Numerical Modeling of a Two-Dimensional Electrohydrodynamic Plume between a Blade and a

Flat Plate

A.T. Pérez1, Ph. Traoré 2, D. Koulova-Nenova 3, H. Romat 2

1- Dpto. De Electronica y Electromagnetismo,Facultad de Fisica Avda. Reina Mercedes s/n, 41012 Sevilla, Spain.

2 – Laboratoire d’Etudes Aérodynamiques, Boulevard Pierre et Marie Curie, BP 30179, 86962 Futuroscope-Chasseneuil, France

3 - Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

Abstract: This paper deals with two-dimensional EHD flow occurring between a blade and a plate electrode a distance H apart. This type of EHD flow, known as electro-hydrodynamic plumes, arises when sharp metallic contours submerged in non conducting liquids support high electrostatic potential resulting in charge injection. The aim of this paper is to analyze, from a numerical point of view, the evolution of velocity profiles, flow pattern and other relevant parameters characterizing plumes flows. Numerical results are compared with those obtained from experimental or theoretical works found in literature and show that the numerical technique used is suitable to study EHD plumes.

I. INTRODUCTION When ions are injected from a wire or a blade in a dielectric liquid under the influence of an applied electric field, convective motion in form of a jet arises. These particular flows have been referred to as EHD plumes and are present in most of industrial devices exploiting electric forces. These flows have been observed experimentally [ A---] and the comparability of such flows with what people call thermal plumes has conducted many authors [1-xx] to analyze EHD plumes by the mean of self-similar analysis. Under some simplifying assumptions the governing EHD equations are made tractable and the mathematical model is closed to the one of thermal plumes for Pr → ∞ (where /Pr kν= is the Prandtl number ν is the kinematic viscosity of the fluid and k its thermal diffusivity) In this paper, a different way is proposed and full numerical investigations have been undertaken to study EHD plumes. We solve the entire set of conservation equations (mass, momentum, charge and potential) directly. The appropriate electrodynamics equations coupled with the Navier-Stokes equations are solved numerically using an efficient finite volume method.

II. FORMULATION OF THE PROBLEM

A. Electro-hydrodynamic equations

The system under consideration in this article is a closed rectangular cavity of length L and height H which is filled with a dielectric liquid. An applied difference of potential

10 VVV −=∆ between the blade and the upper wall where electrodes are placed generates a vertical electric field. The z-axis is taken perpendicular to the electrodes. On the bottom wall the injection region is reduced to one grid point in order to be as much as possible closed to experimental conditions generally observed.

Fig. 1 Physical domain and boundary conditions

The problem is formulated considering the usual hypotheses of a Newtonian and incompressible fluid of kinematic viscosity ν, density ρ. For universality in the description of such studies it is particularly convenient to work with non dimensional equations for mass, momentum, and EHD.

0. =∇U (1)

21( . )U U U P U CM qEt R

∂ + ∇ = −∇ + ∆ +∂

(2)

( )/ . ( ) 0q t q U E∂ ∂ + ∇ + = (3)

CqV −=∆ (4)

VE −∇= (5)

U is the fluid velocity, P – the pressure, q - the charge

density, V the potential and E the electric field. The reference length and velocity for scaling are the height H of the cavity and 0 /U K V H= ∆ , where K is the ionic mobility. The

pressure is non dimensionalized by 20Uρ . These previous

choices give rise to the following dimensionless parameters: /( )T V Kε ρν= ∆ , 2

0 /C q H Vε= ∆ , 2 2/M Kε ρ= , 2/ MTR =

T is the instability parameter, C is a measure of the injection level, M is the mobility parameter, R is Electrical Reynolds number. Here K is the ionic mobility, ε the dielectric constant. B. Numerical method The numerical procedure is based on the Augmented Lagrangian method [9], which transforms the Navier-Stokes equations into a variation problem. Here the primitive variables U and P are determined by the algorithm of Uzawa [10]. These equations are integrated over finite volume [11] using a staggered grid and a semi-implicit second order in time a space accurate schemes. The linear systems are solved using the Bi-CGSTAB method [12] with a preconditioning based on a modified and incomplete Gauss factorization MILU [13]. The calculation is fully transient. Special care must be provided while solving (3) because of its hyperbolic nature. The SMART algorithm [14] has been employed in that way.

