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Electro-Hydro-Dynamic plumes due to autonomous and non autonomous charge

injection by a sharp blade electrode in a dielectric liquid.

Philippe Traore Institut PPRIME

Jian Wu Institut PPRIME

Christophe Louste Institut PPRIME

11 Boulevard M. Curie 86962 Futuroscope Chasseneuil

France philippe.traore@univ-poitiers.fr

11 Boulevard M. Curie 86962 Futuroscope Chasseneuil

France xikuanghit@gmail.com

11 Boulevard M. Curie 86962 Futuroscope Chasseneuil

France Christophe.louste@univ-poitiers.fr

Quentin Pelletier Institut PPRIME

11 Boulevard M. Curie 86962 Futuroscope Chasseneuil, France

quentin.pelletier.qp@gmail.com

Abstract - This paper addresses the numerical investigation of

an Electro-Hydro-Dynamic (EHD) plume induced by a sharp

blade electrode in a dielectric liquid. Different injection laws

autonomous and non-autonomous ones are considered and the

importance of the blade shape is emphasized especially for non

autonomous injection laws. Different flow regimes arise

according to the value of the Reynolds number based on the ionic

mobility and the distance d between the blade and the vertical

plate. It is found that the critical Reynolds number for which the

transition between steady and unsteady regimes occurs depends

both on the injection law and the blade shape. For much higher

Reynolds number the flow turns to be turbulent and behaves as

jet flow with the development of a Kelvin-Helmoltz instability.

The use of different sharp blades indicates that higher electric

field at the tip of the blade plays a significant effect on the

development of the EHD plume. The plumes have been

characterized by their charge density distribution which shows,

in the early stages of the simulation, that its dynamic is

significantly dependent on the blade shape as well as on the non­

autonomous injection law.

Index Terms-Numerical simulation, Finite volume method,

EHD plumes, Blade electrode, injection law, non autonomous.

I. INTRODUCTION

When high electrostatic potential is applied to a sharp electrode in a dielectric liquid, charge injection occurs at the emitter electrode into the surrounding liquid. The Coulomb force acting upon the injected space charge sets the liquid into motion in the form of a jet like flow referred as Electro­Hydro-Dynamic plume. This type of flow has been thoroughly studied in the past, fIrstly experimentally [1]-[3] and later numerically [4]-[9]. The aim and originality of this paper is to present a numerical study of the flow induced by a sharp electrode where the whole set of the governing equations including Navier-Stokes equations coupled with charge density transport equation and Poisson equation for the electric potential are solved with a fInite volume approach. Due to the extra complexity in the meshing of the computational domain including the blade shape, a fmite-

Lucian Dascalescu Institut PPRIME

11 Boulevard M. Curie 86962 Futuroscope Chasseneuil, France

lucian.dascalescu@univ-poitiers.fr

volume method specifIcally designed for non-orthogonal structured grid has been employed. Another important feature of this problem deals with the way electric charges are injected from the blade in the bulk [12]-[11]. The charge injection process results from a complex chemical reaction at the interface between the blade surface and the liquid. In this study we did not aim to account for all these complex chemical reactions. Several injection laws have been proposed in the past by several authors [12],[11], and in the excellent review of [12]. These injection laws, which drive the amount of charge injected in the liquid, may be autonomous or non-autonomous and both are considered in this study. In non-autonomous injection law the electric charge at the blade surface, used as boundary condition for the charge transport equation, is linked to the local electric fIeld intensity at the blade surface. This induces a stronger coupling between all the different variables of such problem. In autonomous injection law the charge density and electric fIeld are completely decoupled. We have chosen a classical autonomous injection law where the charge density at the blade surface remains independent of the electric fIeld and which may serve as reference case. Two non-autonomous laws are selected from the available literature resources. The reminder of this paper is organized as follows. In the following section the governing equations are given and the numerical method is described. Then in section III we provide the details of the numerical computations which have been conducted. In section IV the main results are presented and the effect of the different injecting laws as well as different blade shapes is highlighted. Finally the conclusion is drawn in section V.

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II. FORMULATION OF THE PROBLEM

A. Governing equations

The system under consideration in this article is depicted on figure l. A dielectric liquid fills a cavity where two electrodes are immerged. A blade shaped one and a vertical one.

