Giuseppe Petrone

Eric Chénier
email: eric.chenier@univ-mlv.fr

Guy Lauriat

STABILITY ANALYSIS OF NATURAL CONVECTIVE FLOWS IN HORIZONTAL ANNULI: EFFECTS OF THE AXIAL AND RADIAL ASPECT RATIOS

Keywords: numbers: 44.25.+f Natural convection, 47.20.Bp Buoyancy-driven instabilities, 47.55.Iv Core-annular flows

Linear stability analyses for two-dimensional natural convection in horizontal air-filled annuli are performed for three-dimensional perturbations and radius ratios in the range 1.2 ≤ R ≤ 3. Flow transitions from moderate to large gap annuli, which have not been reported before, are thoroughly investigated. As a result, stability diagrams are obtained for finite and for infinite length annuli.

The leading disturbances and threshold values are found to agree well with experimental data and three-dimensional numerical solutions. Three-dimensional simulations were also carried out to examine the influence on the flow stability of no-slip boundary conditions at the end walls.

I. INTRODUCTION

Free convective flow of air in horizontal annuli bounded by two cylinders held at different temperatures has been intensely studied in the last past decades as well experimentally as theoretically and numerically. It is established that the basic flow at small Rayleigh numbers, Ra, consists of two large cells, symmetrically located with respect to the vertical plane containing the cylinder axis. When increasing Ra, this flow becomes unstable and leads to different patterns according to the value of the radius ratio, R.

Extensive numerical simulations for the two-dimensional model of flows in transversal sections of annular spaces were conducted. Under the 2D assumption, dual steady solutions were highlighted when the Rayleigh number exceeds a critical value: one being the crescentshaped eddy flow commonly observed and the other flow consisting of two counter-rotating eddies and their mirror images [START_REF] Yoo | Dual steady solutions in natural convection between horizontal concentric cylinders[END_REF] . The appearance of dual solutions is due to an imperfect trans-critical bifurcation [START_REF] Mizushima | Transitions of natural convection in a horizontal annulus[END_REF] . When the radius ratio tends to 1, the imperfect bifurcation gets closer to the classical pitchfork bifurcation of the Rayleigh-Bénard convection between to infinite parallel plates, with a threshold value Ra c = 1708. A recent 2D stability analysis showed that dual but also triple solutions are indeed stable for certain (Ra, R)-gaps [START_REF] Petrone | Stability of free convection in air-filled horizontal annuli: Influence of the radius ratio[END_REF] . Moreover, the flows consisting of two or three counter-rotating eddies and their mirror images are linearly unstable for R > 2 (and Ra < 10, 000) due to the onset of a disturbance breaking the vertical reflection symmetry of the basic flow. Thus, only the common couple of counter-rotating cells remains stable for large radius ratios.

The multi-cellular flow patterns obtained with the 2D-flow model and made of longitudinal rolls confined in the top region of the annular space were observed in experiments for a narrow gap annulus (R = 1.15) [START_REF] Liu | Natural convection heat transfer in long horizontal cylindrical annuli[END_REF] . These patterns were also shown in three-dimensional numerical simulations in combination with transverse rolls in the upper region of the annulus (R < 1.2) [START_REF] Dyko | On the presence of odd transverse convective rolls in narrow-gap horizontal annuli[END_REF] . For moderate radius ratios, a three-dimensional spiral motion occurs. It consists of the basic crescent-shaped flow with transverse convective cells at the top of the annular space. This spiral flow was obtained as well in numerical works [START_REF] Dyko | A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus[END_REF][START_REF] Dyko | Three-dimensional natural convective states in a narrow-gap horizontal annulus[END_REF][START_REF] Petrone | Three-dimensional study of multiple transitions for natural convection in horizontal annuli[END_REF] as in experiments [START_REF] Grigull | Natural convection in horizontal cylindrical annuli[END_REF][START_REF] Powe | Free convective flow patterns in cylindrical annuli[END_REF][11] . For wide gap annuli, an unsteady régime was experimentally observed by Bishop and Carley 12 (R = 3.69) and Labonia et al. 13 (R = 2.38). This non steady flow behavior was recently confirmed by 3D-numerical simulations for R = 2.4: the thermal plume oscillated in the each of the transversal section of the annulus [START_REF] Petrone | Etude numérique des écoulements de convection naturelle et mixte dans un espace annulaire horizontal : approche par analyse de stabilité linéaire[END_REF] .

A large number of experimental results for narrow (R ≤ 1.23), moderate (1.23 ≤ R ≤ 1.71) and wide gap annuli (R ≥ 1.71) had been put together by Powe et al. [START_REF] Powe | Free convective flow patterns in cylindrical annuli[END_REF] in order to draw a stability chart showing the critical Grashof number as a function of the modified radius ratio σ = 2/(R -1). This chart clearly depicts the regions where 2D multi-cellular, 3D-spiral and 2D-oscillatory flows respectively occur. An analytical stability analysis of 2D-flows with respect to 3D-disturbances in annuli of infinite length was performed by Mojtabi and Caltagirone [START_REF] Mojtabi | Etude de la stabilité d'un écoulement de convection naturelle dans un espace annulaire horizontal[END_REF][START_REF] Mojtabi | Energy stability of a natural convective flow in a horizontal annular space[END_REF] for infinitesimal (1 ≤ R ≤ 4) and finite amplitude (1 ≤ R ≤ 2) perturbations. The basic flow was approximated in the vertical upper section of the annulus and the disturbances considered as axisymmetric.

For R = 1 (i.e. parallel plate limit), the stability thresholds evaluated either by linear stability analysis or by an energetic method are equal to Ra c = 1708. These thresholds augment and diverge from each other when increasing the radius ratio. The linear stability analysis was later on improved by taking into account the azimuthal direction [START_REF] Dyko | A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus[END_REF] and, as foreseen, the most unstable disturbance was found to occur in the upper part of the annulus. A numerical three-dimensional linear stability analysis of the crescent-shaped flows was carried-out by Choi and Kim [START_REF] Choi | Three-dimensional linear stability of natural convective flow between concentric horizontal cylinders[END_REF] for infinite length annuli and 1.2 ≤ R ≤ 2.

They demonstrated a weak dependence of the Ra-threshold (1714 ≤ Ra c ≤ 2514) with the radius ratio (1.2 ≤ R ≤ 1.9). The onset of this instability is due to the growing of transverse rolls at the top of the annulus what qualitatively corresponds to numerical and experimental observations [START_REF] Dyko | A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus[END_REF][START_REF] Dyko | Three-dimensional natural convective states in a narrow-gap horizontal annulus[END_REF][START_REF] Petrone | Three-dimensional study of multiple transitions for natural convection in horizontal annuli[END_REF][START_REF] Grigull | Natural convection in horizontal cylindrical annuli[END_REF][START_REF] Powe | Free convective flow patterns in cylindrical annuli[END_REF][11] . For R = 2, no critical value was detected for Ra ≤ 100, 000. This lack of a threshold value was attributed to its sharp increase with R.

The main purpose of this paper is to improve the knowledge of the stability thresholds as a function of the radius ratio, for air-filled horizontal annuli. A particular attention will be paid on the sudden disappearance beyond R = 2 of the transition due to the onset of transverse rolls as described by Choi and Kim 17 . To this end, a three-dimensional linear stability analysis was carried-out for two-dimensional basic flows with 1.2 ≤ R ≤ 3 and P r = 0.7. The first part of the present paper describes briefly the numerical methods whereas the results are discussed in the second part, divided into six sections. The basic solutions, which are used to perform the linear stability analyses, are first defined. Afterward, neutral stability curves are described as a function of the wavenumber of the disturbances.

Stability diagrams are then established for annuli of finite length, and the effect of no-slip boundary conditions are studied by performing three-dimensional non-linear simulations.

Next, the transitions for infinite length annuli with various radius ratios are investigated.

To end, comparisons with numerical and experimental studies are performed and discussed.

II. NUMERICAL METHODS

A. Mathematical model and numerical method

Linear and non-linear equations

The air-filled annulus of length L is formed by two coaxial and horizontal cylinders of radii r * i and

r * o = R r * i , held at temperatures T * (r * i ) = T * i and T * (r * o ) = T * o < T * i as illustrated in
= (T * -T * ref )/(T * i -T * o ) where T * ref = (T * i + T * o )/2.
The dimensionless Navier-Stokes and energy equations, expressed in the Boussinesq approximation, write:

           ∂ u ∂t = -∇ • ( u ⊗ u) + P r∇ 2 u -∇p + RaP rT k ∇ • u = 0 ∂T ∂t = -∇ • ( uT ) + ∇ 2 T (1)
where Ra = (gβ(T * i -T * o )d 3 )/(να) and P r = ν/α are the Rayleigh and Prandtl numbers with β the thermal expansion coefficient and ν the kinematic viscosity.

The equations are treated in the cylindrical coordinates (r, θ, z) in the domain 0 Consider the two-dimensional steady velocity u 0 = u 0 e r + v 0 e θ , temperature T 0 , and pressure p 0 fields and the following three-dimensional disturbances δ u = δu e r + δv e θ + δw e z , δT and δp satisfying slip boundary conditions on the adiabatic end-walls at z = 0, A:

≤ r ≤ 1, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ A,
                   δu = ū(r, θ, t) cos (kz) δv = v(r, θ, t) cos (kz) δw = w(r, θ, t) sin (kz) δp = p(r, θ, t) cos (kz) δT = T (r, θ, t) cos (kz) (2)
where k = nπ/A is the dimensionless wavenumber and n is an integer. By substituting in the governing equations (1) the velocity, pressure and temperature fields by u 0 + δ u, p 0 + δp and T 0 + δT and by neglecting the second-order transport terms, the variables ū, v, w, p and T satisfy the time-dependent equations:

∂(f ū) ∂r + η ∂v ∂θ + f k w = 0 (3a) ∂(f ū) ∂t = - ∂(f ūu 0 ) ∂r - ∂(ηvu 0 ) ∂θ -f k wu 0 + ηvv 0 - ∂(f u 0 ū) ∂r - ∂(ηvū) ∂θ + ηv 0 v -f ∂ p ∂r + P r ∂ ∂r f ∂ ū ∂r + ∂ ∂θ η 2 f ∂ ū ∂θ -f k 2 ū - 2η 2 f ∂v ∂θ - η 2 ū f (3b) -RaP r f T cos θ ∂(f v) ∂t = - ∂(f ūv 0 ) ∂r - ∂(ηvv 0 ) ∂θ -f k wv 0 -ηūv 0 - ∂(f u 0 v) ∂r - ∂(ηv 0 v) ∂θ -ηu 0 v -f ∂ p ∂θ + P r ∂ ∂r f ∂v ∂r + ∂ ∂θ η 2 f ∂v ∂θ -f k 2 v + 2η 2 f ∂ ū ∂θ - η 2 v f (3c) +RaP r f T sin θ ∂(f w) ∂t = - ∂(f u 0 w) ∂r - ∂(ηv 0 w) ∂θ + f k p + P r ∂ ∂r f ∂ w ∂r + ∂ ∂θ η 2 f ∂ w ∂θ -f k 2 w (3d) ∂(f T ) ∂t = - ∂(f ūT 0 ) ∂r - ∂(ηvT 0 ) ∂θ -f k wT 0 - ∂(f u 0 T ) ∂r - ∂(ηv 0 T ) ∂θ + ∂ ∂r f ∂ T ∂r + ∂ ∂θ η 2 f ∂ T ∂θ -f k 2 T ( 3e
)
with η = R -1 and f = ηr + 1. The boundary conditions for the unknows ū, v, w and T are homogeneous.

