**8. References**


390 Numerical Simulation – From Theory to Industry Numerical Simulations of Water Waves' Modulational Instability Under the Action of Wind and Dissipation <sup>17</sup> Numerical Simulations of Water Waves' Modulational Instability Under the Action of Wind and Dissipation 391

> [13] E. Fermi, J. Pasta, and S. Ulam. Studies of non linear problems. *Amer. Math. Month.*, 74 (1), 1967.

16 Will-be-set-by-IN-TECH

unstable solutions, a norm based on the potential energy of the perturbations was introduced. A nonlinear stability diagram resulting from the numerical simulations of the fully nonlinear equation has been given in the (A, Ω)-plane which coincides with the linear stability analysis of [4]. In the presence of wind, dissipation and modulational instability it is found that wind energy goes to the subharmonic sideband whereas dissipation lowers the amplitude of the fundamental mode of the wave train yielding to a permanent frequency-downshifting. This permanent frequency downshift is strongly influenced by the wind and dissipation parameter. If the wave group is at equilibrium in energy input and dissipation, the fdNLS equation reduces to the classical NLS equation, and predict no influence of the wind. However, by considering the asymmetry between wave components, induced by strong nonlinearity (higher than fourth order), a strong influence of the wind and dissipation is observed.

*Mediterranean Institute of Oceanography (MIO), Aix-Marseille Univ., Université du Sud Toulon-Var,*

*Institut de Recherche sur les phénomènes hors équilibre (IRPHE), Aix-Marseille Univ., Ecole Centrale*

[1] T. B. Benjamin and J. E. Feir. The disintegration of wave trains on deep water. *J. Fluid*

[2] H. Segur, D. Henderson, J. Carter, J. Hammack, C. M. Li, D. Pheiff, and K. Socha.

[3] G. Wu, Y. Liu, and D. K. P. Yue. A note on stabilizing the benjamin-feir instability. *J. Fluid*

[4] C. Kharif, R. Kraenkel, M. Manna, and R. Thomas. The modulational instability in deep water under the action of wind and dissipation. *J. Fluid Mech.*, 664:417–430, 2010. [5] G. G. Stokes. On the theory of oscillatory waves. *Trans. Camb. Phil. Soc.*, 8:441–455, 1847. [6] M. J. Lighthill. Contributions to the theory of waves in non-linear dispersive systems. *J.*

[7] G. B. Whitham. Linear and nonlinear waves. Wiley-Interscience (New-York), 1974. [8] V. E. Zakharov. Satbility of periodic waves of finite amplitude on the surface of a deep

[9] C. Kharif and E. Pelinovsky. Physical mechanisms of the rogue wave phenomenon. *Eur.*

[10] C. Kharif, E. Pelinovsky, and A. Slunyaev. Rogue waves in the ocean. In *Mathematical*

[11] J. W. Dold and D. H. Peregrine. Water-wave modulation. In *Proc. 20th Intl. Conf. Coastal*

[12] M. L. Banner and X. Tian. On the determination of the onset of breaking for modulating

*Aspects of Vortex Dynamics*. Springer (ISBN: 978-3-540-88418-7), 2009.

surface gravity water waves. *J. Fluid Mech.*, 367:107–137, 1998.

Stabilizing the benfamin-feir instability. *J. Fluid Mech.*, 539:229–271, 2005.

**Author details**

*CNRS/INSU, UMR 7294, IRD, UMR235, France*

*Marseille, CNRS/INSIS UMR 7342, France*

*Mech.*, 27:417–430, 1967.

*Mech.*, 556:45–54, 2006.

*Inst. Math. Appl.*, 1:269–306, 1965.

fluid. *J. Appl. Tech. Phys.*, 9:190–194, 1968.

*Eng. (Taipei)*, volume 1, pages 163–175. 1986.

*J. Mech. B Fluids*, 22 (6):603–633, 2003.

Julien Touboul

Christian Kharif

**8. References**


© 2012 Riahi and Lili, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Spectral Modeling and Numerical Simulation of** 

Mohamed Riahi and Taieb Lili

http://dx.doi.org/10.5772/48617

**1. Introduction** 

Additional information is available at the end of the chapter

**Compressible Homogeneous Sheared Turbulence** 

This chapter is mainly in the area of the use of Rapid Distortion Theory (RDT) to clarify and to well better increase our understanding of the physics of the compressible turbulent flows. This theory is a computationally viable option for examining linear compressible flow physics in the absence of inertial effects. In this linear limit, the statistical evolution of incompressible homogeneous turbulence can be described completely in terms of closed spectral covariance equations (see Refs. (Hunt, 1990 & Savill, 1987) and references therein). Many papers in literature deal with homogeneous compressible turbulence and RDT solution (Cambon et al., 1993; Coleman & Mansour, 1991; Blaisdell et al., 1993, 1996; Durbin & Zeman, 1992; Jacquin et al., 1993; Livescu & Madnia, 2004; Riahi et al., 2007; Riahi, 2008; Riahi & Lili, 2011; Sarkar, 1995; Simone, 1995; Simone et al., 1997). These studies have yielded very valuable physical insight and closure model suggestions. In all the above works, the fluctuation equations are solved directly to infer turbulence physics. For the case of viscous compressible homogeneous shear flow in the RDT limit no analytical solutions are known. Simone et al. (1997) performed RDT simulations of homogeneous shear flow and showed that the role of the distortion Mach number, *Md*, on the time variation of the turbulent kinetic energy is consistent with that found in the direct numerical simulation (DNS) results. In this chapter, numerical solutions to the RDT equations for the special case of mean shear is described completely by finding numerical solutions obtained by solving linear double point correlations equations. Numerical integration of these equations is carried out using a second-order simple and accurate scheme (Riahi & Lili, 2011). Indeed, this numerical method is proved more stable and faster than the previous one which use linear transfer matrix (Riahi et al., 2007 & Riahi, 2008) and allows in particular to obtain accurately the asymptotic behavior of the turbulence parameters (for large values of the non-dimensional times *St*) characteristic of equilibrium states. To perform this work, RDT code solving
