Jean-Paul Thibault
email: jean-paul.thibault@inpg.fr

Lionel Rossi
email: lionel.rossi@netcourrier.com

Electromagnetic flow control: characteristic numbers and flow regimes of a wall-normal actuator

Electromagnetic (EM) flow control of boundary layer refers to the use of 'wall-flush' electrodes ( j, current density) and 'sub-surface' magnets (B, magnetic induction) used in combination to create local Lorentz body forces ( j × B). In the present application the working fluid is seawater. Close to the boundary wall, these j × B forces can act directly on velocity and vorticity. In this paper, the characterization of a wall-normal EM actuator (i.e. j × B forces are mainly wall-normal above the central axis of the actuator) is considered. An idealized inertial and integral approach leads to the definition of characteristic EM numbers in term of velocity, time, acceleration and length-scales. These numbers are useful in introducing an EM parameter similar to the Froude number. Furthermore, two asymptotic EM flow regimes, which depend on flow velocity and on EM forces intensity, are also discussed.

Introduction

The work presented in this paper was developed within the context of electromagnetic (EM) flow control in seawater where Lorentz forces are imposed near the wall by means of EM actuators. It is known from the literature that EM flow control can reduce turbulent intensity and drag [START_REF] Nosenchuck | The direct control of wall shear stress in a turbulent boundary layer[END_REF] as well as prevent flow separation [START_REF] Henoch | Experimental investigation of a salt water turbulent boundary layer modified by an applied streamwise magnetohydrodynamic body force[END_REF][START_REF] Weier | Boundary layer control by means of electromagnetic forces[END_REF].

The EM actuator is a novel concept that permits the direct application of local three-dimensional Lorentz forces within the flow. These local EM body forces are associated with additional forcing terms ( j × B) in Navier-Stokes equations. Typically, an EM actuator comprises a pair of wall-flush electrodes and a pair of sub-surface magnets. The configuration of electrodes and magnets on the wall surface is such that the curl of j × B is non-zero in the vicinity of the actuator [START_REF] Rossi | Contrôle électromagnétique d'écoulement en eau de mer[END_REF][START_REF] Rossi | ElectroMagnetic forcing in turbulence and flow control: analytical definitions of EM parameters and hydrodynamic characterisations of the flow[END_REF]. This means that the EM forces acting on the fluid near the wall can pump or deflect the flow as well as impose vorticity sources. Velocity and/or vorticity fields are therefore modified by EM control either directly during activation or indirectly due to the persistence of induced velocity (wall-normal component and wall jets) and vorticity [START_REF] Rossi | Investigation of wall normal electromagnetic actuator for seawater flow control[END_REF].

One of the key questions regarding this concept of flow control is the determination of length-scales and timescales appropriate to the EM forcing and their comparison to the mean flow scales. In the following the description of an EM actuator and the basic equations governing EM flow control in seawater are given. Following this, a number of characteristic parameters, derived from an idealized model are computed, and an EM Froude similarity is suggested and discussed. Finally, the EM Froude similarity is validated with experimental data using various electrical duty cycles and two different EM actuators.

Description of the EM actuator and basic equations

The combination 'wall-flush' electrodes ( j, current density) and 'sub-surface' magnets (B, magnetic induction) allows to create local Lorentz body forces ( j × B) within seawater boundary layers (see figure 1(a)). Close to the wall, these j × B forces are able to act directly on velocity and vorticity components. Grosso modo, above the centre of the EM actuator, the magnetic field lines and electric field lines intersect each other perpendicularly and are parallel to the wall. Therefore, the Lorentz forces generated by the interaction of these two fields are mainly normal to the wall (as in figure 1(b)). In most of the volume above the actuator, curl( j × B)i s non-zero, which results in the imposition of vorticity sources within the flow. These sources are distributed all around the edges of the EM actuator (figure 1(c)) [START_REF] Rossi | Investigation of wall normal electromagnetic actuator for seawater flow control[END_REF]. By using permanent magnets the intensity and sign of j × B forces is directly controlled by the electrical power supplied to the electrodes.

To begin the analysis, the appropriate equations are given in table 1. Seawater is an electrolyte, but it is idealized here as a medium with a poor bulk conductivity σ . The governing equations for the fluid are (1) continuity and (2) the Navier-Stokes equations including the extra EM term due to the Lorentz forces. The vorticity equation ( 3) is nothing more than the curl of (2). The existence of the right-hand side term: curl( j × B) demonstrates that EM forces can act as a vorticity source. Equation (4) for the magnetic induction, B, reduces to the Laplace equation in the steady state when µσ is very small. This corresponds to the use of permanent magnets and the very poor conductivity of seawater which gives a very low value to the magnetic Reynolds number (the ratio of magnetic convection to magnetic diffusion). The

div u = 0 (1) ∂B ∂t = curl(u × B) + 1 µσ ∇ 2 B ⇒∇ 2 B ≈ 0 (4) ρ du dt + ∇P + ρg = µ∇ 2 u + j × B (2) j = σ(E + u × B) → j ≈ σ E (5) ρ dω dt = ρ(ω •∇)u + µ∇ 2 ω + curl( j × B) (3) div B = 0 and div j = 0 (6)
constitutive equation for the current density j is given by Ohm's law [START_REF] Rossi | ElectroMagnetic forcing in turbulence and flow control: analytical definitions of EM parameters and hydrodynamic characterisations of the flow[END_REF] where u × B is the electromotive field and E the electric field imposed at the electrodes. In the present case, the current density must be relatively high in order to produce strong EM forces. In fact, due to the moderate induction offered by the permanent magnets, the imposed electric field E has to be much larger than the induced electric field u × B leading to the indicated simplifications. Finally, equations ( 6) express the conservation of magnetic induction and of electric current.

Both j and B are independent of the flow and consequently the EM force j × B also has the same property. The EM force distribution therefore depends only on the actuator geometry. In addition, B is produced by the permanent magnets and the EM force intensity is fixed by the electric power supply (applied current and time of activation).

The remarkable fact that the EM force distribution is independent of the flow, places EM flow control out of the conventional problem of magnetohydrodynamics (where J×B forces are directly dependent on the flow). In addition, this independence offers the possibility to design EM actuators aiming specific goals. For example, the size of an EM actuator might also be fitted to the size of a typical structure present in turbulent boundary layers (micro-actuator) (see Robinson [START_REF]Coherent motions in the turbulent boundary layer[END_REF], Adrian et al [START_REF] Tomkins | Vortex organization in the outer region of the boundary layer[END_REF]), as to a larger scale like the spacing between packets of structures (macro-actuator) (see Meng [START_REF] Meng | Engineering insight of near-wall microturbulence for drag reduction and derivation of a design map for seawater electromagnetic turbulence control[END_REF], Zoo [START_REF] Zhou | Mechanisms for generating coherent packets of hairpin vortices in channel flow[END_REF], Rossi [START_REF] Rossi | Contrôle électromagnétique d'écoulement en eau de mer[END_REF]).

Description of the EM forces and characteristic scales

A description of the EM force for an actuator acting normal to the wall

Due to the special influence of the magnets and electrodes, the EM force field has a rather complicated three-dimensional shape. To a good approximation, the force distribution can be computed using the analytical solution given by Akoun and Yonnet [START_REF] Akoun | 3D analytical calculation of the forces exerted between two cuboidal magnets[END_REF]. Basically, it assumes a uniform distribution of electric charge on the electrodes and a uniform distribution of magnetic charge density on the magnets. The model also supposes a uniform electrochemical potential at the surface of the electrodes [START_REF] Boissonneau | Experimental analysis of couplings between electrolysis and hydrodynamics in the context of MHD in seawater[END_REF]. The electrochemical reaction results in a difference between the potential of the electrode and the potential of the flow very close to the electrode (over a distance corresponding to the diffusion layer). This over-potential depends on the electrode material and the local concentration of the reacting species and current density. Thus, in the present case a constant over-potential is assumed.

The EM forces computed via this analytical solution are three-dimensional and decrease rapidly with distance from the maximum value at the wall [START_REF] Rossi | Contrôle électromagnétique d'écoulement en eau de mer[END_REF]. More precisely, the computed three-dimensional EM force distribution (see figure 2(a)) shows that the forces are wall-normal above the centre of actuator and are three-dimensionally centripetal all around. At the centre of the actuator, the intensity of the EM forces decreases rapidly with distance from the wall. Quantitative values across the actuator are shown on figure 2(b) for different y values.

Characteristic scales

The characteristic length-scales of an EM actuator are clearly L E (electrode spacing) and L M (magnet spacing) on the wall (figure 3). However, unambiguous definition of the wallnormal length-scale h EM is difficult. It represents the vertical extent of the volume within which the EM forces act directly on the flow.

The approach of this question here involves the integration of the EM force over a volume bounded by L E , L M and a variable height y (as illustrated in figure 3). Symmetry considerations are such that the resultant of EM forces is normal to the wall. More precisely, the force distribution presents two planes of symmetry normal to the wall, namely: the one equidistant from electrodes and the one equidistant from the magnets. Three characteristic parameters of an EM actuator are defined and computed as follows:

(i) Mean EM acceleration g EM : [START_REF]Coherent motions in the turbulent boundary layer[END_REF] υ is the volume of integration (figure 3); f EM is the EM body force, ρ the fluid density, g EM is the mean integral EM acceleration and e y the unit vector perpendicular to the wall. (ii) EM velocity, V EM :

g EM = 1 ρυ υ f EM dυ = 1 ρυ υ j × B dυ = g EM e y
V EM = 2g EM y (8)
This represents the velocity that EM force could produce at a height y from the wall but neglecting viscosity or wall effects. (iii) Characteristic times:

T EM = 2y g EM (9)
This T EM corresponds to the time at which pumping sets in at a height y.

