A Obeidat

S Haider

K E Meyer

T Schnipper

S Mayer

J H Walther

Instabilities and transition in three-dimensional flows with rotation INFLUENCE OF PISTON POSITION ON THE SCAVENGING AND SWIRLING FLOW IN TWO-STROKE DIESEL ENGINES

Introduction

We study the effect of piston position on the in-cylinder swirling flow in a low speed large two-stroke marine diesel engine model. We are using Large Eddy Simulations in OpenFOAM, with three different models for the turbulent flow: a one equation model (OEM), a dynamic one equation model (DOEM) and Ta Phuoc Loc's model (TPLM). The simulated flows are grid-independent and they are computed in situations analogous to two different piston positions where the air intake ports are uncovered 100% and 50%, respectively. We find that the average flow inside the cylinder changes qualitatively with port closure from a Burgers vortex profile to a solid body rotation while the axial velocity changes from a wake-like profile to a jet-like profile. The numerical results are compared with measurements in a similar geometry [START_REF] Haider | Particle image velocimetry of swirling flow in a pipe[END_REF] and we find a good agreement between simulations and measurements. Furthermore, we consider the unsteady flow and identify a dominant frequency in a power spectrum based on velocity which we show is due to precession of the vortex core, and compare with measurements of the unsteady flow obtained with Laser Doppler Anemometry.

Numerical methods

We study the swirling flow numerically using large eddy simulations [START_REF] Schiestel | Modeling and Simulation of Turbulent Flows[END_REF][START_REF] Lesieur | Large Eddy Simulations of Turbulence[END_REF]. We apply three different models. The one equation eddy model where the kinetic energy k is solved while another scale is estimated [START_REF] Schumann | Subgrid Scale Model for Finite Difference Simulations of Turbulent Flows in Plane Channels and Annuli[END_REF][START_REF] Davidson | Large-Eddy Simulations: A dynamic one-equation subgrid model for three-dimensional recalculating flows[END_REF][START_REF] Lesieur | Large Eddy Simulations of Turbulence[END_REF][START_REF] Thomas | The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence[END_REF][START_REF] Schiestel | Modeling and Simulation of Turbulent Flows[END_REF], the dynamic k-equation eddy-viscosity model, where the model constants are recalculated during the simulation rather than to be pre-calculated [START_REF] Germano | A dynamic subgrid-scale eddy viscosity model[END_REF][START_REF] Lesieur | Large Eddy Simulations of Turbulence[END_REF][START_REF] Schiestel | Modeling and Simulation of Turbulent Flows[END_REF] and the Ta Phouc Loc model, which is based on the velocity-vorticity (v-ω) formulation of the Navier-Stokes equations, where two spatial filters are used [START_REF] Phuoc | Modèles de sous maille appliqúes aux ecoulements instationnaires décollés[END_REF][START_REF] Tenaud | Large eddy simulations of a spatially developing incompressible 3D mixing layer using the v-ω formulation[END_REF].

Computational domain and results

The computational domain is shown in Fig. 1

  (a). Flow enters uniformly at an oblique angle which ensures the overall in-cylinder swirling flow as shown in Fig.1(b)[START_REF] Haider | Particle image velocimetry of swirling flow in a pipe[END_REF]. All the results are obtained by simulating the flow on a grid with 8 million cells.

Figure 1 :

 1 Figure 1: (a) Sketch of the cylindrical computational domain shown in grey shades, in the case where the piston covers the intake ports by 50%. Flow enters radially through the horizontal inlet section and exits through the vertical outlet. Notice that the exhaust is shortened in the figure. Data are extracted at the enumerated cross-sectional planes marked by z i , and R = 9.5 cm. (b) Visualization of the three-dimensional time average streamlines (in blue and green colors) of the mean-field for the 100% open port case. Notice the 'braided' separation region shown by red streamlines and the modest pitch of the flow in the main cylinder. The small inset shows the swirling motion of the streamlines

Figure 2 :

 2 Figure 2: (a,b) Time averaged axial and tangential velocity profiles for the 100% open ports, (c,d) Time axial and tangential velocity profiles for the 50% open ports, at the axial position z1/2R = 0.96.


Anthony Randriamampianina

Instabilities and transition in three-dimensional flows with rotation INERTIA-GRAVITY WAVES DURING THE TRANSITION TOWARDS GEOSTROPHIC TURBULENCE WITHIN A BAROCLINIC CAVITY

Introduction

The transition to disordered behaviour in the form of 'Baroclinic Chaos' provides an important prototypical form of chaotic transition in fluid dynamics. This is of particular geophysical relevance in the context of understanding the origins of chaotic behaviour and limited predictability in the large-scale atmospheres of terrestrial planets and in the oceans. For many years, the differentially-heated, rotating cylindrical annulus, "the baroclinic cavity", has proved a fruitful means of studying fully-developed, nonlinear baroclinic instability in the laboratory. Transitions within the regular wave regime follow canonical bifurcations to low-dimensional chaos [START_REF] Randriamampianina | Direct Numerical Simulation of bifurcations to low-dimensional chaos in an air-filled rotating baroclinic annulus[END_REF], but disordered flow appears to emerge via a different mechanism involving small-scale secondary instabilities [START_REF] Read | Direct Numerical Simulation of transitions towards Structural Vacillation in an air-filled, rotating, baroclinic annulus[END_REF]. In previous laboratory experiments of baroclinic waves, such fluctuations have been associated with a flow regime termed Structural Vacillation which is regarded as the first step in the transition to fullydeveloped geostrophic turbulence. Subsequent development within this so-called 'transition zone' leads to the gradual and progressive breakdown of the initially regular wave pattern into an increasingly disordered flow, ultimately leading to a form of stratified 'geostrophic turbulence'. Here we present an analysis which focusses on the small-scale features.

The numerical model and the case discussed

The physical model is that of the experimental rig used at the university of Oxford, UK ( [START_REF] Wordsworth | Turbulence, waves and jets in a differentially heated annulus experiment[END_REF], [START_REF] Wordsworth | Theoretical and experimental investigations of turbulent jet formation in planetary fluid dynamics[END_REF]), filled with a liquid defined by a Prandtl number P r = 16. It is composed of two vertical coaxial cylinders of inner radius a = 4.5cm and outer radius b = 15.cm, held at constant temperature difference, ∆T = T b -T a = 2K and enclosed by two horizontal insulating rigid lids separated by a distance d = 26.cm. The whole cavity rotates around the central axis with Ω = 1.25rad/s. The Navier-Stokes and energy equations coupled via the Boussinesq approximation are solved using a pseudo-spectral collocation-Chebyshev Fourier method associated with a second order time scheme.

Results

Structural Vacillation

The results are consistent with the structural vacillation observed in the laboratory, characterized by a spatiotemporal behaviour. While the large scale of the flow persists, here with a dominant azimuthal wavenumber m = 3, the flow does show noticeable irregular fluctuations at small scales which are going to progressively destroy the regularity of these main cells. This is illustrated by the streaklines shown at different heights of the cavity in figure 1.

Inertia gravity waves

Unlike a previous study using air as working fluid [START_REF] Read | Direct Numerical Simulation of transitions towards Structural Vacillation in an air-filled, rotating, baroclinic annulus[END_REF] where the driven mechanism resulted from a radial Rayleigh-Bénard rotating convection due to high centrifugal accelerations, the dynamics of the observed structural vacillation is mainly governed by inertia-gravity waves developing on top of large-scale baroclinic instabilities [START_REF] Plougonven | Inertia-gravity waves spontaneously generated by jets end fronts. Part I: different baroclinic life cycles[END_REF]. The small-scale structures appear spontaneously as soon as the large-scale regular waves occur, and indeed satisfy the characteristic dispersion relation given in [START_REF] Fritts | Gravity wave dynamics and effects in the middle atmosphere[END_REF]. These fluctuations are found to behave like the travelling waves emitted by thermal boundary layer instability determined as inertia-gravity waves for the value of Prandtl number considered here, according to analytical studies [START_REF] Gill | Instabilities of a buoyancy-driven system[END_REF]- [START_REF] Bergholz | Instability of steady natural convection in a vertical fluid layer[END_REF]. Moreover, a recent similar numerical study carried out at the university of Oxford UK [START_REF] Jacoby | Generation of inertia-gravity waves in the rotating thermal annulus by a localised boundary layer instability[END_REF], but considering a higher Prandtl number fluid, confirmed that these small-scale fluctuations correspond to inertia-gravity waves. A space-time map of the temperature along the azimuth at fixed radius and height during about three fundamental periods of the large-scale motion is displayed in figure 2, showing the perturbations attached to the m = 3 large-scale structure.

EUROMECH

  Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation

Figure 1 :

 1 Figure 1: Instantaneous streaklines at different heights of the cavity

Figure 2 :

 2 Figure 2: Space-time map of the temperature along the azimuth at fixed radius and height.


Farzaneh Samsami

Andre Thess

Yuri Kolesnikov

Instabilities and transition in three-dimensional flows with rotation EXPERIMENTAL INVESTIGATION OF TRANSITION TO TURBULENCE IN A MAGNETIC OBSTACLE

We present an experimental study about the structure of the flow in the movement of an electrically conducting fluid past a localized magnetic field,which has been called magnetic obstacle. We studied the sequences of vortex generation and the structure of bifurcations in the flow of eutectic alloy GaInSn. The flow is shown to undergo really complicated phenomena. Here we will try to introduce these phenomena.