Discretization and solvers

The equations were discretized on a structured and staggered grid by a finite volume method. All spatial derivatives were approximated by a second-order centered scheme. The mesh [r i , r i+1 ] × [θ j , θ j+1 ] is defined by

r i = tanh c r 2 i-1 Nr -1 + tanh(c r ) 2 tanh(c r ) , i = 1, N r + 1 θ j =        π exp 2πc θ j-1 N θ -1 exp (πc θ ) -1 , j = 1, N θ 2 + 1 2π -θ N θ +2-j , j = N θ 2 + 1, N θ + 1
where N r is any integer and N θ an even integer. To adjust the grid distribution at the upper part of the annulus (near θ = π) and in the wall regions, two strictly negative real parameters c r and c θ were introduced. If one of these two parameters was zero, the grid distribution was set uniform in the corresponding direction.

The computations of the 2D steady states ( u 0 , T 0 , p 0 ) and of the leading perturbations were already described in details for two-dimensional disturbances 3 . Briefly, the numerical procedure is as follows: first, the basic solution is obtained by solving the 2D equations

(1) at the steady state with the Newton algorithm; the dominant disturbances, solutions of equations (3a-3e), their growth rates λ and frequencies ν f are then evaluated by the Arnoldi method with a first-order time differencing.

For a given wavenumber, the critical parameter Ra c (k), which corresponds to a growth rate λ = 0, is computed iteratively by solving the implicit non-linear equation λ(Ra c ) = 0 with the Lagrange algorithm. This method simply consists in successive approximations of the threshold value by linear interpolations of the Rayleigh number as a function of the growth rates 8 .

Three-dimensional computations are also performed for finite length annuli with dimensionless length A = L/d and by applying no-slip boundary conditions at the adiabatic end-walls. These simulations were conducted for half-annuli with symmetry conditions in the azimuthal direction at θ = 0 and θ = π. The three-dimensional scheme was thoroughly described in a previous paper [START_REF] Petrone | Three-dimensional study of multiple transitions for natural convection in horizontal annuli[END_REF] .

B. Influence of the discretization parameters on Ra c (k)

The validity of the numerical results was checked in a previous paper 3 for 2D-disturbances (k = 0): the relative differences between our threshold values for the saddle-node bifurcation and the critical values published by Mizushima et al. [START_REF] Mizushima | Transitions of natural convection in a horizontal annulus[END_REF] are less than 0.25% for 1.2 ≤ R ≤ 3.

A good agreement is also shown with the Choi and Kim's results 17 for 3D-disturbances: in infinite length annuli and R < 2, the critical Rayleigh numbers and wavenumbers differ from less than 1.5 % and 1 % respectively. In order to better underline the agreements and explain the few observed disagreements between our results and the 3D-linear stability results of the archival literature, the detailed comparisons are deferred for the Results section III E.

To ensure the convergence of solutions, numerous checks were also conducted to measure the sensitivity of the growth rate λ and frequency ν f of the dominant perturbation as well as the dependency of the critical Rayleigh number as a function of the time step and mesh size. In the next paragraphs, the radius ratio, wavenumber and Rayleigh number values are chosen to be representative of each of the transitions discussed in the Results section.

Tests were performed for (R, k) = (1.2, 3) and 1700 ≤ Ra ≤ 1800, for (R, k) = (2.6, 5)

and 124, 500 ≤ Ra ≤ 129, 500, for (R, k) = (1.6, 4.5) and 10, 000 ≤ Ra ≤ 10, 250 and for (R, k) = (1.6, 3.2) and 19, 000 ≤ Ra ≤ 19, 500.

Temporal accuracy

The growth rates λ were evaluated by the Arnoldi method based on a first-order time differencing 3 of the linearized Navier-Stokes and energy equations (Eq. 3). Let ∆t the time step of the temporal scheme. For each set of R, Ra and k and for the 60 × 240 mesh, the points (∆t, λ) are basically aligned along a straight line. This result assess the order of the time approximation. The reference growth rate, λ ref , is thus defined as the extrapolated value corresponding to ∆t = 0.

Tables Ia-Ic illustrate the convergence of λ as a function of the time step for three sets of radius ratio, wavenumber and Rayleigh number. For ∆t = 5 or by using a linear interpolation of the reference growth rates between Ra = 19, 000 and Ra = 19, 500.

In order to preserve a sufficient accuracy in the critical parameter evaluation, the maximum value of the time step was ∆t = 5 • 10 -5 .

Spatial Accuracy

The effect of the grid size on the growth rates and threshold values are shown in Tabs.

IIIa-IIIc and IV. Taking the spatial second-order approximation of the Navier-Stokes and energy equations into account, the growth rates and frequencies were plotted as a function of 1/N 2 r or 1/N 2 θ , where N r and N θ are the number of meshes in the radial and azimuthal directions. As foreseen, the data fall into a straight line. Therefore, linear fits yield the reference values corresponding to N r and N θ → ∞.

Except for R = 1.6 and k = 4.5, tables III and IV show that the growth rates may strongly depend on the grid size. However, the interpolated thresholds are almost insensitive to N r and N θ . For example, all the critical Rayleigh number values calculated with a 60 × 240 meshes differ by less than 1% from Ra c ref . Thus, this mesh size was retained for most of the computations discussed in the present paper.

III. RESULTS

The stability results are presented for 1.2 ≤ R ≤ 3, Ra ≤ 200, 000 and P r = 0.7.

A. Basic steady flows

The basic steady flows are two-dimensional and were carefully examined in a previous paper [START_REF] Petrone | Stability of free convection in air-filled horizontal annuli: Influence of the radius ratio[END_REF] for Ra < 10, 000. These different flow patterns are now briefly depicted.

3(d)).

As mentioned just above, numerous basic flows may exist for a set of the Rayleigh number and radius ratio, and subsequently the 3D-stability analysis should be performed for all these 2D-stable solutions.

We first inquire into the stability of the C -C + flow pattern. As indicated in the chart (Fig. 2), this kind of flow is only 2D-stable for R < 2. The growth rates of the dominant perturbations of wavelength k = 1 are computed and are then copied out on figure 4(a) as a function of Ra and for R = 1.2,1.4, 1.6 and 1.8. Figure 4 Consequently, the only basic solutions used in the next sections consist of one large cell C + (Fig. 3(a)) eventually topped by two small eddies when R ≤ 1.25, with the mirror counterparts.

B. Neutral stability curves

For R = 1.2, the linear stability analysis is based upon three kinds of flow patterns: the basic solution evolves continuously from C + (Fig. 3 Notice that the discontinuities observed along each marginal curve are fictive: it is indeed reasonable to suppose that they are linked by "S"-shaped curves, unfortunately not easy to track with the numerical methods used. 6, whether the transitions are destabilizing or not, for odd (dashed lines) and even (continuous lines) integers n. If both odd and even multiples of the wavenumber π/A can develop in an annulus of axial aspect ratio A, the effective threshold for the 2D-steady flow destabilization is given by the lowest value of each transition. In Figure 6, the filled circles and squares located along the even and odd stability curves delimit the gap for the onset of the various marginal modes. For convenience's sakes, the notations used to refer to the marginal modes consist in placing side-by-side one or two characters followed by an integer.

The rules are:

• "S" or "O" marks out the temporal régime: steady or oscillatory;

• "s" is eventually added if the marginal mode satisfies the same spatial property as the basic flow, namely the mirror symmetry with respect to the vertical (r, z) mid-plane;

• the number differentiates the disturbances with identical temporal régime and spatial property.

In order to distinguish the various transitions, the thresholds are labeled as following:

• Ra c 1 is for the first transition, 1.2 ≤ R ≤ 1.95;
• Ra c 2 corresponds to the reversal transition, 1.95 ≥ R ≥ 1.5;

• Ra c 3 is for the second transition, 1.5 ≤ R ≤ 2.24;

• Ra c 4 is for the last instability, 2.3 ≤ R ≤ 3.

For R = 1.2 (Fig. 6(a)), let us compare first the even stability curve, namely the continuous line, with the results by Dyko et al. [START_REF] Dyko | A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus[END_REF] displayed by the dotted line. A quite good agreement is highlighted except close to A = 3 where the onset of a steady symmetrical disturbance labeled "Ss3" occurs below their marginal curve. In fact, a better agreement can be recovered by considering the second even dominant disturbance, namely the steady symmetrical mode "Ss2" not drawn on Fig. 6(a).

Therefore, it can be assumed that Dyko et al. [START_REF] Dyko | A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus[END_REF] have detected the mode "Ss2" but not the mode "Ss3". Notice that different non-linear flow patterns were identified by Dyko and Vafai 5,7 in narrow-gap annuli, using three-dimensional numerical simulations: five different supercritical states, including three states with even or odd numbers of transversal rolls in the upper part of the annulus, and two states with longitudinal rolls in the upper region in combination with transversal rolls located between the longitudinal rolls and the primary flow. The maximal values of the axial velocity are slightly shifted in the azimuthal direction from modes "Ss2" to "Ss1", and the mode "Ss3" is characterized by numerous but smaller structures. For 2 ≤ R ≤ 3, the bounded unstable regions vanish. The marginal curve is shown in Fig. 6(i) for R = 2.4. For 2.24 < R ≤ 3 (transition labeled as Ra c 4 ), the marginal mode "S1" (Fig. 7(d)) is steady but contrary to "Ss1", "Ss2" and "Ss3", it breaks the mirror symmetry of the basic flow. In accordance with the study of the leading perturbations, the second mode is oscillatory and corresponds to the mode "O1".

D. Three-dimensional simulations

Comparisons with linear stability results

Flow fields computed by solving the unsteady, three-dimensional governing equations (Eq. 1) with slip boundary conditions at the end-walls are discussed in this section.

The growth rates of the axial component of the velocity obtained either by 3D simulations or by the linear stability analysis are reported in Tab. V for R = 1.8. Computations were performed for k = π and A = 2 (n = 2) at Ra c 1 ± 5%, Ra c 2 ± 5% and 0.5

× (Ra c 1 + Ra c 2 ),
and they were initialized by the fluid at rest conditions when the growth rates of the linear stability analysis were negative and, otherwise by the steady three-dimensional flow obtained either at Ra c 1 -5% or at Ra c 2 +5%. The succession of 2D-stable, 2D-unstable and the reverse transition to 2D stable flows was confirmed. The relative difference between the growth rates λ evaluated by the two methods was systematically less than 5.5%. It is worth noting that the evaluation of λ through 3D-computations is much more expensive and, generally provides less accurate values owing to the small but finite amplitude perturbations.

