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doi: 10.3934/dcdsb.2020032

Interior structural bifurcation of 2D symmetric incompressible flows

Deniz Bozkurt 1, , Ali Deliceoğlu 1, and  Taylan Şengül 2,,

1. 

Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey
2. 

Department of Mathematics, Marmara University, 34722 Istanbul, Turkey

* Corresponding author: Taylan Şengül

Received  May 2019 Published  February 2020

The structural bifurcation of a 2D divergence free vector field $ \mathbf{u}(\cdot, t) $ when $ \mathbf{u}(\cdot, t_0) $ has an interior isolated singular point $ \mathbf{x}_0 $ of zero index has been studied by Ma and Wang [23]. Although in the class of divergence free fields which undergo a local bifurcation around a singular point, the ones with index zero singular points are generic, this class excludes some important families of symmetric flows. In particular, when $ \mathbf{u}(\cdot, t_0) $ is anti-symmetric with respect to $ \mathbf{x}_0 $, or symmetric with respect to the axis located on $ \mathbf{x}_0 $ and normal to the unique eigendirection of the Jacobian $ D\mathbf{u}(\cdot, t_0) $, the vector field must have index 1 or -1 at the singular point. Thus we study the structural bifurcation when $ \mathbf{u}(\cdot, t_0) $ has an interior isolated singular point $ \mathbf{x}_0 $ with index -1, 1. In particular, we show that if such a vector field with its acceleration at $ t_0 $ both satisfy the aforementioned symmetries then generically the flow will undergo a local bifurcation. Under these generic conditions, we rigorously prove the existence of flow patterns such as pairs of co-rotating vortices and double saddle connections. We also present numerical evidence of the Stokes flow in a rectangular cavity showing that the bifurcation scenarios we present are indeed realizable.

Keywords: Flow structures, structural stability, divergence-free vector field and bifurcation.
Mathematics Subject Classification: Primary: 34D30, 34C23; Secondary: 37N10.
Citation: Deniz Bozkurt, Ali Deliceoğlu, Taylan Şengül. Interior structural bifurcation of 2D symmetric incompressible flows. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020032
References:
[1]

A. A. Andronov, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press, New York-Toronto, Ont., 1973.  Google Scholar

[2]

P. Bakker, Bifurcations in Flow Patterns., Nonlinear Topics in the Mathematical Sciences, 2, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3512-2.  Google Scholar

[3]

A. Bisgaard, M. Brøns and J. N. Sørensen, Vortex breakdown generated by off-axis bifurcation in a cylinder with rotating covers, Acta Mechanica, 187 (2006), 75-83.  doi: 10.1007/s00707-006-0367-y.  Google Scholar

[4]

M. Brøns and A. V. Bisgaard, Bifurcation of vortex breakdown patterns in a circular cylinder with two rotating covers, J. Fluid Mech., 568 (2006), 329-349.  doi: 10.1017/S0022112006002424.  Google Scholar

[5]

M. Brøns and J. N. Hartnack, Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries, Phys. Fluids, 11 (1999), 314-324.  doi: 10.1063/1.869881.  Google Scholar

[6]

M. Brøns, B. Jakobsen, K. Niss, A. V. Bisgaard and L. K. Voigt, Streamline topology in the near wake of a circular cylinder at moderate Reynolds numbers, J. Fluid Mech., 584 (2007), 23-43.  doi: 10.1017/S0022112007006234.  Google Scholar

[7]

M. Brøns, L. K. Voigt and J. N. Sørensen, Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface, J. Fluid Mech., 428 (2001), 133-148.  doi: 10.1017/S0022112000002512.  Google Scholar

[8]

C. H. Chan, M. Czubak and T. Yoneda, An ODE for boundary layer separation on a sphere and a hyperbolic space, Phys. D, 282 (2014), 34-38.  doi: 10.1016/j.physd.2014.05.004.  Google Scholar

[9]

M. Dam, J. J. Rasmussen, V. Naulin and M. Brøns, Topological bifurcations in the evolution of coherent structures in a convection model, Phys. Plasmas, 24 (2017). doi: 10.1063/1.4993613.  Google Scholar

[10]

P. Gaskell, M. Savage and M. Wilson, Stokes flow in a half-filled annulus between rotating coaxial cylinders, J. Fluid Mech., 337 (1997), 263-282.  doi: 10.1017/S0022112097005028.  Google Scholar

[11]