Introduction

In practical applications we always deal with the nonideal, i.e. confined and nonhomogenous magnetic fields. There are many applications in which we combine such a magnetic field with the liquid metal flow and benefits from the interaction force which is called Lorentz force. This force can be used as an active force to drive the fluid flow, in a pump or stirrer, or as a passive force in measurement devices. The liquid metal flows under the influence of confined non-homogeneous magnetic fields, so called magnetic obstacle [START_REF] Votyakov | Structure of the wake of a magnetic obstacle[END_REF] is hardly understood. There are some literatures who try to discover the secrets of the magnetic obstacle [START_REF] Votyakov | Structure of the wake of a magnetic obstacle[END_REF]- [START_REF] Votyakov | On the analogy between streamlined magnetic and solid obstacles[END_REF] but they all suffer from the lack of experimental evidences and the transition to turbulence is still obscure. In the obove references the authors demonstrated that the flow pattern in a magnetic obstacle is really more complex than in the wake behind an ordinary hydrodynamical body. Votyakov et al. [START_REF] Votyakov | Structure of the wake of a magnetic obstacle[END_REF] showed that The flow will undergo two bifurcations (rather than one) and will involve up to six (rather than just two) vortices. The fluid flow is predicted to be stable and free of vortices at high velocities. By decreasing the flow rate, inner vortices and later on the six-vortex pattern will appear. This is in contrast to what we know from ordinary hydrodynamics in which vortex generation will be amplified by increasing flow rate. They also predicted the appearence of two bifurcation and related them to the appearence of the vortices, see figure 1. In this letter we will present the experimental results of the transition to turbulence in a magnetic obstacle.

Experimental set up

A 1200mm long channel with a rectangular 100mm × 25mm cross section made of Plexiglas is used in this experiment. The channel is filled up with 10mm GaInSn and has one Hartmann wall, one free surface and two side walls.The maximum intensity of the magnetic field in the Hartmann wall and in the free surface will be 0.4T and 0.2T accordingly and the related Hartmann number will be Ha = 110.6. The 20mm × 30mm × 40mm permanent magnet is installed on a moving rail beneath the channel which has the course of 850mm (figure 2). The distance between the magnet and the Hartmann wall is 8 mm. The experimental set up enables us to move the magnet with constant velocities varying from 5mm.s -1 to 80mm.s -1 , i.e. Re = 125 -2000. The streamlines has been illuminated using gas bubbles generated due to the reaction of the acid HCl and GaInSn oxide on the free surface. A camera which moves with the magnet has been used to record the vortex generation sequences.

Result and discussion

As the velocity and hence the Re Number changed, we observed different behaviours inside the magnetic field region and in downstream. These includes formation of bifurcations or their suppression, vortex shedding behind the magnet, symmetry breakdown, dupplication of vortices and also a behaviour that looks like intermittency. Two of the pictures taken during this investigation is shown in figures 3 and 4. In this figures a magnet indicatior shows the end position of the magnet. Notice that all the pictures do not have the same scale.

Conclusion

A new complicated phenomena that shows the features of the turbulent flow has been discovered. The vast number of turbulent phenomana that are visible in the fluid flow past a magnetic obstacle make it worth investigating in order to achieve a better understanding of magnetic obstacle or even turbulence.

Acknowledgment

The authors express their gratitude for financial support by the Deutsche Forschungsgemeinschaft.

Figure 1 :

 1 Figure 1: Topology of the predicted fluid flow as mentioned in [1].

EUROMECH

  Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation

Figure 2 :

 2 Figure 2: Sketch of channel and permanent magnet arrangement for this experiment.

Figure 3 :

 3 Figure 3: Transient 6 vortex structure at Re = 1000 Figure 4: A street of vortices at Re = 1500


Presenting Orlandi

Instabilities and transition in three-dimensional flows with rotation GRID TURBULENCE IN SOLID BODY ROTATION

Introduction

Isotropic turbulence is generated in wind tunnels by an uniform stream going through a solid grid. After an anisotropic region the flow becomes isotropic, and decays in the streamwise direction with a power law (x 1 /M ) -m . In a laboratory it is rather difficult to investigate the effects of the solid body rotation on the power law decay. The study is easier in numerical experiments with three periodic directions and a synthetic isotropic turbulence at t = 0 which decays as t -m . The power law m changes with the rotation number N = 2ΩL/U , the inverse of the Rossby number used in geophysical flows. In this setup the initial conditions are different from those in real experiments. However in the experiments even the grid rotates and therefore the inlet conditions may change with the rotation rate.

In the present study the Navier-Stokes equations in a rotating frame are solved in a domain with dimensions 12π × 2π × 2π respectively in the streamwise, and in the other two homogeneous directions. Radiative boundary conditions at L 1 = 12π allow the disturbance to exit from the computational domain. At the inlet an analytical velocity distribution U l = δ 1l + A l sin(f x 2 ) sin(f x 3 ) + A R (x 2 , x 3 ) is assigned, with A 1 = 0, A 2 = A 3 = 0.25 and A R = 0.01. The solidity of the grid is M = 2π/f . This inlet condition is not isotropic having a turbulent stress u 2 u 3 = 0 which should go to zero at a distance x 1 /M to have an isotropic turbulent flow, further downstream decreasing as (x 1 /M ) -m . The simulations are performed at a Reynolds number, which, for non rotating conditions close to the exit, has R λ = O(10) and m = 2.5. Several simulations were performed with fixed f = 8 and N varying from 1 to 10. Note that these values correspond to rotation rates higher than those achieved in laboratory experiments (Jacquin et al. [START_REF] Jacquin | Mathieu Homogeneous turbulence in the presence of rotation[END_REF]). Preliminary simulations show a large reduction of m with N (Fig. 1a). The streamwise oscillations of q, not so evident in this figure because of the log scale, in a normal scale show a decrease of the amplitude and an increase of the wave length with the increase of N . This is an evidence that the vortical structures become longer when N raises. This effect has been recently observed in rotating tank experiments by Moisy et al. [START_REF] Moisy | Sommeria Decay laws, anisotropy and cycloneanticyclone asymmetry in decaying rotating turbulence[END_REF] where the turbulence is generated at t = 0 and then decays in time. Here the conditions are different being the small scales generated at the inlet, and survive together with those downstream which are modified by the rotation. The additional effect of rotation is to reduce the energy transfer from large to small scales and therefore the rate of energy dissipation. This is demonstrated by the streamwise evolution of R λ in Fig. 1 where, at high N , R λ is constant (m ≈ 1).

To understand in more detail the streamwise evolution across the anisotropic and isotropic regions it is worth looking at the u 1 equation in rotational form, where the kinetic energy q appears together with the components of the Lamb vector (λ = v × ω), related to the energy transfer from large to small scales

∂q ∂x 1 -u 2 ω 3 + u 3 ω 2 = - ∂ p ∂x 1 = ∂ u 2 1 ∂x 1 (1)
In this equation the direct effect of N does not enter, which instead acts on the other two velocity components modifying λ 1 . To understand the effects of N on the streamwise evolution of the velocity rms u 2 l , which for N = 10 are compared in Fig. 2a with those for N = 0. For N = 0 u 2 1 grows for the positive contribution of the λ 1 component (Fig. 2b), and the other two equal stresses decrease in x 1 . At the end of the anisotropic region (x 1 /M ≈ 3) all the stresses decrease, although u 2 1 is slightly smaller than the other two. The transport equations for u 2 l (l = 1) stresses show a positive contribution of N for one and negative for the other, this is visible in Fig. 2a, where, for N = 10 in the anisotropic region, u 2 2 and u 2 3 are different. Through this imbalance a greater u 2 1 than that for N = 0 is obtained. The cause is visible in Fig. 2b where positive values of λ 1 occur in correspondence of the u 2 1 bump. By increasing the solid body rotation the helicity density h = v • ω increases, which from the identity |v × ω| 2 + |v • ω| 2 = |v| 2 |ω| 2 reduces the transfer from large to small scales, and the energy remains at the large scales. The helicity distribution is zero in the non-rotating case and grows with the rotation rate, with a positive contribution up to x 1 /M ≈ 1 and negative downstream. The analysis of the Lamb vector and the helicity density will be presented in more detail at the conference. The transfer of energy to the large scales, characteristic of rotating flows, can be observed by the spectra together with streamwise vorticity visualizations at the same location. In particular at x 1 /M = 2 and at X 1 /M = 10 the transverse spectrum E 33 (k 2 ), for all values of N , and, for N = 0 and N = 10, contour plots of ω 1 (x 2 , x 3 ) are considered. The spectra in Fig. 3a together with the ω 1 contours in Fig. 3b show that the influence of the inlet disturbance at x 1 /M = 2 is large for N = 0 and that it reduces by increasing N . In Fig. 3c, at high N , the inlet disturbances are less visible, while larger structures appear. The spectra show better the transfer of energy to large scales. At x 1 /M = 10 the figures 3d-f clearly demonstrate the formation of energetic large scales at N = 10, which are those giving m = 1.

Figure 1 :

 1 Figure 1: Profiles of a) q and b) R λ versus x 1 /M at different values of N .

EUROMECHFigure 2 :

 2 Figure 2: Profiles of a) the velocity rms ( , , l = 1 , • , l = 2 , , l = 3 and b) the component λ 1 ( , ) and ∂ p ∂x1 ( , ) versus x 1 /M ; lines N = 0, solid symbols N = 10.

