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 [1] as well as prevent flow separation, [2] [3] .
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 (jxB) in Navier Stokes equations. Typically, an EM actuator comprises a pair of wall flush electrodes and a pair of subsurface magnets. The configuration of electrodes and magnets on the wall surface is such that the curl of jxB is non-zero in the vicinity of the actuator [4] [5]. 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 the persistence of induced velocity (wall-normal component and wall jets) and vorticity, [6].
One of the key questions regarding this concept of flow control is the determination of lengthscales and time-scales 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 on idealised 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. 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 1b. In most of the volume above the actuator Curl(jxB) is 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 1c) [6]. By using permanent magnets the intensity and sign of jxB 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 idealised 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 electromagnetic 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(JxB) 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 constitutive equation for the current density j is given by Ohm's law (5) where uxB 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 uxB 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 jxB 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).

Fluid's equations Magnetic induction equation and Ohm's law
The outstanding fact that the EM force distribution is independent of the flow places EM flow control out of the conventional problematic of Magnetohydrodynamics (where JxB forces are directly depending 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 be fitted as well to the size of a typical structure present in turbulent boundary layers (micro-actuator), see Robinson [7], Adrian et al. [8], as to a larger scale like the spacing between packets of structures (macro-actuator), see Meng [9], Zoo et al [10], and [4].

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 have a rather complicated 3-D shape. To a good approximation, the force distribution can be computed using the analytical solution given by Akoun & Yonnet [11]. 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 [13]. 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 in 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 [4]. More precisely, the computed three-

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 wall-normal 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 plans of symmetry normal to the wall namely: the one equidistant from electrodes and the one equidistant from the magnets. (i) Mean EM acceleration g EM : 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 electromagnetic acceleration and e y the unit vector perpendicular to the wall.
(ii) EM velocity, V EM : 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,: This T EM corresponds to the time at which pumping sets in at a height y.
Equation (8) can be reorganised as in equation (10). This non-dimensional ratio can be interpreted as the EM equivalent of the Froude number with a value of 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 un-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. It allows here to 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 EM actuator geometry. The characteristic wall normal length is therefore independent of current or magnetic intensities, assuming EM forces are still above viscosity damping.   [5]. 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. Table 3: Estimation of EM characteristic numbers for any current I or induction B deduced from the values computed for I=1A, B surf =1T (i.e. EM1 stands for I=1A and B=1T) and identical geometries of actuators.

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 electromagnetic field (effectively the work of EM forces). This expression is: A similarity law can be constructed by taking the EM Froude number, F rEM , as a constant in equation (11). 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 electromagnetic 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.
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)): 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 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. 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 smaller than T EM too. α is a dissipation coefficient (α≤1) mostly due to wall effects during the activation. For activation time larger than T EM , via [4] it is possible to give numerical values for α : i.e. α~0.26 for 1999 actuator and α~0.29 for 2000 actuator.

Multiple activation
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: 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 un-equality: In addition to the flow acceleration, local EM actuators are also a source of vorticity corresponding to the curl(jxB) see [4]. In multi-activation system, like in the network of 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 (1m/s, Rex=10 7 , U ∞ =28.6u τ ) comparing two classes of actuator sizing: macro and micro actuator. For each case the typical length-scales in wall units including streaks spacing (100 + ), streaks size (40 + ) and streaks vorticity (123 s -1 ), see [12], are 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 2 asymptotic EM regimes. In the experiments the magnetization of permanent magnets is 1.3T and their typical longitudinal length-scale spacing, L EM , is 30 mm.

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 were carried out on a flow initially at rest in an aquarium large enough to avoid confinement effects.
The second series of experiments were performed in a seawater tunnel, with and without external flow.

EM activation on a flow initially at rest (PTV measurements)
The aquarium has dimensions 50cm x 50cm x 60cm and is filled with salt-water (35g NaCl/l). The EM actuator is situated in the centre of the side vertical wall of the aquarium (figure 8a). Figure 8b shows

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 x 100 mm and a test section length of 1.3 m [4]. 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 realised just above the EM actuator, which is inserted (wall flush) in the top wall of the seawater tunnel (see figure 8c). Only one wall-normal plane of measurement equidistant from the two electrodes is studied here. This plane, 0° on figure 8, is also streamwise to the external flow. A double pulse Yag laser and a digital/numerical camera (1000x1000 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.

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: 0.5A 0.8A and 1.1A (see figure 10). The curves show the same behaviour. Flow starts from rest far from the wall (y>60mm) 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 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 [4] [5].  The measurements reported on 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 x0°, x45°and x90° respectively (cf. figure 8).
Measurements taken at different current intensities are normalised to the corrected current base I*=1A via the 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 x0°, x45° and x90° 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 & 2000) and 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 10s) 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 the one of the aquarium consequently the duration of activation was limited to 3 seconds. For longer activation confinement effects grew strong. Measurements are done by PIV in this case. Even with these notably different experimental conditions, the normalised 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 and 10 seconds 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.

Electromagnetic flow regime
For the actuator 2000, the flow (of seawater) in the tunnel was set at two different velocity values: "high velocity" (U ext~1 00 mm/s) and "low velocity" (U ext~1 0 mm/s). Figure 14 shows axial velocity profiles taken from PIV measurements. In each case the velocity profile of the flow without activation is superposed to an EM activated. The very slight drop in the velocity from the maximum value in the unactivated profile for y>15mm, 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 electromagnetic time, T EM , to the transit time, T flow in each case. The ratio of these two time scales is that given in equation (13). 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, 17. Finally, the time ratio EM R can be generalised to asymptotic domains: EM R>>1 corresponds to "pumping mechanism" and EM R<<1 corresponds to "deflecting mechanism" of EM force effects.

Conclusion
The (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 one) 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 one) 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.