Boundary layer separation of 2-D incompressible Dirichlet flows

Previous Article
Global estimates and blow-up criteria for the generalized Hunter-Saxton system
DCDS-B Home
This Issue
Next Article
Optimal harvesting for a stochastic N-dimensional competitive Lotka-Volterra model with jumps

March 2015, 20(2): 675-682. doi: 10.3934/dcdsb.2015.20.675

Boundary layer separation of 2-D incompressible Dirichlet flows

Quan Wang ^1, , Hong Luo ^2, and Tian Ma ^2,

1.	Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2.	College of Mathematics and Software Science, Sichuan Normal University, Chengdu,Sichuan 610066, China, China

Received March 2014 Revised June 2014 Published January 2015

Abstract
Related Papers

In this paper, the solutions of Navier-Stokes equations governing 2-D incompressible flows with the Dirichlet boundary condition are analyzed. We derive a condition for boundary layer separation, and the condition is determined by initial values and external forces. More importantly, the condition can predict when and where the boundary layer separation occurs directly. In addition, we also get an algebraic equation for the separation point and the separation time. The algebraic equation can tell us where the boundary layer separation does not occur in a short period of time. The main technical tool is the geometric theory of incompressible flows developed by T. Ma and S. Wang in [15].

Keywords: Boundary layer separation, Dirichlet boundary condition., predicable condition, Navier-Stokes Equations, 2-D incompressible fluid flows.

Mathematics Subject Classification: Primary: 35Q30, 35Q35, 76D10, 76M9.

Citation: Quan Wang, Hong Luo, Tian Ma. Boundary layer separation of 2-D incompressible Dirichlet flows. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 675-682. doi: 10.3934/dcdsb.2015.20.675

References:

[1]	P. Blanchonette and M. A. Page, Boundary-Layer separation in the two-layer flow past a cylinder in a rotating frame,, Theoret. Comput. Fluid Dynamics, 11 (1998), 95.
[2]	Chorin, J. Alexandre and J. E. Marsden, A mathematical Introduction to Fluid Mechanics,, Third edition. Texts in Applied Mathematics, (1993). doi: 10.1007/978-1-4612-0883-9.
[3]	W. E and B. Engquist, Blow up of solutions of the unsteady Prandtl's equation,, Commun. Pure Appl. Math, 50 (1997), 1287. doi: 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4.
[4]	M. Ghil, T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows,, Indiana Univ. Math. J, 50 (2001), 159. doi: 10.1512/iumj.2001.50.2183.
[5]	M. Ghil, J. Liu, C. Wang and S. Wang, Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow,, Physica D, 197 (2004), 149. doi: 10.1016/j.physd.2004.06.012.
[6]	M. Ghil and T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows with Dirichlet boundary conditions: Applications to boundary-layer separation,, SIAM J. Applied Math, 65 (2005), 1576. doi: 10.1137/S0036139903438818.
[7]	H. Luo, Q. Wang and T. Ma, A predicable condition for boundary layer separation of 2-D incompressible fluid flows,, Nonlinear Analysis: Real World Applications, 22 (2015), 336. doi: 10.1016/j.nonrwa.2014.09.007.
[8]	O. B. Larin and V. A. Levin, Boundary layer separation in a laminar supersonic flow with energy supply source,, Technical Physics Letters, 34 (2008), 181. doi: 10.1134/S1063785008030012.
[9]	O. B. Larin and V. A. Levin, Effect of energy supply to a gas on laminar boundary layer separation,, Journal of Applied Mechanics and Technical Physics, 51 (2010), 11. doi: 10.1007/s10808-010-0003-4.
[10]	J. Liu and Z. Xin, Boundary layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation,, Arch. Rat. Mech. Anal, 135 (1996), 61. doi: 10.1007/BF02198435.
[11]	T. Ma and S. Wang, Rigorous characterization of boundary layer separations,, Proc. of the Second MIT Conference on Computational Fluid and Solid Mechanics, (2003), 1008. doi: 10.1016/B978-008044046-0/50246-3.
[12]	T. Ma and S. Wang, Interior structural bifurcation and separation of $2-D$ incompressible flows,, J. Math. Phy, 45 (2004), 1762. doi: 10.1063/1.1689005.
[13]	T. Ma and S. Wang, Asymptotic structure for solutions of the Navier-Stokes equations,, Discrete and Continuous Dynamical Systems, 11 (2004), 189. doi: 10.3934/dcds.2004.11.189.
[14]	T. Ma and S. Wang, Boundary Layer Separation and Structural Bifurcation for 2-D Incompressible Fluid Flows,, Discrete and Continuous Dynamical Systems, 10 (2004), 459. doi: 10.3934/dcds.2004.10.459.
[15]	T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics,, Mathematical Surveys and Monographs, (2005). doi: 10.1090/surv/119.
[16]	T. Ma and S. Wang, Boundary layer and interior separations in the Taylor-Couette-Poiseuille flow,, Journal of Mathematical Physics, 50 (2009), 1. doi: 10.1063/1.3093268.
[17]	O. A. Oleinik, On the mathematical theory of boundary layer for unsteady flow of incompressible fluid,, J. Appl. Math. Mech, 30 (1966), 951. doi: 10.1016/0021-8928(66)90001-3.
[18]	O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory,, Chapman and Hall/CRC, (1999).
[19]	L. Prandtl, On the motion of fluids with very little friction,, in Verhandlungen des dritten internationalen Mathematiker-Konggresses, (1905), 484.
[20]	H. Schlichting, Boundary Layer Theory,, eighth ed., (2000). doi: 10.1007/978-3-642-85829-1.
[21]	F.T. Smith and S.N. Brown, Boundary-Layer Separation,, Proceedings of the IUTAM Symposium London, (1986). doi: 10.1007/978-3-642-83000-6.
[22]	Quan. Wang, Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders,, Discrete and Continuous Dynamical Systems-B, 19 (2014), 543. doi: 10.3934/dcdsb.2014.19.543.