Figure 3 :

 3 Figure 3: Transverse spectra, a) x 1 /M = 2 , d) x 1 /M = 10 different N lines as Fig.1; contours plot of ω 1 b) for N = 0 c) N = 10 at x 1 /M = 2 with ∆ω 1 = 1 e) for N = 0 f) N = 10 at x 1 /M = 10 with ∆ω 1 = 0.1


TURBULENT FLOW IN ROTATING RIBBED CHANNEL WITH CORIOLIS FORCES AND CENTRIPETAL BUOYANCY

René VAN In turbomachinery, one of the applications in which rotational effects are most relevant is the coolant flows in the internal cavities of rotating turbine blades. The flow in internal cooling channels in stator blades is already complex due to the intricate shape of the passages and the presence of turbulence promoters such as ribs. In rotor blades Coriolis forces affect the stability of the shear layers and induce secondary flows. Moreover, because of the thermal boundary layers along the hot channel surfaces, the centrifugal acceleration induces radial buoyancy in the vicinity of the walls. The latter turns the typical forced convection regime in a mixed convection regime.

In most past experimental studies concerned with rotating channel flows, point-wise velocimetry techniques are used (e.g. [START_REF] Nakabayashi | Turbulence characteristics of two-dimensional channel flow with system rotation[END_REF]), which cannot identify coherent instantaneous flow structures and velocity gradients. Few studies in literature present full field data in rotating channels using PIV, but typically the imaging system is attached to the fixed frame [START_REF] Elfert | Detailed Flow Investigation Using PIV in a Rotating Square-Sectioned Two-Pass Cooling System with Ribbed Walls[END_REF]. The relative velocity inside the channel is obtained by subtracting the peripheral velocity from the measured absolute velocity, resulting in low temporal and spatial resolution and in large uncertainties, especially at high rotation rates. Due to the lack of detailed experimental data, often numerical models need to be validated against more accurate simulations for validation, namely LES and DNS. The latter are anyway extremely expensive, if not unaffordable, due to the complexity of the cooling geometries.

Present experimental approach

The present contribution addresses the effect of Coriolis and buoyancy forces in a rotating channel with one ribbed wall, reproducing a turbine blade cooling channel. Two-dimensional PIV measurements are performed in a rotating facility where both the test section and the measurement chain rotate on a turntable up to 130 rpm [START_REF] Di Sante | A new facility for time resolved PIV measurements in rotating channels[END_REF]. The Reynolds number value based on the bulk velocity (2.8 m/s) and the hydraulic diameter (79 mm) is 1, 5 • 10 4 . A rotation number value equal to 0.35 is achieved both in clockwise and counter-clockwise sense. The ribbed wall, machined out of copper, is heated to a uniform temperature of about 100 o C by means of electrical resistances powered by an on-board PDI controller. This allows obtaining a buoyancy number equal to 0.3. As the flow is outward the buoyancy is centripetal. In-plane velocity fields are measured along the channel symmetry plane by means of a 25 W continuous laser diode and a high-speed CMOS camera. Both ensembleaveraged and time-resolved measurements are performed. In the latter case the realizations are acquired at 3.3 kHz, allowing to resolve the resolution of the fine temporal flow scales. To the authors' best knowledge, these are the first time-resolved PIV measurements in a channel flow under rotational buoyancy.

Results

Figure 1 displays the mean streamlines for the non-rotating and rotating cases. With respect to the nonrotating case (fig. 1a), the anti-cyclonic rotation (fig. 1b) leads to the augmentation of free shear entrainment, which produces a stronger streamline curvature, and hence an earlier reattachment. The opposite holds for cyclonic rotation (fig. 1c). Adding the effect of buoyancy forces in the cyclonic rotating case (1d) produces an even larger recirculation spanning the whole inter-rib area, which enhances mixing and transport. Figure 2 depicts the spatio-temporal travelling of the spanwise vortices shed by the rib in different rotating conditions. The vortical structures are identified by regions exceeding a positive threshold of swirling strength. Vortices which have a significant extension in the space-time domain satisfy the requirement of temporal coherence, which cannot generally be imposed analyzing instantaneous realizations. With anti-cyclonic rotation (fig. 2a) higher convective velocities and fewer counter-clockwise rotating vortices are found with respect to cyclonic rotation (fig. 2b). Furthermore the spanwise rollers are blurred by the destabilizing effect of anti-cyclonic rotation. In cyclonic rotation with centripetal buoyancy (fig. 2c) the vortical structures are even neater, and counter-clockwise rotating vortices are fewer. It is argued that this is due to the buoyant flow streaming backward (negative Xdirection), which interacts with the free shear flow past the rib: both streams create a pattern which is little affected by curvature, reminiscent of a mixing layer. The present data illustrate the great potential of the chosen EUROMECH Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation approach for exploring unsteady turbulent flows in rotating frames, both for complex configurations and for fundamental studies.

  DEN BRAEMBUSSCHE, Filippo COLETTI, Irene CRESCI & Tony ARTS Turbomachinery and Propulsion Department, von Karman Institute for Fluid Dynamics Chausse de Waterloo 72, 1640 Rhode-Saint-Gense, Belgium 1 Background

Figure 1 :

 1 Figure 1: Mean streamlines. Flow from left to right.

Figure 2 :

 2 Figure 2: Iso-contours of signed swirling strength in the space-time domain. Flow in the positive X-direction. Time advances in negative Z-direction. Blue (red) indicates vortices which rotate clockwise (counter-clockwise) in the X-Y plane.


Cheick Tidjane Kone

Michael David
email: michael.david@cran.uhp-nancy.fr

Francis Lepage
email: francis.lepage@cran.uhp-nancy.fr

Multi-channel Clustering Algorithm for improving Performance of Large-Scale Wireless Multi-Sink Sensor Networks

Keywords: C.2.1 [Computer-Communication Networks]: Network Architecture and Design -Network topology; C.2.3 [Computer-Communication Networks]: Network Operations -Network management Algorithms, Performance Wireless sensor networks, Multiple sinks, Multiple channels, Clustering algorithm, Network performance

This paper presents a simple and distributed clustering algorithm suitable for large-scale wireless sensor networks (WSNs) consisting of several thousands of sensor nodes and few sink nodes. A two-tiered hierarchical architecture is used to increase scalability and ensure performances and durability of such a system: Level 1 called sensor network is partitioned into several equilibrate clusters with one leader or sink by cluster; Level 2 also called sink network is composed by N sink nodes placed in planned manner into monitored region and is connected through IEEE 802.11 radio interfaces. A multi-channel system is used to create a cellular structure by assigning one frequency channel per cluster. We use simulation technique to evaluate and compare the impact of two distributed schemes (a single channel one and a multi-channel one) on network capacity like traffic load, energy consumption, medium access delay, end-to-end delay and data delivery ratio.

INTRODUCTION

Wireless Sensor Networks (WSNs) typically have little or no fixed infrastructure as compared with wired network. Their deployment is often random and could be done into large monitoring region. Assuming for example that a chemical or nuclear plant explodes and that civil security Department want to quickly and accurately monitor the area. A spreading of sensor nodes with an average density of one per 100 m2 leads to deploy 10000 nodes in a monitoring area of one square kilometer but one million nodes if this area is 10 square kilometers. In such an application case where the network density is high, the whole network traffic must remain limited to be able to ensure a high availability of such large-scale networks. Moreover, the lifetime duration, the connectivity and the scalability of the network are important design goals to ensure network performance over time and provide services at every part of the deployment area [START_REF] Akyildiz | Wireless sensor networks: a survey[END_REF], [START_REF] Verdone | Wireless Sensor and Actuator Networks: Technologies, Analysis and Design[END_REF]. Hierarchical architectures or clustering techniques are used to solve the scalability problem. Moreover, the use of multiple sink nodes in large-scale WSNs have been advocated as one possible way to facilitate the manageability of the network, to shorten the communication path in number of hops and to scale the network capacity and its lifetime duration. So, a WSN with multiple sinks could be partitioned into several groups or clusters with one leader (a sink node) per cluster. Also, a lot of works focus on the number and/or positions of sinks according to the application's needs in order to maximize the network capacity and its lifetime [START_REF] Mclaughlan | Coverage-based clustering of wireless sensor and actor networks[END_REF], [START_REF] Oyman | Multiple sink network design problem in large scale wireless sensor network networks[END_REF].

For significantly optimizing energy and communication efficiency in the clusters, the multi-channel system can be used because it reduces the radio interferences, the contention to access the wireless channel and the bandwidth occupation [START_REF] Wang | Spatial channel reuse in wireless sensor networks[END_REF]. So, we can use the cellular network approach that assigns one frequency channel per cluster while respecting the spatial channel reuse. So, all radio transceivers within a cluster operate on the same channel.

The main contribution of this paper is to propose a simple and distributed clustering algorithm which use multiple sinks and frequency channels to improve the performance and ensure a high availability of large-scale WSNs (in order of ten thousands of nodes). It creates a cellular structure by assigning one sink and one frequency channel per cluster. Then, we show that this algorithm by using multiple channels provides better network performances (like traffic load, energy consumption, medium access delay, end-to-end delay and packet delivery ratio) compared to the single channel scheme.