M. Ghil, T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows, Indiana Univ. Math. J., 50 (2001), 159-180.  doi: 10.1512/iumj.2001.50.2183.  Google Scholar

[12]

F. Gürcan and A. Deliceoğlu, Streamline topologies near nonsimple degenerate points in two-dimensional flows with double symmetry away from boundaries and an application, Phys. Fluids, 17 (2005), 7pp. doi: 10.1063/1.2055527.  Google Scholar

[13]

F. Gürcan and A. Deliceoğlu, Saddle connections near degenerate critical points in Stokes flow within cavities, Appl. Math. Comput., 172 (2006), 1133-1144.  doi: 10.1016/j.amc.2005.03.012.  Google Scholar

[14]

F. Gürcan, A. Deliceoğlu and P. Bakker, Streamline topologies near a non-simple degenerate critical point close to a stationary wall using normal forms, J. Fluid Mech., 539 (2005), 299-311.  doi: 10.1017/S0022112005005689.  Google Scholar

[15]

F. Gürcan, Flow Bifurcations in Rectangular, Lid-Driven, Cavity Flows, Ph.D thesis, University of Leeds, 1997. Google Scholar

[16]

J. N. Hartnack, Streamline topologies near a fixed wall using normal forms, Acta Mech., 136 (1999), 55-75.  doi: 10.1007/BF01292298.  Google Scholar

[17]

M. Heil, J. Rosso, A. L. Hazel and M. Brøns, Topological fluid mechanics of the formation of the Kármán-vortex street, J. Fluid Mech., 812 (2017), 199-221.  doi: 10.1017/jfm.2016.792.  Google Scholar

[18]

C.-H. Hsia, J.-G. Liu and C. Wang et al., Structural stability and bifurcation for 2-D incompressible flows with symmetry, Methods Appl. Anal., 15 (2008), 495-512.  doi: 10.4310/MAA.2008.v15.n4.a6.  Google Scholar

[19]

D. D. Joseph and L. Sturges, The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. Part Ⅱ, SIAM J. Appl. Math., 34 (1978), 7-26.  doi: 10.1137/0134002.  Google Scholar

[20]

R. Liu, T. Ma, S. Wang and J. Yang, Topological phase transition Ⅴ: Interior separations and cyclone formation theory, preprint. Available from: https://hal.archives-ouvertes.fr/hal-01673496. Google Scholar

[21]

H. Luo, Q. Wang and T. Ma, A predicable condition for boundary layer separation of 2-D incompressible fluid flows, Nonlinear Anal. Real World Appl., 22 (2015), 336-341.  doi: 10.1016/j.nonrwa.2014.09.007.  Google Scholar

[22]

T. Ma and S. Wang, Structural classification and stability of divergence-free vector fields, Phys. D, 171 (2002), 107-126.  doi: 10.1016/S0167-2789(02)00587-0.  Google Scholar

[23]

T. Ma and S. Wang, Interior structural bifurcation and separation of 2D incompressible flows, J. Math. Phys., 45 (2004), 1762-1776.  doi: 10.1063/1.1689005.  Google Scholar

[24]

T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, Mathematical Surveys and Monographs, 119, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/119.  Google Scholar

[25]

T. Ma and S. Wang, Topological phase transitions Ⅳ: Dynamic theory of boundary-layer separations. preprint, https://hal.archives-ouvertes.fr/hal-01672759 Google Scholar

[26]

M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 2 (1963), 179-180.  doi: 10.1016/0040-9383(65)90018-2.  Google Scholar

[27]

A. Perry, M. Chong and T. Lim, The vortex-shedding process behind two-dimensional bluff bodies, J. Fluid Mech., 116 (1982), 77-90.  doi: 10.1017/S0022112082000378.  Google Scholar

[28]

R. C. T. Smith, The bending of a semi-infinite strip, Australian J. Sci. Res. Ser. A, 5 (1952), 227-237.   Google Scholar

[29]

L. Tophøj, S. Møller and M. Brøns, Streamline patterns and their bifurcations near a wall with Navier slip boundary conditions, Phys. Fluids, 18 (2006), 8pp. doi: 10.1063/1.2337660.  Google Scholar

[30]

Q. Wang, H. Luo and T. Ma, Boundary layer separation of 2-D incompressible Dirichlet flows, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015) 675–682. doi: 10.3934/dcdsb.2015.20.675.  Google Scholar

[31]