The second destabilization is experienced for Ra = 22, 000 > Ra c 3 = 19, 962, R = 1.7

and A = 1.96 which corresponds to the critical wavenumber k = 2π/A ≈ 3.2 (n = 2). In accordance with the stability analysis, the flow is three-dimensional and oscillatory with a frequency ν f ≈ 7.96, a value very close to ν fc 3 = 8.03. Favorable qualitative comparisons of the axial components of velocity, obtained either by 3D-computations or by linear stability analysis at the transition (pattern "Os1") are shown in Fig. 8. Note that the numerical simulation was carried out for a Rayleigh number slightly greater than Ra c 3 because the growth rate at the transition threshold being negligibly small, the computation time should have been highly expensive.

Influence of the boundary conditions at the end walls

This section deals with the effects of no-slip boundary conditions at the end walls on the onset of thermal instabilities for moderate aspect ratio. The aim is to assess the relevance of the stability results provided by the perturbation analysis conducted for slip conditions at the end-walls. The first transition is particularly interesting because of the existence of disconnected instability regions for large enough radius ratios. Four computations, initialized by rest conditions, were thus performed, for R = 1.9 and various couples (Ra,

A).

The simulations B, D and E shown in Fig. 9 It evolves alternatively between two and three couples of counter-rotating cells.

The flow structures exhibit thermal instabilities at the annulus top region, except for case D. The two rolls due to the dynamical boundary conditions as well as the small annular length enforce the fluid flow to satisfy the symmetry with respect to the mid (r, θ)-plane.

Consequently, the disturbances with 5 cells cannot develop. The present simulations show therefore fully 3D-patterns for parameters lying in regions bounded by the continuous lines whereas the flow is essentially 2D outside of these regions. This finding may suggest that the stability of the 3D flows with no-slip conditions at the end-walls satisfy the stability requirements displayed in Fig. 9, but only for perturbations with an even number of cells along the axial direction. These results are consistent with the earlier findings of Dyko et al. [START_REF] Dyko | A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus[END_REF] for moderate gap annuli, specifically that due to the viscous shearing effects of the end-walls, only an even number of rolls forms in the annulus.

Two remarks are added to conclude this section. First, the oscillatory behavior of flow E was not predicted by the steady marginal disturbance "Ss2" at the transition. Secondly, the end-walls in finite length annuli do not generate finite amplitude perturbations able to systematically destabilize 2D-steady flows for parameter sets close to case D.

E. Stability diagram for infinite length annuli

The transition thresholds for infinite length annuli are now presented as a function of the radius ratio. The basic flows at the transitions only consist of the C + pattern (see Fig. For R < 2 and Ra < Ra c 1 , low speed velocities are seen at the top region of the annulus so that a conductive régime is nearly expected. Since the radius ratio is small, the cylinder surfaces are locally almost parallel and horizontal at the top of the annular space where a Rayleigh-Bénard like instability occurs. This instability is illustrated by the axial velocity of the marginal mode "Ss2" at the first transition, for R = 1.6 (Fig. 12(a)). This disturbance consists of transverse rolls with an axial wavenumber close to π (see Fig. 13). The onset of the transversal upper eddies depends not only on the local curvature of the walls but also on the Rayleigh number. At the annulus top region, the azimuthal velocity increases indeed with Ra and then stretches out the transversal convective perturbations towards the lateral region when the flow particles are close to the outer cylinder (Fig. 12(b)). This stretching effect produced by the basic flow probably explains the reason for the instability damping. On the other hand, the increase of the radius ratio deteriorates the conditions where the Rayleigh-Bénard like instability may occur by decreasing the angular sector where the wall cylinders may reasonably be considered as parallel and horizontal. Finally, just beyond R = 1.95, the curvature is so pronounced that the onset of the thermal instability can no more develop under the shape of two counter-rotating transversal eddies. The axial wavenumber of the disturbance for the reversal transition increases linearly with σ = 2/(R -1) from k c 2 = 3.60 at R = 1.9 to k c 2 = 5.09 at R = 1.5 (Fig. 13). Above Ra c 2 , the two-dimensional flow is anew stable.

The transition at Ra c 3 (Fig. 11) is due to the onset of the oscillatory mode "Os1" (see for instance Fig. 7(f)). The frequency (Fig. 14) is almost doubled from ν fc 3 = 7.09 at R = 1.5 to ν fc 3 = 13.06 at R = 2.24. The wavenumber k c 3 does not change monotonously with R but remains close to π (Fig. 13): the maximum value is reached for R ≈ 2 (k c 3 = 3.43).

The last bifurcation Ra c 4 (Fig. 11) is associated with the dissymmetrical steady perturbation "S1" (see for instance Fig. 7(d)). The wavenumber k c 4 = 5.25 for R ≈ 2.3 decreases slowly with R (k c 4 = 5.06 for R = 3).

The instability threshold Ra c 1 was previously discussed in the archival literature. It was firstly studied by Mojtabi and Caltagirone [START_REF] Mojtabi | Etude de la stabilité d'un écoulement de convection naturelle dans un espace annulaire horizontal[END_REF][START_REF] Mojtabi | Energy stability of a natural convective flow in a horizontal annular space[END_REF] and more recently by Choi and Kim 17 and Dyko et al. [START_REF] Dyko | A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus[END_REF] . Figure 15 shows comparison between Ra c 1 and the critical Rayleigh numbers reported in [START_REF] Mojtabi | Etude de la stabilité d'un écoulement de convection naturelle dans un espace annulaire horizontal[END_REF][START_REF] Mojtabi | Energy stability of a natural convective flow in a horizontal annular space[END_REF] by using linear and energetic stability analyses. Except for the smallest radius ratios, the present critical values differ substantially from those reported in [START_REF] Mojtabi | Etude de la stabilité d'un écoulement de convection naturelle dans un espace annulaire horizontal[END_REF][START_REF] Mojtabi | Energy stability of a natural convective flow in a horizontal annular space[END_REF] when R increases. The discrepancy is due to the assumptions introduced in their analytical model.

An approximation function is also displayed in Fig. 15. It reads

Ra 1 (R) = 1072 R -2398 0.7069 R -1.485 (4)
and fits Ra c 1 with a relative error not exceeding 1.4% for 1.2 ≤ R < 2. With this approximated function, the Rayleigh-Bénard threshold corresponding to horizontal differentially heated plates (R = 1) is recovered within 0.25%, only. calculated by using a spectral method. The relative difference does not exceed 1.5% and 1%, respectively. For R = 2, Choi and Kim 17 did not obtain any critical threshold in accordance with the present work showing that the first transition disappears just before R = 2. According to [START_REF] Choi | Three-dimensional linear stability of natural convective flow between concentric horizontal cylinders[END_REF] , the disturbances are oscillatory and are damped for all Ra-values studied (Ra ≤ 100, 000). In particular, the growth rate and the frequency were found equal to λ = -1.055 and ν f = 60.9/(2π) = 9.69 for Ra = 25, 000 and k = 3.3. This last result does not agree with our findings. We obtained indeed another transition at Ra c 3 = 24, 621 (Fig. 11) with k c 3 = 3.43 (Fig. 13). To investigate the reason of such a difference, the same computation as in Choi and Kim 17 was carried out. Although the frequency ν f = 9.64 of our computed leading disturbance is very close to that found by Choi and Kim 17 , the growth rate is positive λ = 8.9 10 -2 and thus very different from the growth rate reported in their paper (λ = -1.055).

Simple analytical functions were sought to define relations between the critical Rayleigh numbers and the radius ratios. The critical Rayleigh numbers Ra c i (R) are approximated by the function Ra i (R), for i = 1 to 4:

• for 1.2 ≤ R ≤ 1.95, Ra 1 (R) = (1072 R -2398)/(0.7069 R -1.485) with |Ra c 1 - Ra 1 |/Ra c 1 < 1.4%; • for 1.95 ≥ R ≥ 1.5, Ra 2 (R) = 10, 440/(R -1) -7300 with |Ra c 2 -Ra 2 |/Ra c 2 < 2%;
• for 1.5 ≤ R ≤ 2.24, Ra 3 (R) = 52, 000 -37, 400 × (2.24 -R) 0.244 with |Ra c 3 -

Ra 3 |/Ra c 3 < 4%; • for 2.3 ≤ R ≤ 3, Ra 4 (R) = 59, 100 R -27, 300 with |Ra c 4 -Ra 4 |/Ra c 4 < 1.4%.

F. Comparisons with numerical and experimental studies

Many experimental results but few three-dimensional numerical solutions were reported in the literature. The most quoted results are drawn in the present stability chart (Fig. 11). The filled and open symbols correspond to three-and two-dimensional flows, respectively, while the half-filled circles are for parameter values for which both 2Dand 3D-solutions were observed. The stability chart by Powe et al. [START_REF] Powe | Free convective flow patterns in cylindrical annuli[END_REF] is also drawn in Fig. 11.

The present linear stability analysis is in excellent agreement with the Choi and Kim's results 17 for 1.2 ≤ R < 2 and also with the works by Mojtabi and Caltagirone [START_REF] Mojtabi | Etude de la stabilité d'un écoulement de convection naturelle dans un espace annulaire horizontal[END_REF][START_REF] Mojtabi | Energy stability of a natural convective flow in a horizontal annular space[END_REF] and by Dyko et al. [START_REF] Dyko | A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus[END_REF] for narrow-gap annuli. Beyond the critical value Ra c 1 , the transverse eddies at the top of the annular space grow and the solution converges toward a non-linear state consisting of a spiral flow as observed in many numerical and experimental studies [START_REF] Dyko | A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus[END_REF][START_REF] Dyko | Three-dimensional natural convective states in a narrow-gap horizontal annulus[END_REF][START_REF] Petrone | Three-dimensional study of multiple transitions for natural convection in horizontal annuli[END_REF]11 .

Although the Ra c 1 -curve is in agreement with the Powe's chart for moderate radius ratios (Fig. 11). According to Powe et al. [START_REF] Powe | Free convective flow patterns in cylindrical annuli[END_REF] , multi-cellular flows take place above the transition.

Although our linear stability analysis indicates that the 2D multi-cellular flows are unstable, Dyko and Vafai 5 showed longitudinal rolls in the top portion of the annuli for R ≤ 1.15, but in combination with transverse rolls between the longitudinal rolls and the main flow.

As a matter of fact, the curves in the Powe's chart delimit non-linear states and then the comparison with the linear stability thresholds can only be qualitative. Both sub-critical bifurcations and the possible existence of multiple non-linear flows for the same set of parameters may explain such discrepancies.

For moderate radius ratios (1.5 < R < 2), the successive transitions reported in the present study, and more specifically the two-dimensional stable region for large Rayleigh number values (Ra c 2 < Ra < Ra c 3 ), are not mentioned in the chart by Powe et al. [START_REF] Powe | Free convective flow patterns in cylindrical annuli[END_REF] , except the first transition at Ra c 1 . However, the experimental works by Grigull and Hauf 9 reported that three-or two-dimensional flows can exist in the ranges 1.3 ≤ R ≤ 6.3 and 1680 ≤ Ra ≤ 21, 000. Recent three-dimensional numerical simulations for R = 1.7 and Ra = 10, 000 8

(half-filled circle in Fig. 11) also showed flows either essentially two-dimensional or fully three-dimensional as a function of the initial conditions.