The rest of the paper is organized as follows: Section 2 gives an overview of the related work and specifies our objectives. Section 3 defines the network model. Section 4 describes our clustering algorithm using multiple sink nodes and multiple channels. Section 5 evaluates the performance of our algorithm. Conclusion and future work are given in Section 6.

RELATED WORK AND OBJECTIVES

In this section, we summarize the related work regarding the problem of heterogeneous WSNs and distinguish our approach from other works. Some works focus on a multi-sink WSN, consisting of sensor and sink nodes. In a multi-sink WSN, the number of sinks represents the number of clusters. Authors discuss the optimal number and the sink nodes' location for maximizing the coverage and the lifetime of the network. Reference [START_REF] Oyman | Multiple sink network design problem in large scale wireless sensor network networks[END_REF] proposes an off-line and centralized solution for calculating the sinks' optimal number and their location while having a prior knowledge on the network's lifetime constraint and all sensors' locations. Another solution based on a linear programming formulation is proposed in [START_REF] Kim | Optimal Multisink Positionning and Energy-efficient Routing in Wireless Sensor Networks[END_REF]. By using a centralized graph partitioning algorithm, the reference [START_REF] Slama | Multiple Mobile Sinks Deployment for Energy Efficiency in Large Scale Wireless Sensor Networks[END_REF] proposes to partition the network into clusters of size relatively equal in number of nodes while having also a prior knowledge of all sensors' positions. Then, they randomly define one sink per cluster and each sink can move in the area of its cluster.

For maximizing the network lifetime too, another type of heterogeneous WSN called "Wireless Sensor and Actor Network" is used in [START_REF] Akkaya | Clustering of wireless sensor and actor networks based on sensor distribution and connectivity[END_REF]. Actor nodes collect the data of sensor nodes and have two radio transceivers like sinks. Unlike sink nodes, actor nodes can perform other actions (such as mechanical actions) and can move. They propose to determine the number of cluster-heads (actor nodes) and their location through a mechanism based on the k-hop Independent Dominating Set (k-IDS). They suppose that all sensor and actor nodes know their location (with a mechanism like GPS). After having constructed a k-IDS, each actor must move in order to be relocated close to a dominator node. This process is highly expensive in terms of message exchanges.

We remark that the main objectives of these above proposals are to find the locations and/or the number of sink/actor nodes which maximize the network's lifetime. Moreover, their approach is centralized and the location system of all sensor nodes is a strong assumption. Such a location system integrated on each node is highly expensive for large-scale WSNs. Note that most of these proposals are tested for hundreds of nodes and that no paper describes a test bed for larger WSNs (our target is 10000 nodes).

Our approach is different from previous ones because our goal is to improve the performance of large-scale networks and also extend its lifetime duration but in a distributed manner and regardless of the location of sensor and sink nodes. In addition, data traffic is generated over network to address the network operation issue. We propose a distributed clustering algorithm which considerably reduces the traffic in large-scale WSNs. Then, we combine a multi-channel system to our approach because it has been advocated as one way of significantly scaling capacity and lifetime of the network [START_REF] Vassis | Performance evaluation of single and multi-channel actor communication for wireless sensor actor networks[END_REF]. Indeed, if the node density is very high, the level of interference is also very high, and the overall network capacity is compromised [START_REF] Gupta | The capacity of wireless networks[END_REF]. Thus, a cellular structure can be created by assigning one frequency channel per cluster while respecting the spatial channel reuse [START_REF] Katzela | Channel assignment schemes for cellular mobile telecommunication systems: a comprehensive survey[END_REF]. Note that a single radio transceiver might suffice for using dynamic channel switching [START_REF] Darties | A Divide-and-Conquer Scheme for Assigning Roles in Multi-Channel Wireless Mesh Networks[END_REF] and a radio transceiver switching delay has no impact on the network performance [START_REF] Kyasanur | Capacity of multichannel wireless networks under the protocol model[END_REF]. For instance, the latency of CC2420 radio transceiver to switch channel is about 200 μs.

NETWORK MODEL / ASSUMPTIONS

We consider a two-tiered heterogeneous WSN as illustrated in Figure 1. The level 1 called sensor network operates on 802.15.4 channels and the level 2 called sink network on a single 802.11 channel. All radio transceivers of network of level 1 have the same radio range. Unlike sensor nodes, we assume that sink nodes have less energy-constrained and have two radio transceivers: one for level 1 and the other for level 2. We suppose that the sensor nodes are randomly deployed to detect, track and transmit/relay some physical events to an end-user through the sink network. The sink nodes are assumed to be placed in planned manner on the area while forming a connected sink network. We assume that the sink network uses a MANET routing protocol such as OLSR to route the physical events to an end-user. We also assumed that all nodes in overall network are stationary, which is typical for WSNs. Each sink can construct a cluster at most k hops. For combining the multi-channel system to our clustering protocol, we suppose only sinks know their frequency channel Cx which will be assigned to their cluster. This will enable sensors of a same cluster to switch their radio transceiver on the channel dedicated to cluster in order to have different intra-clusters communications (Figure 1). Note that four 802.15.4 frequency channels (Cx) suffice to create a pattern and facilitate the spatial channel reuse.

PROPOSED ALGORITHM 4.1 Definition of Parent Rules

Let us consider Id(v) defining the identifier of node v. We use the MAC address as identifier because it is unique. Parent(v) represents the current parent of node v in a cluster-tree. It value is -1 when a node does not belong to a cluster-tree. Hops(v) denotes the current distance in number of hops between the node v and its clusterhead (a sink); Hops(v) = -1 if node v does not belong to a cluster-tree; Energy(v) represents the energy consumed rate of node v, i.e., the ratio between the consumed amount of energy and the initial amount of energy. The parameter λ ∈ ] 0; 1[ defines the predefined threshold for the energy consumed rate of nodes. The 1-Density(v) (also noted Density(v) defining the link density of node v, i.e., the ratio between the number of links between v and its 1-neigbors, the number of links between v's 1-neighbors and the number of node inside v's 1-neighborhood. We choose the link density criteria because, in [START_REF] Mitton | Self-organization in large scale ad hoc networks[END_REF], the authors showed that it provides a better stability of the virtual topology than other metrics (degree, identity) facing the topological changes (i.e., a slight evolution in the neighborhood of a node). Weight(v) defining the weight of node v is a linear combination of Energy(), Hops(), Density(), Parent(), Id(). For comparing the weights, we use a lexicographical order of Energy(), Hops(), Density(), Parent(), Id(). So, this Weight function guarantees that two nodes have different weights. Given two nodes (u and w) inside the neighborhood table of a node v. Then, the node v selects as parent between both nodes u and w the node having the highest weight. For example the node v will pick u as parent if Weight(u) > Weight(w), i.e. if and only if : •

• Energy(u) ≤ λ ∧ Energy(w) > λ, ∨ • Energy(u) ≤ λ ∧ Energy(w) ≤ λ ∧ Hops(u) < Hops(w), ∨ • Energy(u) ≤ λ ∧ Energy(w) ≤ λ ∧
Energy(u) > λ ∧ Energy(w) > λ ∧ (Parent(v) = -1, i.e, node v had no parent) ∧ Id(u) < Id(v)

Cluster-tree Construction

We propose a clustering algorithm suitable for large-scale WSNs exploiting the capabilities of multiple sinks and channels to enhance the network performances. To construct a cluster-tree of the network, we define four states for a node (see Figure 2):

• Ordinary: the initial state of any node (except sink node) or a node does not belong to a cluster-tree. • Leader: the state of a sink. It is the root node or the chief of a cluster. • Member: node belonging to a cluster-tree.

• Border: node belonging to a cluster-tree and intermediate between clusters.

Figure 2. States changes' diagram.

The proposed clustering algorithm constructs k-cluster-tree where each sensor node within a cluster can reach the Leader of this cluster within at most k hops. The cluster-tree construction proceeds by waves and is message-triggered. We use only four broadcast messages types namely: CLUSTER_INITIATE, CLUSTER_ACCEPT, CLUSTER_FINISH and HELLO. Our heuristic proceeds as follows. First, each node performs the neighborhood discovery process by periodically sending a packet HELLO to signal its presence to close nodes, construct and maintain its neighborhood table. Thus, each node will be able to compute its own metric (as the link density) and update the information (identifier, weight, etc.) of its close neighbors. After a delay, each leader (i.e., sink node) passes through transition 1 (Figure 2) and initiates the construction of its cluster-tree by inviting (through a broadcast of a message CLUSTER_INITIATE) its close neighbors to join the cluster to which it is the chief. Then, it triggers a timer to retransmit this message if it has received no messages CLUSTER_ACCEPT during this time. The message CLUSTER_INITIATE contains the information about the sink node (its identifier that is also the identifier of its cluster, state, channel assigned to its cluster).