J. Yang and R. Liu, A new fluid dynamical model coupling heat with application to interior separations, preprint, arXiv: 1606.07152. Google Scholar

show all references

References:
[1]

A. A. Andronov, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press, New York-Toronto, Ont., 1973.  Google Scholar

[2]

P. Bakker, Bifurcations in Flow Patterns., Nonlinear Topics in the Mathematical Sciences, 2, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3512-2.  Google Scholar

[3]

A. Bisgaard, M. Brøns and J. N. Sørensen, Vortex breakdown generated by off-axis bifurcation in a cylinder with rotating covers, Acta Mechanica, 187 (2006), 75-83.  doi: 10.1007/s00707-006-0367-y.  Google Scholar

[4]

M. Brøns and A. V. Bisgaard, Bifurcation of vortex breakdown patterns in a circular cylinder with two rotating covers, J. Fluid Mech., 568 (2006), 329-349.  doi: 10.1017/S0022112006002424.  Google Scholar

[5]

M. Brøns and J. N. Hartnack, Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries, Phys. Fluids, 11 (1999), 314-324.  doi: 10.1063/1.869881.  Google Scholar

[6]

M. Brøns, B. Jakobsen, K. Niss, A. V. Bisgaard and L. K. Voigt, Streamline topology in the near wake of a circular cylinder at moderate Reynolds numbers, J. Fluid Mech., 584 (2007), 23-43.  doi: 10.1017/S0022112007006234.  Google Scholar

[7]

M. Brøns, L. K. Voigt and J. N. Sørensen, Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface, J. Fluid Mech., 428 (2001), 133-148.  doi: 10.1017/S0022112000002512.  Google Scholar

[8]

C. H. Chan, M. Czubak and T. Yoneda, An ODE for boundary layer separation on a sphere and a hyperbolic space, Phys. D, 282 (2014), 34-38.  doi: 10.1016/j.physd.2014.05.004.  Google Scholar

[9]

M. Dam, J. J. Rasmussen, V. Naulin and M. Brøns, Topological bifurcations in the evolution of coherent structures in a convection model, Phys. Plasmas, 24 (2017). doi: 10.1063/1.4993613.  Google Scholar

[10]

P. Gaskell, M. Savage and M. Wilson, Stokes flow in a half-filled annulus between rotating coaxial cylinders, J. Fluid Mech., 337 (1997), 263-282.  doi: 10.1017/S0022112097005028.  Google Scholar

[11]

M. Ghil, T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows, Indiana Univ. Math. J., 50 (2001), 159-180.  doi: 10.1512/iumj.2001.50.2183.  Google Scholar

[12]

F. Gürcan and A. Deliceoğlu, Streamline topologies near nonsimple degenerate points in two-dimensional flows with double symmetry away from boundaries and an application, Phys. Fluids, 17 (2005), 7pp. doi: 10.1063/1.2055527.  Google Scholar

[13]

F. Gürcan and A. Deliceoğlu, Saddle connections near degenerate critical points in Stokes flow within cavities, Appl. Math. Comput., 172 (2006), 1133-1144.  doi: 10.1016/j.amc.2005.03.012.  Google Scholar

[14]

F. Gürcan, A. Deliceoğlu and P. Bakker, Streamline topologies near a non-simple degenerate critical point close to a stationary wall using normal forms, J. Fluid Mech., 539 (2005), 299-311.  doi: 10.1017/S0022112005005689.  Google Scholar

[15]

F. Gürcan, Flow Bifurcations in Rectangular, Lid-Driven, Cavity Flows, Ph.D thesis, University of Leeds, 1997. Google Scholar

[16]

J. N. Hartnack, Streamline topologies near a fixed wall using normal forms, Acta Mech., 136 (1999), 55-75.  doi: 10.1007/BF01292298.  Google Scholar

[17]

M. Heil, J. Rosso, A. L. Hazel and M. Brøns, Topological fluid mechanics of the formation of the Kármán-vortex street, J. Fluid Mech., 812 (2017), 199-221.  doi: 10.1017/jfm.2016.792.  Google Scholar

[18]

C.-H. Hsia, J.-G. Liu and C. Wang et al., Structural stability and bifurcation for 2-D incompressible flows with symmetry, Methods Appl. Anal., 15 (2008), 495-512.  doi: 10.4310/MAA.2008.v15.n4.a6.  Google Scholar

[19]