At a first sight, the onset of the second instability at Ra c 3 seems to be in disagreement with the experimental results by Grigull and Hauf 9 who reported a reversal transition to two-dimensional flows occuring above Ra = 21, 000, for 1.3 ≤ R ≤ 6.3. This noticeable contradiction can be raised by considering the three-dimensional numerical simulations performed by Petrone et al. [START_REF] Petrone | Three-dimensional study of multiple transitions for natural convection in horizontal annuli[END_REF] for R = 1.7 and Ra = 22, 000 > Ra c 3 (half-filled circle in Fig. 11). Indeed, they proved that the choice of a too small aspect ratio (A = 6) damps the instability onset and therefore the flow remains steady and essentially two-dimensional in the core flow region. Although for a larger aspect ratio (A = 20), the flow is oscillatory with a frequency πν f = 25 and a characteristic wavelength k = πn/A = 3.45, both in good agreement with the frequency πν fc 3 = 25.23 (Fig. 14) and the wavelength k c 3 = 3.2 (Fig. 13) at the transition, the axial component of the velocity is very small. Thus, either a too small aspect ratio or the existence of a small axial velocity, probably hard to be experimentally observed or measured, may explain the results by Grigull and Hauf 9 , namely the assumed two-dimensionality of the flows.

The last transition Ra c 4 is correctly represented by the marginal curve proposed by Powe et al. [START_REF] Powe | Free convective flow patterns in cylindrical annuli[END_REF] in their stability chart. Despite this qualitative agreement, the transitions differs:

2D-oscillatory flows versus 3D-steady disturbances. However, in both cases, perturbations and experimental flows break the mirror symmetry of the counter-rotating basic flow cells.

The resulting flow consists in an oscillatory thermal plume in each (r, θ)-plane as recently confirmed in the experiments by Labonia et al. 13 and in three-dimensional numerical simulation by Petrone et al. [START_REF] Petrone | Etude numérique des écoulements de convection naturelle et mixte dans un espace annulaire horizontal : approche par analyse de stabilité linéaire[END_REF] (filled squares above Ra c 4 in Fig. 13). It can be noticed that the second leading mode at the transition Ra c 4 is oscillatory, breaks the mirror symmetry of the basic flow and corresponds to the mode labeled "O1". The superimposition of this disturbance and of the basic flow permits to recover the oscillation of the thermal plume.

The bifurcation at Ra c 4 is therefore probably subcritical.

IV. CONCLUSION

Three-dimensional linear stability analyses of two-dimensional natural convection flows in air-filled horizontal annuli were performed for radius ratios in the range 1.2 ≤ R ≤ 3. New transitions were highlighted for moderate and large radius ratios. It was emphasized that 2D-flows may be stable for 1.5 ≤ R < 2, i.e. above the first instability threshold reported in the literature. The basic solution is therefore destabilized by an oscillatory disturbance for a Rayleigh number value one order of magnitude higher than the first transition. For larger radius ratios (R ≥ 2.3), the crescent-shaped flow is stable for large Ra-values, up to 10 5 . The flow transitions are thus through disturbances breaking the symmetry of the base flow. Such transitions were evidenced for infinite and finite length annuli with slip boundary conditions at the end-walls. The effect of no-slip conditions on the first instability region was also investigated by three-dimensional simulations. The wavelengths of disturbances corresponding to odd submultiples of the annular length did not develop in small length annuli.

Finally, the disturbances and critical Rayleigh numbers were compared and discussed with experimental results and three-dimensional numerical simulations reported in the literature.

To conclude, natural convection flows in horizontal annuli can be classified into three (a) R = 1.2 and k = 3

λ |λ -λ ref | |λ ref | Ra c ≈ Ra (3) Ra c -Ra c ref Ra c ref λ N r × N θ Ra (1)
= 1700 Ra [START_REF] Mizushima | Transitions of natural convection in a horizontal annulus[END_REF] = 1800 Ra [START_REF] Petrone | Stability of free convection in air-filled horizontal annuli: Influence of the radius ratio[END_REF] 30 × 120 -1.734

N r → ∞ λ → λ ref Ra c → Ra c ref N θ → ∞ -2.374 • 10 -1 3.917 • 10 -1 1737.74 (b) R = 2.6 and k = 5 λ |λ -λ ref | |λ ref | Ra c ≈ Ra (3) Ra c -Ra c ref Ra c ref λ N r × N θ Ra (1) =
-3 N r → ∞ λ → λ ref Ra c → Ra c ref N θ → ∞ 2.320 • 10 -1 -2.140 • 10 -1 10, 130.20
Table III: Influence of the meshes on the growth rate and Ra c .

λ + i2πν f |λ -λ ref | |λ ref | + i ν f -ν f ref ν f ref Ra c ≈ Ra (3) Ra c -Ra c ref Ra c ref λ N r × N θ Ra (
N r → ∞ λ + i2πν f → λ ref + i2πν f ref Ra c → Ra c ref N θ → ∞ -2.

Fig. 1 .

 1 Lengths and velocity are scaled by the annulus gap d = r * o -r * i and the thermal diffusivity velocity α/d, respectively, where α is the thermal diffusivity. The dimensionless temperature is defined by T

  where r = (r * -r * i )/d and A = L/d is the axial aspect ratio. No-slip boundary conditions are imposed on the heated inner cylinder (T i = 0.5) and cooled outer cylinder (T o = -0.5).

Figure 2

 2 Figure 2 sums up the different stable flow patterns occurring for Ra < 10, 000 and 1.2 ≤ R ≤ 3 with respect to 2D-disturbances. Dual 1-3 and triple 3 stable solutions are met in the parameter space (Ra, R). They consist in the common large cell (labeled C + , Fig. 3(a)), and possibly one or two superimposed small eddies (flows referenced by C -C + , Fig. 3(b), and C + C -C + , Fig. 3(c)) with their mirror counterparts. It is worth pointing out that the C -C + flow is destabilized by the onset of a disturbance that breaks the mirror symmetry of the basic solution 3 . Therefore, it is essential to simulate the two-dimensional flows in full annular spaces to observe this instability, without using any symmetrical assumption. The last mentioned flow pattern c + C + is intermediate between C + and C + C -C + and exists in a narrow (Ra, R)-domain, for small gap annuli (1.2 ≤ R ≤ 1.35) and Rayleigh numbers (2000 < Ra <≈ 5500). It consists of a co-rotating eddy at the top of the large cell (Fig.

  (a) also indicates the stability thresholds obtained for two-dimensional disturbances by the Rayleigh numbers corresponding to the right edges of the rectangular hachured boxes. Although the flows with the C -C + pattern are stable for two-dimensional disturbances (k = 0), it exists at least one three-dimensional perturbation with a positive growth rate, for example the perturbation with a wavelength k = 1. Thus, the flows with one large cell topped by one eddy, with the mirror images, are obviously always unstable. Before studying the linear stability of the C + C -C + flow pattern, it is necessary to remind us of the origin of this structure (details are presented in 3 ). For narrow gap annuli, R ≤ 1.25, the multicellular flows are a result of a continuous transformation from C + to c + C + and then to C + C -C + patterns with the increase of the Rayleigh number value. Just before R = 1.26, the stability diagram for the two-dimensional disturbances is highly modified by the appearance of a new branch of stable solutions on which the C + C -C + flow pattern now takes place: the continuous change from C + to C + C -C + as a function of Ra is now impossible, on the other hand multiple solutions (C + and C + C -C + ) may now be expected. Figure 4(b) presents the growth rates of the dominant perturbation for the wavelength k = 1 as well as the stability regions for 2D disturbances. For R ≥ 1.3, the C + C -C + flows are clearly always unstable.

  (a)) to c + C + (Fig. 3(d)) slightly above Ra = 1917 and to C + C -C + (Fig. 3(c)) beyond Ra ≈ 2270. For R ≥ 1.3, only the common couple of counter-rotating cells composes the basic flows about which the perturbation analysis is performed. For 1.2 ≤ R ≤ 1.4, the two-dimensional steady flow is destabilized when Ra > Ra c (k) as illustrated in Fig. 5(a) for R = 1.2 and R = 1.3. When the wavelength of the threedimensional disturbances tends towards zero, the stability thresholds for 2D-disturbances 3 are recovered. Indeed for k = 0, the pitchfork bifurcations at Ra = 2068 and Ra = 2456 for R = 1.2, and then the Hopf bifurcation at Ra = 6008 for R = 1.3, are well found.

For 1 . 5 ≤

 15 R ≤ 1.95 (Fig. 5(b)), the previous neutral stability curves are fold back to give rise to bounded regions of instability. Above the closed curves, the two-dimensional flows are stable again until the next transition, for which the critical Rayleigh numbers are one order of magnitude higher than those of the first bifurcation. The area of the bounded instability region decreases with R to finally collapse for 1.95 < R < 2. Only the upper marginal curve persists for larger radius ratios. For 2 ≤ R ≤ 3, Ra c (k) increases with R and reaches 151, 330 at R = 3. C. Stability diagrams for finite length annuli Consider now horizontal annuli of finite axial length A = L/d, with slip conditions at the end-walls. In such configurations, the basic flow is still two-dimensional. Its stability property is deduced from the neutral curves Ra c (k) in Figs. 5(a) and 5(b) by extracting the critical Rayleigh number values corresponding to the discrete wavenumbers k = nπ/A, n being an integer. The curves min n (Ra(k = nπ/A)) or max n (Ra(k = nπ/A)) are plotted in Fig.

From R = 1 .2 to 1 . 4 (

 114 Figs. 6(a)-6(c)), the shapes of the neutral stability curves are qualitatively similar. The three marginal modes are steady and symmetric. The mode "Ss1" is obtained for very small A-values only, and it collapses when increasing R. The mode "Ss2" is the most often found and it remains almost alone for R = 1.4. The mode "Ss3" is obtained within small ranges of A. By considering simultaneously the odd and even multiples of the elementary wavenumber π/A, "Ss2" is the mode which generally drives the onset of the instability.The three patterns of the marginal disturbances "Ss1", "Ss2" and "Ss3" are characterized by high values of the axial component of the velocity ( w) which are located at annulus top region (Figs.7(a)-7(c)). The patterns of the mode "Ss2" (Fig.7(b)) are in qualitative good agreement with those presented in the linear stability study performed by Choi and Kim 17 .

For 1 . 5 ≤

 15 R ≤ 1.95, a bounded instability region (Ra c 1 < Ra < Ra c 2 ) clearly appears, firstly in only one domain (R = 1.5, Fig. 6(d)), and then split into several independent sub-domains (1.8 ≤ R ≤ 1.95). For intermediate radius ratios, 1.6 ≤ R ≤ 1.7 (Figs. 6(e)and 6(f)), the sub-domains are delimited by odd neutral curves. The marginal modes of the first instability region are mainly governed by the steady symmetrical disturbance "Ss2", except for R = 1.5 (Fig.6(d)) where the oscillatory perturbation "O1", breaking the mirror symmetry of the basic flow, occurs. The disturbance "O1" (Fig.7(e)) is also evidenced for the second instability region (the upper curves at Ra c 3 ) with the symmetrical oscillatory mode labeled "Os1" (Fig.7(f)). These two unsteady perturbations have very similar growth rates and alternatively emerge as the leading marginal mode, depending on both R and A.