Equation ( 8) can be reorganized as in equation [START_REF] Zhou | Mechanisms for generating coherent packets of hairpin vortices in channel flow[END_REF]. This non-dimensional ratio can be interpreted as the EM equivalent of the Froude number with a value of 1.

V 2 EM 2g EM y = 1 ( 10
)
Due to the sharp drop in the strength of the EM forces, the integrated mean EM acceleration (g EM ) decreases with y. Figure 4 gives g EM values as a function of y.

With this definition, the EM parameters depend on the height of integration, y. As a result of the fact that the EM forces decrease rapidly with the distance from the wall (i.e. y) but never equal zero, the velocity is asymptotically limited as y increases. The EM velocity profile versus y is shown in figure 5. By taking 99% of this velocity limit, it is possible to define an non-arbitrary integration height and hence a characteristic wall-normal length, h EM , for an EM actuator. This formulation is similar to the one used to define the boundary layer thickness in ordinary fluid mechanics.

Here one can give an objective value to height of forcing while the force distribution presents, theoretically, an infinite extent decreasing with increasing distance from the wall. In addition, due to the nature of the EM force, this characteristic length depends only on the EM actuator geometry. The characteristic wall-normal length is therefore independent of current or magnetic intensities, assuming EM forces are still above viscosity damping.

Given this definition of h EM , the EM mean acceleration g EM and the EM time T EM can be based on the integration volume with h EM as height. Since h EM depends only on the actuator geometry, g EM , T EM and V EM are direct functions of The calculated value of 0.8 s for the characteristic timescale of EM pumping (actuator 1999) is in good agreement with the duration of the transition phase experimentally observed for EM pumping on a flow initially at rest [START_REF] Rossi | Contrôle électromagnétique d'écoulement en eau de mer[END_REF][START_REF] Rossi | ElectroMagnetic forcing in turbulence and flow control: analytical definitions of EM parameters and hydrodynamic characterisations of the flow[END_REF].

Table 3 gives the laws of dependence of the characteristic EM numbers on the control parameters of an actuator, i.e. I imposed current and B applied magnetic induction.

EM Froude similarity

The pumping effect grows with the intensity of the EM forcing. Using the EM parameters, it is possible to define an EM Froude number appropriate to the flow. This EM Froude number is the ratio of the inertia present in the normal component of the flow (effectively the kinetic energy) to the potential energy in the EM field (effectively the work of EM forces). This expression is

F rEM = V 2 2g EM h EM = V 2 V 2 EM (11)
A similarity law can be constructed by taking the EM Froude number, F rEM , as a constant in equation [START_REF] Akoun | 3D analytical calculation of the forces exerted between two cuboidal magnets[END_REF]. When EM forces are above the damping of viscosity, this similarity might be extended to other geometries of actuators considering the pertinent energy and work of forces.

Characteristics parameter of various EM regimes

An EM actuator is designed to act on a flow as it passes through the EM force field. Its actual effect on the flow is expected to depend on the velocity of the fluid as it approaches the domain of action above the actuator.

The characteristic time of the undisturbed flow T flow can be defined as the transit time of fluid particles over the EM actuator. This is given as

T flow = L EM U mean ( 12
)
Where U mean is the undisturbed mean flow velocity and L EM is the corresponding length of the actuator in the mean flow direction. It is interesting to compare T flow , to the characteristic EM forcing time T EM (equation ( 8)):

EM R = T flow T EM [START_REF] Boissonneau | Experimental analysis of couplings between electrolysis and hydrodynamics in the context of MHD in seawater[END_REF] This non-dimensional time ratio, EM R, presents two asymptotic domains: (i) EM R ≫ 1 corresponds to strongly dominant EM forcing, i.e. the imposition of an EM pumping Table 3. Estimation of EM characteristic numbers for any current I or induction B deduced from the values computed for I = 1A, B surf = 1 T (i.e. subscript EM1 stands for I = 1 A and B = 1 T) and identical geometries of actuators. regime with a single actuator. (ii) EM R ≪ 1 corresponds to week EM forcing, i.e. a pumping regime cannot be established using only a single actuator. In this case, the flow will simply be deflected by the generation of a normal velocity component.

I = 1A,B surf = 1T h EM g EM1 V EM1 T EM1 I , B surf h EM g EM = IBg EM1 V EM = √ IBV EM1 T EM = T EM1 √ IB

Multiple activation

As the flow passes over an EM actuator network (see [START_REF] Nosenchuck | The direct control of wall shear stress in a turbulent boundary layer[END_REF] and figure 6) it undergoes the influence of multiple activations.

Each of them has an activation time T act , which cannot be larger than T flow . Between each activations, the flow can recover from the effects of the EM forcing. The following model defines the wall-normal velocity on the (n +1)th actuators as the result of the competition between the relaxation in the flow, represented by ηV n , and the EM forcing, represented by αg EM T act with T act <T EM . η is a relaxation coefficient (η 1) which is mostly due to viscous dissipation, transport and diffusion. Note that if T flow is smaller than T EM (which is the case usually) then T act is also smaller than T EM . α is a dissipation coefficient (α 1) mostly due to wall effects during the activation. For activation time larger than T EM , via [START_REF] Rossi | Contrôle électromagnétique d'écoulement en eau de mer[END_REF] it is possible to give numerical values for α: i.e. α ∼ 0.26 for the 1999 actuator and α ∼ 0.29 for the 2000 actuator.

V n+1 = ηV n + αg EM T act ,
with T act <T EM and T act <T flow (14)

In the case of a network (figure 6) with a number of successive actuators, e.g. for an application aimed at drag reduction, a stationary limit velocity (V lim ) can be defined. This is given by

V lim = 1 1 -η αg EM T act (15)
Clearly, this limit depends on the η coefficient. From the energy point of view, the electrical power consumption of an EM actuator network increases with T act . From the physical point of view, the limit of EM flow regime possible corresponds to the time T EM . Consequently, it is not possible to have V lim > V EM . Finally, minimum energy consumption implies the following inequality:

T act (1 -η)T EM α ( 16
)
In addition to the flow acceleration, local EM actuators are also a source of vorticity corresponding to the curl( j × B) (see [START_REF] Rossi | Contrôle électromagnétique d'écoulement en eau de mer[END_REF]). In a multi-activation system, like in the network in figure 6, the flow experiences successive activations that are able to modify its vorticity. This alteration (or control) of vorticity is strongly linked to the design of EM actuators as well as to the duty-cycle of the network power supply. Figure 7 shows an illustration (1 m s -1 , R ex = 10 7 , U ∞ = 28.6u τ ) comparing two classes of actuator sizing: macro-and microactuator. For each case the typical length-scales in wall units including streaks spacing (100 + ), streaks size (40 + ) and streaks vorticity (123 s -1 ) (see [START_REF] Mengjcs | Wall layer microturbulence phenomenological model and a semi markov probability predictive model for active control of turbulent boundary layers NUWC Division Newport Technical Digest[END_REF]) are compared to the typical intensity of the EM vorticity source. This EM angular acceleration source ωEM expressed in s -2 is defined as follows:

ωEM = 1 ρυ υ curl( f EM ) dυ (17)
Comparing these two extreme situations, it is clear that the possible modes of action are different. For the macro-actuator, about 1000 wall units (see figure 7(a)), the EM vorticity source (±200 s -2 ) results in an additional vorticity (typically ±7s -1 ), which is of smaller intensity than the pre-existing vorticity.

This means that macro-actuators have to be used in multiple actuation modes. For the micro-actuator, about 100 wall units (figure7(b)) the EM vorticity source (±20 000 s -2 ) results in an additional vorticity (typically ±700 s -1 ) which is extremely intense compared to the pre-existing vorticity. This means that a single actuation applied locally is capable of a local control.

Experiments in aquarium and tunnel

The aim of the experimental investigation is to verify the hypothesis of Froude similarity and to check for the existence of two asymptotic EM regimes. In the experiments, the magnetization of permanent magnets is 1.3 T and their typical longitudinal length-scale spacing, L EM ,is30mm.

Description of experimental facilities and measurements

Two measurement methods are used to quantify the flow induced by EM forcing: particle tracking velocimetry (PTV) and particle image velocimetry (PIV). The first series of experiments was carried out on a flow initially at rest in an aquarium large enough to avoid confinement effects. The second series of experiments was performed in a seawater tunnel, with and without external flow.

EM activation on a flow initially at rest (PTV measurements).

The aquarium has dimensions of 50 cm × 50 cm × 60 cm and is filled with salt-water (35 g NaCl/l). The EM actuator is situated in the centre of the side vertical wall of the aquarium (figure 8(a)). Figure 9 shows a triple exposure (superposition) image of the flow after a 10 s activation. This superposition is used for PTV measurements, based on intervals between images of: 0.08 s and 0.12 s, respectively. Three different electric current intensities of 0.5, 0.8 and 1.1 A are used. It is then possible to check the Froude similarity concerning currents.

EM activation on a seawater wall bounded flow (PIV measurements)

. These experiments were carried out in a seawater tunnel with a cross sectional area of 100 mm × 100 mm and a test section length of 1.3 m [START_REF] Rossi | Contrôle électromagnétique d'écoulement en eau de mer[END_REF]. For all reported PIV measurements, y = 0 at the wall and x = 0 corresponds the centre of the EM actuator. Consequently, the centres of the magnets are situated at x =± 15 mm. Activation is carried out either on originally static fluid and then or on flows with an imposed external velocity up to a Reynolds number R ex ∼ 3 × 10 5 . The PIV measurements are realized just above the EM actuator, which is inserted (wall-flush) in the top wall of the seawater tunnel (see figure 8(c)). Only one wall-normal plane of measurement equidistant from the two electrodes is studied here. This plane, 0˚in figure 8, is also streamwise to the external flow. A double pulse YAG laser and a digital/numerical camera (1000 × 1000 pixels 2 , double frame) are used to take frames. Typically, the delay of acquisition is adjusted for a typical displacement close to 25% of the size of the cross-correlation window. The cross-correlation windows overlap is 75% and the ratio of primary to secondary cross correlation peaks is better than 1.2. The various PIV measurements windows are: (i) above the actuator, 40 mm × 40 mm with a resolution of 24.8 pixels mm -1 . The size of the correlation windows is 32 pixels, i.e. about 1.3 mm. (ii) Right downstream the actuator, 25 mm×25 mm, with a resolution of 39.9 pixels mm -1 . The typical size of the correlation window downstream the actuator is about 0.8 mm.

