Numerical simulation and self-similar analysis of singular solutions of Prandtl equations

Previous Article
Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models
DCDS-B Home
This Issue
Next Article
Topological stability of hyperbolic sets of flows under random perturbations

January 2010, 13(1): 101-116. doi: 10.3934/dcdsb.2010.13.101

Numerical simulation and self-similar analysis of singular solutions of Prandtl equations

1.	Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

Received May 2009 Revised September 2009 Published October 2009

Abstract
Related Papers

We use the iterative grid redistribution method(IGR) to solve Prandtl equations and study the self-similar behavior of the singular solutions of them. The IGR method enables us to get more accurate solutions of Prandtl equations when they develop singularity. We also study the self-similar behavior of the singular solutions. Blow up rate and blow up profiles are derived and the results are verified by the numerical solutions.

Keywords: Prandtl equations, singularity, self-similarity..

Mathematics Subject Classification: Primary: 35Q30, 14E15; Secondary: 60G18, 33F0.

Citation: Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101

[1]	Rogelio Valdez. Self-similarity of the Mandelbrot set for real essentially bounded combinatorics. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 897-922. doi: 10.3934/dcds.2006.16.897
[2]	Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3239-3252. doi: 10.3934/dcds.2015.35.3239
[3]	Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313
[4]	José Ignacio Alvarez-Hamelin, Luca Dall'Asta, Alain Barrat, Alessandro Vespignani. K-core decomposition of Internet graphs: hierarchies, self-similarity and measurement biases. Networks & Heterogeneous Media, 2008, 3 (2) : 371-393. doi: 10.3934/nhm.2008.3.371
[5]	Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767
[6]	Congming Li, Jisun Lim. The singularity analysis of solutions to some integral equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 453-464. doi: 10.3934/cpaa.2007.6.453
[7]	Veronica Felli, Elsa M. Marchini, Susanna Terracini. On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 91-119. doi: 10.3934/dcds.2008.21.91
[8]	Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure & Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549
[9]	Cheng-Jie Liu, Ya-Guang Wang, Tong Yang. Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2011-2029. doi: 10.3934/dcdss.2016082
[10]	Joachim Naumann, Jörg Wolf. On Prandtl's turbulence model: Existence of weak solutions to the equations of stationary turbulent pipe-flow. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1371-1390. doi: 10.3934/dcdss.2013.6.1371
[11]	Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002
[12]	Takahiro Hashimoto. Nonexistence of positive solutions of quasilinear elliptic equations with singularity on the boundary in strip-like domains. Conference Publications, 2005, 2005 (Special) : 376-385. doi: 10.3934/proc.2005.2005.376
[13]	Xiangdi Huang, Zhouping Xin. On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4477-4493. doi: 10.3934/dcds.2016.36.4477
[14]	F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91
[15]	Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003
[16]	Nemanja Kosovalić, Brian Pigott. Self-excited vibrations for damped and delayed higher dimensional wave equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2413-2435. doi: 10.3934/dcds.2019102
[17]	Vassilis G. Papanicolaou, Kyriaki Vasilakopoulou. Similarity solutions of a multidimensional replicator dynamics integrodifferential equation. Journal of Dynamics & Games, 2016, 3 (1) : 51-74. doi: 10.3934/jdg.2016003
[18]	Jose Carlos Camacho, Maria de los Santos Bruzon. Similarity reductions of a nonlinear model for vibrations of beams. Conference Publications, 2015, 2015 (special) : 176-184. doi: 10.3934/proc.2015.0176
[19]	A. M. Vershik. Polymorphisms, Markov processes, quasi-similarity. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1305-1324. doi: 10.3934/dcds.2005.13.1305
[20]	Boran Hu, Zehui Cheng, Zhangbing Zhou. Web services recommendation leveraging semantic similarity computing. Mathematical Foundations of Computing, 2018, 1 (2) : 101-119. doi: 10.3934/mfc.2018006

2017 Impact Factor: 0.972

American Institute of Mathematical Sciences