  (enlargement of Fig. 6(h)) were carried out for Ra = 4000 and A = 3.5, A = 4.4 and A = 4.9. The simulation C was also performed for A = 4.4 but for Ra = 3000. The flow fields are plotted in Fig. 10 through velocity fields and isothermal patterns in the vertical upper (r, z)-plane. A strong thermal convection motion, produced by two couples of counter-rotating rolls in the top region of the annulus, is seen for the steady-flow cases B and C. In contrast, the isotherms exhibit a weak 3D-structure in the case A = 4.4 and Ra = 4000 (flow D) for which the solution can be assumed twodimensional far enough from the end-walls. Notice that this solution was also achieved by starting the computation from the 3D-flow pattern of case C. The boundary rolls are due to the viscous shear layers associated with the no-slip boundary conditions but not to a thermal instability. The E-flow structure is oscillatory with a dimensionless period τ = 30.

  3(a)) with its mirror counterpart. For a fixed R-value, the critical parameter Ra c (R) is analytically calculated by using a quadratic polynomial approximation of Ra c (k) in the vicinity of the absolute extrema such that Ra c (R) = min k (Ra c (k)) or Ra c (R) = max k (Ra c (k)) for thresholds corresponding to direct transitions or to reverse transitions to stable flows. The critical Rayleigh number values Ra c (R) are plotted in Fig. 11 as a function of σ = 2r * i /d = 2/(R-1) (bottom abscissa axis) or as a function of R (top abscissa axis). It should be noted that parameter σ was widely used in previous works instead of R. The ranges of the radius ratio and Rayleigh number values where the four transitions occur are as follows: • for 1.2 ≤ R ≤ 1.95 and 1734 ≤ Ra c 1 ≤ 2894; • for 1.95 ≥ R ≥ 1.5 and 3643 ≤ Ra c 2 ≤ 13, 540; • for 1.5 ≤ R ≤ 2.24 and 17, 130 ≤ Ra c 3 ≤ 52, 000; • for 2.3 ≤ R ≤ 3 and 110, 040 ≤ Ra c 4 ≤ 151, 330. Notice that a curve similar to the Ra c 1 -curve shown in Fig. 11 was already obtained by Yoo 1 . It indicated the Rayleigh number beyond which the C -C + steady solutions occur, that is the 2D-transition threshold corresponding to a saddle-node bifurcation as demonstrated by Mizushima et al. 2 and later by Petrone et al. 3 . Let also recall that these C -C + steady solutions are always unstable with respect to three-dimensional disturbances (see Sec. III A). Thus, although these Rayleigh number values are close to Ra c 1 , they characterize in no way the onset of a disturbance for the usual couple of counter-rotating cells.

( 1 .

 1 23 ≤ R ≤ 1.71), discrepancies appear for smaller-gap annuli (R ≤ 1.23) where the threshold values are from twice to almost four times larger compared with our results

  categories according to the value of the radius ratio. Transverse, steady roll perturbations are amplified at Ra c ≈ 2000 in narrow-gap annuli with R < 1.5. Moderate-gap annuli are defined by radius ratios at which multiple transitions occur for the two-dimensional flows, namely for 1.5 < R < 2. Large-gap annuli, R > 2, are associated with onset of oscillatory disturbances for 2 < R < 2.24 and Ra = O(10 4 ), and with steady, symmetrical breaking disturbances for R > 2.24 and Ra = O(10 5 ).

(a) R = 1 . 2 ,R = 1 . 6 ,λ-λ ref λ ref 7 . 5 •

 121675 Ra = 2000 and k = Ra = 10, 000 and k = 4.5 ∆t λ 10 -5

Figure 1 :Figure 2 :

 12 Figure 1: Giuseppe Petrone, Physics of Fluids

Figure 3 :Figure 6 :

 36 Figure 3: Giuseppe Petrone, Physics of Fluids

Figure 7 :Figure 8 :Figure 10 :Figure 11 :Figure 13 :Figure 14 :

 7810111314 Figure 7: Giuseppe Petrone, Physics of Fluids

Figure 15 :

 15 Figure 15: Giuseppe Petrone, Physics of Fluids

  For the smallest time step (∆t = 5 • 10 -6 ), the relative difference between λ and λ ref is still larger than 5% whereas the relative discrepancy in the frequency ν f never exceeds 0.2%. Despite these unsatisfactory results, the critical Rayleigh number, Ra c , predicted with only one step of the Lagrange method, namely a linear interpolation, provides an accurate guess of the reference threshold Ra c ref . It should be noted that Ra c ref is almost identical by computing it with a linear approximation of the (Ra c , ∆t)-data for ∆t → 0

	affected by ∆t.
	• 10 -5 , the relative discrepancies
	with λ ref are systematically less than 0.5%. For R = 1.6, k = 3.2, Ra = 19, 000 and
	Ra = 19, 500 the dominant perturbation is oscillatory with a frequency ν f . As shown in
	Tab. II, the growth rates are not properly evaluated whereas the frequencies are weakly

  Table VI shows the excellent agreement of Ra c 1 and k c 1 between the present results and those by Choi and Kim 17

Table I :

 I Influence of the time step on the growth rate. Ra c -Ra c ref + i2πν f → λ ref + i2πν f ref Ra c → Ra c ref -7.540 • 10 -2 + 47.75i 2.000 • 10 -1 + 48.32i 19, 136.88

		λ + i2πν f		Ra c
		|λ -λ ref | |λ ref |	+ i	ν f -ν f ref ν f ref		Ra c ref
	∆t	Ra = 19, 000			Ra = 19, 0500
	7.5 • 10 -5 5.516 • 10 -3 + 47.68i (107.32% + 0.16%i)	2.810 • 10 -1 + 48.25i (40.48% + 0.16%i)	18, 989.99 (0.768%)
	5 • 10 -5 -2.110 • 10 -2 + 47.70i (72.02% + 0.10%i)	2.539 • 10 -1 + 48.27i (26.93% + 0.10%i)	19, 038.36 (0.515%)
	2.309 • 10 -1 5 • 10 -5 2.310 • 10 -1 2.5 • 10 -5 2.311 • 10 -1 1.5 • 10 -5 2.312 • 10 -1 10 -5 2.313 • 10 -1 2.5 • 10 -5 -4.777 • 10 -2 + 47.74i 0.17% 0.12% 0.07% 0.03% 0.02% (36.65% + 0.05%i) 1.5 • 10 -5 -5.849 • 10 -2 + 47.74i (22.43% + 0.03%i) -6.415 • 10 -2 + 47.74i 10 -5 (14.92% + 0.02%i) 5 • 10 -6 -7.136 • 10 -2 + 47.75i (5.36% + 0.01%i)	(c) R = 2.6, Ra = 124, 500 and k = 5 ∆t λ-λ ref 2.267 • 10 -1 + 48.31i 19, 087.01 (13.36% + 0.05%i) (0.261%) λ λ ref 7.5 • 10 -5 -9.520 • 10 -2 0.40% 2.157 • 10 -1 + 48.31i 19, 106.66 (7.84% + 0.03%i) (0.158%) 5 • 10 -5 -9.532 • 10 -2 0.27% 2.5 • 10 -5 -9.545 • 10 -2 0.13% 2.113 • 10 -1 + 48.31i 19, 116.46 (5.62% + 0.02%i) (0.107%) 1.5 • 10 -5 -9.550 • 10 -2 0.08% 10 -5 -9.553 • 10 -2 0.05% 2.044 • 10 -1 + 48.32i 19, 129.38 (2.19% + 0.01%i) (0.039%)
	∆t → 0 ∆t → 0	λ → λ ref 2.313 • 10 -1 λ	0%	∆t → 0	λ → λ ref -9.558 • 10 -2	0%

Table II :

 II Influence of the time step on the growth rate, frequency and Ra c for R = 1.6 and k = 3.2.

Table IV :

 IV 160 • 10 -2 + 47.75i 2.540 • 10 -1 + 48.32i 19, 039.16 Influence of the meshes on the growth rate, frequency and Ra c for R = 1.6 and k = 3.2.

		λ
	Ra linear stability 3D-simulation relative gap
	2099.5 -3.5497 • 10 -1 -3.5578 • 10 -1	0.23%
	2320.5 3.0947 • 10 -1 3.0930 • 10 -1	0.05%
	3700	1.5894	1.6071	1.10%
	4930.5 4.0314 • 10 -1 4.1827 • 10 -1	3.62%
	5449.5 -4.3943 • 10 -1 -4.6380 • 10 -1	5.25%

Table V :

 V Comparisons of the growth rate, λ, achieved by stability analyses and by three-dimensional simulations for R = 1.8.

	R	Ra c 1	Ra ref c 1	k c 1	k ref c 1
		Rac 1 -Ra ref c 1		kc 1 -k ref c 1
		Ra ref c 1		k ref c 1
	1.2	1734.6	1714.1	3.04	3.05
		1.2%		0.15%
	1.3	1764.9	1741.1	3.04	3.04
		1.4%		0.06%
	1.5	1861.5	1842.2	3.04	3.03
		1.1%		0.47%
	1.8	2207.6	2178.3	3.13	3.10
		1.4%		0.87%
	1.9	2514.7	2514.5	3.20	3.19
		0.2%		0.35%

Table VI :

 VI Comparisons of Ra c 1 and k c 1 with values reported by Choi and Kim 17 (Ra ref c 1 , k ref c 1 ).

Acknowledgments

This work was supported in part by the french computational center (Institute for Development and Resources in Intensive Scientific computing) through Grant N o 051265.

List of figures 1. Geometry of the annular space.

2. Chart 18 of the two-dimensional flow patterns in the (R, Ra) plane.

Neutral stability curves as a function of

Marginal modes are noted: "O" for oscillatory and "S" for steady régimes, "s" for patterns satisfying the same spatial property as the basic flow (the mirror symmetry) and the integer distinguishes perturbations with the identical temporal régime and spatial property.


Angy Geerts

Nathalie Veg-Sala

EVIDENCE ON INTERNET COMMUNICATION MANAGEMENT STRATEGIES FOR LUXURY BRANDS

Keywords: Brand management, Internet, Communication, Luxury, Semiotic analyses

Luxury brands and the Internet communication media seem to be inconsistent at first sight. On the one hand, luxury brands have to maintain the control of their elitism and their image; on the other hand, Internet is defined as a mass media of communication that is basically not selective. However, some years ago, all the luxury brands started to set up their own websites and some of them have gone as far as opening online shops. Then, regarding these paradoxes and a literature review, the purpose of this article is to analyze how luxury brands can manage scarcity and the use of Internet.

To answer this question, a structural semiotic methodology on luxury brands and their communication is used. The purpose is to bring about a better understanding of how luxury brands use Internet in their communication strategy and how this media may or may not fit into a luxury brand's management of the values of rarity. The methodology analyses factors that can impact the success or the failure of Internet communication for luxury brands.

The results show that Internet is not entirely incompatible with luxury. It all depends on how it is used. Internet can strengthen the core values of brands thanks to original and interactive applications. But sometimes if not enough attention is paid, Internet may dilute the brand values. So many precautions have to be taken to avoid the dilution of brand narratives and some managerial implications are developed in this article, especially concerning e-shopping.

