Ginzburg-Landau Model : an Amplitude Evolution Equation for Shallow Wake Flows
Linear and weakly nonlinear analysis of shallow wake
flows is presented in the present paper. The evolution of the most
unstable linear mode is described by the complex Ginzburg-Landau
equation (CGLE). The coefficients of the CGLE are calculated
numerically from the solution of the corresponding linear stability
problem for a one-parametric family of shallow wake flows. It is
shown that the coefficients of the CGLE are not so sensitive to the
variation of the base flow profile.
[1] M. Van Dyke, An Album of Fluid Motion. New York: The Parabolic
Press, 1982, Photograph No. 173.
[2] E. J. Wolansky, J. Imberger, and M.L. Heron, "Island wakes in shallow
coastal waters," J. Geophys. Research, vol. 89, , pp. 10553-10559,
1984.
[3] R. G. Ingram, and V.H. Chu, "Flow around islands in Rupert Bay: An
investigation of the bottom friction effect," J. Geophys. Research, vol.
92, pp. 14521-14533, 1987.
[4] G. H. Jirka, "Large scale flow structures and mixing processes in
shallow flows," J. Hydr. Research, vol. 39 , pp. 567-573, 2001.
[5] S. A. Socolofsky, and G. H. Jirka, "Large scale flow structures and
stability in shallow flows," J. Environ. Eng. Sci.., vol. 3 , pp. 451-462,
2004.
[6] D. Chen, and G. H. Jirka, "Experimental study of a plane turbulent in a
shallow water layer," Fluid Dyn. Res.., vol. 16 , pp. 11-41, 1995.
[7] B. L. White, and H. M. Nepf, "Shear instability and coherent structures
in shallow flow adjacent to a porous layer," J. Fluid Mech.., vol. 593 ,
pp. 1-32, 2007.
[8] M. E. Negretti, S. A. Socolofsky, A. C. Rummel, and G. H. Jirka,
"Stabilization of cylinder wakes in shallow water flow by means of
roughness elements: an experimental study," Exp. Fluids., vol. 38 , pp.
403-414, 2005.
[9] D. Chen, and G. H. Jirka, "Absolute and convective instabilities of plane
turbulent wakes in a shallow water layer," J. Fluid Mech.., vol. 338 , pp.
157-172, 1997.
[10] A. A. Kolyshkin, and M. S. Ghidaoui, "Stability analysis of shallow
wake flows," J. Fluid Mech.., vol. 494 , pp. 355-37, 2003.
[11] M. S. Ghidaoui, A. A. Kolyshkin, J. H. Liang, F. C. Chan, Q. Li, and K.
Xu, "Linear and nonlinear analysis of shallow wake flows," J. Fluid
Mech.., vol. 548 , pp. 309-340, 2006.
[12] A. A. Kolyshkin, and S. Nazarovs, "Influence of averaging coefficients
on weakly nonlinear stability of shallow flows," IASME Transactions.,
vol. 2, no. 1, pp. 86-91, 2005.
[13] K. Stewartson, and J. T. Stuart, "A non-linear instability theory for a
wave system in plane Poiseuille flow", J. Fluid Mech., vol. 48, pp. 529-
545, 1971.
[14] F. Feddersen, "Weakly nonlinear shear waves", J. Fluid Mech., vol. 371,
pp. 71-91, 1998.
[15] L. S. Aranson, and L. Kramer, "The world of the complex Ginzburg-
Landau equation", Reviews in Modern Phys., v. 74, pp. 99 - 143, 2002.
[16] M. C. Cross, and P. C. Honenberg, "Pattern formation outside the
equilibrium", Reviews in Modern Phys., v. 65, pp. 851 - 1112, 1993.
[17] T. Leveke, and M. Provansal, "The flow behind rings: bluff body wakes
without end effects", J. Fluid Mech., vol. 288, pp. 265-310, 1995.
[18] P. Le Gal, J. F. Ravoux, E. Floriani, and T. Dudok de Wit, "Recovering
coefficients of the complex Ginzburg-Landau equation from
experimental spatio-temporal data: two examples from hydrodynamics",
Physica D., vol. 174, pp. 114-133, 2003.
[19] P. J. Blennerhassett, "On the generation of waves by wind", Proc. of the
Royal Soc. London. Ser. A: Mathematical and Physical Sciences., vol.
298, pp. 451-494, 1980.
[20] M. S. Ghidaoui, and A. A. Kolyshkin, "A quasi-steady approach to the
instability of time-dependent flows in pipes", J. Fluid Mech., vol. 465,
pp. 301-330, 2002.
[21] A. A. Kolyshkin, R. Vaillancourt, and I. Volodko, "Weakly nonlinear
analysis of rapidly decelerated channel flow", IASME Transactions., vol.
2, no. 7, pp. 1157-1165, 2005.
[22] F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin, "Stability and
vortex structures of quasi-two-dimensional shear flows", Sov. Phys.
Uspekhi., vol. 33, pp. 495-520, 1990.
[23] V. L. Streeter, E. B. Wylie, and K. W. Bedford, Fluid Mechanics (ninth
edition), New York: McGraw Hill., 1998.
[24] P. A. Monkiewitz, "The absolute and convective nature of instability in
two-dimensional wakes at low Reynolds numbers", Phys. Fluids., vol.
31, pp. 999-1006, 1988.
[1] M. Van Dyke, An Album of Fluid Motion. New York: The Parabolic
Press, 1982, Photograph No. 173.