EM Froude similarity 4.2.1. Similarity for the 1999 EM actuator and different current intensities.

Wall-normal velocity (i.e. velocity component perpendicular to the wall) measurements (PTV) are taken along the wall-normal central axis y of the 1999 EM actuator for three electric current intensities: 0.5, 0.8 and 1.1 A (see figure 10). The curves show the same behaviour. Flow starts from rest far from the wall (y>60 mm) and is progressively accelerated due to Lorentz force pumping. This acceleration increases with the electric current. The velocity has a plateau type maximum value between 8 and 12 mm. This is attributed to wall effects. Furthermore, V must vanish at the wall, i.e. y = 0. The intensity of the pumped flow increases with proximity to the wall and with the electric current intensity. Given that the distance over which the Lorentz forces are significant is about 20 mm far for the actuator, the measurements confirm that the flow resulting from EM activation is able to extend over a larger volume than the forces depending on EM R [START_REF] Rossi | Contrôle électromagnétique d'écoulement en eau de mer[END_REF][START_REF] Rossi | ElectroMagnetic forcing in turbulence and flow control: analytical definitions of EM parameters and hydrodynamic characterisations of the flow[END_REF].

Figure 11 presents a simple transposition of the previous curves (figure 10) obtained by dividing the measured velocities by the square root of their electric current intensities, i.e. V * = V √ I * /I with I * = 1 A. All values collapse onto a single curve. This transposition clearly demonstrates the existence of Froude similarity.

The measurements reported in figure 12 are taken at various y positions: 12 mm, 18 mm and 24 mm, and they refer to the three wall-normal planes at ×0˚, ×45˚and ×90˚, respectively (cf figure 8). Measurements taken at different current intensities are normalized to the corrected current base I * = 1 A via the V * = V √ I * /I similarity law. These wall-normal velocity profiles offer a first quantitative characterization of the wall-normal flow above the actuator. It indicates the intensity of the EM pumping and validates the use of the similarity proposed to various intensity of the forces. By extension, the EM characteristic velocity (V EM ) scales the potential pumping effect due to Lorentz forces above the actuator. The symmetry of the velocity profile versus ×0˚, ×45˚and ×90˚can be noticed. This symmetry is due to the acceleration of the flow and to the axial symmetry of the EM forces (see figure 2).

Comparison between the 1999 and 2000 actuators.

Different series of experiments with two actuators (1999 and 2000) and under various experimental conditions were carried out both for comparative purposes and for validation of the similarity law. The 1999 actuator was used in an aquarium large enough to allow 'long' activation (up to 10 s) before confinement effects became significant. The measurements were taken using PTV. The 2000 actuator was used in the seawater tunnel. The dimensions of the tunnel are necessarily smaller than those of the aquarium. Consequently, the duration of activation was limited to 3 s. For longer activation confinement effects grew strong. Measurements are done by PIV in this case.

Even with these notably different experimental conditions, the normalized measurements (with equation ( 11)) presented in figure 13 show that the V * profiles effectively collapse onto universal curves for both actuators. This confirms the similarity law proposed for the flow induced by the Lorentz forces. The activation times, respectively, of 3 s and 10 s in these experiments, are both larger than T EM . In both cases; behaviour is quite similar confirming that the pumping regime mainly depends on the EM time.

EM flow regime

For the actuator 2000, the flow (of seawater) in the tunnel was set at two different velocity values: 'high velocity' (U ext ∼ 100 mm s -1 ) and 'low velocity' (U ext ∼ 10 mm s -1 ). Figure 14 shows the axial velocity profiles taken from PIV measurements. In each case, the velocity profile of the flow without activation is superposed with an EM activated profile.

Clearly, EM activation changes the flow near the wall. The EM forces induce a new flow component in the nearwall flow, which is ascribed to the presence of wall jets. The 'EM' profiles show the presence of wall jets downstream of the actuator. They are characterized by an increase in velocity in the vicinity of the wall and a deficit in axial velocity further out. Figure 15 gives a schematic illustration of this local mechanism.

The very slight drop in the velocity from the maximum value in the unactivated profile for y>15 mm (seen in figure 13) is thought to be due to some imperfections in the upstream damping chamber of the seawater loop. This will be modified for future experiments but is not thought to have any influence over the EM flow profiles here.

The wall jets are observed for both 'low velocity' and 'high velocity' flows. In order to give a more accurate meaning to this type of flow classification, with respect to EM pumping, it is interesting to compare the EM time, T EM , with the transit time, T flow in each case. The ratio of these two timescales is that given in equation [START_REF] Boissonneau | Experimental analysis of couplings between electrolysis and hydrodynamics in the context of MHD in seawater[END_REF]. When EM R ≫ 1 the flow is considered as a case of low velocity and when EM R ≪ 1 the flow is considered as a case of high velocity. Classification of the flows according to the EM R values is used on the series of measurements presented in figures 16 and 17.

EM R>1: (this case corresponds to an external velocity of 16.6 mm s -1 ) (see figure 17 wall, is found to be significantly larger than the mean flow velocity defined by the flow rate/cross-sectional area of the tunnel (see figure 16). Its maximum is at y = 1.3 mm from the wall. Clearly, in this case the wall jet dominates the flow near wall.

EM R<1: (this case corresponds to the velocity of 101.5 mm s -1 ) (see figure 17(b)). The transit time of the fluid particles is about T flow = 0.3 s and the EM forcing time is about T EM = 0.64 s (I = 1.78 A). The competition between the axial mean flow and the EM pumping (normal to the wall) is such that wall jets are less well defined and the flow is mainly deflected to the wall.

Finally, the time ratio EM R can be generalized to asymptotic domains: EM R ≫ 1 corresponds to 'pumping mechanism' and EM R ≪ 1 corresponds to 'deflecting mechanism' of EM force effects.

Conclusion

The ability of EM actuators to generate novel flows inducing strong changes in the near wall flow has been demonstrated. The forces developed above the EM actuator considered in this paper are mainly normal to the wall above the central axis of the actuator; in addition, the curl of these forces acts as a vorticity source for the flow. Depending on the intensity of the EM forces relative to the initial flow conditions, the flow can be deflected or effectively pumped to the wall.

The inertial model proposed here is based on an integrated EM acceleration. This model leads to the identification of EM characteristic numbers such as EM velocity, EM forcing time and EM normal length. It is remarkable that these numbers are independent of the flow length-scales depending only on the actuators length-scales. A group of non-dimensional parameters is constructed from these characteristic numbers:

(i) The EM Froude number is identified as the most important regarding similarity between various experiments. This is well confirmed by measurements where this EM Froude number appears to control the pumping of the flow by Lorentz forces. (ii) The EM time ratio allows to distinguish between EM forcing regimes. When the EM time is smaller than the transit time (i.e. EM ratio larger than 1), the flow is strongly pumped to the wall and is entirely dominated by the EM forces. The relative energetic price of this kind of activation is quite high. When the EM time is larger than the transit time (i.e. EM ratio smaller than 1), the flow is deflected to the wall. In both cases, a resulting novel wall flow is observed (i.e. wall jets), which is capable of reorganizing the near-wall flow and consequently modifying the production of turbulence.

The analysis presented here is validated by measurements. It has to be considered as a guideline for any attempt to optimize the use of single or multiple EM actuators.

Figure 1 .

 1 Figure 1. Wall-normal actuator: (a) photograph of the 1999 EM actuator, (b) schematic of magnet and electrode arrangement and associated EM forces, (c) sources of EM vorticy in the boundary layer due to Lorentz forces.

Figure 2 .

 2 Figure 2. (a) Three-dimensional view of computed EM force lines above an EM actuator set in the x-z plane for y = 0; (b) variation of the normal component of EM forces, f y , as a function of y and x in the plane equidistant to the electrodes. For these computations: B surf = 1T and I = 1A.

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Volume of integration above an EM actuator.

Figure 6 .

 6 Figure 6. Top view of an EM actuator network with four phases of power supply on a same board.

Figure 7 .Figure 8 .

 78 Figure 7. Comparison in wall units of the typical length-scales of an ordinary turbulent boundary layer (streaks spacing and axial vorticity) to the EM vorticity source, ωEM above the electrodes, for an external velocity of 1 m s -1 , R ex = 107, U ∞ = 28.6u τ . Two typical actuator length-scales are considered: (a) macro-actuator (about 1000 wall units) with B = 1 T and 1 = 1A,(b) micro-actuator (about 100 wall units) optimized for T flow = T EM . (a) (b) (c)

Figure 8 (

 8 b) shows the relative positions of the laser and light sheets normal to the wall. The dimensions of the aquarium are quite large compared to the size of the EM actuator, i.e. a factor of 20 with L EM = 30 mm. The EM actuator is activated with a 1 A DC power supply for at least 10 s, a time far greater than the characteristic time of EM forcing T EM ∼ 0.8s.Three planes of measurement (shown in figure8(b)) are studied in order to specify the flow above the actuator. These wall-normal planes are: (i) (0˚) the plane equidistant from the electrodes, (ii) (45˚) the plane in the diagonal of the EM actuator and (iii) (90˚) the plane equidistant from the magnets.