INTRODUCTION

Brands are cultural imagery whose consumption is mainly based on desire and emotions [START_REF] Belk | The fire of desire: A multisided inquiry into consumer passion[END_REF]. They are narratives and values shared by groups of consumers [START_REF] Levy | Interpreting consumer mythology: a structural approach to consumer behavior[END_REF][START_REF] Stern | Consumer myths: Frye's taxonomy and the structural analysis of consumption text[END_REF][START_REF] Thompson | Consuming desire and desirous consumption: toward a deeper understanding of the social construction of consumer wants and the nature of consumption symbolism[END_REF]. They represent the entrance into legends or myths. Their implementation is made through all kind of supports that help to give meaning, that is to say, through all their communication tools. Brand communication can be carried out through multiple channels and is not limited to advertising (Hasting, 1990). Particularly, it seems important to consider new media to follow socio-cultural changes. Brands have to stay in step with their times to avoid becoming outmoded. While evaluation of communication techniques has to be taken into account in both cases, the thinking, the strategies and the purposes are not the same for brands in the mass market and brands in the luxury market. Mass market is synonymous with profitability and effective values. It focuses on a large number of products and low prices. For brands in this mass market, the purpose of communication is visibility. Accordingly, the Internet appears to be the media that consumers use the most. Conversely, the luxury market has to show an image of selectivity and rarity. The challenge of these brands is to appear to send a message accessible only to a very small part of the population [START_REF] Vigneron | Measuring perception of brand luxury[END_REF]. The choice of the communication tools is very important and has to fit with that. Along those lines, the Internet, defined as a mass media, does not seem to be consistent with the luxury universe. However, for some years, all the luxury brands have set up their own website and some of them have gone as far as opening online shops.

Previous research has mainly set the focus on the one hand, on the incompatibility between luxury and Internet and on the other hand on the opportunity this new media brings up [START_REF] Roux | Gérer l'ingérable : la contradiction interne de toute maison de luxe[END_REF][START_REF] Dall | Luxury branding on the Internet: lost opportunity or impossibility?[END_REF][START_REF] Nyeck | Luxury brands online and offline: the case of French brands[END_REF]. However few have brought a better understanding over the way to optimize Internet communication for luxury brands and how to minimize the incompatibility.

Then, faced with these paradoxes, the purpose of this article is to analyze how luxury brands can manage scarcity and the use of Internet. Particularly, we want to answer the following questions: (1) how do luxury brands develop their values on the websites? (2) Are these value in line with those developed on traditional communication support? ( 3) And what are the factors that may reinforce the value of these brands on the Internet in order to keep the consistency of the brand? To answer these questions, this paper is structured around five sections. The first part is about the literature review. It highlights the concept of luxury and its paradoxes. Research focusing on links between communication, Internet and luxury are explained. This literature review leads to research propositions. In the second part, the research methodology is developed. A structural semiotic analysis is used to understand how luxury brands use Internet in their communication strategy and how this media may or may not fit into a luxury brand's management of the values of selectivity and rarity. The third part of this paper presents the results of tested brands. Then, we discuss the main contributions of this article, and in the last part of concluding comments the managerial implications, the research limits and further research are developed.

LITERATURE REVIEW

The definition of luxury is very subjective and includes various connotations. However, it can be broadly defined today as a way of being, a way of life but also a way to buy, consume and use, rather than know-how or a way to do something [START_REF] Roux | Gérer l'ingérable : la contradiction interne de toute maison de luxe[END_REF][START_REF] Nyeck | Luxury brands online and offline: the case of French brands[END_REF]. In luxury, intangible values are very important [START_REF] Bechtold | Le paradoxe du luxe[END_REF] To be more precise, several types of luxury have been defined. Lipovetsky identifies two broad categories: eternal luxury and emotional luxury [START_REF] Lipovetsky | Le luxe éternel: de l'âge du sacré au temps des marques[END_REF]. While traditional luxury includes values such as superfluousness, knowhow, and the desire to make a certain impression, emotional luxury emphasises values of personal pleasure, feelings and affective and experiential features. This distinction is close to those established between interpersonal luxury and personal luxury [START_REF] Vigneron | Measuring perception of brand luxury[END_REF]. Interpersonal luxury includes strong social motivations.

Personal luxury appeals to the values of pleasure seeking and emotion [START_REF] Dubois | Observations: understanding the world of international luxury brands: the "dream formula[END_REF]. Luxury is therefore plural and addresses several aspects, primarily related to personal values and interpersonal values.

Luxury brands, as other type of brand needs to be managed with consistency [START_REF] Kapferer | The new strategic brand management[END_REF]. It is required the brand to broadcast the same message through the different media [START_REF] Aaker | Building strong brands[END_REF][START_REF] Keller | Managing brand for the long run: brand reinforcement and revitalization strategies[END_REF][START_REF] Kapferer | The new strategic brand management[END_REF]. The continuity of its value is essential for the brand development and the brand strength [START_REF] Semprini | Analyser la communication : comment analyser les images, les médias, la publicité, Editions L'Harmattan, collection Champs Visuels[END_REF][START_REF] Keller | Strategic brand management: building, measuring, and managing brand equity[END_REF][START_REF] Kapferer | The new strategic brand management[END_REF]. Consistency is at the heart of luxury brand management as well as selectivity, legitimacy, creation and perfect quality [START_REF] Roux | Comment se positionnent les marques de luxe[END_REF] Other special features at the heart of luxury brand management, including those related to the inherent paradoxes, are to be analysed and highlighted. The management of luxury brands has contradictions that other brands do not [START_REF] Roux | Gérer l'ingérable : la contradiction interne de toute maison de luxe[END_REF][START_REF] Nuemo | The mass marketing of luxury[END_REF]. These include the paradoxes related to brand management and the various elements of the marketing mix, which do not fit with the basic principles of profitability. For example, in terms of product distribution, their goal is not to be the more accessible but to control distribution and choose their locations for their symbolism. The communication of luxury brands is perhaps the most important and most difficult paradox to manage. Indeed, like all brands, those in the luxury sector have to develop awareness in the buying public. They need to communicate to show they exist, that they are present and strong in the competitive world. However, at the same time, luxury brands try to protect their selectivity and remain discreet [START_REF] Bechtold | Le paradoxe du luxe[END_REF]. Thus, in particular through their communication policy, luxury brands need to combine selectivity and diffusion, rarity and enlargement, elitism and increased awareness.

According to this perspective and this observation, in the following section we want to analyse the main characteristics of this new channel; consider its place in the luxury sector; understand how this mass media may be part of a strategy for luxury brands. Internet has become an essential part of branding strategies and communication for companies whose products range from consumer goods to luxury goods [START_REF] Dall | Luxury branding on the Internet: lost opportunity or impossibility?[END_REF]. Parallel to the increasing number of brands with their own website, there has been an increased interest in the literature regarding the use of the Internet in brand management. This interest has led many authors to develop their interest in brand management online [START_REF] Chen | Assessing the impact of the Internet on Brands[END_REF][START_REF] Chernatony | Succeeding with Brands on the Internet[END_REF][START_REF] Clauser | Offline rules, Online Tools[END_REF]. They agree that the basic rules of branding and the essence of the brand itself are the same online and offline [START_REF] Dall | Luxury branding on the Internet: lost opportunity or impossibility?[END_REF]. According to [START_REF] Nyeck | Luxury brands online and offline: the case of French brands[END_REF], the Internet is a complementary tool in communication strategies and its triple objective is to inform, remind and persuade. The web can help to establish links between consumers and brands. In contrast to other media, the Internet allows a dual interaction with and through the tool itself [START_REF] Nyeck | Luxury brands online and offline: the case of French brands[END_REF]. The web creates a central positioning among other communication tools, with, on the one hand, interpersonal and personal aspects and on the other hand, a communication mode that is somewhere between static and dynamic. Internet can have different impacts on brand. This technology can be used in different ways [START_REF] Chen | Assessing the impact of the Internet on Brands[END_REF]. Marketing practices that are made possible by the Internet can be classified into three types [START_REF] Chen | Assessing the impact of the Internet on Brands[END_REF]. First, browsers and search engines provide access to a wide range of information on the Internet in terms of products and services. Then by integrating websites with consumer databases, it becomes possible to use more information on consumers and their needs. Finally, the Internet is distinctive because not only it enables consumers to communicate with suppliers but also it allows them to communicate with each other. Firms that have success on the Internet are those that have proved they are capable of getting into virtual communities [START_REF] Chen | Assessing the impact of the Internet on Brands[END_REF]). However, with these forums, brands must accept some loss of control because consumers become active co-producers of value (de [START_REF] Chernatony | Succeeding with Brands on the Internet[END_REF][START_REF] Ind | Branding on the Web: A Real Revolution[END_REF]. Internet constitutes a very wide communication tool with a very strong power in terms of worldwide diffusion, able to reach a large audience and enabling to implement a bilateral communication with the stakeholders of the brand. According to these characteristics and their impact on brand, it is well founded to wonder if the use of such a tool is appropriate in the luxury market, especially regarding the e-market.

Demand for the sale of luxury on line is growing at a global level. Recent studies show that the "rich" are now likely to be connected and are happy to make purchases online [START_REF] Okonkwo | Luxury Fashion Branding[END_REF]. However, luxury goods, clothing, leather goods and fine jewlery are categorized as products with a good deal of "sensory" character. Their aesthetic features can be assessed through the five senses. Several researches have shown a resistance from luxury brands to sell online [START_REF] Seringhaus | Selling Luxury Brands Online[END_REF][START_REF] Kapferer | The Luxury Strategy: Break The Rules Of Marketing To Build Luxury Brands[END_REF]. However, more and more brands seem to take the challenge by opening their own e-shop on their website. Then, in the process of buying luxury products, point of sale plays a crucial role [START_REF] Okonkwo | Can the Luxury Fashion Brand Store Atmosphere be transferred to the Internet[END_REF][START_REF] Okonkwo | Luxury Fashion Branding[END_REF]. Nevertheless, is it realistic to think that the magic of these places can be reproduced on the web? According to [START_REF] Kapferer | The new strategic brand management[END_REF] and [START_REF] Dall | Luxury branding on the Internet: lost opportunity or impossibility?[END_REF], the e-market represents at the same time an opportunity, an inevitable development, and a threat to luxury brands. Indeed, the Internet offers purchasing opportunities to consumers who have no close access to one of the few distributors of the luxury brand, or to those who feel intimidated by the fact of entering these stores [START_REF] Seringhaus | Selling Luxury Brands Online[END_REF]. Furthermore, for brands whose images are based on a very selective distribution, there is a real threat with this democratization. Internet brings together the "high class" and "mass class". This is why previous research has shown an apprehension of the brand regarding online sells, and some author recommends avoiding E-shopping for some type of luxury brands [START_REF] Seringhaus | Selling Luxury Brands Online[END_REF][START_REF] Kapferer | The Luxury Strategy: Break The Rules Of Marketing To Build Luxury Brands[END_REF] According to this literature review, oppositions between traditional luxury universe and the Internet world have been pinpointed (Bechtod, 1991;[START_REF] Roux | Le luxe : entre prestige et marché de masse[END_REF][START_REF] Nuemo | The mass marketing of luxury[END_REF]. However, regarding socio-cultural changes, luxury brands had no other option than integrating this new media in their communication policy. It is required that all communication support send the same message, the same values in order to stay clear in the consumer's mind. This is what authors named consistency of the brand value and this is largely emphasized in the literature [START_REF] Aaker | Building strong brands[END_REF][START_REF] Keller | Managing brand for the long run: brand reinforcement and revitalization strategies[END_REF][START_REF] Kapferer | The new strategic brand management[END_REF]. These elements should also be taken into account regarding e-shopping [START_REF] Okonkwo | Can the Luxury Fashion Brand Store Atmosphere be transferred to the Internet[END_REF]. This leads us to the following research propositions: P1: Maintaining the consistency of the luxury brand value between traditional communication support and Internet support could minimize the paradoxes between selectivity and diffusion P2: For e-commerce, the paradoxes between diffusion and selectivity could be mitigated by maintaining the codes of the brand and its universe.

