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Non-normal stability analysis of a shear current under surface gravity waves

Published online by Cambridge University Press:  31 July 2008

D. AMBROSI  and
M. ONORATO
D. AMBROSI
Affiliation:
Dip. di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy
M. ONORATO
Affiliation:
Dip. di Fisica Generale, Università di Torino, via Pietro Giuria 1, 10125 Torino, Italy
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Abstract

The stability of a horizontal shear current under surface gravity waves is investigated on the basis of the Rayleigh equation. As the differential operator is non-normal, a standard modal analysis is not effective in capturing the transient growth of a perturbation. The representation of the stream function by a suitable basis of bi-orthogonal eigenfunctions allows one to determine the maximum growth rate of a perturbation. It turns out that, in the considered range of parameters, such a growth rate can be two orders of magnitude larger than the maximum eigenvalue obtained by standard modal analysis.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 609 , 25 August 2008 , pp. 49 - 58
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Burns, J. C. 1953 Long waves in running water. Proc. Camb. Phil. Soc. 49, 695–706.Google Scholar
Caponi, E. A., Yuen, H. C., Milinazzo, F. A. & Saffman, P. G. 1991 Water wave instability induced by a drift layer. J. Fluid Mech. 222, 297–313.Google Scholar
Drazin, P. J. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Farrell, F. & Ioannou, P. J. 2008 The stochastic parametric mechanism for growth of wind-driven surface water waves. J. Phys. Oceanogr. 38, 862–879.Google Scholar
Farrell, F. & Moore, A. M. 1992 An adjoint method for obtaining the most rapidly growing perturbation to oceanic flows. J. Phys. Oceanogr. 22, 338–349.Google Scholar
van Gastel, K., Janssen, P. A. E. M. & Komen, G. J. 1985 On phase velocity and growth rate of wind-induced gravity-capillary waves. J. Fluid Mech. 161, 199–216.Google Scholar
Kawai, S. 1979 Generation of initial wavelets by instability of a coupled shear flow and their evolution to wind waves. J. Fluid Mech. 93, 661–703.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1998 Instabilities of a horizontal shear flow with a free surface. J. Fluid Mech. 364, 147–162.Google Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185–204.CrossRefGoogle Scholar
Miles, J. W. 2001 a Gravity waves on shear flows. J. Fluid Mech. 443, 293–299.Google Scholar
Miles, J. W. 2001 b A note on surface waves generated by shear-flow instability. J. Fluid Mech. 447, 173–177.Google Scholar
Morland, L. C., Saffman, P. G. & Yuen, H. C. 1991 Waves generated by shear layers instabilities. Proc. R. Soc. Math. Phy. Sci. 433, 441–450.Google Scholar
Olsson, P. J. & Henningson, D. S. 1994 Optimal disturbances in watertable flow. Stud. Appl. Maths 94, 183–210.CrossRefGoogle Scholar
Quarteroni, A., Sacco, R. & Saleri, F. 2007 Numerical Mathematics. Springer.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Shemdin, O. H. 1972 Wind generated current and phase speed of wind waves. J. Phys. Oceanogr. 2.2.0.CO;2>CrossRefGoogle Scholar
Shrira, V. I. 1993 Surface waves on shear currents: solution of the boundary-value problem. J. Fluid Mech. 252, 565–565.Google Scholar
Valenzuela, G. R. 1976 The growth of gravity–capillary waves in a coupled shear flow. J. Fluid Mech. 76, 229–250.Google Scholar
Yih, C. S. 1972 Surface waves in flowing water. J. Fluid Mech. 51, 209–220.CrossRefGoogle Scholar