D. D. Joseph and L. Sturges, The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. Part Ⅱ, SIAM J. Appl. Math., 34 (1978), 7-26.  doi: 10.1137/0134002.  Google Scholar

[20]

R. Liu, T. Ma, S. Wang and J. Yang, Topological phase transition Ⅴ: Interior separations and cyclone formation theory, preprint. Available from: https://hal.archives-ouvertes.fr/hal-01673496. Google Scholar

[21]

H. Luo, Q. Wang and T. Ma, A predicable condition for boundary layer separation of 2-D incompressible fluid flows, Nonlinear Anal. Real World Appl., 22 (2015), 336-341.  doi: 10.1016/j.nonrwa.2014.09.007.  Google Scholar

[22]

T. Ma and S. Wang, Structural classification and stability of divergence-free vector fields, Phys. D, 171 (2002), 107-126.  doi: 10.1016/S0167-2789(02)00587-0.  Google Scholar

[23]

T. Ma and S. Wang, Interior structural bifurcation and separation of 2D incompressible flows, J. Math. Phys., 45 (2004), 1762-1776.  doi: 10.1063/1.1689005.  Google Scholar

[24]

T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, Mathematical Surveys and Monographs, 119, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/119.  Google Scholar

[25]

T. Ma and S. Wang, Topological phase transitions Ⅳ: Dynamic theory of boundary-layer separations. preprint, https://hal.archives-ouvertes.fr/hal-01672759 Google Scholar

[26]

M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 2 (1963), 179-180.  doi: 10.1016/0040-9383(65)90018-2.  Google Scholar

[27]

A. Perry, M. Chong and T. Lim, The vortex-shedding process behind two-dimensional bluff bodies, J. Fluid Mech., 116 (1982), 77-90.  doi: 10.1017/S0022112082000378.  Google Scholar

[28]

R. C. T. Smith, The bending of a semi-infinite strip, Australian J. Sci. Res. Ser. A, 5 (1952), 227-237.   Google Scholar

[29]

L. Tophøj, S. Møller and M. Brøns, Streamline patterns and their bifurcations near a wall with Navier slip boundary conditions, Phys. Fluids, 18 (2006), 8pp. doi: 10.1063/1.2337660.  Google Scholar

[30]

Q. Wang, H. Luo and T. Ma, Boundary layer separation of 2-D incompressible Dirichlet flows, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015) 675–682. doi: 10.3934/dcdsb.2015.20.675.  Google Scholar

[31]

J. Yang and R. Liu, A new fluid dynamical model coupling heat with application to interior separations, preprint, arXiv: 1606.07152. Google Scholar

Figure 1.  The topological structure of $ {\mathbf u}_0 $ near the origin, under the assumptions: (a) (S2), (b) (S7), $ k $ is even, $ \alpha \lambda < 0 $, (c) (S7), $ k $ is even, $ \alpha \lambda > 0 $, (d) (S7), $ k $ is odd
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Figure 2.  Structural bifurcation diagram of the unfolding of codimension-one singularities for the $ \operatorname{ind}( {\mathbf u}^0, {\mathbf x}_0) = -1 $ case in Theorem 3.2: (a) $ t = t_0+\epsilon $, (b) $ t = t_0 $, (c) $ t = t_0 - \epsilon $
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Figure 3.  Structural bifurcation diagram of the unfolding of codimension-one singularities for the $ \operatorname{ind}( {\mathbf u}^0, {\mathbf x}_0) = 1 $ case in Theorem 3.2: (a) $ t = t_0+\epsilon $, (b) $ t = t_0 $, (c) $ t = t_0 - \epsilon $
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Figure 4.  Schematic illustration of the unfolding of codimension-one singularities for flows with reflectional symmetry (18) in the $ \operatorname{ind}( {\mathbf u}^0, {\mathbf x}_0) = -1 $ case. (a) $ t = t_{0}-\epsilon $, (b) $ t = t_{0} $, (c) $ t = t_{0}+\epsilon $
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Figure 5.  Schematic illustration of the unfolding of codimension-one singularities for flows with anti-symmetry (19) in the $ \operatorname{ind}( {\mathbf u}^0, {\mathbf x}_0) = -1 $ case. (a) $ t = t_{0}-\epsilon $, (b) $ t = t_{0} $, (c) $ t = t_{0}+\epsilon $
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Figure 6.  The illustration of the dimensionless boundary value problem
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Figure 7.  The streamlines patterns in rectangular cavities of various aspect ratios $ A $ and various lid speed ratios $ S $. (a) $ (A, S) = (0.5, -1) $, (b) $ (A, S) = (1, -1) $, (c) $ (A, S) = (0.3, -1) $, (d) $ (A, S) = (0.16, -1). $
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Table 1.  The bifurcated solutions for $ k=2 $, $ n=3 $, $ 2 \lambda \lambda_1 + \alpha \lambda_2 > 0 $
$ \epsilon<0 $ $ \epsilon>0 $
$ \alpha \beta + 2 \lambda^2> 0 $ $ {\mathbf x}_{0}(\epsilon) $ is a saddle $ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
$ \alpha \beta + 2 \lambda^2< 0 $ $ {\mathbf x}_{0}(\epsilon) $ is a center $ {\mathbf x}_{\pm}(\epsilon) $ are centers, $ {\mathbf x}_{0}(\epsilon) $ is a saddle
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$ \epsilon<0 $ $ \epsilon>0 $
$ \alpha \beta + 2 \lambda^2> 0 $ $ {\mathbf x}_{0}(\epsilon) $ is a saddle $ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
$ \alpha \beta + 2 \lambda^2< 0 $ $ {\mathbf x}_{0}(\epsilon) $ is a center $ {\mathbf x}_{\pm}(\epsilon) $ are centers, $ {\mathbf x}_{0}(\epsilon) $ is a saddle
Table 2.  The bifurcated solutions for $ k=2 $, $ n>3 $, $ 2 \lambda \lambda_1 + \alpha \lambda_2 > 0 $
$ \epsilon<0 $ $ \epsilon>0 $
$ {\mathbf x}_{0}(\epsilon) $ is a saddle $ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
Table Options