Figure 9 .Figure 10 .

 910 Figure 9. Superposition of three successive images used for PTV measurements.

Figure 11 .

 11 Figure 11. Variation of V * versus y along the axis of the EM actuator. V * = V √ I * /I is estimated for I * = 1 A from experimental points (corresponding to I = 0.5, 0.8 and 1.1 A).

Figure 12 .

 12 Figure 12. Variation of V * as a function of y for different measurements planes: (a) ×0˚, (b) ×45˚, (c) ×90˚. V * = V √ I * /I is estimated for I * = 1 A from experimental points (corresponding to I = 0.5, 0.8 and 1.1 A).

Figure 13 .

 13 Figure 13. EM Froude similarity between the 2000 and 1999 actuators. V * is estimated from experimental points (corresponding to I = 0.5, 0.8 and 1.1 A for the 1999 actuator and I = 1.8 A for the 2000 actuator); for I * = 1 A and various heights y: 12, 18 and 24 mm. The measurement plane is equidistant from the electrodes and perpendicular to the wall (see figure 8(b), 0˚).

Figure 14 .

 14 Figure 14. Axial velocity profiles downstream of the EM actuator (x ∼ 37 mm from the axis). : with EM forces and : without EM forces. (a) Mean velocity (without EM forces) is about 101.5 mm s -1 . Activation current is 1.78 A; (b) mean velocity (without EM forces) is about 16.6 mm s -1 , activation current is ∼1.73 A.

Figure 15 .Figure 16 .

 1516 Figure 15. Illustration of wall jets downstream of the EM actuator.

Figure 17 .

 17 Figure 17. PIV plot of the velocity difference uu ref in mm s -1 , for two different mean velocities U mean .( a) Low velocity: U mean = 16.6mms -1 and current: I ∼ 1.73 A; (b) high velocity: U mean = 101.5mms -1 and current: I ∼ 1.78 A.

Table 1 .

 1 Fluid, magnetic and electric equations used in seawater EM flow control.

	Fluid equations	Magnetic induction equation and Ohm's law

Table 2 .

 2 T EM1 , g EM1 , V EM1 and h EM for actuators 1999 and 2000 and B surf = 1 T and I = 1A

		V EM1	T EM1	g EM1	h EM
		(m s -1 )	(s)	(m s -2 )	(m)
	Actuator 2000 Actuator 1999	0.0599 0.0495	0.6911 0.8076	-0.086 69 -0.061 33	0.0207 0.02

I (total current) and B (magnetic induction). In table 2, the values with subscript EM1 are reported for a nominal current I = 1 A and a nominal induction B = 1T.

LEGI is a joint laboratory of the Centre National de la Recherche Scientifique (CNRS), Université Joseph Fourier (UJF) and the Institut National Polytechnique de Grenoble (INPG).

Acknowledgments

This work has been carried out with the support from DGA/BEC and DGA/DSP (departments of the French Ministry of Defense). Many thanks for extensive discussions and contribution to DrFMJMcCluskey, Dr S Tardu, P Carrechio and M Kusulga of LEGI; 'service repro-visu' of the ENSHMG and his director Mr F Bonnel; Dr O Cugat and Dr J Delamare of LEG-ENSIEG.


Gabriel Gamrat
email: g.gamrat@imc.pcz.czest.pl

Michel Favre-Marinet
email: michel.favre-marinet@hmg.inpg.fr

Dariusz Asendrych

NUMERICAL MODELLING OF HEAT TRANSFER IN RECTANGULAR MICROCHANNELS

The paper presents both three and two-dimensional numerical analysis of convective heat transfer in microchannels. The three-dimensional geometry of the microchannel heat sink followed the details of the experimental facility used during a previous research step. The heat sink consisted of a very high aspect ratio rectangular microchannel. Two channel heights, namely 1mm and 0.3mm (0.1mm), were used for 3D (2D) numerical model respectively. Water was employed as the cooling liquid. The Reynolds number ranged from 200 to 3000. In the paper, the thermal entrance effect is analyzed in terms of heat transfer efficiency. Finally, the comparison between measured and computed heat flux and temperature fields is presented.

Contrary to the experimental work, the numerical analysis did not reveal any significant scale effect in heat transfer in microchannel heat sink up to the smallest size considered (0.1 mm).

OBJECTIVES

Microchannel heat sinks are of special interest as high efficiency cooling devices due to undeniable advantages such as small coolant demand and dimensions, which directly result in high convective heat transfer coefficient. Some experimental studies [START_REF] Peng | Heat transfer characteristics of water flowing through microchannels[END_REF], [START_REF] Qu | Heat transfer for water flow in trapezoidal silicon microchannels[END_REF], [START_REF] Gao | Scale effects on hydrodynamics and heat transfer in two-dimensional mini and micro-channels[END_REF]] indicate that the reduction of size may affect heat transfer in microchannels. These scale effects are difficult to interpret and the physical mechanisms are still not clarified [START_REF] Guo | Size effect on microscale singlephase flow and heat transfer[END_REF]. Several numerical studies [START_REF] Weisberg | Analysis of micro-channels for integrated cooling[END_REF], [START_REF] Fedorov | Three-dimensional conjugate heat transfer in the micro-channel heat sink for electronic packaging[END_REF], Qu and Mudawar (2002a, b), [START_REF] Toh | Numerical computation of fluid flow and heat transfer in microchannels[END_REF], [START_REF] Tunc | Heat transfer in rectangular microchannels[END_REF]] have been published in recent years on this subject. Among the possible sources of deviation from the laws governing the macroscale flows, effects of axial heat conduction in the walls and entrance effects in the microchannel test section are seriously suspected to contribute to the observed scale effect. The present paper devoted to the numerical modelling of the flow and heat transfer in microchannels is focused on the coupling of axial heat conduction in the walls with convection in the microchannels. Associated entrance effects are of special interest, because they are probably significant in most published data.

This study pertains to experimental investigations [START_REF] Gao | Scale effects on hydrodynamics and heat transfer in two-dimensional mini and micro-channels[END_REF] conducted in large-span rectangular microchannels, which have revealed a significant reduction in the Nusselt number for microchannel height less than 500 µm. The geometry considered in the present numerical simulations is as close to the real situation as possible. It is therefore possible to compare numerical and experimental results and to test the relevance of conduction/convection coupling effects and associated entrance effects in the deviation of heat transfer results from the macroscale laws.

EXPERIMENTAL SET-UP

In the experimental set-up, the active channel walls were two hand-polished plane bronze blocks, which were separated by a stainless plate (of thickness e) with a hollowed out central part of width w (see dimensions in Fig. 1). Details of the facility may be found in [START_REF] Gao | Scale effects on hydrodynamics and heat transfer in two-dimensional mini and micro-channels[END_REF]. The height e of the microchannel test section was determined by the plate thickness, which could be varied in the range mm 1 1 . 0 ÷ by steps of 0.1 mm. Heating was provided by four electric cartridges, which were inserted inside the two blocks and surrounded by an insulating material. The inlet/outlet water temperatures (T in , T out ) were measured inside the upstream/ downstream containers, then the fluid bulk temperature T f distribution in the microchannel was approximated as a linear one. It is worth noting that the heater is slightly shorter than the microchannel length. Four thermocouples were mounted along the microchannel in the plane of symmetry with their tips located 1mm above the microchannel surface. Two pressure transducers were flushmounted in the wall of the inlet/outlet containers. Thus, the pressure at the microchannel entrance was corrected by the inertia term accounting for flow acceleration in the converging channel entrance. The data were interpreted by using the Reynolds number Re and the Nusselt number Nu, both based on the hydraulic diameter D h .

Since measurements of the local heat flux ϕ were not possible in the experimental study of [START_REF] Gao | Scale effects on hydrodynamics and heat transfer in two-dimensional mini and micro-channels[END_REF], the authors assumed that the power density dissipated into the test section was uniformly distributed along the active microchannel surfaces. Results from the experimental investigations showed that the longitudinal distribution of Nu as given by Shah and London (1978, referenced as SL hereafter) is well recovered for the channels of largest height in the laminar regime. However, for e < 500 µm, a significant decrease in Nu was observed for reduced channel size.

NUMERICAL MODEL

The geometry of microchannel heat sink shown in Fig. 1 followed the details of the experimental setup.

Taking advantage of the symmetry of the experimental facility, the numerical model was represented by half of the actual geometry. Heat transfer process occurring in such a complex geometry consists of coupled conduction in the solid and convection to the cooling fluid. The conventional Navier-Stokes and energy equations were simplified by the following assumptions:

(1) fluid incompressibility (2) laminar flow character (3) negligible radiation heat transfer (4) negligible buoyancy (5) constant fluid and solid properties (6) negligible viscous heating Because of the assumptions of constant fluid properties and negligible buoyancy, the mass and momentum equations were not coupled with the energy equation. Therefore, the temperature field was calculated by solving the energy equation after the converged solution for the flow field was obtained. Such an approach allowed to significantly speed up the computational convergence. The momentum and continuity equations were solved for the following hydraulic boundary conditions:

(1) uniform velocity profile was set at the inlet to the test section, (2) the flow was assumed to be fully developed at the test section exit plane (3D model) or at the channel exit plane (2D model).

The thermal boundary conditions are given as follows:

(1) constant heat flux density at the walls of the heater cartridges (3D model, total heat flux Φ = 180 W), constant energy source inside the cartridge volume (2D model), (2) free convection heat transfer with surrounding air at the outside walls of the test section; the assumed values of the heat transfer coefficient and ambient temperature were equal to 10 W/m 2 K and 300 K, respectively.