METHODOLOGY

Faced with the research problem and the analysis of the literature, the objective of this study is to explore how luxury brands use Internet in their communication strategy and how this medium can or cannot fit to manage the values of selectivity of luxury brands. More precisely, we want to compare Internet communication with other tools like posters, the press and window-shopping. In order to do that, a structural semiotic methodology is implemented.

Structural analysis tries to understand the relationship of a number of elements under the principle of solidarity of the terms of a structure [START_REF] Barthes | Rhétorique de l'image[END_REF][START_REF] Eco | Sémiologie des messages visuels[END_REF][START_REF] Floch | Sémiotique, marketing et communication:sous les signes, les stratégies[END_REF]. It analyses procedures to describe systems of meanings. Structural semiotics is based on the concept of sign. This is formed by the relationship between a noticeable element, the signifier (i.e. expression), and the meaning given to that signifier in a more or less structured code, the signified (i.e. content) [START_REF] Barthes | Rhétorique de l'image[END_REF][START_REF] Mick | Consumer research and semiotics: exploring the morphology of signs, symbols, and significance[END_REF][START_REF] Courtès | Analyse sémiotique du discours: de l'énoncé à l'énonciation[END_REF]Greimas and Courtès, 1993;[START_REF] Alreck | Commentary: A new formula for gendering products and brands[END_REF]. All symbols relevant to the subject of study should be considered: the words and language, all types of visuals, gestures, concepts [START_REF] Barthes | Rhétorique de l'image[END_REF][START_REF] Eco | Sémiologie des messages visuels[END_REF]. The study of the confrontation between these signs and symbols and their interactions helps to create a structure of meaning. [START_REF] Floch | Sémiotique, marketing et communication:sous les signes, les stratégies[END_REF] speaks of the principle of "bricolage" (from the French -"do-it-yourself" or "handiwork"). The structural semiotic approach is particularly relevant when studying the deeper signification of brand narratives and their invariants [START_REF] Floch | Sémiotique, marketing et communication:sous les signes, les stratégies[END_REF][START_REF] Courtès | Analyse sémiotique du discours: de l'énoncé à l'énonciation[END_REF].

The process of the analysis is structured in several parts. First, corpuses of communication are established for each brand studied. The corpuses are divided into two parts: one on traditional media (press advertisements and posters -old and new -, images of storefronts and shops and images of the product) and one on the Internet. . Data were collected during 2009, thanks to an extensive research on the Internet and via screenshots. Many communications have been sent via managers of luxury brands. Then, respectively, with traditional media and on the Internet, an analysis grid is prepared. It serves to highlight the elements that shape the meaning. The grid is divided into three parts: an analysis of the plastic message (medium of communication, frame, composition, shapes, colours, lighting and texture), an analysis of the pictorial message (motifs, figures, objects and characters) and an analysis of the linguistic message (typography, colours, shapes of letters and meaning) [START_REF] Courtès | Analyse sémiotique du discours: de l'énoncé à l'énonciation[END_REF]. To highlight the structure of meanings of each brand, the meanings of the various communication media are grouped together. Their relationships, overlaps and redundancies are analysed. The invariant meanings are put forward in forming the narrative of each brand. This analysis is done in parallel for the traditional media of communication and for the Internet. In a final step, a comparison between these two parts of the communication policy is made. To improve the internal validity of the study, all the analyses are first made by the two authors of this paper. Then their results are compared and confronted. A match rate of about 90% was found. Following this confrontation, slight adjustments were made to refine the analysis.

Six luxury brands are considered in the methodology: Goyard, Hermès, Louis Vuitton, Boucheron, Cartier and Van Cleef & Arpels. This choice was motivated by several factors. First, to improve the external validity of the study, two sectors of activity were considered: leather goods and jewlery. Moreover, we chose brands with a high level of awareness and the same level of development according to a small study on 30 luxury consumers and previous research on luxury brands [START_REF] Dubois | Y a-t-il un euro consommateur du luxe? Une analyse comparative des profils sociodémographiques des acheteurs européens[END_REF][START_REF] Okonkwo | Luxury Fashion Branding[END_REF]. Finally, selected brands have all developed a website for over two years. To make comparisons, some of these brands have developed an e-commerce, while others do not.

RESULTS

The results of the study are presented in the following way. First, the analysis of each brand is made. Then, a discussion is developed and some implications are deduced.

Hermès Table 1: Hermès communication

Brand values Traditional communication Internet communication The Know-How related to the job of Saddle maker

Stores: sign bearing the effigy of a horse and saddle maker Product Names: "Amazon" Text: "Under the sign, the hand", "Everything changes, nothing changes" Hermes Magazine: illustration of the craft Home Page: Logo with the carriage and horse Hermès "Universe" and "Expertise" Presentation of 4 tabs related to the saddle maker (Leather, Silk, Horse, Know-how)

Contribution of Internet: Interactivity of know-how and originality

Tab "Expertise": Videos shifted that focus somewhat / not on the mastery of traditional brand craft Presentation of figurative elements out of step with traditional know-how (helicopter, kiwi, pink flamingos, slot machines) « Savoir Vivre à la française » Text: "Elegance and comfort with Hermès" "24, Faubourg Saint Honoré" "Paris" Products related to « Savoir Vivre à la française »: tableware, scarves, gloves reflecting elegance Visual symbols of Paris, cobblestone streets, golf, picnic in the park

This table shows the values of the brand identified through both traditional and internet communication

Brand values:

The brand narrative of Hermes is « savoir-vivre à la française », a state or a way to grasp the moments of life, with delicacy and distinction. It also highlights product quality, with two noble materials: silk and leather, symbols of the French equestrian spirit. This Know-how is related to the original skills of the brand: the saddler. The brand maintain its craftsmanship tradition, and handmade its products. We can see some pictures of craftsmen working on leather, which reinforce the place of the brand in the leather sector.

Consistency of means of communication:

There is an inconsistency in this narrative between the classic communication channels and the Internet. Although there are a few evocations of traditional expertise of the brand through a video on Internet, other applications are not coherent with the values of the brand and generate more confusion. The website architecture is very complex, its contents is shifted, somewhat traditional and modern at the same time.

Internet contribution:

In the case of Hermès, the Internet does not reinforce the brand values. It goes against the overall communication policy of the brand. Namely, the Internet does not refer to the « savoir-vivre à la française », a fundamental value of the brand since its creation. The use of Internet in this case is creative but leads to confusion regarding Hermès values. E-shopping: E-shopping is present on the Hermes website. Already on the home page, you can click either on "Hermes Universe" or on "E-Shopping". Therefore, there is an immediate focus on the purchase as the viewpoint of the mass market. Also as before (i.e. for about one year), the site was only focused on e-shopping and did not have the part about the universe of the brand. The presentation of e-shopping is like a product catalogue. It is relatively limited. The vocabulary and architecture are those of a site for the mass market. There are expressions like "your account", "buy", "How to pay for your purchases". The inconsistency between the brand values in the traditional channels of communication and the Internet is reinforced through e-shopping.

Louis Vuitton Table 2: Louis Vuitton communication

This table shows the values of the brand identified through both traditional and internet communication

Brand values:

The narrative of the Louis Vuitton brand is the spirit of travel (both real and spiritual). The brand offers its expertise to attain this state of escape through both trunks and leather goods. This also refers to the original skills of the brand. The brand expose trunks and purse in these shops (in the entrance) and in its advertising. The Brand sets the focus on its "tradition" and its "innovation".

Consistency of communication support:

A real coherence emerges between the values expressed by the brand's traditional channels of communication and those conveyed via the Internet. We find the same types of staging of the brand's know-how and its narrative of travel.

Internet contribution:

In the case of Louis Vuitton, the Internet strengthens the message of the brand and its values. Internet offers the opportunity to develop interactive applications. The use of explanatory videos to support the expertise of the brand, and the "Soundwalk" (audio guide application for major cities around the world) to bring the cybernaut into the world of travel are some examples.

E-shopping: E-shopping is on the Louis Vuitton website but it is not dominant in its first display (the e-shopping only appears in the last tab of five). Indeed, there is a desire to stage the universe and the brand's identity. Concerning the specific e-shopping pages, the format is like a product catalogue with descriptions of items and prices. The products for sale online are mainly those related to leather, which reflects the will to focus on its main expertise. This consistency in setting forth the know-how of the brand is also enhanced by the possibility to customize products ordered. The e-shopping is fully integrated into the site and does not give rise to a new Internet window when clicking on the tab. This sequencing helps to strengthen the consistency of the brand and its commitment to integrate e-shopping in the larger universe of the brand's identity.

Goyard Table 3: Goyard communication

Brand values Traditional communication Internet communication The expertise trunk

Advertising: presentation of products in large size Focus on quality Text: "Malletier since 1853" Goyard shop: Rue Saint Honoré in Paris Authenticity of the store decorations Focus on brand expertise with special orders Presentation of the brand history Brown color to sow the brand authenticity

Contribution of Internet

Only a reinforcement of the values

This table shows the values of the brand identified through both traditional and internet communication

Brand values:

The Maison Goyard puts forward in its communication expertise trunk maker "Malletier since 1853". All bags, accessories and bags are the finest materials. Its shop is located on Rue Saint Honore in Paris since its creation and strengthens its history. The decorations demonstrate the authenticity of the brand. The presentation bags close to the center of advertising highlights the details of products and their quality.

Consistency of means of communication:

There is a consistency in this narrative between the classic communication channels and the Internet. Indeed, the brand know-how is highlighted not only through the press advertising and other traditional media (shopping) but also on the website with the presentation of the history of the brand and special orders.

Internet contribution:

For Goyard, the Internet allows it to strengthen its values (related to its know-how). However, the website is not particularly developed and is very simple and traditional. This element can fit with the brand authenticity and the willingness of not being too modern. E-shopping: Goyard website does not have an e-commerce part. The brand does not sell online at this time.

Cartier

This table shows the values of the brand identified through both traditional and internet communication

Brand values:

The narrative of Cartier is based on jewelry know-how (extended to watch making) in a world of elite social status. Prestige, domination and demonstration of social values are highlighted.

Consistency of means of communication:

There is a consistency between the values communicated via traditional media and the Internet. The visual codes and the figurative language are the same. Jewelry know-how and high social status are dominant in all means of communication.

