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Ginzburg-Landau Model for Curved Two-Phase Shallow Mixing Layers

Authors: Andrei A. Kolyshkin, Irina Eglite

Abstract:

Method of multiple scales is used in the paper in order to derive an amplitude evolution equation for the most unstable mode from two-dimensional shallow water equations under the rigid-lid assumption. It is assumed that shallow mixing layer is slightly curved in the longitudinal direction and contains small particles. Dynamic interaction between carrier fluid and particles is neglected. It is shown that the evolution equation is the complex Ginzburg-Landau equation. Explicit formulas for the computation of the coefficients of the equation are obtained.

Keywords: mixing layer, shallow water equations, Ginzburg-Landau equation, weakly nonlinear analysis

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1083363

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References:


[1] G.H. Jirka, "Large scale flow structures and mixing processes in shallow flows", J. Hydr. Res., vol. 39, pp. 567-573, 2001.
[2] V.H. Chu and S. Babarutsi, "Confinement and bed-friction effects in shallow turbulent mixing layers", J. Hydr. Eng., vol. 114, pp. 1257- 1274, 1988.
[3] W.S.J. Uijttewaal and R. Booij, "Effect of shallowness on the development of free-surface mixing layers", Phys. Fluids., vol. 12, pp. 1257-1274, 1988.
[4] W.S.J. Uijttewaal and J. Tukker, "Development of quasi twodimensional structures in a shallow free-surface mixing layer", Exp. Fluids., vol. 24, pp. 192-200, 1998.
[5] B.C. Prooijen and W.S.J. Uijttewaal, "A linear approach for the evolution of coherent structures in shallow mixing layers", Phys. Fluids., vol. 14, pp. 4105-4114, 2002.
[6] V.H. Chu, J.H. Wu, and R.E. Khayat, "Stability of transverse shear flows in shallow open channels", J. Hydr. Eng., vol. 117, pp. 1370- 1388, 1991.
[7] D. Chen and G.H. Jirka, "Linear stability analysis of turbulent mixing layers and jets in shallow water layers", J. Hydr. Res., vol. 36, pp. 815- 830, 1998.
[8] M.S. Ghidaoui and A.A. Kolyshkin, "Linear stability analysis of lateral motions in compound open channels", J. Hydr. Eng., vol. 125, pp. 871- 880, 1999.
[9] A.A. Kolyshkin and M.S. Ghidaoui, "Gravitational and shear instabilities in compound and composite channels", J. Hydr. Eng., vol. 128, pp. 1076-1086, 2002.
[10] Y. Yang, J.N. Chung, T.R. Troutt, and C.T. Crowe, "The influence of particles on the stability of two-phase mixing layers", Phys. Fluids., vol. A2, pp. 1839-1845, 1990.
[11] Y. Yang, J.N. Chung, T.R. Troutt, and C.T. Crowe, "The effect of particles on the stability of a two-phase wake flow", Int. J. Multiphase Flow, vol. 19, pp. 137-149, 1993.
[12] 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.
[13] F. Feddersen, "Weakly nonlinear shear waves", J. Fluid Mech., vol. 372, pp. 71-91, 1998.
[14] P.J. Blennerhassett, "On the generation of waves by wind", Philosophical Transactions of the Royal Society of London, Ser. A., vol. 298, pp. 451-494, 1980.
[15] A.A. Kolyshkin and M.S. Ghidaoui, "Stability analysis of shallow wake flows", J. Fluid Mech.., vol. 494, pp. 355-377, 2003.
[16] A.A. Kolyshkin and S. Nazarovs, "Linear and weakly nonlinear analysis of two-phase shallow wake flows", WSEAS Transactions on Mathematics., vol. 6, pp. 1-8, 2007.
[17] I. Eglite and A.A. Kolyshkin, "Linear and weakly nonlinear instability of slightly curved shallow mixing layers", WSEAS Transactions on Fluid Mechanics., vol. 6, pp. 123-132, 2011.
[18] D. Zwillinger, Handbook of differential equations. New York: Academic Press, 1998.