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Non-modal stability analysis of the boundary layer under solitary waves

  • Joris C. G. Verschaeve (a1), Geir K. Pedersen (a1) and Cameron Tropea (a2)


In the present work, a stability analysis of the bottom boundary layer under solitary waves based on energy bounds and non-modal theory is performed. The instability mechanism of this flow consists of a competition between streamwise streaks and two-dimensional perturbations. For lower Reynolds numbers and early times, streamwise streaks display larger amplification due to their quadratic dependence on the Reynolds number, whereas two-dimensional perturbations become dominant for larger Reynolds numbers and later times in the deceleration region of this flow, as the maximum amplification of two-dimensional perturbations grows exponentially with the Reynolds number. By means of the present findings, we can give some indications on the physical mechanism and on the interpretation of the results by direct numerical simulation in Vittori & Blondeaux (J. Fluid Mech., vol. 615, 2008, pp. 433–443) and Özdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) and by experiments in Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231). In addition, three critical Reynolds numbers can be defined for which the stability properties of the flow change. In particular, it is shown that this boundary layer changes from a monotonically stable to a non-monotonically stable flow at a Reynolds number of $Re_{\unicode[STIX]{x1D6FF}}=18$ .


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Andersson, P., Brandt, L., Bottaro, A. & Henningson, D. S. 2001 On the breakdown of boundary layer streaks. J. Fluid Mech. 428, 2960.
Benjamin, T. B. 1966 Internal waves of finite amplitude and permanent form. J. Fluid Mech. 25, 241270.
Bertolotti, F., Herbert, T. & Spalart, P. 1992 Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441474.
Biau, D. 2016 Transient growth of perturbations in Stokes oscillatory flows. J. Fluid Mech. 794, R4.
Blondeaux, P., Pralits, J. & Vittori, G. 2012 Transition to turbulence at the bottom of a solitary wave. J. Fluid Mech. 709, 396407.
Brandt, L., Schlatter, P. & Henningson, D. S. 2004 Transition in boundary layers subject to free-stream turbulence. J. Fluid Mech. 517, 167198.
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 16371650.
Carr, M. & Davies, P. A. 2006 The motion of an internal solitary wave of depression over a fixed bottom boundary in a shallow, two-layer fluid. Phys. Fluids 18, 016601.
Carr, M. & Davies, P. A. 2010 Boundary layer flow beneath an internal solitary wave of elevation. Phys. Fluids 22, 026601.
Corbett, P. & Bottaro, A. 2000 Optimal perturbations for boundary layers subject to stream-wise pressure gradient. Phys. Fluids 12 (1), 120130.
Corbett, P. & Bottaro, A. 2001 Optimal linear growth in swept boundary layers. J. Fluid Mech. 435, 123.
Davis, S. H. & von Kerczek, C. 1973 A reformulation of energy stability theory. Arch. Rat. Mech. Anal. 52, 112117.
Ellingsen, T. & Palm, E. 1975 Hydrodynamic stability. Phys. Fluids 18, 487.
Fenton, J. 1972 A ninth-order solution for the solitary wave. J. Fluid Mech. 53, 257271.
Frigo, M. & Johnson, S. G. 2005 The design and implementation of FFTW3. In Proceedings of the IEEE, vol. 93, pp. 216231.
Galassi, M., Davies, J., Theiler, B., Gough, B., Jungman, G., Alken, P., Booth, M. & Rossi, F. 2009 GNU Scientific Library Reference Manual. Network Theory Ltd.
Gaster, M. 2016 Boundary layer transition initiated by a random excitation. In Book of Papers 24th International Congress of Theoretical and Applied Mechanics. International Union of Theoretical and Applied Mechanics (IUTAM).
Grimshaw, R. 1971 The solitary wave in water of variable depth. Part 2. J. Fluid Mech. 46, 611622.
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241260.
Herbert, T. 1988 Secondary instability of boundary layers. Annu. Rev. Fluid Mech. 20, 487526.
Jimenez, J. 2013 How linear is wall-bounded turbulence? Phys. Fluids 25, 110814.
Joseph, D. D. 1966 Nonlinear stability of the Boussinesq equations by the method of energy. Arch. Rat. Mech. Anal. 22, 163.
von Kerczek, C. & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753773.
Levin, O. & Henningson, D. S. 2003 Exponential versus algebra growth and transition prediction in boundary layer flow. Flow Turbul. Combust. 70, 183210.
Liu, P. L.-F. & Orfila, A. 2004 Viscous effects on transient long-wave propagation. J. Fluid Mech. 520, 8392.
Liu, P. L.-F., Park, Y. S. & Cowen, E. A. 2007 Boundary layer flow and bed shear stress under a solitary wave. J. Fluid Mech. 574, 449463.
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493517.
Luo, J. & Wu, X. 2010 On the linear instability of a finite Stokes layer: instantaneous versus floquet modes. Phys. Fluids 22, 113.
Miles, J. W. 1980 Solitary waves. Annu. Rev. Fluid Mech. 12, 1143.
Özdemir, C. E., Hsu, T.-J. & Balachandar, S. 2013 Direct numerical simulations of instability and boundary layer turbulence under a solitary wave. J. Fluid Mech. 731, 545578.
Park, Y. S., Verschaeve, J. C. G., Pedersen, G. K. & Liu, P. L.-F. 2014 Corrigendum and addendum for boundary layer flow and bed shear stress under a solitary wave. J. Fluid Mech. 753, 554559.
Sadek, M. M., Parras, L., Diamessis, P. J. & Liu, P. L.-F. 2015 Two-dimensional instability of the bottom boundary layer under a solitary wave. Phys. Fluids 27, 044101.
Sanderson, C. & Curtin, R. 2016 Armadillo: a template-based C++ library for linear algebra. J. Open Source Softw. 1, 26.
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.
Shaikh, F. N. & Gaster, M. 1994 The non-linear evolution of modulated waves in a boundary layer. J. Engng Maths 28, 5571.
Shen, J. 1994 Efficient spectral-Galerkin method I. Direct solvers for the second and fourth order equations using Legendre polynomials. SIAM J. Sci. Comput. 15, 14891505.
Shen, J. 1995 Efficient spectral-Galerkin method II. Direct solvers of second fourth order equations by using Chebyshev polynomials. SIAM J. Sci. Comput. 16 (1), 7487.
Shuto, N. 1976 Transformation of nonlinear long waves. In Proceedings of 15th Conference on Coastal Engineering. American Society of Civil Engineers.
Sumer, B. M., Jensen, P. M., Sørensen, L. B., Fredsøe, J., Liu, P. L.-F. & Carstensen, S. 2010 Coherent structures in wave boundary layers. Part 2. Solitary motion. J. Fluid Mech. 646, 207231.
Tanaka, H., Winarta, B., Suntoyo & Yamaji, H. 2011 Validation of a new generation system for bottom boundary layer beneath solitary wave. Coast. Engng 59, 4656.
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability with eigenvalues. Science 261, 578584.
Verschaeve, J. C. G. & Pedersen, G. K. 2014 Linear stability of boundary layers under solitary waves. J. Fluid Mech. 761, 62104.
Vittori, G. & Blondeaux, P. 2008 Turbulent boundary layer under a solitary wave. J. Fluid Mech. 615, 433443.
Vittori, G. & Blondeaux, P. 2011 Characteristics of the boundary layer at the bottom of a solitary wave. Coast. Engng 58, 206213.
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