III. RESULTS AND DISCUSSION In this kind of flow 3 different flow regimes can be highlighted according the value of the T parameter. Under a first value T1 that we have found to be close to 2350, the regime is laminar and stationary. The plume is perfectly stable. Between T1 and another value T2, the regime becomes periodic and the plume starts to oscillate due to the interaction of the two vortices on each side. Beyond T2 the regime becomes fully turbulent and the plume oscillates in chaotic manner. First we are interested in the stationary laminar regime. On Figure 2, 3 and 4 we have plotted the typical vertical velocity profiles for different z distances from the blade, the characteristic half-width ∆ of the plume taken from velocity profiles and the maximum vertical velocity in the plume. These results are fully in agreement with those found in experiment of Mc Cluskey et al [1].

Fig. 2 Vertical velocity profile according z distance from blade

Fig. 3 Characteristic half-width of the plume

Fig. 4 Maximum vertical velocity in the plume as a function of distance from

the blade

On Figure 5 and 6 we can see velocity profiles in the z middle plane respectively for different values of the T and M parameters. As expected the flow is speeded up with the increase of T but the dynamic in the stationary regime is completely independent of M. Looking at equation (2) we see

that in the stationary regime, and neglecting the inertial term, we are left only with two parameters 2RM T= and C. "

The cusp visible in Figure 2 is a consequence of the absence of charge diffusion. Since the charge is injected only in a point, and it does not diffuse, it remains at the center line of the plume. Therefore, the electric force is confined to this center line and, from equation (2), the derivative of the velocity is not continuous at this line.

Figure 3 and figure 4 show that, between z=0.2 and z=0.8, the half-width of the plume and the maximum vertical velocity have a potential dependence on the plume length. In this region the plume is well developed and it corresponds to the region where self-similar analysis [2,3] may be applied. For z<0.2 and z>0.8 the presence of the walls dominates the dynamics and a boundary layer approach is not of application.

Fig. 5: Velocity profile in the z middle plane for different T values

Fig. 6: Velocity profile in the z middle plane for different M values

The Figure 7 shows the evolution of the vertical velocity at a given monitoring point Pm versus time for different values of the T parameter. For T=2000 we are entirely stationary. For the values 5000, 10000 and 20000 the flow is unsteady but

periodic and the EHD plume oscillates slowly. For the value T=120000 showed here, the chaotic behavior of the signal becomes more evident.

Fig. 7: Time evolution of the vertical velocity taken at a monitoring point Pm(xm,ym)

Figure 8 shows the time evolution of the velocity and the isocontours of the vertical velocity which highlight the formation and the behavior of the plume with the two sided vortices. After its growth the plume is stable until time t=1.32. After this time, the plume starts to oscillate longitudinally because of the unsteadiness of the two sided vortices which interact with it. After t=5.28 secondary vortices appear in the flow which shows an increase of the turbulent activity. At time t=10 we can see not less than 6 different vortices those interactions with the plume will dictate its proper dynamic.

IV. CONCLUSION

Numerical simulations have been undertaken to analyze EHD plumes obtained between a blade and a plane. The existence of three different regimes, steady laminar, periodic and fully turbulent has been highlighted according the value of T. First results on velocity profiles, characteristic half-width of the plume and maximum velocity show total agreement with previous experiments and theoretical works. It has been shown that the dynamic of the plume is not affected by the value of the M parameter. Finally captions of the isocontours of the vertical velocity versus time show the oscillating behavior of the plume for high value of T in the turbulent regime.

t=0.165

t=0.33

t=0.825

t=1.32

t=5.28

t=6.27

t=6.6

t=7.755

t=9.57

t=10

Fig 8 Evolution of the isocontours of the vertical velocity and velocity versus time

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