Upon receiving a CLUSTER_INITIATE message from a neighbor leader, sensor records the information about leader, selects it as its parent, passes through the transition 2 (Figure 2) and becomes member node. Then, it broadcasts a message CLUSTER_ACCEPT in order to advertise to the leader its membership to the cluster and also invite other close nodes to join this cluster. Then, it triggers a timer to retransmit its message CLUSTER_ACCEPT if during this time, it has received no messages CLUSTER_ACCEPT coming from nodes which are farther (in number of hops) of leader of this cluster than it. CLUSTER_ACCEPT message contains the information about the source (identifier, state, density, identifier of cluster, channel assigned to cluster, parent, number of hops from clusterhead and consumed energy rate). Based on the received information of CLUSTER_ACCEPT message, any node (sink or sensor) executes a procedure according its state and passes through any transition of figure 2: • If it is a leader (or sink) then it updates the node's information which has sent this message. It stops the retransmission of message CLUSTER_INITIATE. Finally, it passes through transition 4 and switches the current channel of its radio transceiver on the frequency channel assigned to its cluster if it had not yet changed. • If it is a sensor node and is member or border then:

-It updates the node's information which has sent this message. Then, it will select as parent, the node having the highest weight between its current parent and the node which has sent this message. the end to this cluster-tree. Upon receiving the message CLUSTER_FINISH, any node executes a procedure according its state and passes through any transition. For example:

• If it is a leader node then it updates the information of the node which has sent this message. • If it is a sensor node and is member or border then:

-It updates the node's information which has sent this message. Then, it stops, according its state, the retransmission of messages CLUSTER_ACCEPT or CLUSTER_FINISH. -It passes through transition (transition 5 if it is member or through transition 6 if it is border). Then, it switches the current channel of its radio transceiver on the frequency channel assigned to the cluster if it had not still changed.

Note that some nodes (member or border) cannot switch their radio transceivers on the channels assigned to the clusters owning them. Thus, these nodes will disconnect of the clusters over time and will become Ordinaries. As these nodes know the frequency channels assigned to their clusters, they must switch their radio transceivers on these channels before their disconnection. Then, they can reconnect to a cluster upon receiving of HELLO messages and thereafter can become member or border nodes.

PERFORMANCE EVALUATION OF THE SINGLE AND MULTI-CHANNEL SCHEMES 5.1 Simulation Parameters

We assess the performance of our approach on the test bed developed using OPNET Modeler 15.0 [START_REF][END_REF]. In the network of level 1, we use the physical layer defined in the IEEE 802.15.4-2003 standard which operates at the 2.4 GHz frequency bands with a data rate of 250 Kb/s, a radio range of 50 meters. The Mac sub-layer does not implement the 802.15.4 MAC protocol but a non-slotted CSMA/CA non-beacon mode that performs better in such configuration. We use the power consumption model from the Open-ZB web page [16]. This model is based on the Crossbow MICAz motes' specifications [START_REF]Crossbow motes and Stargate-Xscale[END_REF]. The network of level 2 implements the IEEE 802.11g-2003 standard with a data rate of 1Mb/s, a radio range of 500 m, in DCF mode and OLSR as routing protocol. The network consists of one end-user, 9 sinks uniformly placed and several static sensors (between 1000 and 10000 to address the scalability issue) randomly distributed over a same area of 1000 m by 1000 m. Each sink must construct a cluster at most 5 hops. All simulations are run during 7200s and over the same area. HELLO packets are sent periodically within a time interval [f HELLO -5, f HELLO + 5]s with f HELLO = 60s. For reducing network congestion, we use a crossing time of packets within the queue of the MAC layer of sensors to remove the packets delayed. Its value is 2s. The timeout for a sensor to switch from a state to another is 5s. In order to estimate the end-to-end delay and the packets delivery ratio of both schemes, we only consider the network consisting of 10000 sensors, 9 sinks and one end-user. Among the 10000 sensors, we randomly pick sources (between 200 and 1000 to address the network operation issue) which perform independent event readings. The packet generation process performed independently on each source follows an exponential distribution of average 300s, where an event-driven monitoring application is assumed. Application packet payload is 5 bytes. The results obtained are within 95% confidence interval. We study the performance of our algorithm on both schemes (single channel one and multi-channel one). Figure 3 depicts the control traffic in bits received per sensor per second versus the network density. The control traffic consists of messages HELLO, CLUSTER_INITIATE, CLUSTER_ACCEPT and CLUSTER_ FINISH. We remark that the control traffic received per each sensor remains acceptable in both schemes, even with an increase of the network density. However, the control traffic observed in both schemes rapidly grows when the number of sensors increases. This suggests saturation for larger network (100000 nodes) used in the same conditions. The network traffic must remain limited to be able to ensure a high availability of such large-scale networks (low losses of packets and low delays). So, if the network density increases, the radio range of sensors or the number of clusters (sinks) should be adjust to reduce the network traffic. On the same figure, we notice that the control traffic received per each sensor is significantly lower (~18% less) in the multi-channel scheme compared to the single-channel scheme. In fact, after having switched on another frequency channel, a node has fewer neighbors (~12% less of neighbors than a node in a single channel network in our simulation results) and therefore receives fewer packets. So, the multi-channel scheme minimizes better than the single-channel scheme the bandwidth consumption, the energy consumption (Figure 4) and the medium access delay (Figure 5), since these above properties depend of the network load and the node degree (number of neighbors) over time. On the Figure 5, we remark that during 1 hour, a node in a multi-channel network consumes on average 7% less energy than a node in a single channel network. On the Figure 5, the media access delay is approximately similar in both schemes for the networks with low density. But, we also observe a difference of media access delay when the network density is high. The media access delay varies very little in the multi-channel scheme even with an increase of the network density, whilst it rapidly grows in the single channel scheme. Thus, it is appropriate to implement a multi-channel system unlike a single channel when the network density is high. Figure 6 illustrates the average end-to-end delay of data traffic from some sensors to an end-user. This delay in the multi-channel network is better (at almost 2 ms whatever the number of source nodes) than in the single-channel network because the multichannel scheme assures a low media access delay compared to the single channel scheme (Figure 5). This shows that multi-channel scheme performs better than the single channel one. However, the overall performances observed on both schemes are good for such a WSN (10000 nodes) since they guarantee a small overall system delay (under 12 ms) and a good delivery packets ratio over 93%.

Simulation results and analysis

CONLUSION AND PERSPECTIVES

In this paper, we used multiple sinks and channels to ensure the performance of the large-scale WSN (our target was 10000 sensor nodes). We propose a simple and distributed clustering algorithm which creates a cellular system by assigning one sink and one frequency channel per cluster. Then, we discuss the performance of both schemes (a single channel one and a multi-channel one) based on this algorithm. Simulation results have shown that the multi-channel scheme performs generally better than the single channel scheme. The multi-channel scheme has a considerable influence on the energy consumption of the overall network. Moreover, it assures a low media access delay compared to the single channel scheme, even with a high network density. It also guarantees a smaller overall network delay than the single channel scheme. Thus, the multi-channel approach contributes in enhancing the performance and the lifetime duration of network.

As future works, we plan to evaluate the performance of our algorithm on a largest network topology (100000 sensor nodes). A theoretical study should be done to determine the upper working limit of such large networks. We also intend to propose a distributed algorithm which dynamically assigns one frequency channel per cluster when the sinks are assumed randomly deployed into a monitoring region.

Figure 1 .

 1 Figure 1. Proposed topology inside a large-scale WSN with multiple sinks and multiple channels.

  Hops(u) = Hops(w) ∧ Density(u) > Density(w), ∨ • Energy(u) ≤ λ ∧ Energy(w) ≤ λ ∧ Hops(u) = Hops(w) ∧ Density(u) = Density(w) ∧ (u = Parent(v), i.e, node u was the parent of node v just before this comparison), ∨ • Energy(u) ≤ λ ∧ Energy(w) ≤ λ ∧ Hops(u) = Hops(w) ∧ Density(u) = Density(w) ∧ (Parent(v) = -1, i.e, node v had no parent) ∧ Id(u) < Id(v), ∨ • Energy(u) > λ ∧ Energy(w) > λ ∧ (u = Parent(v), i.e, node u was the parent of node v just before this comparison) , ∨

Figure 3 .

 3 Figure 3. Average control traffic received per sensor node per second versus the network density: single channel vs multichannel.

Figure 4 .

 4 Figure 4. Average energy consumed per sensor node per hour versus the network density: single channel vs multi-channel.

Figure 5 .

 5 Figure 5. Average medium access delay of sensors network versus the network density: single channel vs multi-channel.

Figure 6 .

 6 Figure 6. Average end-to-end delay: single channel vs multichannel.

ACKNOWLEDGMENTS

This work was supported by OPNET Technologies, Inc. Through "Teaching with OPNET" program.


George Khujadze

Jean-Pierre Hickey

Martin Oberlack

Instabilities and transition in three-dimensional flows with rotation STREAMWISE ROTATING POISEUILLE FLOW: MODAL AND NON-MODAL STABILITY ANALYSES

A better understanding of transition to turbulence in a Poiseuille flow rotating about the streamwise axis is sought by investigating the stability of the flow. Using the classical linear modal analysis, we define the instability envelop and find that the rotation increases the exponential growth of the most unstable mode and, for high levels of rotation, we observe a re-stabilization of the flow. The influence of rotation on transient energy growth is also investigated as the set of linear equations is non-normal. We show that the energetic growth can be of the order of O 10 3 in the sub-critical region but for high Ro the non-normality of the equation set is reduced, resulting in a decrease of the maximal energetic growth. The maximal growth is achieved by the purely spanwise disturbances until a certain point where the maximal energy growth is caused by oblique ones.