The x-direction runs along the channel. The origin is at the channel entrance after the converging region. Numerical computations of the flow were carried out by using the commercial code Fluent 5.4. The equations were discretized by means of a second order upwinding finite volume method. Meshing was performed by using Gambit 1.3. Due to very complex geometry, the computation domain was split into several regions.

The domain corresponding to solid block was meshed using unstructured mesh.

Structured mesh was utilized for inlet/outlet containers and microchannel. The total number of cells was optimized from series of tests and equal to about 2x10 6 (2.7x10 6 ) for e = 1 (0.3) mm. The number of mesh nodes used in the microchannel was equal to 164 (300) x 20 x 30 for e = 1 (0.3) mm, respectively. Special care had to be taken with the cell aspect ratio especially in the entrance region of the test section, i.e. for the rapidly developing flow. In order to avoid a too large number of nodes, the grid applied in the microchannel entrance was non-uniformly distributed in x-direction.

Model validation

The numerical model was checked for laminar simultaneously thermally and hydraulically developing flow along a two-dimensional channel with symmetrical uniform heat flux surfaces. We used for comparison the results of [START_REF] Hwang | Finite difference analysis of forced-convection heat transfer in entrance region of a flat rectangular duct[END_REF] reported by SL and fitted by [START_REF] Bejan | The optimal spacing of parallel plates cooled by forced convection[END_REF] using empirical formulas of the [START_REF] Churchill | A general expression for the correlation of rates of transfer and other phenomena[END_REF] type. For Pr = 0.7, the local Nusselt number is given by ( )

3 1 3 3 2 1 * f w h 235 . 8 x 587 . 0 ) T T ( D Nu         +         = λ - ⋅ ϕ = - (1)
where the dimensionless abscissa is Pr

1 Re D x x h * = (2)
Equation ( 1) assumes uniform inlet velocity and temperature profiles, i.e. it takes into consideration the socalled entrance effect.

Figure 2 shows an excellent agreement of the present simulation with Eq. ( 1). This provides confidence in the numerical technique and therefore in the results produced with the current numerical approach.

RESULTS AND DISCUSSION

Hydraulic properties [START_REF] Shah | Laminar flow forced convection in ducts[END_REF] presented a comprehensive review of laminar forced convection in ducts of various cross-section, including rectangular channels. Their results will be used as the reference in this study.

The Fanning friction factor f is related to the pressure drop ∆p over the hydrodynamically developing length x via

x D V 2 p f h 2 ρ ∆ = (3)
The Poiseuille number is defined as

Po = f Re (4)
For laminar developing flow in two-dimensional channels of usual size, [START_REF] Shah | Laminar flow forced convection in ducts[END_REF] proposed the following formula Figure 3 shows the variations of the Poiseuille number against x + for the two microchannel heights considered in this study.

For the complex geometry of the numerical simulation (including inlet container), Po values are significantly smaller than predicted by SL. This is obviously due to the finite thickness of the boundary layers at the channel entrance, as opposed to uniform velocity profile in the SL reference case. Furthermore, the distribution of Po depends Fig. 2. Variation of Nusselt number along the channel (Pr=0.7). Comparison with the formula of [START_REF] Bejan | The optimal spacing of parallel plates cooled by forced convection[END_REF].

on Re and channel height separately, contrary to SL reference case where the proper dimensionless group is x + . As a result, the distributions of Po(x + ) for both channel heights are different as shown in Fig. 3. It can also be observed that the pressure drop increases with channel height, although both distributions tend to the asymptotic value of laminar fully developed flow (Po = 24).

Heat transfer Temperature field. Figure 4 shows the temperature distribution in the plane of symmetry. The region of the highest temperature gradients corresponds to the presence of the microchannel insulation. It is worth emphasizing that insulation is not complete around the bronze blocks. There is a gap near the microchannel inlet allowing a part of the heat flux to be conducted to the surrounding blocks. That in turn leads to: (i) additional heat transfer to water in the inlet container and (ii) increased heat losses to ambient air.

Figure 4 shows that the wall temperature decreases in the last part of the microchannel (x/L > 0.73) due to the lack of heating in this region (see Fig. 1 to compare heater cartridge and microchannel lengths).

Nusselt number. The Nu number is defined as

λ h D h Nu ⋅ = (7)
where the convective heat transfer coefficient h is given by

f w T T h - ϕ = (8)
Two-dimensionality checks showed that the wall heat flux ϕ num varied less than 5 % over two-third of the span. As a consequence, the wall heat flux, the wall and local fluid bulk temperatures (respectively T w , T f ) are taken in the plane of symmetry of the flow.

Figure 5 illustrates the variations of Nu against the dimensionless coordinate defined by Eq. ( 2). The numerical results are compared with the data obtained for uniform inlet velocity and temperature profiles and Pr = 6. This reference curve allows to identify and quantify the entrance effect. The Nusselt number was calculated along the channel for three values of the Reynolds number. Each curve in Fig. 5 corresponds to a particular value of Re. As can be seen in Fig. 5, the particular curves are clearly Re-dependent revealing relative shifts and intersections. The differences between these curves can obviously be explained by the Re- The data may also be presented for a fixed value of x and varying Re, as in the experimental procedure. As an example, Figure 5 shows the variation of Nu for x/L = 0.44 (solid symbols). The comparison between the reference curve and the marked numerical points for this particular position in the microchannel allows for the following observations:

(1) the entrance effect is weaker for the present results when e = 1 mm (2) the convergence of data points and the reference curve when e = 0.3 mm indicates that the entrance effect is so weak at this position in the channel that Nu is no more sensitive to the inlet conditions (3) the entrance effect is reduced for lower channel height Further explanation should be given for the first observation.

A non-uniform temperature profile was observed at channel inlet because a proportion of the total heating power was delivered to water in the converging part of the inlet container.

The close agreement of the data with the reference curve for x* > 0.005 suggests that the numerical results achieved for complex geometry can be physically interpreted and justified. In consequence, the numerical results were recognized as correct.

Experiment vs simulation

Figure 6 shows the distribution of the dimensionless temperature θ, defined as

λ ⋅ ϕ - = θ h num in D T T (9)
along the channel both for fluid and solid domains. The numerical solution for θ is considered at the location of the thermocouples.

Comparison between experimental results and numerical predictions for the water bulk temperature (squares in Fig. 6) shows that discrepancies are relatively small. The assumption of linear variation of T f is rather well verified. One can see that greater discrepancies occur in the solid region (circles in Fig. 6) resulting in higher values of the difference f w T Tfor experimental results. For e = 1 mm, these discrepancies vary from about 5 % to 18 % with reference to experimental data, while for e = 0.3 mm they are larger and reach about 40 %. In this latter case, they account for the decrease in Nu observed in the experiments.

The numerical heat flux ϕ num was normalized by the value ϕ exp corresponding to uniform distribution over the heating surface, which was assumed during experiments. The three following regions may be observed on Fig. 7: (1) the microchannel entrance where strong values of ϕ num occur owing to very narrow velocity and temperature boundary layers in this region (2) the middle part of the microchannel where the ϕ num distribution can be assumed as approximately uniform (3) the exit part (around 30 % of the channel length)

where ϕ num decreases due to the lack of heating and to conduction in the solid blocks The assumption used in the experiments neglected heat transfer in inlet and outlet containers. This explains why the numerical heat flux is smaller than the experimental one and, moreover, is non-uniformly distributed.

The location of the thermocouple tips could noticeably decrease the values of Nu, especially for small microchannel heights, owing to finite temperature gradient in the solid blocks. Nu was calculated in the experiments [START_REF] Gao | Scale effects on hydrodynamics and heat transfer in two-dimensional mini and micro-channels[END_REF] with the temperature given by the thermocouples, which were about 1 mm away from the liquid/solid interface. In order to eliminate the influence of the thermocouples position in the determination of Nu, the experimental temperature was corrected using the numerical solution. The experimental wall temperature was estimated by using the wall temperature gradient in the solid. Furthermore, the actual wall heat flux is probably close to the numerical value.

Consequently, the experimental Nusselt number was recalculated with the numerical heat flux for want of a better estimation of ϕ. For e = 1 mm, which may be considered as a conventional size for a channel flow, above comparisons indicate that:

(1) the wall temperature is slightly underestimated by the numerical model (Fig. 6a) (2) the heat flux is significantly overestimated (about 25 %) by the assumption used in the experiments due to neglected heat transfer in inlet/outlet containers (Fig. 7) (3) Nu is slightly overestimated (about 10 to 20 %) by the numerical model when Nu exp is calculated with ϕ num .

For e = 0.3 mm, comparisons show that:

(1) the wall temperature is much higher than the numerical solution (Fig. 6b) (2) the heat flux ϕ num distribution is the same as in the previous case (3) The Nusselt number Nu exp is much lower (about 40 to 80 %) than the numerical value.

An important conclusion is therefore that the observed reduction in Nu exp cannot be accounted for by conduction effects, which would have distorted the temperature field and the corresponding heat flux distribution.

As a consequence, the reduction in Nu exp is not explained by the present numerical simulation. Further investigations are planned to take into account the variations of the fluid properties with tempearature in the numerical model.

2D NUMERICAL SIMULATIONS

The simulation performed for 2D geometry could bring obvious advantages as reduction in computer running time and simplification of modeled geometry. This section compares results obtained in 2D and 3D numerical simulations. The computations were achieved for the two extreme channel heights used in the experiments, i.e. 0.1 mm and 1 mm, with the corresponding Re number ranges:

Re ∈ (200 ÷ 2166) for e = 1 mm Re ∈ (500 ÷ 3000) for e = 0.1 mm Figure 9 shows an excellent agreement between Nu distributions obtained with the 2D and 3D models. This suggests an application of 2D model for further quantitative investigations.

