Interior structural bifurcation of 2D symmetric incompressible flows

Deniz Bozkurt; Ali Deliceoğlu; Taylan Şengül

doi:10.3934/dcdsb.2020032

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July 2020, 25(7): 2775-2791. 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 July 2020 Early access April 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 and Continuous Dynamical Systems - B, 2020, 25 (7) : 2775-2791. doi: 10.3934/dcdsb.2020032

References:

[1]	A. A. Andronov, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press, New York-Toronto, Ont., 1973.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	F. Gürcan, Flow Bifurcations in Rectangular, Lid-Driven, Cavity Flows, Ph.D thesis, University of Leeds, 1997.
[16]	J. N. Hartnack, Streamline topologies near a fixed wall using normal forms, Acta Mech., 136 (1999), 55-75. doi: 10.1007/BF01292298.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	T. Ma and S. Wang, Topological phase transitions Ⅳ: Dynamic theory of boundary-layer separations. preprint, https://hal.archives-ouvertes.fr/hal-01672759
[26]	M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 2 (1963), 179-180. doi: 10.1016/0040-9383(65)90018-2.
[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.
[28]	R. C. T. Smith, The bending of a semi-infinite strip, Australian J. Sci. Res. Ser. A, 5 (1952), 227-237.
[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.
[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.
[31]	J. Yang and R. Liu, A new fluid dynamical model coupling heat with application to interior separations, preprint, arXiv: 1606.07152.

show all references

References:

[1]	A. A. Andronov, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press, New York-Toronto, Ont., 1973.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	F. Gürcan, Flow Bifurcations in Rectangular, Lid-Driven, Cavity Flows, Ph.D thesis, University of Leeds, 1997.
[16]	J. N. Hartnack, Streamline topologies near a fixed wall using normal forms, Acta Mech., 136 (1999), 55-75. doi: 10.1007/BF01292298.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	T. Ma and S. Wang, Topological phase transitions Ⅳ: Dynamic theory of boundary-layer separations. preprint, https://hal.archives-ouvertes.fr/hal-01672759
[26]	M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 2 (1963), 179-180. doi: 10.1016/0040-9383(65)90018-2.
[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.
[28]	R. C. T. Smith, The bending of a semi-infinite strip, Australian J. Sci. Res. Ser. A, 5 (1952), 227-237.
[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.
[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.
[31]	J. Yang and R. Liu, A new fluid dynamical model coupling heat with application to interior separations, preprint, arXiv: 1606.07152.

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

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$ \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

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$ \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

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	$ \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

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	$ \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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Interior structural bifurcation of 2D symmetric incompressible flows

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