Download as excel

$ \epsilon<0 $ $ \epsilon>0 $
$ {\mathbf x}_{0}(\epsilon) $ is a saddle $ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
Table 3.  The bifurcated solutions for $ k>2 $, $ 2k<n+1 $, $ \alpha \lambda_2 >0 $
$ \epsilon<0 $ $ \epsilon>0 $
$ {\mathbf x}_{0}(\epsilon) $ is a saddle $ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
Table Options

Download as excel

$ \epsilon<0 $ $ \epsilon>0 $
$ {\mathbf x}_{0}(\epsilon) $ is a saddle $ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
Table 4.  The bifurcated solutions for $ k>2 $, $ 2k>n+1 $, $ \alpha \lambda_2>0 $
$ \epsilon<0 $ $ \epsilon>0 $
$ \alpha \beta> 0 $ $ {\mathbf x}_{0}(\epsilon) $ is a saddle $ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
$ \alpha \beta< 0 $ $ {\mathbf x}_{0}(\epsilon) $ is a center $ {\mathbf x}_{\pm}(\epsilon) $ are centers, $ {\mathbf x}_{0}(\epsilon) $ is a saddle
Table Options

Download as excel

$ \epsilon<0 $ $ \epsilon>0 $
$ \alpha \beta> 0 $ $ {\mathbf x}_{0}(\epsilon) $ is a saddle $ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
$ \alpha \beta< 0 $ $ {\mathbf x}_{0}(\epsilon) $ is a center $ {\mathbf x}_{\pm}(\epsilon) $ are centers, $ {\mathbf x}_{0}(\epsilon) $ is a saddle
Table 5.  The bifurcated solutions for $ k>2 $, $ 2k=n+1 $, $ \alpha \lambda_2 > 0 $
$ \epsilon<0 $ $ \epsilon>0 $
$ \alpha \beta + k \lambda^2> 0 $ $ {\mathbf x}_{0}(\epsilon) $ is a saddle $ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
$ \alpha \beta + k \lambda^2< 0 $ $ {\mathbf x}_{0}(\epsilon) $ is a center $ {\mathbf x}_{\pm}(\epsilon) $ are centers, $ {\mathbf x}_{0}(\epsilon) $ is a saddle
Table Options

Download as excel

$ \epsilon<0 $ $ \epsilon>0 $
$ \alpha \beta + k \lambda^2> 0 $ $ {\mathbf x}_{0}(\epsilon) $ is a saddle $ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
$ \alpha \beta + k \lambda^2< 0 $ $ {\mathbf x}_{0}(\epsilon) $ is a center $ {\mathbf x}_{\pm}(\epsilon) $ are centers, $ {\mathbf x}_{0}(\epsilon) $ is a saddle
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