Hydraulic properties

3D simulations have shown that the Poiseuille number should be determined as a function of Re for given channel length L and height e, since the entrance effect was found to be Re-dependent. As a consequence, the computations were achieved for given values of L and e and varying Re, contrary to 3D simulations. It is, however, presented in Fig. 10 against L + , in consistency with the reduction of experimental data. Comparison between numerical and experimental data.

As can be seen in Fig. 10, the numerical results show a consistent distribution of Po(L + ) which is located below SL curve, as predicted by the 3D model and explained previously.

The entrance effects are found to be negligible when L + is higher than about 0.1 with corresponding values of the Re number: Re = 300 for e = 1 mm Re = 3000 for e = 0.1 mm The two sets of numerical results then overlap.

Heat transfer

Figure 11 shows the variations of Nu along the channel for the same two extreme channel heights (e = 1 mm and e = 0.1 mm). Computations were achieved by varying the Reynolds number as for hydrodynamics (ranges are given in previous subsection). Nu was estimated in the sections of the four thermocouples with the temperature at the fluid/solid interface. One can observe that Nu values are well fitted to the reference curve except for data corresponding to the section of the first thermocouple (x/L = 0.073, open circles in Fig. 11). For the conditions of the numerical simulation, Nu is therefore affected on a short distance by the finite thickness of the thermal boundary layer at the channel entrance.

CONCLUSIONS

The three-dimensional flow and associated heat transfer in a rectangular micro-channel heat sink were analyzed numerically. The geometry of the test section used in the experiments of [START_REF] Gao | Scale effects on hydrodynamics and heat transfer in two-dimensional mini and micro-channels[END_REF] was taken into account in 3D computations. The numerical simulation considered the coupling between convection in microchannels and conduction in the walls and in the complete solid material. The local heat transfer characteristics i.e. temperature, heat flux and Nusselt number, were obtained. The key findings of the present study are as follows:

(1) solutions of numerical simulation using the continuum model (conventional Navier-Stokes equations and conventional energy equation) are in satisfactory agreement with published data on flow and heat transfer in two-dimensional channels (2) thermal properties show that entrance effects are dependent on Reynolds number and channel height separately, contrary to [START_REF] Shah | Laminar flow forced convection in ducts[END_REF] reference case (3) the numerical simulations turned out to be very helpful as a complement to the interpretation of experimental data, where complex measurements of heat flux and temperature field are obviously not possible (4) the numerical models used in this work assumed some simplifications e.g. did not take into account viscous heating, which could decrease the Nusselt number especially for channels of smaller heights than those investigated in the current work (5) the numerical simulations show that there is no size effect on heat transfer when the channel height is reduced from 1 mm down to 0.1 mm. As a result, the strong reduction in the Nusselt number observed in experiments cannot be explained by conduction effects due to the complex geometry, like axial conduction in the walls or lack of twodimensionality of the heat flux distribution.

Fig. 1 .

 1 Fig. 1. Sketch of the test section

Fig. 3 .

 3 Fig. 3. Variation of Poiseuille number along the channel. Influence of inlet velocity profile.

Fig. 4 .

 4 Fig. 4. Temperature distribution in the plane of symmetry. e = 1 mm. Re = 2166. Φ = 180 W.Scale given in K.

Fig. 5 .

 5 Fig. 5. Variation of Nusselt number along the channel for two channel heights: a) e = 1 mm, b) e = 0.3 mm.

Fig. 6 .

 6 Fig. 6. Variation of wall and fluid temperatures along the channel. Comparison between numerical and experimental results for two channel heights: a) e = 1 mm, b) e = 0.3 mm.

  Figure 8 compares the numerical and experimental results of Nu. It should be noted that only the second and third thermocouples (x/L = 0.25 and 0.44) were considered in the experimental Nu values because they correspond to the region of nearly uniform heat flux.

Fig. 7 .

 7 Fig. 7. Variation of dimensionless wall heat flux along the channel.

Fig. 10 .Fig. 9 .

 109 Fig. 10. Poiseuille number as a function of L + (Re).

Fig. 11 .

 11 Fig. 11. Variation of Nusselt number along the channel for two channel heights: a) e = 1 mm b) e = 0.1 mm.

ACKNOWLEDGEMENTS

This research was supported by the CNRS and Rhône-Alpes Region. The authors would like to gratefully acknowledge Socrates organization for supporting the visit of Gabriel Gamrat in Grenoble.


Mariusz Niklas

Michel Favre-Marinet
email: michel.favre-marinet@hmg.inpg.fr

Pr Michel Favre-Marinet

The flow characteristics of a network of parallel microchannels (hydraulic diameter: 110 µm) are investigated both experimentally and numerically. The microchannel cross-section was triangular, as in the case of microheat pipes. The pressure drop across the microchannel network showed a dramatic increase with a departure from the law of fully developed flow in ducts as soon as the Reynolds number of the flow exceeded about 10. Numerical computation of the flow was carried out using the classical laws of hydrodynamics in an attempt to explain this surprising result. There was a good agreement with experimental results, which suggests that there are no size effects at the length scales used in the experiments. Moreover, the mechanisms responsible for the large pressure drop for higher Reynolds numbers were identified in the numerical analysis as being extra head losses due to separation in several parts of the test section.

INTRODUCTION Background

Much work in recent years has gone into the development of microchannel technology. This has been especially useful in electronics where ever faster and smaller microchips are built. Heat removal, however, is a major problem, that limits the capabilities of such circuits. High speeds are accompanied by the intense generation of heat, so the development of higher density and faster microchips requires significant heat removal from these components. This trend in miniaturization requires the development of very small and efficient heat exchangers. As a result, exchangers based on microchannel technology have begun to be used. Enhancement of the heat transfer coefficient is obviously required, along with a reduction in length scales. This has been observed many times since the first experiments of Tuckerman and Pease [START_REF] Tuckerman | High-Performance Heat Sinking for VLSI[END_REF]. Li [2] reported that the major advantage of microchannel heat sinks is related to the high heat transfer coefficients, up to sixty times higher than in conventional, macroscale heat exchangers.

Many experimental studies on microchannels are reported in the literature and show that some results cannot be explained by conventional theories of transport phenomena (for a review, see [START_REF] Obot | Toward a Better Understanding of Friction and Heat/Mass Transfer in Microchannels-A Literature Review[END_REF][START_REF] Guo | Size Effect on Microscale Single-Phase Flow and Heat Transfer[END_REF][START_REF] Sobhan | A Comparative Analysis of Studies on Heat Transfer and Fluid Flow in Microchannels[END_REF]). Even if the discussion is restricted to papers related to single phase flows in trapezoidal microchannels, contradictory conclusions appear regarding the flow hydrodynamics. Wu and Little [START_REF] Wu | Measurement of Friction Factors for the Flow of Gases in Very Fine Channels Used for Microminiature Joule-Thomson Refrigerators[END_REF][START_REF] Wu | Measurement of the Heat Transfer Characteristics of Gas Flow in Fine Channel Heat Exchangers Used for Microminiature Refrigerators[END_REF] investigated flows in microchannels of a trapezoidal cross-section (hydraulic diameter between 45.5 to 83.1 µm) etched in a silicon substrate and abraded into glass. They measured friction factors larger than predicted on the basis of conventional laws for circular pipes. Urbanek et al. [START_REF] Urbanek | An Investigation of the Temperature Dependence of Poiseuille Numbers in Microchannel Flow[END_REF] observed a temperature dependence of the Poiseuille number in triangular and trapezoidal microchannels. Flockhart and Dhariwal [START_REF] Flockhart | Experimental and Numerical Investigation into the Flow Characteristics of Channels Etched in <100> silicon[END_REF] found no scale effects on the hydraulics in trapezoidal channels etched in silicon wafers (27 or 63 microns in height, Reynolds number <240). However, Qu et al. [START_REF] Qu | Pressure-Driven Water Flows in Trapezoidal Silicon Microchannels[END_REF] observed a significant increase in the friction factor f in the same type of channels (height 28 to 111 microns) over the whole range of Re investigated (up to 1600).

1

An Experimental Study and Numerical Modeling of the Flow in a Network of Triangular Microchannels

A discussion of the possible microscale effects can be found in Mala et al. [START_REF] Mala | Heat Transfer and Fluid Flow in Microchannels[END_REF], Guo and Li [START_REF] Guo | Size Effect on Microscale Single-Phase Flow and Heat Transfer[END_REF], and Koo and Kleinstreuer [START_REF] Koo | Liquid Flow in Microchannels: Experimental Observations and Computational Analyses of Microfluidics Effects[END_REF]. These analyses took into account the effects due to channel entrance, wall slip, surface roughness, electrokinetics, and variable fluid properties. At the moment, there is no agreement concerning the phenomena involved in the physics of these flows. It is therefore necessary to carry out a number of fundamental studies to identify the differences between experimental data and conventional flow characteristics and to detect at which length scale such differences appear.

Motivation of the Present Work

This study forms part of a joint project involving several laboratories, with the goal of designing new heat pipes of a very small size (typical length scale of 200 µm). Since hydrodynamics, and in particular the resisting wall friction, play an important role in the working principles of heat pipes, it was decided to investigate single-phase isothermal flows across microchannels of triangular cross-section identical to those that will be used for heat pipe operation. The present study concerns flows across a network of 55 parallel microchannels [START_REF] Niklas | Pressure Losses in a Network of Triangular Microchannels[END_REF]. Experiments were conducted on a test section identical to the microheat pipe prototype, except that the network was connected to an external closed-loop circuit. Measurements of the pressure drop across the network as a function of the flow rate were complemented by a numerical computation of the flow in order to interpret the data and detect possible size effects on the results.