Governing equations and geometry of the flow

The application of system rotation to canonical wall-bounded flows is of great academic and engineering interest. From an academic perspective, these flows provide unique insight into the influence of body forces on the flow. From an engineering point of view, the presence of rotating flow is ubiquitous in most turbo-machinery applications making the study of these flows of great practical relevance. To the knowledge of the authors, [START_REF] Oberlack | Group analysis, DNS and modeling of a turbulent channel flow with streamwise rotation[END_REF] were the first to study the streamwise rotating Poisseuille flow using Lie group analysis and direct numerical simulations (DNS). Althought the flow setup share some similarities with the classical spanwise rotating channel flow, the streamwise rotation, in the present case, does not break the centerline symmetry. More recently, an experimental investigation was conducted at the RWTH-Aachen ( [START_REF] Recktenwald | Comparison of DNS and PIV data of turbulent channel flow rotating about the streamwise axis[END_REF]) and a DNS study was performed to verify the results by [START_REF] Oberlack | Group analysis, direct numerical simulation and modeling of a turbulent channel flow with streamwise rotation[END_REF]. A study conduced by [START_REF] Masuda | Instabilities of plane Poiseuille flow with a stream-wise system rotation[END_REF] shed light on the non-modal stability characteristics of this flow. However, their study overlooked the importance of non-modal transient growth in the flow. In the present work, we investigate the modal and nonmodal stability characteristics of a streamwise rotating Poiseuille flow (figure 1). The non-dimensionalised Navier-Stokes and the continuity equations for the incompressible rotating channel flow are:

∂u ∂t + (u • ∇) u = -∇P + 1 Re ∇ 2 u -Ro × u ∇ • u = 0
Reynolds and Rossby numbers are defined in the following form: Re = U max h/ν, Ro = 2Ω 1 h/U max . ν is the kinematic viscosity, U max is the maximal base flow velocity and h is the half-height of the channel. We consider only a streamwise rotation (see figure 1), hence Ω = (Ω 1 , 0, 0).

Exponential and transient growths

The numerical calculations were performed in MATLAB using a spectral collocation method based on Chebyshev polynomials modified from the code developed by S. Reddy ( [START_REF] Schmid | Stability and transition in shear flows[END_REF]). The original code was extended to include the additional system rotation.

By setting Re = 100 and evaluating the effective perturbations for different rates of rotation in the figures 2 we observe the appearance of spectral instabilities from a Rossby number of Ro = 0.246 at α ≈ 0.55 and β ≈ 2.0. By increasing the Rossby number to Ro = 0.5 (right plot in the figure 2), one can see a great range of perturbations (represented by the shaded area) that are able to destabilise the flow.

In modal analysis, a conclusion on the stability must be made without regards to the evolution of the problem at the near initial state. We present the results of non-modal analysis of the flow and determine the possible energetic growth the the perturbations. Non-modal analysis reveals that streamwise rotation allows for a significant level of transient energy growth to occur (up to O(10 3 )) in exponentially stable flow. At high Ro, the transient growth is significantly reduced as the linearised governing equation set reaches an (nearly) anti-symmetric form for which the non-normality of the operator set is greatly reduced. For a given Ro, the initial conditions leading to the maximal energy growth change from a 2D (streamwise independant) to a 3D perturbation at a given Reynolds number. As the rotational terms are streamwise dependant, the shift represents the minimal Reynolds number at which the transient growth is caused primarily by the rotational body force.

Figure 1 :

 1 Figure 1: Streamwise rotating Poiseuille flow.

EUROMECHFigure 2 :Figure 3 :

 23 Figure 2: Re = 100. Results corresponding to Rossby numbers Ro = 0, 0.246, 0.5 are presented on left, middle and right plots correspondingly. The contour lines represent the value of the maximal imaginary eigenvalue for every perturbation couple. The first spectral instability for Re = 100 is seen at a rotation of Ro = 0.246.


Marc Buffat

Lionel Le Penven

Anne Cadiou

Julien Montagnier

BY-PASS TRANSITION DESCRIPTION USING AN ORTHOGONAL DECOMPOSITION OF THE VELOCITY FIELD

In the last two decades, theory and computer simulations have led to significant advances in the understanding of boundary layer bypass transition induced by free-stream disturbances [5]. It has been shown that transition can be initiated by the growth of velocity perturbations taking the form of streaks elongated in the streamwise direction and modulated in the spanwise direction. As the streaks grow downstream, they become susceptible to high-frequency secondary instability and breakdown to turbulence finally occurs. The physical process explaining the emergence of streaks is known as the lift-up effect and is the result of interactions between streamwise vorticity and the boundary layer shear. Streaks can be generated by perturbations inside the boundary layer or by free-stream turbulence. The secondary instability can be initiated by free-stream turbulence, by selecting a high-frequency perturbation mode [8] or by streak interactions [2]. Numerical analysis of bypass transition can be found in [6], [7]. The main objective here is to study by numerical simulations (and to describe) boundary layer bypass transition between two parallel walls using an orthogonal decomposition of the velocity into two solenoidal fields.

The numerical simulations are performed using a spectral Galerkin projection method based on an orthogonal decomposition of the velocity into two solenoidal vector fields [3]. Since, by construction, the two orthogonal vector fields of the decomposition are associated with the Orr-Sommerfeld and the Squire modes of the linear stability theory respectively, the method makes it possible to evaluate kinetic energy transfers due to the coupling between these two modes and their interactions with the base flow. The code, developed in our team, is parallelized by an hybrid OpenMP/MPI approach and runs efficiently on ten thousand CPU cores. The studied problem is a plane channel flow with thin boundary layers and the computational domain is supposed to be far enough from the inlet section such that the potential effect of the leading edges can be neglected.

Different approaches can be used to select the disturbance that generates the streaks. Using the variational approach of [4], the optimal perturbation which sustains maximum temporal growth in the linear regime can be determined. This optimal perturbation consists of a spanwise wave with a streamwise wavenumber β ≈ 2/δ inside the boundary layer and is zero outside the boundary layer. Such an optimal perturbation can be used to model disturbances inside the boundary layer. Free-stream turbulence can be described as a superposition of modes of the continuous spectrum of the Orr-Sommerfeld-Squire equations. To characterise the modes that generate streaks, Zaki et al [8] studied the forced Squire response to an Orr-Sommerfeld eigenfunction. They defined a coupling coefficient between an Orr-Sommerfeld eigenfunction and the Squire modes to characterize their interaction and the ability of the Orr-Sommerfeld mode to generate streaks.

To model the generation of streaks by free-stream turbulence, we follow a similar methodology adopted by [8], and study the transient response of the linearised equations to an Orr-Sommerfeld mode, chosen as initial condition. Using the same orthogonal decomposition of the velocity field as in the DNS code, the linear system of two equations is obtained using a weak formulation, and is equivalent to the classical Orr-Sommerfeld-Squire equations. Instead of using the two classical scalar fields: the normal velocity v and the normal vorticity η, this formulation allows us to analyse the solution in terms of two orthogonal velocity fields: the Orr-Sommerfeld (OS) u os (v) and the Squire (SQ) u sq (η) velocities, associated with two orthogonal velocity spaces W os and W sq . In our numerical approach, the eigenvectors are splitted into two distinct sets, depending on their orthogonality with the subspace W os : the set {u -} ⊂ W sq , associated with the Squire modes of the homogeneous Squire equation, and the complementary set {u + }. The projection u + os of u + in W os is associated with an Orr-Sommerfeld eigenmode and its projection u + sq in W sq with the coupled Squire eigenmode. The ratio r ⊥ = ||u + sq ||/||u + os || of these two projections characterize the non-orthogonality of an eigenvector u + with the set {u -}, a large value of r ⊥ indicating an important nonorthogonality (non normality). By analysing the transient response with an initial condition equal to u + os , we found that the most amplified u + os are associated to eigenvectors u + r with a normal to the wall wavelength (outside the boundary layer) of the order of 2δ. These eigenvectors u + r have large value of r ⊥ , indicating an important non-orthoganlity with {u -}. An optimal problem can be solved for this transient problem, and the optimal initial condition is similar to the optimal perturbation calculated with the optimal growth theory. This optimal perturbation consists mainly of an OS velocity (streamwise vortices inside the boundary layer) at the initial time, transformed after the transient time into a streamwise SQ velocity. However, as this optimal perturbation is zero ouside the boundary layer, it cannot be chosen as a model of free-stream turbulence. By analysing the expansion of this optimal perturbation in the eigenvectors basis, the optimal modes are found to be a combination of eigenmodes around the eigenvectors u + r having the maximum value of r ⊥ . A remarkable finding is that the shape of the transient response is nearly identical for an initial condition equal to the optimal initial condition (or the optimal mode) and for an initial condition equal to the W os projection of modes u + r , and that only the amplitude of the transient response depends on the particular initial condition. For all these initial conditions, the transient response corresponds to SQ velocity or streaks characterised by a large peak of the streamwise velocity component inside the boundary layer. Based on this analysis, we use henceforth the ratio r ⊥ to select the inlet perturbation in our simulations.