[2] E. J. Wolansky, J. Imberger, and M.L. Heron, "Island wakes in shallow
coastal waters," J. Geophys. Research, vol. 89, , pp. 10553-10559,
1984.
[3] R. G. Ingram, and V.H. Chu, "Flow around islands in Rupert Bay: An
investigation of the bottom friction effect," J. Geophys. Research, vol.
92, pp. 14521-14533, 1987.
[4] G. H. Jirka, "Large scale flow structures and mixing processes in
shallow flows," J. Hydr. Research, vol. 39 , pp. 567-573, 2001.
[5] S. A. Socolofsky, and G. H. Jirka, "Large scale flow structures and
stability in shallow flows," J. Environ. Eng. Sci.., vol. 3 , pp. 451-462,
2004.
[6] D. Chen, and G. H. Jirka, "Experimental study of a plane turbulent in a
shallow water layer," Fluid Dyn. Res.., vol. 16 , pp. 11-41, 1995.
[7] B. L. White, and H. M. Nepf, "Shear instability and coherent structures
in shallow flow adjacent to a porous layer," J. Fluid Mech.., vol. 593 ,
pp. 1-32, 2007.
[8] M. E. Negretti, S. A. Socolofsky, A. C. Rummel, and G. H. Jirka,
"Stabilization of cylinder wakes in shallow water flow by means of
roughness elements: an experimental study," Exp. Fluids., vol. 38 , pp.
403-414, 2005.
[9] D. Chen, and G. H. Jirka, "Absolute and convective instabilities of plane
turbulent wakes in a shallow water layer," J. Fluid Mech.., vol. 338 , pp.
157-172, 1997.
[10] A. A. Kolyshkin, and M. S. Ghidaoui, "Stability analysis of shallow
wake flows," J. Fluid Mech.., vol. 494 , pp. 355-37, 2003.
[11] M. S. Ghidaoui, A. A. Kolyshkin, J. H. Liang, F. C. Chan, Q. Li, and K.
Xu, "Linear and nonlinear analysis of shallow wake flows," J. Fluid
Mech.., vol. 548 , pp. 309-340, 2006.
[12] A. A. Kolyshkin, and S. Nazarovs, "Influence of averaging coefficients
on weakly nonlinear stability of shallow flows," IASME Transactions.,
vol. 2, no. 1, pp. 86-91, 2005.
[13] K. Stewartson, and J. T. Stuart, "A non-linear instability theory for a
wave system in plane Poiseuille flow", J. Fluid Mech., vol. 48, pp. 529-
545, 1971.
[14] F. Feddersen, "Weakly nonlinear shear waves", J. Fluid Mech., vol. 371,
pp. 71-91, 1998.
[15] L. S. Aranson, and L. Kramer, "The world of the complex Ginzburg-
Landau equation", Reviews in Modern Phys., v. 74, pp. 99 - 143, 2002.
[16] M. C. Cross, and P. C. Honenberg, "Pattern formation outside the
equilibrium", Reviews in Modern Phys., v. 65, pp. 851 - 1112, 1993.
[17] T. Leveke, and M. Provansal, "The flow behind rings: bluff body wakes
without end effects", J. Fluid Mech., vol. 288, pp. 265-310, 1995.
[18] P. Le Gal, J. F. Ravoux, E. Floriani, and T. Dudok de Wit, "Recovering
coefficients of the complex Ginzburg-Landau equation from
experimental spatio-temporal data: two examples from hydrodynamics",
Physica D., vol. 174, pp. 114-133, 2003.
[19] P. J. Blennerhassett, "On the generation of waves by wind", Proc. of the
Royal Soc. London. Ser. A: Mathematical and Physical Sciences., vol.
298, pp. 451-494, 1980.
[20] M. S. Ghidaoui, and A. A. Kolyshkin, "A quasi-steady approach to the
instability of time-dependent flows in pipes", J. Fluid Mech., vol. 465,
pp. 301-330, 2002.
[21] A. A. Kolyshkin, R. Vaillancourt, and I. Volodko, "Weakly nonlinear
analysis of rapidly decelerated channel flow", IASME Transactions., vol.
2, no. 7, pp. 1157-1165, 2005.
[22] F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin, "Stability and
vortex structures of quasi-two-dimensional shear flows", Sov. Phys.
Uspekhi., vol. 33, pp. 495-520, 1990.
[23] V. L. Streeter, E. B. Wylie, and K. W. Bedford, Fluid Mechanics (ninth
edition), New York: McGraw Hill., 1998.
[24] P. A. Monkiewitz, "The absolute and convective nature of instability in
two-dimensional wakes at low Reynolds numbers", Phys. Fluids., vol.
31, pp. 999-1006, 1988.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:54354", author = "Imad Chaddad and Andrei A. Kolyshkin", title = "Ginzburg-Landau Model : an Amplitude Evolution Equation for Shallow Wake Flows", abstract = "Linear and weakly nonlinear analysis of shallow wake
flows is presented in the present paper. The evolution of the most
unstable linear mode is described by the complex Ginzburg-Landau
equation (CGLE). The coefficients of the CGLE are calculated
numerically from the solution of the corresponding linear stability
problem for a one-parametric family of shallow wake flows. It is
shown that the coefficients of the CGLE are not so sensitive to the
variation of the base flow profile.", keywords = "Ginzburg-Landau equation, shallow wake flow,weakly nonlinear theory.", volume = "2", number = "2", pages = "97-5", }