Another issue of this study is the distribution of the flow rate and pressure losses in a network of parallel microchannels as is relevant to the problems of microheat exchangers. Hetsroni et al. [START_REF] Hetsroni | Two-Phase Flow Patterns in Parallel Micro-Channels[END_REF] recently investigated a system of parallel microchannels etched in a silicon substrate. The cross-section of each channel was an isosceles triangle with a base of 200-310 µm. Experiments were performed with air-water and steamwater two-phase flows. Adiabatic single-phase flows were studied by Yin et al. [START_REF] Yin | Single-Phase Pressure Drop Measurements in a Microchannel Heat Exchanger[END_REF] in parallel microchannels with a working fluid of nitrogen. Each microchannel had a round cross-section 0.787 mm in diameter. The authors linked the pressure measurements with conventional published correlations on head losses to analyze the total pressure drop across a microchannel heat exchanger. This approach enabled them to identify and quantify internal manufacturing defects. In the present study, we consider single-phase flows in parallel triangular microchannels, 110 µm in hydraulic diameter. To our knowledge, there is no published work on single-phase flows of such small size in this geometric configuration.

EXPERIMENTAL SETUP

Geometry of the Test Section

The main element of the experimental setup was a cell consisting of two silicon wafers bonded together. One of the wafers was etched via a wet anisotropic technique to form a system of 55 parallel microchannels of triangular cross-section. The configuration of the channels network is depicted in Figure 1; the dimensions are given in Table 1. The channels were 215 µmin ). All of the channels were connected to an upstream distributor channel, called the header and a downstream collector channel of identical rectangular cross-section (width 800 µm, height 188 µm). The system was covered with a second silicon wafer fixed by direct wafer bonding. Two holes (1.5 × 1.5 mm 2 ) were made in the cover in order to connect the network of microchannels to a demineralized water circuit. They were located in the plane of symmetry of the test section.

The surface area of the channel network was equal to 2 × 2cm 2 . The cell was inserted and maintained tightly between two large blocks fastened with bolts. Large diameter holes were tapped into one of the blocks to connect the cell to the external loop. Tee-shaped connectors (internal diameter = 3 mm) were then screwed onto the block and used to take pressure measurements. Since the cross-section of the connecting parts of the system was much larger than the cell channel dimensions (47 times the collector/distributor cross-section), it was assumed that the pressure at the connector taps was the same as at the inlet/outlet cell ports. The x-direction runs along the axis of symmetry of the system and the z-axis across the distribution channel. The origin is at the distribution channel inlet on the axis of symmetry of the system (see Figure 1).

The Reynolds number of the flow is based on the average bulk velocity V across the channels (flow rate/total cross-sectional area) and the hydraulic diameter D h of one channel. The range of Reynolds number investigated was 1 to 700. The pressure drop p across the complete system of channels is related to the Fanning friction factor f via

f = p 2ρV 2 D h L (1)
The Poiseuille number is defined as

Po = f Re (2)

Instrumentation

The pressure drop p was measured either using piezoresistive strain gauge transmitters placed upstream and downstream of the cell (full scale, 10 6 Pa; accuracy, 10 4 Pa) or a differential gauge (full scale, 10 4 Pa; accuracy, 10 2 Pa). The flow rate was measured by precision weighing or using a highaccuracy flowmeter (full scale, 0.25 l min -1 ; accuracy, 0.001 l min -1 ) for the highest flow rates. The main uncertainties in the measurements are due to the determination of the microchannel dimensions, which play a crucial role in the pressure losses. The dimensions were measured by a mechanical profilometer. Uncertainties are given in Table 1.

NUMERICAL COMPUTATIONS Numerical Scheme

Numerical computations of the flow were carried out by using the commercial code Fluent 5.4. The flow regime was assumed to be laminar. Continuity and Navier-Stokes equations were discretized by means of a second-order upwinding finite volume method. As these equations are non-linear, a SIMPLEC (Semi-Implicit Pressure-Linked Equations-Consistent) algorithm was used. This algorithm is based on a prediction-correction method that allows the equations to be linearized and solved iteratively. The pressure under-relaxation factor was set to one. Meshing was performed by using Gambit 1.3. Boundary conditions consisted of a no-slip condition at the walls and a fixed pressure at the domain outlet. The mesh was unstructured in the crosssection of the microchannels (triangular cells) and structured in the longitudinal direction (rectangular cells). It was unstructured in the other parts of the flow. Several cases were computed by varying the flow rate across the system. The solution was treated as converged when the residual level was lower than 10 -5 . The evolution of the pressure drop as a function of the number of iterations was also checked.

Numerical Accuracy

Preliminary tests on accuracy were performed during the computations of a single channel flow with periodic conditions, which corresponds to fully developed flow in the channel. During these tests, the cells were equally distributed along the channel. Convergence was obtained after approximately 1000 iterations (level of normalized residuals less than 10 -5 ). Grid influence was tested on the calculated pressure drop across a single channel. It was observed that about 200 cells in the crosswise direction are sufficient to have a satisfactory result, the asymptotic value being approached to within an accuracy of 0.5% with this number of cells. As a result, 214 cells were used in further computations. The mesh was refined near the inlet in the streamwise direction to compute the developing boundary layers. It was then checked that the pressure drop was independent of the number of cells along the channel when more than thirteen cells were considered. For the complete system, the mesh consisted of about 600,000 prism/wedge elements. Grid independence was carefully checked on the solution. Comparisons showed that small differences (less than 3%) were obtained for 300,000, 600,000, and 900,000 elements.

RESULTS

Figure 2 shows the experimental results. Uncertainties on Po and Re were estimated by standard methods [START_REF] Holman | Experimental Methods for Engineer[END_REF] from precision errors on the various parameters and are reported on the graph. Precision errors on the microchannel dimensions accounted for the largest contribution to these uncertainties. The other data plotted in Figure 2 correspond to numerical results and will be detailed later. The experimental pressure drop deviates from the theoretical result for fully developed flow in triangular channels at low values of the Reynolds number (between 10 and 20). This deviation corresponds to between eight and sixteen dimensionless channel lengths L + (= L/D h 1/Re), which is much larger than expected in duct flows (≈ 0.1). However, it must be noted that the measured pressure drop corresponds to the whole system, including the triangular microchannels. The aim of the following section is to indicate the contribution of the various parts of the system to the total pressure drop, deduced from numerical computations.

Single Channel Flow

Preliminary calculations were divided into three steps for a single triangular microchannel. These were:

1. fully developed flow in a channel 2. flow with entrance effects due to the development of boundary layers 3. flow with local losses due to a sudden change in the crosssectional area at the inlet of a microchannel.

Streamwise-periodic conditions over the microchannel length were assumed for the velocity field for the first step. The computations of the fully developed flow provided excellent agreement with the results of the literature (Po = 13.3 [START_REF] Bejan | Convection Heat Transfer[END_REF]). The velocity was assumed uniform at the channel inlet, and the development of boundary layers was computed for the second step. The final step considered a channel of rectangular cross-section (height 188 µm, width 215 µm) followed by a microchannel with a triangular cross-section of the same maximal width. For the two latter cases, the grid was carefully refined near the microchannel inlet, where the boundary layers are developing. Figure 2 compares the numerical results to the experimental data. It clearly shows that the entrance effects in the microchannels have little influence on the flow resistance and cannot account for the rapid rise in pressure drop when the Reynolds number grows beyond about twenty.

Flow Through the System of Microchannels

In the next phase of the work, the computational domain was similar to the real network of channels but was reduced to onehalf of the whole system owing to the symmetry of the flow (see Figure 3). The inlet condition was simplified in comparison with the actual arrangement, which corresponds to a flow inlet perpendicular to the cell plane. It was modeled as a uniform velocity distribution at the entrance of a small channel of rectangular cross-section, denoted as the main inlet channel in Figure 1. A symmetrical outlet channel was considered at the flow exit (Figure 1 and Table 1). A condition of symmetry in the mid-plane of the cell was used in the computations.

Pressure Losses

When the complete model is considered, the numerical results closely follow the tendency of the experimental data (see Figure 2). The pressure drop across the system is slightly overestimated by the computations. These differences may be due to uncertainties in the numerical simulations and the simplification of the actual inlet/outlet system. There are also possible additional errors in the measurements due to, for example, the presence of air trapped in the corners of the triangular crosssection, something that cannot be completely excluded. Guo and Li [START_REF] Guo | Size Effect on Microscale Single-Phase Flow and Heat Transfer[END_REF] suggested that this phenomenon could give rise to a reduction in the actual cross-section and to subsequent local acceleration of the flow. They also claimed that the reduction of the wetted area should be the prevailing phenomenon, giving rise to a reduction in the measured friction factor. This could partly account for the differences between the present experimental and numerical results. On the other hand, specific small size effects were discussed by Koo and Kleinstreuer [START_REF] Koo | Liquid Flow in Microchannels: Experimental Observations and Computational Analyses of Microfluidics Effects[END_REF]. These authors indicate that among possible microscale effects, the wall slip velocity is negligible for liquid flows in microchannels, though this effect can probably be discarded in the present case.