show all references

References:

[1]	P. Blanchonette and M. A. Page, Boundary-Layer separation in the two-layer flow past a cylinder in a rotating frame,, Theoret. Comput. Fluid Dynamics, 11 (1998), 95.
[2]	Chorin, J. Alexandre and J. E. Marsden, A mathematical Introduction to Fluid Mechanics,, Third edition. Texts in Applied Mathematics, (1993). doi: 10.1007/978-1-4612-0883-9.
[3]	W. E and B. Engquist, Blow up of solutions of the unsteady Prandtl's equation,, Commun. Pure Appl. Math, 50 (1997), 1287. doi: 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4.
[4]	M. Ghil, T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows,, Indiana Univ. Math. J, 50 (2001), 159. doi: 10.1512/iumj.2001.50.2183.
[5]	M. Ghil, J. Liu, C. Wang and S. Wang, Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow,, Physica D, 197 (2004), 149. doi: 10.1016/j.physd.2004.06.012.
[6]	M. Ghil and T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows with Dirichlet boundary conditions: Applications to boundary-layer separation,, SIAM J. Applied Math, 65 (2005), 1576. doi: 10.1137/S0036139903438818.
[7]	H. Luo, Q. Wang and T. Ma, A predicable condition for boundary layer separation of 2-D incompressible fluid flows,, Nonlinear Analysis: Real World Applications, 22 (2015), 336. doi: 10.1016/j.nonrwa.2014.09.007.
[8]	O. B. Larin and V. A. Levin, Boundary layer separation in a laminar supersonic flow with energy supply source,, Technical Physics Letters, 34 (2008), 181. doi: 10.1134/S1063785008030012.
[9]	O. B. Larin and V. A. Levin, Effect of energy supply to a gas on laminar boundary layer separation,, Journal of Applied Mechanics and Technical Physics, 51 (2010), 11. doi: 10.1007/s10808-010-0003-4.
[10]	J. Liu and Z. Xin, Boundary layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation,, Arch. Rat. Mech. Anal, 135 (1996), 61. doi: 10.1007/BF02198435.
[11]	T. Ma and S. Wang, Rigorous characterization of boundary layer separations,, Proc. of the Second MIT Conference on Computational Fluid and Solid Mechanics, (2003), 1008. doi: 10.1016/B978-008044046-0/50246-3.
[12]	T. Ma and S. Wang, Interior structural bifurcation and separation of $2-D$ incompressible flows,, J. Math. Phy, 45 (2004), 1762. doi: 10.1063/1.1689005.
[13]	T. Ma and S. Wang, Asymptotic structure for solutions of the Navier-Stokes equations,, Discrete and Continuous Dynamical Systems, 11 (2004), 189. doi: 10.3934/dcds.2004.11.189.
[14]	T. Ma and S. Wang, Boundary Layer Separation and Structural Bifurcation for 2-D Incompressible Fluid Flows,, Discrete and Continuous Dynamical Systems, 10 (2004), 459. doi: 10.3934/dcds.2004.10.459.
[15]	T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics,, Mathematical Surveys and Monographs, (2005). doi: 10.1090/surv/119.
[16]	T. Ma and S. Wang, Boundary layer and interior separations in the Taylor-Couette-Poiseuille flow,, Journal of Mathematical Physics, 50 (2009), 1. doi: 10.1063/1.3093268.
[17]	O. A. Oleinik, On the mathematical theory of boundary layer for unsteady flow of incompressible fluid,, J. Appl. Math. Mech, 30 (1966), 951. doi: 10.1016/0021-8928(66)90001-3.
[18]	O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory,, Chapman and Hall/CRC, (1999).
[19]	L. Prandtl, On the motion of fluids with very little friction,, in Verhandlungen des dritten internationalen Mathematiker-Konggresses, (1905), 484.
[20]	H. Schlichting, Boundary Layer Theory,, eighth ed., (2000). doi: 10.1007/978-3-642-85829-1.
[21]	F.T. Smith and S.N. Brown, Boundary-Layer Separation,, Proceedings of the IUTAM Symposium London, (1986). doi: 10.1007/978-3-642-83000-6.
[22]	Quan. Wang, Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders,, Discrete and Continuous Dynamical Systems-B, 19 (2014), 543. doi: 10.3934/dcdsb.2014.19.543.

