Group theoretic approach and analytical solutions of the compressible Navier-Stokes equations. (English) Zbl 07893914
Summary: A group theoretic analysis of the compressible Navier-Stokes equations of an ideal gas is carried out. The 12-dimensional Lie symmetry group is computed. The commutation table and the Levi decomposition of its Lie algebra are presented. The equations are reduced and self-similar one-, two- and three-dimensional solutions are computed.
MSC:
| 76M60 | Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics |
| 76N06 | Compressible Navier-Stokes equations |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
References:
| [1] | M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, United States National Bureau of Standards, 1964. · Zbl 0171.38503 |
| [2] | N. A. E. D. Al Sayed Hamdouni Liberge Razafindralandy, The symmetry group of the non-isothermal Navier-Stokes equations and turbulence modelling, Symmetry, 2, 848-867, 2010 · Zbl 1302.35299 · doi:10.3390/sym2020848 |
| [3] | G. K. Batchelor, An Introduction to Fluid Dynamics, 1999 · Zbl 0958.76001 |
| [4] | G. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations, volume 154 of Applied Mathematical Sciences, Springer, 2002. · Zbl 1013.34004 |
| [5] | A. Cheviakov, Symbolic computation of nonlocal symmetries and nonlocal conservation laws of partial differential equations using the GeM package for Maple, in J.-F. Ganghoffer and I. Mladenov, editors, Similarity and Symmetry Methods: Applications in Elasticity and Mechanics of Materials, volume 73 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing, (2014), 165-184. · Zbl 1309.74075 |
| [6] | M. A. Chhay Hamdouni, A new construction for invariant numerical schemes using moving frames, Comptes Rendus Mécanique, 338, 97-101, 2010 · Zbl 1300.76021 · doi:10.1016/j.crme.2010.01.001 |
| [7] | M. A. Chhay Hamdouni, Lie symmetry preservation by finite difference schemes for the burgers equation, Symmetry, 2, 868-883, 2010 · Zbl 1287.65061 · doi:10.3390/sym2020868 |
| [8] | M. E. A. P. Chhay Hoarau Hamdouni Sagaut, Comparison of some Lie-symmetry-based integrators, Journal of Computational Physics, 230, 2174-2188, 2011 · Zbl 1416.65398 · doi:10.1016/j.jcp.2010.12.015 |
| [9] | T. S. P. Colonius Lele Moin, The free compressible viscous vortex, Journal of Fluid Mechanics, 230, 45-73, 1991 · Zbl 0732.76068 · doi:10.1017/S0022112091000708 |
| [10] | N. Curle and H. Davies, Modern Fluid Dynamics. Volume 2: Compressible Flow, Van Nostrand Reinhold Company, 1971. · Zbl 0231.76002 |
| [11] | V. A. Dorodnitsyn, Finite difference models entirely inheriting continuous symmetry of original differential equations, International Journal of Modern Physics C, 5, 723-734, 1994 · Zbl 0940.65528 · doi:10.1142/S0129183194000830 |
| [12] | W. R. Fushchych Popowych, Symmetry reduction and exact solutions of the Navier-Stokes equations. Ⅰ, Journal of Nonlinear Mathematical Physics, 1, 75-113, 1994 · Zbl 0956.35099 · doi:10.2991/jnmp.1994.1.1.6 |
| [13] | W. R. Fushchych Popowych, Symmetry reduction and exact solutions of the Navier-Stokes equations. Ⅱ, Journal of Nonlinear Mathematical Physics, 1, 158-188, 1994 · Zbl 0956.35099 · doi:10.2991/jnmp.1994.1.2.3 |
| [14] | E. Garnier, N. Adams and P. Sagaut, Large Eddy Simulation for Compressible Flows, Scientific Computation, Springer, 2006. |
| [15] | V. R. A. G. P. Grassi Leo Soliani Tempesta, Vorticies and invariant surfaces generated by symmetries for the 3D Navier-Stokes equation, Physica A, 286, 79-108, 2000 · Zbl 1052.35510 · doi:10.1016/S0378-4371(00)00223-5 |
| [16] | P. E. Hydon, Symmetry Methods for Differential Equations. A Beginner’s Guide, 2000 · Zbl 0951.34001 · doi:10.1017/CBO9780511623967 |
| [17] | N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations. Vol 1-3, 1996 · Zbl 0864.35003 |
| [18] | G. M. Khujadze Oberlack, DNS and scaling laws from new symmetry groups of ZPG turbulent boundary layer flow, Theoretical and Computational Fluid Dynamics, 18, 391-411, 2004 · Zbl 1148.76320 |
| [19] | M. Oberlack, A unified approach for symmetries in plane parallel turbulent shear flows, Journal of Fluid Mechanics, 427, 299-328, 2001 · Zbl 1007.76067 · doi:10.1017/S0022112000002408 |
| [20] | P. Olver, Applications of Lie Groups to Differential Equations, volume 107 of Graduate Texts in Mathematics, Springer-Verlag, 1986. · Zbl 0588.22001 |
| [21] | F. D. R. C. Olver Lozier Boisvert Clark, NIST Handbook of Mathematical Functions, 2010 · Zbl 1198.00002 |
| [22] | D. Razafindralandy and A. Hamdouni, Consequences of symmetries on the analysis and construction of turbulence models, Symmetry, Integrability and Geometry: Methods and Applications, 2 (2006), Paper 052, 20 pp. · Zbl 1092.22014 |
| [23] | D. A. Razafindralandy Hamdouni, Analysis of subgrid models of heat convection by symmetry group theory, Comptes Rendus Mécanique, 335, 225-230, 2007 · Zbl 1121.76018 |
| [24] | D. A. N. Razafindralandy Hamdouni Al Sayed, Lie-symmetry group and modeling in non-isothermal fluid mechanics, Physica A: Statistical Mechanics and its Applications, 391, 4624-4636, 2012 · doi:10.1016/j.physa.2012.05.063 |
| [25] | D. Razafindralandy, A. Hamdouni and M. Chhay, Numerical simulation research progress, Chapter Symmetry in Turbulence Simulation, Nova Publishers, (2009), 161-207. · Zbl 1195.65002 |
| [26] | D. A. M. Razafindralandy Hamdouni Oberlack, Analysis and development of subgrid turbulence models preserving the symmetry properties of the Navier-Stokes equations, European Journal of Mechanics/B, 26, 531-550, 2007 · Zbl 1112.76044 · doi:10.1016/j.euromechflu.2006.10.003 |
| [27] | P. K. M. Sachdev Joseph Haque, Exact solutions of compressible flow equations with spherical symmetry, Studies in Applied Mathematics, 114, 325-342, 2005 · Zbl 1145.76399 · doi:10.1111/j.0022-2526.2005.01552.x |
| [28] | A. Toutant, General and exact pressure evolution equation, Physics Letters A, 381, 3739-3742, 2017 · Zbl 1374.76008 · doi:10.1016/j.physleta.2017.10.008 |
| [29] | S. Tsangaris and T. Pappou, Analytical Solutions for the Unsteady Compressible Flow Equations Serving as Test Cases for the Verification of Numerical Schemes, Technical report, Defense Technical Information Center, 2000. |
| [30] | G. Ünal, Constitutive equation of turbulence and the Lie symmetries of Navier-Stokes equations, in N. H. Ibragimov, K. Razi Naqvi and E. Straume, editors, Modern Group Analysis VII, Mars Publishers, 1997,317-323. |
| [31] | T. A. R. B. J. P. Zang Dahlburg Dahlburg, Direct and large-eddy simulations of three-dimensional compressible Navier-Stokes turbulence, Physics of Fluids A: Fluid Dynamics, 4, 127-140, 1992 · Zbl 0742.76060 · doi:10.1063/1.858491 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