Figure 4 shows the contribution of the various phenomena to the total pressure losses as a function of the Reynolds number. In this figure, the different sources of pressure losses in the flow are separated. The curves correspond to the various assumptions used for the computations and reproduce the results associated with each assumption. The bottom part of the figure corresponds to a fully developed flow along the entire length of the microchannels. Additional losses due to entrance effects and losses associated with the sudden change of the cross-sectional area at the beginning of the microchannels correspond to the next two regions. Finally, the upper part of the figure corresponds to the sum of the pressure losses in the main inlet/outlet channels and in the distributor/collector. This figure demonstrates that two main phenomena have a significant contribution to the total losses in the system. The regular losses in the microchannels account for a very large part of the total losses at low values of the Reynolds number (80% for Re = 5). This contribution obviously decreases when the Reynolds number is increased (20% for Re = 700). This may be understood since the regular pressure drop in a microchannel is proportional to the flow rate whereas the pressure losses in the other parts of the system are due to separation and mixing, as presented hereafter, and are likely to be proportional to the flow rate squared. This latter scaling law is confirmed by Figure 5 As can be seen in Figure 4, the pressure drop is caused mainly by losses in the distributor and collector channels for the high range of Reynolds numbers. For Re higher than 200, the pressure drop caused by friction in the microchannels is less than half of the total losses. Due to the good agreement between the present numerical and experimental results, it may be concluded that the deviation of Po from the fully-developed solution in long channels in the laminar regime is due to inertial pressure losses in the distributor and the collector channels. It is worth recalling that the present numerical computations were based on the assumption that the fluid is a continuum and on the classical laws of hydrodynamics, which predict the flow reasonably well. It seems therefore that no size effects are present in the flow investigated, which was characterized by a length scale of 110 µm. The same conclusion was found by Gao et al. [START_REF] Gao | Scale Effects on Hydrodynamics and Heat Transfer in Two-Dimensional Mini and Microchannels[END_REF] for flows in two-dimensional microchannels 100 µm in height.

Pressure and Velocity Fields

Further insight into the flow may be gained from analysis of the pressure and velocity fields deduced from the numerical results. For the following presentation of results, the pressure and velocity fields are considered in the plane located at y = 100 µm above the bottom of the microchannels. The pressure field is illustrated on Figures 678for one typical value of the Reynolds number (Re = 390). The pressure is normalized in The normalized pressure exhibits a sharp drop near the inlet of the distribution channel and only slight variations further downstream (see Figure 6). The variations of p * are more uniformly distributed along the collector channel. These results may be compared to those of Yin et al. [START_REF] Yin | Single-Phase Pressure Drop Measurements in a Microchannel Heat Exchanger[END_REF], who measured the pressure drop in a microchannel heat exchanger. Their heat exchanger consisted of 41 tubes, each tube having eleven ports 0.787 mm in diameter. Contrary to the present arrangement, the inlet and outlet were placed at opposite sides of the distributor and collector channels. Despite this difference, the trend for the pressure drop is the same in both studies. Yin et al. [START_REF] Yin | Single-Phase Pressure Drop Measurements in a Microchannel Heat Exchanger[END_REF] found that the pressure drop in the exit header was much higher than that in the inlet header. Figure 6 shows that the pressure variation is also much higher in the collector channel than that in the distributor channel if we exclude the strong pressure drop over the first mm of this channel. These larger pressure losses may be due to jet mixing in the collector channel, as discussed later. Figure 7 shows the pressure variations along the axis of symmetry of the system. The left part of the curve corresponds to the main inlet channel (x < 0 mm) and the distribution channel. The triangular microchannel starts at x = 0.8 mm and ends at x = 19.2 mm. The collector channel and the main outlet channel correspond to x > 19.2 mm symmetrically from the inlet part. The linear pressure distribution from x = 0.8 mm to x = 19.2 mm shows a regular pressure drop along the microchannel located in the plane of symmetry.

The distribution of bulk velocities in the system of parallel microchannels is plotted in Figure 9. The results of the computations were completed using symmetry to depict the complete system. The mean velocities vary significantly from channel to channel according to the pressure differences shown in Figure 6. The highest velocities and mass flow rates are observed near the axis of symmetry of the system of channels, as expected. It is worth noting that the ratio of the highest to lowest bulk velocities along the system is slightly Reynolds-dependent (equal to 2.5 for Re = 195 and 3 for Re = 390). This maldistribution of the total flow rate is associated with the present arrangement of the flow inlet and outlet. It should be noted that the objective of the research was to investigate the flow characteristics of a test section with the same geometry as that of a microheat pipe. A better arrangement from the point of view of microheat exchangers would probably have been obtained by placing the inlet and outlet at opposite sides of the distributor and collector channels. Preliminary computations of this arrangement confirm this tendency. Yin et al. [START_REF] Yin | Single-Phase Pressure Drop Measurements in a Microchannel Heat Exchanger[END_REF] also investigated this configuration and found a significantly smaller mass flow maldistribution.

The explanation of the pressure distribution of Figures 6 and8 is clearly visible in Figure 10, which shows contours of the z-velocity component in the inlet region. Figure 10b shows these contours over several cross-sections of the distributor channel. The z-direction is perpendicular the axis of symmetry of the system so that Figure 10 shows the flow along the distributor channel. The sudden change in the flow direction at the end of the main inlet channel gives rise to a strong separation at the beginning of the distribution channel. This reverse flow causes a significant reduction of the distribution channel cross-section and consequently to a high pressure drop in this region (see Figure 6). Low pressures are associated with the region of separation (see Figure 8a). Sufficiently far from the region of reversed flow, the pressure is practically constant.

Figure 11 shows the contours of the x-velocity component in the outlet region. The x-axis corresponds to the outlet channel direction. The flow pattern in the collector channel is quite different from that in the distributor channel. This region is dominated by the merging of jets exiting the triangular microchannels. These jets are responsible for flow mixing in the collector channel and for significant head losses in this region. These head losses cannot be easily avoided because they are related to the sudden change of the section at the end of the triangular microchannels (Figure 11, left). Again, a region of separation is found at the beginning of the main outlet channel (Figure 11, right). This is a source of additional pressure losses. The pressure reaches low values in the recirculation region (see Figure 8b), but they are far from the saturation pressure so that cavitation effects are not expected for this value of Re. However, cavitation cannot be completely excluded for the highest flow rates.

CONCLUSIONS

The main purpose of the present study was to separate geometric effects from possible scale effects on hydrodynamics in microchannels. Experimental results on the pressure losses across a network of triangular microchannels 110 µmi nh ydraulic diameter were compared to numerical computations based upon the classical laws of hydraulics. The numerical computations were performed in two steps. In the first, the flow was computed in a single channel where fully developed flow, entrance effects due to the development of boundary layers, and local losses due to a sudden change of section were treated. In the second part of the work, the flow was computed for the entire system of channels corresponding to the experiments.

The main conclusion is that no scale effects are observed in the tested system of channels for the size used in the experiments (hydraulic diameter = 110 µm). Numerical computations show that geometric effects are responsible for the deviation from the fully-developed solution in long channels. An analysis of the results shows that the additional pressure losses are mainly due to the following sources:

1. in the collector channel, the pressure losses are caused by the jets issuing from the microchannels. They cannot be avoided because it is impossible to end the microchannels by slow diverging sections owing to the lack of space between two consecutive channels. 2. separation is present in the inlet region and the main outlet channel. The passage from the main inlet channel to the distribution channel should be appropriately profiled to avoid separation. The same can be said for the outlet channel.

This research also underlines the care that must be taken in interpreting pressure measurements during the study of microchannel flows. An estimation of the pressure drop across one microchannel requires a careful separation of pressure variations in the region of interest and in the parts of the system used to connect the microchannel to an external hydraulic circuit.

Finally, it should also be stressed that the careful design of a complex network of microchannels is necessary to avoid excessive pressure losses in the system.

Figure 1

 1 System of parallel triangular channels on the silicon plate: a) top view, b) cross-sectional view.

Figure 2

 2 Figure 2Pressure drop across the system of microchannels.

Figure 3

 3 Figure 3 Computational domain of the complete model.

Figure 4

 4 Figure 4 Contribution to total pressure losses versus Re number.

  , which shows the difference between the actual value of the Poiseuille number and the value (Po viscous ) reached for very low values of the Reynolds number. Logarithmic scales are used to highlight the Re-dependence of the modified Poiseuille number Po -Po viscous . Figure 5 clearly demonstrates the linear dependence of Po -Po viscous versus Re for large values of this parameter. The corresponding pressure losses are therefore characterized by a constant friction factor (Eq. 2).

Figure 5

 5 Figure 5 Modified pressure drop across the system of microchannels: experimental results, △ numerical results.

Figure 6

 6 Figure 6 Pressure along the distributor and collector channels for Re = 390. Pressure is given in the planes of symmetry parallel to Oz-axis of both channels.

Figure 7

 7 Figure 7 Pressure on the center line across the system of channels for Re = 390.

Figures 6

 6 Figures 6 and 7 by

Figure 8

 8 Contours of static pressure for Re = 390: a) in the inlet region, b) in the outlet region.

Figure 9

 9 Figure 9 Distribution of bulk velocities in the system of microchannels for Re = 390.

Figure 10

 10 Contours of z velocity component for Re = 390: a) in the inlet region, b) in cross-sections of the distributor channel.

Figure 11

 11 Figure 11 Contours of x velocity component in the outflow region for Re = 390.

  diameter, m f Fanning friction factor (Eq.1), dimensionless L microchannel length, m p pressure, Pa p * normalized pressure, dimensionless Po Poiseuille number (=fRe), dimensionless Po viscous limit of Po when Re →0, dimensionless Re Reynolds number (= ρVD h µ ), dimensionless V average bulk velocity, m.s -1 Greek Symbols p pressure drop, Pa µ dynamic viscosity, kg.m -1 .s -1 ρ density, m 3 .s -1

Table 1

 1 Geometric parameters of the cell

	Triangular microchannels	Microchannel width	215 µm ± 10 µm
		Microchannel height	150 µm ± 5 µm
		Hydraulic diameter	D h = 110 µm
		Microchannel length	L = 18.4 mm
	Distributor channel	Channel height	h dc = 188 µm
	Collector channel	Channel width	a dc = 800 µm
		Channel length	L dc = 20 mm
	Main inlet and outlet channels	Channel height	h io =188 µm
		Channel width	a io = 835 µm
		Channel length	L io = 1.5 mm

This research was supported by the Rhône-Alpes Region. The authors would like to gratefully acknowledge the Socrates program for partly supporting the visit of Mariusz Niklas to Grenoble. They also wish to thank Nicolas Mazellier for performing the experiments.