Internet contribution:

The use of explanatory videos supports the expertise of the brand. The originality of one application, called "Enter a unique place", reaffirms and reinforces the brand narrative on the prestigious social position. It allows the net surfer to visit the online shops. This application can be perceived as a paradox with respect to the elitist values of the brand. It can be seen as a desecration of the act of entering a fine jewelry store, but may also remind users that the visit on the Internet is still only virtual.

E-shopping:

Cartier's website does not sell jewelry online. There was thus no e-shopping. However, the brand leaves the possibility of requesting additional information and creating a "wish list". This service and these "privileges" are reserved only to Cartier Club members. This exclusivity again reinforces the elitist values of the brand.

Van Cleef & Arpels Table 5: Van Cleef & Arpels communication

Brand values Traditional communication Internet communication Expertise in Jewelry

Visual Elements: close-ups of the jewelry, always staged. Dark backgrounds to highlight the brilliance of the stones Textual elements: "Jeweller" Store: presentation of exceptional pieces Home Page: Focus on one jewel Tab Collection: Focus on products, directed and animated on a dark background Tab « La Maison Van Cleef&Arpels »: "Craft, Memory" "Exceptional Pieces", « watchmaker art and crafts »

Internet contribution: Interaction between the brand and products

Watches tab: focus on the technical characteristics of the brand and its expertise with the possibility to download a manual Existence of an I phone application

Magical and fairylike universe

Visual elements: Decor that focuses on the fairies and nature Predominance of an imaginary world where the jewels are put on display Store: Presence of fairies and flowers Website animated around the theme of magic and nature. Tab « Jewelry Collection »: Poetic product names: "A day in Paris, Midsummer night's dream, Butterflies, Fleurette & Snowflake"

This table shows the values of the brand identified through both traditional and internet communication

Brand values:

The brand narrative of Van Cleef & Arpels is based on jeweller expertise in a universe of magic and fairies. The brand offers a world of unrealistic characters and spaces. In its classical advertising, the brand presents its jewels in close-up in a fairy world to put forward the quality of its jewelry. In the brand shop, few pieces are exposed in order to reinforce the selectivity and the scarcity. We can pinpoint the permanent presence of the nature in the advertising of the brand.

Consistency of means of communication:

In the case of Van Cleef & Arpels, the use of Internet is consistent with the other more traditional communication tools. The Internet brings the user into the know-how and the magical world of the brand. A focus is made already on the home page on nature and fairy world.

Internet contribution:

Internet reinforces the values of Van Cleef & Arpels, by explanatory videos to support the know-how of the brand. You can also enter more easily into the magical world of the brand thanks to lyrical and enchanting animations. The magical world is hence close to the virtual world offered by the various possible applications of the Internet.

E-shopping:

The Van Cleef & Arpels website does not have an e-commerce part. The brand does not sell online at this time. We can also note that there is no possible interaction with the brand except via the newsletter.

At this stage, we can highlight that the jewelry sector uses Internet as a brand image tool and takes advantage of it namely using interactive and original applications to reinforce brand values. However, regarding E-shopping, none of the two brands we have analysed has an Eshopping part. In order to check if this desire not to use e-commerce is related to the sector of these brands we have examined whether there were other jewelry brands that practice e-shopping. Several have emerged including Tiffany and Boucheron. To continue with fine jewelry brands, we decided to analyse Boucheron.

Boucheron

This table shows the values of the brand identified through both traditional and internet communication

After analysing this brand, we can highlight the consistency between the values communicated via traditional media and the Internet. More than for the other jewellers, Internet reinforces the message and the values of Boucheron. E-shopping is very present on the website of Boucheron but, as for Louis Vuitton, the presentation of e-shopping is completely integrated in the website. There is no distinction between the brand universe and the sale of products by Internet. The vocabulary and architecture are those of a site for the mass market. There are expressions like "your shopping bag", "Your favourites". Boucheron sells extravagant, very expensive jewelry (up to 40 000£). Despite this type of strategy, the presentation of the products remains classic and prestigious. Then the e-shopping mixes tradition and innovation, the two narratives of the brands.

RESEARCH CONTRIBUTIONS

The purpose of this article was to analyze how luxury brands can manage scarcity and the use of Internet. The literature review that we have conducted has led us to these research propositions: P1: Maintaining the consistency of the luxury brand value between traditional communication support and Internet support could minimize the paradoxes between selectivity and diffusion P2: For e-commerce, the paradoxes between diffusion and selectivity could be mitigated by maintaining the codes of the brand and its universe. In order to test these propositions a structural semiotic analysis has been implemented through six notorious brands which websites have been created at least 2 years ago.

Here is the discussion of the primary findings resulting from our research.

From a general overview, for five on the six brands we have analysed, their core central values are reinforced through the utilization of the Internet and its interactive applications. Internet is used as a brand image tool for both leather goods and jewelry sectors. Actually using Internet as a brand image tool makes it possible to avoid inconsistency emerging from the use of a mass media in a luxury strategy. This is in line with literature recommendations about maintaining consistency of brand values on the different communication support [START_REF] Kapferer | Les marques, capital de l'entreprise[END_REF][START_REF] Aaker | Building strong brands[END_REF][START_REF] Keller | Managing brand for the long run: brand reinforcement and revitalization strategies[END_REF]. Only Hermès set the focus on the e-commerce part, thus neglecting the brand image aspects and therefore the luxury atmosphere. From a brand sector point of view, Richemont Group's Jewellers choose a risk-free approach by allowing limited interactions with the consumers, which avoids loss of control regarding brand values. This approach seems not to be related to the brand sector since our control case, Boucheron, which does not belong to Richemont, uses original interactivity with its consumers and e-commerce too. This risk-free approach may result from a consistent group strategy. The leather goods sector uses Internet as an interactive platform with consumers and takes advantage of the original opportunities Internet brings in the communication world. They are willing to establish a bilateral communication strategy. This technique is more risky in terms of loss of control regarding the content of information. The dominant craftsmanship aspect in the jewelry could explain this difference between jewelry and leather goods. Overall, this analysis is consistent with the research proposition: P1: Maintaining the consistency of the luxury brand value between traditional communication support and Internet support could minimize the paradoxes between selectivity and diffusion.

Focussing on the E-shopping aspects, three brands (Goyard, Van Cleef & Arpels, Cartier) have not (yet?) taken up the challenge of e-commerce. Three brands (Hermès, Louis Vuitton, Boucheron) have decided to integrate this part as much as possible in the context of the brand image tool so that the core values are omnipresent and the luxury atmosphere is preserved. One brand (Hermès) has positioned itself differently from other brands by offering a particular website where e-commerce is dominant and where mass market flirts with luxury. It means that E-shopping is present for both sectors we have analyzed. Regarding further on the other brands of these sectors like Gucci, Longchamp or ST Dupont (Leather goods) or Tiffany & Co and Mauboussin (Jewelry), E-shopping is also used by these. This leads us to moderate the vision of [START_REF] Kapferer | The Luxury Strategy: Break The Rules Of Marketing To Build Luxury Brands[END_REF] stating that luxury brands should always communicate through the Internet but never sell through it on web 2.0. We think that selling through the Internet is not impossibility but a real opportunity to be considered by luxury brands and their managers if they go for a completely integrated strategy respecting the holy rules of luxury communication, such as highlighting the values of selectivity and rarity. Then, this analysis is consistent with the research proposition: P2: For e-commerce, the paradoxes between diffusion and selectivity could be mitigated by maintaining the codes of the brand and its universe.

CONCLUDING COMMENTS SECTION

The results of this research and the previous discussion lead us to make several recommendations for managers of luxury brands. First, brands have to consider the Internet as a fundamental communications tool. Because of the several applications that can be created, Internet gives brands the opportunity to express all the values of their identity in an original way and to maintain brand narratives consistency. This will allow consumers to better understand the brand values and to discover them through funny applications. In order to achieve that, each part of the website should be created to highlight the brand universe. In order to maintain the consistency of the brand on the Internet, it is important to resume on all pages on the website the same color codes, the same universe, the same word as used in other communication media. The use of many innovative applications can also be a means to strengthen the brand values. Regarding e-shopping, managers should consider the opportunity of Internet as a selling platform as stated above. The success of such a strategy will depend on the way it is implemented. Brands should absolutely maintain their selective image on the E-shopping by namely using luxury codes in terms of content, layout of the website. Brands have to use a mass-market platform and turn it into a luxury world. Moreover, besides the turnover generated by the E-shopping, this really constitutes an open window to the world for the brand, regarding the fact that consumers take now a lot of information online. Another implication of this research concerns structural semiotics. It constitutes a new trend for managers and researchers to analyse and identify their brand values. The description in this research of the several methodological steps can be a basis for further research. Some limits and furthers research can be highlighted. Despite the fact that many cautions have been taken, we could broaden the scope of the research namely by analysing more different sectors like fashion, cosmetics or table arts in order to improve the external validity and to know if the Internet communication strategies selected by the brands are in some way related to the sector they belong to. In addition, in order to achieve an acceptable internal validity, the analysis was conducted by the two researchers independently, after which they confronted their results. A request for an expert point of view has been made. To continue this work, a quantitative analysis will compare the results of this research with consumer perceptions of luxury. The study of their attitudes toward e-shopping and these various arrangements constitutes the next step in this research.

  Text elements: " Louis Vuitton bags surprise 1854" Shop: at the entrance and ground floor, presence of trunks Close-up of bags and suitcase Home Page: Bag put forward on the photo Service tab: Videos and photos of manufacturing and traditional manufacture Tab Universe with: "Heritage, Family House, Know-how" Internet contribution: Personalisation of the know-how Tab Universe: headings "Special Orders; customisation" The travel Text: "Love to learn," "Soul of Travel," "Necessary travel" "The Spirit of Travel" Visuals: sailing, swimming pool, sea, boat, train, taxi Landscapes: New York, desert Products related to travel: trunks, logbook Home Page: Integration of the bag in the travel universe (boat, beach, scooter) Textual elements: "The Art of Travel", "The passenger in history", "Travel with Coppola" Presentation of City guides Internet contribution : Interactivity of travel Creation of an application: MP3 audio to various cities such as Paris, Beijing ...

Table 4 :

 4 Cartier Communication

	Brand values	Traditional communication	Internet communication
	The Expertise of Jewelry	Visual Elements focused on jewelry close-up	Home Page: Focus on products
	(and watchmaker)	Textual elements: "Jeweller since 1847"	Cartier House tab: "Heritage", "Excellence"

Table 6 :

 6 Boucheron communication

	Brand Value	Traditional Communication	Internet Communication
	Expertise of the Jewelry	Visual elements : close up of jewels	Home page : Focus on Place Vendôme
		« Jeweler since 1857 »	« Boucheron Paris », « Haute Joaillerie », « La Maison
		In the shop :minimalist presentation of the	Boucheron », « Bijoux », , « precious moments», (related
		Presentation of Paris and place Vendôme	to the craftsmanship tradition of the brand)
		Dark blue color to improve presentation of the	Focus on products, no staging
		products	Dark color: put the jewels forward
	Innovation	Advertising with staged jewels in the mouth of
		sexy men and women
	Internet contribution :		Home Page: E-shop for each country
	Presentation of all		Sell of jewels « Boucheron and you »
	innovation of Boucheron