8th ICCHMT, Istanbul, 25-28 May 2015 133 ON THE SUBHARMONIC INSTABILITIES OF INTERNATIONAL SHORT-CRESTED WAVES N. Allalou*,1,2, D. Boughazi2 , M. Debiane2 , C. Kharif3, N. Benyahia4,2 * Université de Boumerdes, Faculté des sciences, Siège (Ex INIL), Boumerdes, Algérie Email:[email protected] 2 Faculté de Physique, Université des Sciences et de la Technologie Houari Boumedienne, B.P. 32 El Alia, Alger 16111, Algérie. 3 Institut de Recherche sur les Phénomènes Hors Equilibre,Technopole de Château-Gombert, 49 rue F.Joliot-Curie, B.P.146,13384 Marseille Cedex 13, France. 4 Facult é des Sciences et Sciences Appliquées, Université de Bouira - Rue DRISSI Yahia - Bouira 10000 1 Algérie. Keywords: short crested interfacial wave; Galerkin method; subharmonic instability. ABSTRACT A short crested interfacial waves are threedimensional waves which occurs at the interface of two fluid of different density. This threedimensional patterns are reduced by two progressive interfacial plane waves propagating at an oblique angle,θ, to each other. A numerical Galerkin method is used to study the linear stability of fully short crested interfacial waves on infinite depth. it is shown that: (i) near the twodimensional standing interfacial wave limit, the three-dimensional modulation instability of class I(a) is the dominant; (ii) near the two-dimensional progressive interfacial wave limit, they are unstable to class I(a) perturbations and are developed in the direction of propagation; (iii) fully threedimensional interfacial waves, in which the perturbations of class I(b) are the most instable. INTRODUCTION The present study extends the work of Ioualalen [1] on the stability of three-dimensional surface gravitywaves to three-dimensional interfacial gravity waves.The three-dimensional interfacial waves consideredhere are short crested interfacial waves which may be produced by two progressive interfacial waves at anoblique angle to each other. The limiting cases in this family are the twodimensional progressive interfacialwaves (where the interfacial wavetrains propagate in the same direction) and two-dimensional interfacial standing waves (where the interfacial wavetrains propagate in opposite directions). The properties ofthree- dimensional interfacial waves have been discussed in [2]. Using a perturbation method, the authorsobtained 27th-order solutions.A numerical Galerkin method is used to study the linear stability of fully short crested interfacial waves on infinite depth. MATHEMATICAL FORMULATION OF THE PROBLEM We consider the motion under the influence of gravity of three-dimensional progressive waves on the interface between two homogeneous fluids of infinite layers. Both fluids are assumed to be inviscid andincompressible, and the motion in either fluid is assumed to be irrotational. The subscript (1) will be used for the upper layer and (2) will be used for the lower layer. The governing equations are φixx + φiyy + φizz = 0 z ≥ η for i=1 and z ≤ η for i=2 (1) ηt + φix ηx + φiy η y − φiz = 0 on z=η (i=1,2) (2) 2 1 2 1ω 2 2 µ φ1t + η + φ1x + φ1y + φ1z − − 2 2 α ( ) 2 1 2 1ω 2 2 φ2t + η + φ2 x + φ2 y + φ2 z − = 0 on z=η 2 2 α (3) φ1z = 0 for z → ∞ (4) φ2 z = 0 for z → −∞ (5) ( ) 8th ICCHMT, Istanbul, 25-28 25 May 2015 Where , , , and , , , are the velocity potentials of the upper and lower fluids respectively, , , is the equation of the interface and / is the density ratio. The stability problem consists in superimposing steady of short-crested interfacial and small harmonic perturbations,, modulated by wave numbers p and q in the two horizontal directions. directions The problem is solved then in the frame R* that moving with the celerity of the wave c. In this new frame of reference propagating at a speedc= speed ω/αthe system of equations (1)-(5)) admit doubly periodic solutions of permanent form ̅ and which take the following form dimensional interfacial short crested wave, class I(b) instabilities are dominant for all values of h (figure 3). Is instability is also three-dimensional. three In figure 4,we ,we have shown the maximum amplification rates as function of wave steepness for incident angle θ=80°. =80°. For this value of angle, the short-crested crested interfacial wave behaves like a two-dimensional dimensional interfacial wave. The analysis of these results shown that the dominant instability is of class I(a) and it occurs for 0. Thus, this instability leads to a subharmonic perturbation in x direction and quasi superharmonic perturbation in y-direction. N i (r ) η = ∑ h ∑ A mn cos(mx ) cos(ny ) r =1 mn N α z i (r ) φ 1 = ∑ h ∑ B mn sin(mx ) cos(ny )e mn r =1 mn N −α z i (r ) φ 2 = ∑ h ∑C mn cos( mx ) cos(ny )e mn r =1 mn Progressive interfacial waves which are periodic in two orthogonal directions and are steady relative to a frame of reference eference moving in one of these directions are given Allalou et al. [2] using perturbation method up to 27th order. In the linear stability problem, we look for nontrivial normal forms. A Galerkin spectral method is processed, resulting in a generalized lized eigenvalue problem solved numerically with QZ algorithm. Figure 1: Maxima of growth rates for angle θ=10° and density ratio µ=0.1 =0.1 as function of wave steepness. RESULTS AND DISCUSSION ON When the wavelength of the perturbation in the xx direction (respectively y-direction) direction) is greater than of the unperturbed wavelength in the xx-direction (respectively y-direction), direction), the instability is calledsubharmonic. subharmonic. We investigated three regime of stabilities upon the values ofθ.. The first case correspond to θ=10°,, and the interfacial wave is close to standing waves (figure 1).. For this angle, angle class I(a) remains dominant ominant for any wave wa steepness. The figure 2 shown the diagram stabi stability of the class I(a) for h=0.3.. The wavelength of the perturbation erturbation corresponding to maximum max amplification rate is subharmonic harmonic in the two twox and y directions. Furthermore, the size of instable zone is of order . threeFor θ=40°, representative of fully three Figure 2: Stability diagram of class I(a) for angle θ=10° and density ratio µ=0.1. 8th ICCHMT, Istanbul, 25-28 May 2015 order. Then the problem of stability is solved with Galerkin method. Our results highlight the dominant class is I(a) near the twobidimensional limit. However, for fullythree-dimensional interfacial waves, the dominant class is I(b). REFERENCES [1] Ioualalen, M., and Kharif, C., 1994, On the Figure 3: Maxima of growth rates for angle θ=40° and density ratio µ=0.1 as function of wave steepness. Figure 4: Maxima of growth rates for angle θ=80° and density ratio µ=0.1 as function of wave steepness. CONCLUSIONS This study deals with the stability of the short crested interfacial waves to infinitesimal subharmonic perturbations. The unperturbed fluid is computed with perturbation method up to 27th subharmonic instabilities of steady threedimensional deep water waves, J. Fluid Mech., vol. 262, pp. 265-291. [2] Allalou N., Debiane M. and Kharif C., 2013, “Three-dimensional periodic interfacial gravity waves: analytical and numerical results”, Eur. J. Mech, B/Fluids, vol. 30, no 4, 371-386.
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