American Institute of Mathematical Sciences

Advanced
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

Qiaolin He 1,

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

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]

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, 2021, 15 (3) : 475-498. doi: 10.3934/ipi.2021001
[2]

Rogelio Valdez. Self-similarity of the Mandelbrot set for real essentially bounded combinatorics. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 897-922. doi: 10.3934/dcds.2006.16.897
[3]

Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313
[4]

Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3239-3252. doi: 10.3934/dcds.2015.35.3239
[5]

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
[6]

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
[7]

Changming Song, Yun Wang. Nonlocal latent low rank sparse representation for single image super resolution via self-similarity learning. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021017
[8]

Hillel Furstenberg. From invariance to self-similarity: The work of Michael Hochman on fractal dimension and its aftermath. Journal of Modern Dynamics, 2019, 15: 437-449. doi: 10.3934/jmd.2019027
[9]

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
[10]

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
[11]

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
[12]

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, 2008, 21 (1) : 91-119. doi: 10.3934/dcds.2008.21.91
[13]

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
[14]

Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4921-4941. doi: 10.3934/dcds.2021062
[15]

Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6339-6357. doi: 10.3934/dcdsb.2021021
[16]

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
[17]

Marat Akhmet, Ejaily Milad Alejaily. Abstract similarity, fractals and chaos. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2479-2497. doi: 10.3934/dcdsb.2020191
[18]

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
[19]

Xiangdi Huang, Zhouping Xin. On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4477-4493. doi: 10.3934/dcds.2016.36.4477
[20]

Nemanja Kosovalić, Brian Pigott. Self-excited vibrations for damped and delayed higher dimensional wave equations. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2413-2435. doi: 10.3934/dcds.2019102

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences