Zbl 1248.76150 · doi:10.1017/jfm.2012.121 |
| [72] | J.-M. Roquejoffre, J.-P. Vila, Stability of ZND detonation waves in the Majda combustion model. Asymptot. Anal. 18 (3–4), 329–348 (1998) · Zbl 0931.35026 |
| [73] | B. Sandstede, A. Scheel, Hopf bifurcation from viscous shock waves. SIAM J. Math. Anal. 39, 2033–2052 (2008) · Zbl 1195.35043 · doi:10.1137/060675587 |
| [74] | D. Serre, Systèmes de lois de conservation I–II (Fondations, Diderot Editeur, Paris, 1996), iv+308pp. ISBN:2-84134-072-4 and xii+300pp. ISBN:2-84134-068-6 |
| [75] | M. Short, An Asymptotic Derivation of the Linear Stability of the Square-Wave Detonation using the Newtonian limit. Proc. R. Soc. Lond. A 452, 2203–2224 (1996) · Zbl 0876.76031 · doi:10.1098/rspa.1996.0117 |
| [76] | M. Short, Multidimensional linear stability of a detonation wave at high activation energy. SIAM J. Appl. Math. 57 (2), 307–326 (1997) · Zbl 0882.76100 · doi:10.1137/S0036139995288101 |
| [77] | J. Smoller, Shock Waves and Reaction–Diffusion Equations, 2nd edn. Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, vol. 258 (Springer, New York, 1994), xxiv+632pp. ISBN:0-387-94259-9 · Zbl 0807.35002 · doi:10.1007/978-1-4612-0873-0 |
| [78] | D.S. Stewart, A.R. Kasimov, State of detonation stability theory and its application to propulsion. J. Propuls. Power 22 (6), 1230–1244 (2006) · doi:10.2514/1.21586 |
| [79] | A. Szepessy, Dynamics and stability of a weak detonation wave. Commun. Math. Phys. 202 (3), 547–569 (1999) · Zbl 0947.35019 · doi:10.1007/s002200050595 |
| [80] | B. Texier, K. Zumbrun, Galloping instability of viscous shock waves. Physica D. 237, 1553–1601 (2008) · Zbl 1143.76439 · doi:10.1016/j.physd.2008.03.008 |
| [81] | B. Texier, K. Zumbrun, Hopf bifurcation of viscous shock waves in gas dynamics and MHD. Arch. Ration. Mech. Anal. 190, 107–140 (2008) · Zbl 1155.76037 · doi:10.1007/s00205-008-0112-x |
| [82] | B. Texier, K. Zumbrun, Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions. Commun. Math. Phys. 302 (1), 1–51 (2011) · Zbl 1217.35138 · doi:10.1007/s00220-010-1175-8 |
| [83] | J. von Neumann, Theory of Detonation Waves, Aberdeen Proving Ground, Maryland: Office of Scientific Research and Development, Report No. 549, Ballistic Research Laboratory File No. X-122 Progress Report to the National Defense Research Committee, Division B, OSRD-549 (April 1, 1942. PB 31090). (4 May 1942), 34 pages |
| [84] | J. von Neumann, in John von Neumann, Collected Works, vol. 6. ed. by A.J. Taub (Permagon Press, Elmsford, 1963) [1942], pp. 178–218 |
| [85] | W. Wasow, Linear Turning Point Theory. Applied Mathematical Sciences, vol. 54 (Springer-Verlag, New York, 1985), ix+246pp. · Zbl 0558.34049 · doi:10.1007/978-1-4612-1090-0 |
| [86] | M. Williams, Heteroclinic orbits with fast transitions: a new construction of detonation profiles. Indiana Univ. Math. J. 59 (3), 1145–1209 (2010) · Zbl 1216.37028 · doi:10.1512/iumj.2010.59.3992 |
| [87] | Y.B. Zel’dovich, [On the theory of the propagation of detonation in gaseous systems]. J. Exp. Theor. Phys. 10, 542–568 (1940). Translated into English in: National Advisory Committee for Aeronautics Technical Memorandum No. 1261 (1950) |
| [88] | K. Zumbrun, Multidimensional stability of planar viscous shock waves, in Advances in the Theory of Shock Waves. Progress in Nonlinear Differential Equations and Applications, vol. 47 (Birkhäuser Boston, Boston, 2001), pp. 307–516 · Zbl 0989.35089 · doi:10.1007/978-1-4612-0193-9_5 |
| [89] | K. Zumbrun, Stability of large-amplitude shock waves of compressible navier–stokes equations, with an appendix by Helge Kristian Jenssen and Gregory Lyng, in Handbook of Mathematical Fluid Dynamics, vol. III (North-Holland, Amsterdam, 2004), pp. 311–533 · Zbl 1222.35156 |
| [90] | K. Zumbrun, Stability of detonation waves in the ZND limit. Arch. Ration. Mech. Anal. 200 (1), 141–182 (2011) · Zbl 1233.35030 · doi:10.1007/s00205-010-0342-6 |
| [91] | K. Zumbrun, High-frequency asymptotics and one-dimensional stability of Zel’dovich–von Neumann–Döring detonations in the small-heat release and high-overdrive limits. Arch. Ration. Mech. Anal. 203 (3), 701–717 (2012) · Zbl 1318.76022 · doi:10.1007/s00205-011-0457-4 |
| [92] | K. Zumbrun, P. Howard, Pointwise semigroup methods and stability of viscous shock waves. Indiana Math. J. 47, 741–871 (1998); Errata, Indiana Univ. Math. J. 51 (4), 1017–1021 (2002) · Zbl 0928.35018 |
| [93] | K. Zumbrun, D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ. Math. J. 48, 937–992 (1999) · Zbl 0944.76027 · doi:10.1512/iumj.1999.48.1765 |
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