In the simulations, the inlet boundary layers are thin compared to the distance 2h between the two parallel walls (δ/h = 0.05). The inlet Reynolds number, based on the displacement thickness, is equal to 344 (corresponding to Re x = 40000). The first perturbation is chosen as an oblique wave u + os with β δ = 2 and α δ = 0.04 having the largest value of r ⊥ .To initiate the transition of the streaks, a second perturbation with a larger streamwise wavenumber α δ ≈ 1 and a smaller spanwise wavenumber β δ ≈ 0.3 is introduced at the inlet as in [8]. This second perturbation is a vector u + having a small value of r ⊥ Without external perturbations, the flow remains laminar and the classical Blasius skin friction coefficient is recovered. Using the two previous perturbations, by-pass transition is induced in the boundary layers. As the boundary layer thickness is small compared to the channel height, similar results are obtained to those presented by [8] and [7] for a boundary layer over a flat plate. Using the orthogonal decomposition of the velocity, specific information about the structure of streaks of finite length can be obtained (see Figure 1). Specifically, the streaks are found as the sum of two orthogonal contributions, one part is the SQ velocity field, which is a streamwise oriented contribution representing 60% of the total kinetic energy, the other part is the OS velocity field which is a spanwise invariant term with 40% of the kinetic energy. In the early steps of the transition, the secondary instability affects mainly the SQ streamwise velocity, whereas the OS streamwise velocity remains almost unaffected. In the transition region, strong oscillations of the streaks are observed in the spanwise direction. In that region, the plot of the streamwise perturbation (lower plot in Figure 1b) clearly shows spanwise oscillations of the low speed streaks on top of the boundary layer. Figure 1b shows that the instability mainly affects the SQ streamwise velocity, whereas the OS streamwise velocity remains almost unaffected. At the early stage, before breakdown, this instability occurs through perturbations of the SQ streamwise velocity and looks very similar to the transverse instabilities studied by [1].

EUROMECH

  Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation (a) x = 4h (b) x = 6.9h

Figure 1 :

 1 Figure 1: Instantaneous contours of streamwise velocity perturbation displayed using the orthogonal decomposition in two cross sections: (a) in the streaks region and (b) near the turbulent burst. From top to bottom: SQ velocity, OS velocity, total perturbation.


Jorge A Morales

Wouter Bos

Instabilities and transition in three-dimensional flows with rotation TRANSITION OF MHD FLOWS IN CONFINED GEOMETRIES WITH HELICAL BOUNDARY CONDITIONS

Introduction

The present work reports on a study of flows of conducting fluids in confined geometries, driven by magnetic boundary conditions. The visco-resistive magnetohydrodynamic equations are solved using a pseudo-spectral parallel DNS solver combined with a volume penalization method. The volume penalization technique, which is an immersed boundary method, allows in the present case to use other than periodic boundary conditions in a pseudo-spectral approach. In particular, this type of methods is characterized by a high flexibility in changing the geometry of the considered flow. The use of the volume penalization method was only recently validated in MHD flows [START_REF] Neffaa | The decay of magnetohydrodynamic turbulence in a confined domain[END_REF], and in the present communication we will present results in three-dimensions. As a validation, section 2 contains results of three-dimensional Taylor-Couette flow. Section 3 reports on the flow induced in a conducting liquid by the presence of an imposed helical magnetic field.

2 Validation: simulations of three-dimensional Taylor-Couette flow Three-dimensional simulations of Taylor-Couette flow are performed at different resolutions for different Reynolds numbers. For the geometry studied the critical Reynolds number, above which Taylor-vortices appear, is known to be around 68 [START_REF] Guermond | Nonlinear magnetohydrodynamics in axisymmetric heterogeneous domains using a Fourier/finite element technique and an interior penalty method[END_REF]. We observe this bifurcation at Re = 87 for our simulation at a resolution of 128 3 gridpoints. In Figure 1 we show the vorticity field which clearly illustrates the presence of counter-rotating vortices and their associated boundary layers. The results improve with resolution. For example, a critical Reynolds number is found to be around 72 for a 256 3 grid-point simulation. The bifurcation diagram is shown in Figure 2. Clearly, the method approaches rapidly the analytical value and any precision can be obtained if sufficient computational resources are available.

MHD dynamics in cylindrical geometry

The main goal of our investigation is the understanding of the velocity fields which are induced in plasmas or conducting fluids by the magnetic field boundary conditions. The present approach is thereby an extension of existing studies by Montgomery and coworkers [START_REF] Shan | Nonlinear magnetohydrodynamics by Galerkin-method computation[END_REF]. Using the penalization method we impose no-slip boundary conditions on the velocity field, and the magnetic field at the wall is imposed to be helical. The pinch ratio Θ, defined as the ratio between the imposed poloidal magnetic field and the imposed axial field (Θ = B θ /B z ) is varied and its influence on the flow is monitored. At very low values of the pinch ratio, the velocity field tends to its static limit. At intermediate pinch ratios, a laminar helical velocity field is observed in Figure 4. The flow is essentially axial and it is composed of two pairs of helical structures, moving in the positive and negative axial direction. We also present the kinetic energy evolution which shows that at long times the kinetic energy tends to a finite and constant value (Figure 3). Increasing the pinch-ratio, the laminar flow bifurcates and a multi-mode turbulent flow is observed. A calculation was performed for a pinch ratio Θ = 19.4 (Re rms = Rm rms ≈ 114). The turbulent behavior is illustrated by showing in Figure 5 the energy evolution which shows a nonstationary, fluctuating behavior. Figure 6 displays less organized streamlines than the laminar flow. Helical modes are still present but many other modes are excited as well.

Further work will consider the influence of breaking the symmetry of the geometry on the flow patterns which are generated, both in cylindrical and toroidal geometry.

Figure 1 : 2 ReL2

 12 Figure 1: Taylor counter-rotating vortices, azimuthal vorticity colormap

Figure 2 :

 2 Figure 2: Bifurcation diagram for Taylor-Couette flow

EUROMECH

  Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation

  energy for Θ = 1.94 and N = 128 Ek Emag

Figure 3 :

 3 Figure 3: Energies for Θ = 1.94

  energy for Θ = 19.4 and N = 128 Ek Emag

Figure 5 :Figure 6 :

 56 Figure 5: Energies for Θ = 19.4

Acknowledgments: The authors acknowledge interaction and discussion with Alexandre Delache, Fabien Godeferd, Clément Jause-Labert, David Montgomery and Kai Schneider.


Mukund VASUDEVAN M Ehtisham Siddiqui

Benoît Pier

Julian Scott

Alexandre Azouzi

Roger Michelet

Christian Nicot

Instabilities and transition in three-dimensional flows with rotation EXPERIMENTAL CHARACTERISATION OF TURBULENT FLOW R ÉGIME IN THE THREE-DIMENSIONAL ROTATING-DISK BOUNDARY LAYER

Fully developed turbulent flows are known to display some universal features when bounded by a solid wall. The flow due to a large disk rotating in otherwise still fluid undergoes transition from laminar to turbulent dynamics at a nondimensional radius R ≃ 500. The turbulent régime prevailing beyond this critical radius is unsual in several respects [4,3,2,1]: it is sandwiched between a solid moving surface and the outer fluid at rest, and it develops in the radial direction while the main flow and shear components are in the azimuthal direction.

The aim of the experimental work presented here is to characterise some features of this particular turbulent boundary layer.

Experimental setup

The experimental setup consists of a 50 cm diameter glass disk that is rotated at constant angular speeds, up to 1500 rpm. Disk rotation and viscosity produce a three-dimensional boundary layer of constant thickness δ = ν/Ω d (a typical value is δ = 400 µm), where ν is the kinematic viscosity and Ω d the disk rotation rate. Non-dimensional (based on δ) radial and azimuthal coordinates R and Z are introduced; non-dimensional azimuthal velocities V are used, based on the local linear disk velocity RδΩ d .

Flow measurements are carried out by constant-temperature hot-wire anemometry. The hot wire is positioned by a computer-controlled high-precision two-axes traversing mechanism and provides azimuthal velocity time series. These time-series are processed to obtain the flow characteristics, in particular the mean-flow velocity profiles and the spectral analysis.

Mean-flow velocity profiles

Azimuthal mean velocity profiles are obtained by averaging the hot-wire measurements for 100 disk revolutions at each location. From these measurements it is observed that the mean-flow profiles very closely follow the analytic profile for R ≤ 480 and that weak mean-flow corrections prevail over the range 490 ≤ R ≤ 540. For R ≥ 550, strong mean-flow distortions develop that progressively extend to beyond Z = 15 by R = 600. Such boundary-layer thickening is characteristic of the development of a turbulent boundary layer. In figure 1, meanflow velocities for 550 ≤ R ≤ 650 are plotted against Z using a logarithmic scale. It is found that the velocity profiles are nearly linear in log Z for R ≥ 600, which is the characteristic low of the wall of fully turbulent boundary layers. Velocity profiles for R ≤ 590 are not linear in log Z, suggesting that the flow is not yet fully turbulent and still in the transitional régime. Thus, we conclude from the mean-velocity measurements that the flow begins transition around R ≃ 540 and is fully turbulent by R ≃ 610.

Spectral analysis

Fourier spectra are calculated to show the frequency content of the azimuthal velocity time series at different non-dimensional radii and disk normal positions. In figure 2, spectra corresponding to the transitional and fully turbulent régimes are plotted using a log-log scale to bring out a possible power-law behaviour. For R = 570, the modal peak and its harmonics is still apparent, whereas at R = 600 and above, the peak is absent and there is a linear range extending over a good fraction of a decade in frequency upwards from where the peak would occur. The figure shows linear fits to the data in the range 30 ≤ ω ≤ 200 for R ≥ 600. Such linear behaviour corresponds to spectral power laws with exponents equal to the slopes of the lines, listed in the table.