[1]	Tian Ma, Shouhong Wang. Boundary layer separation and structural bifurcation for 2-D incompressible fluid flows. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 459-472. doi: 10.3934/dcds.2004.10.459
[2]	Roberto Triggiani. Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D, wall-normal boundary controller. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 279-314. doi: 10.3934/dcdsb.2007.8.279
[3]	Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153
[4]	Shirshendu Chowdhury, Debanjana Mitra, Michael Renardy. Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control. Evolution Equations & Control Theory, 2018, 7 (3) : 447-463. doi: 10.3934/eect.2018022
[5]	Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353
[6]	Tian Ma, Shouhong Wang. Structure of 2D incompressible flows with the Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 29-41. doi: 10.3934/dcdsb.2001.1.29
[7]	H. Beirão da Veiga. Vorticity and regularity for flows under the Navier boundary condition. Communications on Pure & Applied Analysis, 2006, 5 (4) : 907-918. doi: 10.3934/cpaa.2006.5.907
[8]	Guangrong Wu, Ping Zhang. The zero diffusion limit of 2-D Navier-Stokes equations with $L^1$ initial vorticity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 631-638. doi: 10.3934/dcds.1999.5.631
[9]	Grzegorz Łukaszewicz. Pullback attractors and statistical solutions for 2-D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 643-659. doi: 10.3934/dcdsb.2008.9.643
[10]	Jean-Pierre Raymond. Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1537-1564. doi: 10.3934/dcdsb.2010.14.1537
[11]	Tian Ma, Shouhong Wang. Global structure of 2-D incompressible flows. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 431-445. doi: 10.3934/dcds.2001.7.431
[12]	Lan Zeng, Guoxi Ni, Yingying Li. Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-20. doi: 10.3934/dcdsb.2019068
[13]	J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647
[14]	Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
[15]	Linjie Xiong. Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary. Kinetic & Related Models, 2018, 11 (3) : 469-490. doi: 10.3934/krm.2018021
[16]	Franck Boyer, Pierre Fabrie. Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 219-250. doi: 10.3934/dcdsb.2007.7.219
[17]	Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277
[18]	Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595
[19]	Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113
[20]	Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769

2017 Impact Factor: 0.972

American Institute of Mathematical Sciences