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8th ICCHMT, Istanbul, 25-28 May 2015
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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:n_allalou2004@yahoo.fr
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
1ω  
2
2
µ φ1t + η + φ1x + φ1y + φ1z −    −
2
2  α  

(
)
2

1 2
1ω  
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.