For reference purposes, the figure also shows a straight line with the Kolmogorov slope -5/3. It is seen that these power-law exponents are not far from the Kolmogorov value, particularly at the largest values of R where the turbulence is most developed. Also apparent from the table, the power-law exponent depends on both R and Z, tending to decrease in absolute value as R increases and to increase with Z.

Discussion

A possible explanation of these variations, more qualitative than quantitative, refers to the physics of the turbulent energy cascade from large to small scales which forms the basis of Kolmogorov's theory. Large scales, here corresponding to frequencies at and below ω ≃ 30, break down to smaller and smaller scales, which leads to the -5/3 law by dimensional analysis when assuming a constant rate of energy supply from the large scales. Imagine following a packet of turbulence as it is convected by the mean flow. If the energy supply rate from the large scales increases with time, the energy flux was lower in the past and, given that the cascade takes a certain time, we expect the spectral energy of the small scales to be less than it would have been had the energy supply been constant, hence a steeper slope on the log-log plot. This suggests larger absolute values of the exponent in the presence of an increasing supply rate, and smaller ones for decreasing supply rate. In the table of figure 2, the absolute value of the Z-averaged exponent is seen to be higher than 5/3 for the first three values of R. This is to be expected since the turbulent intensity, and hence the energy supply rate, increases with R in the transitional region. Both R = 600 and R = 610 have exponents above 5/3 for all Z, suggesting an increasing supply rate at all distances from the disk. This is reasonable because the turbulence is developing up to R = 610. For R = 620 and R = 625, the exponent is less than 5/3 towards the disk and greater than 5/3 at higher Z. This suggests a decreasing supply rate near the disk and an increasing one further out.

Figure 1 :

 1 Figure 1: Mean-flow velocity profiles of the transitional and fully turbulent flow regimes plotted against Z on a logarithmic scale. Solid lines are linear fits to the data in the range 1 ≤ Z ≤ 16 for R = 600, 620 and 650 (slopes are respectively dV /dln(Z) = -0.139, -0.138 and -0.133). The root-mean-square error of each fit is around 0.005.

  Figure 2: Log-log plots of spectra in turbulent régime. For R ≥ 600, linear fits are computed for data in the range 30 ≤ ω ≤ 200 and shown together with a reference line of slope -5/3. The slope values shown in the table have a root-mean-square error of around 0.05.

		94 -2.03 -2.03 -2.09 -2.03 -2.21	-2.06
	610	-1.76 -1.85 -1.81 -1.89 -1.96 -1.96	-1.87
	620	-1.51 -1.59 -1.63 -1.73 -1.79 -1.88	-1.68
	625	-1.42 -1.45 -1.55 -1.68 -1.72 -1.85	-1.61


Shyam Sunder Gopalakrishnan

Benoît Pier

Arie Biesheuvel

Instabilities and transition in three-dimensional flows with rotation GLOBAL STABILITY ANALYSIS OF FLOW THROUGH AN ANEURYSM

The global linear stability analysis of flow through a model fusiform aneurysm is carried out numerically. A gaussian profile was used to model the wall channels to account for the bulge, that gradually straightens out to a uniform circular pipe in the upstream and downstream direction. The model configuration considered in the study is characterised by the bulge length and the maximum diameter of the bulge. These parameters for our model (Model A) were 3.0D and 2.0D consistent with the one used in [START_REF] Sheard | Steady inflow through a model aneurysm: global and transient stability[END_REF] and Model 3 in [START_REF] Salsac | Evolution of the wall shear stresses during the progressive enlargement of symmetric abdominal aortic aneurysms[END_REF], where D is the inlet pipe diameter. A steady parabolic velocity profile was imposed at the inlet along with a swirl prescribed using a sinuous function (w 0 sin(πr/R) where w 0 is a constant parameter used to quantify the inlet swirl and R is the inlet pipe radius) for the azimuthal flow component. The global stability analyses were carried out on base flows that were axisymmetric on which non-axisymmetric perturbations were allowed to grow. For Model A in the absence of inlet swirl (w 0 = 0), the flow was found to be weakly unstable to global eigenmodes of azimuthal modenumbers 4 and 5 at a Reynolds number (based on the maximum centerline velocity and pipe diameter) Re ≈ 9700. The least stable eigenmodes are concentrated at the downstream end of the bulge and were characterised by their slow oscillatory nature. The swirl prescribed at the inlet was varied by changing w 0 and was found to have a stabilizing effect on the stability properties with the flow becoming unstable at a higher Reynolds number. The results obtained are found to be consistent with those presented in [START_REF] Sheard | Steady inflow through a model aneurysm: global and transient stability[END_REF].

Motivation

Aneurysms are localized permanent arterial dilations caused due to disease or other complex processes that result in the weakening of the arterial walls [START_REF] Lasheras | The biomechanics of arterial aneurysms[END_REF]. Among these, fusiform aneurysms, that are commonly found in the abdominal aorta and are characterised by a bulge about the vessel centerline are the focus of the present study. In the recent years, there has been an increase in the studies carried out on blood flows from a fluid mechanics perspective. In particular, the studies on flows in constricted pipes designed to model stenosis have shed light on flow behaviour and stability in such geometries [START_REF] Griffith | Wake behaviour and instability of flow through a partially blocked channel[END_REF]. However, the research on flow through aneurysmal channels has been limited [START_REF] Salsac | Evolution of the wall shear stresses during the progressive enlargement of symmetric abdominal aortic aneurysms[END_REF][START_REF] Sheard | Flow dynamics and wall shear stress variation in a fusiform aneurysm[END_REF]. These studies comprised of experiments [START_REF] Salsac | Evolution of the wall shear stresses during the progressive enlargement of symmetric abdominal aortic aneurysms[END_REF] and numerical simulations [START_REF] Sheard | Flow dynamics and wall shear stress variation in a fusiform aneurysm[END_REF] in axisymmetric aneurysm models investigate the role of wall shear stresses in the rate of enlargement of aneurysms.

The present study aims to explore the stability of the flow in aneurysms for steady inlet flows and to quantify their behaviour by studying the nature of instabilities inherent in these geometries. The analyses were carried out in the same spirit as in the recent article by Sheard & Blackburn [START_REF] Sheard | Steady inflow through a model aneurysm: global and transient stability[END_REF].

Methodology

We use the incompressible Navier-Stokes equations written in cylindrical coordinates to simulate the flows considered in this study. Fully developed Poiseuille flow with maximum centerline velocity U was input upstream at the inlet. To compute the base flow, we adopt the procedure used by Marquet et al. [START_REF] Marquet | Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities[END_REF] wherein a timedependent simulation of the two-dimensional Navier-Stokes equations is first used to obtain an approximate solution of the steady solution at low Reynolds number where it is stable to two-dimensional perturbations. A Newton iteration is then applied to solve the stationary Navier-Stokes equations, starting the procedure with the approximate steady solution as an initial guess. The Newton algorithm requires the inversion of a non-symmetrical sparse matrix which is carried out by a direct method, the UMFPACK library being used to perform a sparse LU factorisation.

For the linear stability analysis, the base flows were assumed axisymmetric due to the general agreement in literature that blood flow in abdominal aortic aneurysms is dominated by axisymmetric flow structures [START_REF] Lasheras | The biomechanics of arterial aneurysms[END_REF]. The three-dimensional perturbation field was discretized in the azimuthal direction using Fourier modes, which are individually decoupled. This permits the stability analyses of individual azimuthal modenumbers (m) to be computed separately on the same axisymmetric domain as used for the base flow computations. The resulting eigenvalue problem was solved using a 'shift-and-invert' strategy as in [START_REF] Marquet | Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities[END_REF]. The two-dimensional stationary Navier-Stokes equations and the general eigenvalue problem are solved numerically by a finite element method. The spatial discretization is a mixed finite-element formulation using P2-P1 Taylor-Hood elements: six-node quadratic triangular elements with quadratic interpolation for velocities (P2) and three-node linear triangular elements for pressure (P1). The meshes as well as the discrete matrices resulting from the variational formulation of the problems and projected onto those meshes are generated with the software FreeFem++ (http://www.freefem.org).

Preliminary Results

The computations shown in figure 1 were carried out on Model A for w 0 = 0. Figure 1a shows the leading global modes characterized by their axial, radial and azimuthal velocity components (top to bottom) for azimuthal modenumber m = 4 at Re = 9700. The spatial distribution of the global modes shows that they are localized at the downstream end of the bulge. Figure 1b shows a typical eigenvalue spectrum obtained from the linear stability analysis at Re = 9700 for various values of m. The temporal growth rate of the eigenmodes σ is plotted in the y-axis and their frequency ω in the x-axis. The least stable eigenmodes are observed for m = 4, 5 with a non-zero frequency. We intend to present the results obtained from our stability analyses along with the effect of varying inlet swirl on Model A during the conference.

EUROMECH

  Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation

  m = 1 m = 2 m = 3 m = 4 m = 5 (a) (b) Figure 1: (a) Global modes visualized by their axial, radial and azimuthal (top to bottom) velocity components. Parameter settings: Model A, m = 4, Re = 9700, w 0 = 0. (b) Typical eigenvalue spectrum in the (ω, σ) plane for different azimuthal modenumbers m. Parameter settings: Model A, Re = 9700, w 0 = 0.