Figure I Sketch of the blade-plane configuration

The two electrodes are at a distance d apart. An applied

difference of potential L1 V = Vo - V; between the blade and

the vertical wall, will generate an electric field towards the plane and will inject electric charges in the bulk. The problem is formulated considering the usual hypothesis of a Newtonian and incompressible fluid of kinematic viscosity v, density

Po' For universality in the description of the problem it is

particularly convenient to work with non-dimensional equations. The basic scales of the problem are the distance d, the injected charge density qo and the velocity

Vo = KH (Vo - V;) / d where KH = �I; / Po represents

the hydrodynamic velocity where I; is the pennittivity. This leads to the following set of dimensionless equations:

v.u=o

au (-0)- 0- I A- E­-+ u.v U = - vP + - D.U +q at Re

aq + v.(q(U +_1 E)) = 0 at M L1V =-q

E=-VV

(1)

(2)

(3)

(4)

(5)

where ii is the fluid velocity, p the modified pressure including the pressure and the scalar from which the electrostriction force derives. As the fluid is homogeneous

and isothermal the dielectric force vanishes and only the

Coulomb force qE acts on the fluid. q is the charge density.

V is the electric potential, E the electric field. These previous scaling choices lead to the following set of dimensionless numbers:

Re is the dimensionless electrical Reynolds number. It is directly proportional to the electric potential difference L1 V = Vo - V; between the two electrodes. M is the ratio

between the so-called hydrodynamic mobility and the true mobility of ions K. C is a dimensionless measure of the injection level. It refers to the amount of charge injected in the bulk through the charge density on the blade qo.

B. Numerical method

A numerical code based on the [mite volume method [13], [14] has been designed to integrate the whole set of partial differential equations (1 )-(4) on a non-orthogonal mesh in order to accurately fit the geometry of the blade (see figure 2). The equations are discretized with second order in time and space schemes. The fluid being assumed to be incompressible, the velocity-pressure coupling algorithm is undertaken by the SIMPLE algorithm [13].

Figure 2 Computational domain with a zoom around the blade tip

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C. Charge density equation treatment

In dielectric liquids the charge density transport equation is purely hyperbolic since charge diffusion may be safely neglected [11]. This equation requires then a special treatment to discretize the convective flux. Indeed as there is no charge diffusion, unsuitable scheme, may generate spurious numerical oscillations dramatically prejudicial for the accuracy and validity of the results. In the same time too diffusive schemes will generate some artificial numerical dissipation which will smear the solution in an unacceptable way. The main desirable and strongest property for such scheme capable to avoid this undesirable numerical behaviours is to be Total Variation Diminishing (TVD) [15]. Many schemes having this property exist and we have chosen the 2nd order Smooth Monotonic Algorithm for Real Transport (SMART) [16]. The interested reader may found additional details in [17] and in [18].

D. Initial and Boundary conditions

All the simulations start from the fluid at rest. The boundary conditions are depicted on figure 3.

oqi on = oV/ = 0

U =0 u'=o y

plane i7q/on=O

v=o U =0 u'=o y

Figure 3 Boundary conditions

No-slip condition for the velocity is considered on all boundaries. On the blade surface, the potential is set to Vo=1 and the charge density is set to

qo(s) = f( C,E(S)inJ,Emax ) where s is the curvilinear

coordinate along the surface blade, f is the function which defmes the injection law. E(S)inJ is the magnitude of local

electric field at the s location on the blade surface and

Emax the maximum magnitude of the electric field located at

the tip of the blade, C the injection strength. Note that the injection strength operates as a parameter in the boundary condition for the charge density. The two points designed by SOl and S02 defme the injecting zone around the tip. They may be set arbitrary or in the contrary being computed by the calculation during the injection law implementation process. In this paper, we should rather let the system computing the location of these two points. Indeed the two points SOl and S02 are defmed such a way that for every

S E [SOpS02] � E(S)inJ � Ec = O.6Emax· The choice of coefficient 0.6 is arbitrary. . When function f is constant or at least independent on the local electric field magnitude then the injection is considered as autonomous. This is the case for a lot of EHD problems and particularly for Coulomb-driven electroconvection between two planes [11]. It means in that case that the amount of charge density injected in the bulk is purely disconnected from the electric field.

E(s\". < 064,."" q(s) =0

...........

SOl

---I I:: E(s\" ;:::: 064,.", q(s)=f(E(s '1F,E�,C

\ Figure 4 Definition of pints SOl and So

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III. NUMERICAL EXPERIMENTS DETAILS

A. Mesh generation

Three different blades with associated meshes have been designed as depicted on figure 5.

0.5

0.4

0.3 a=0.8236

III! 0.2

0.1

>- 0

blade 1 >-

-0.1

-0.2

-0.3

·2 -0.4

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

X

0.02

0.01 a =0.0218 III! �

>-

blade2 0 b=0.00375 >- 0

·2 -0.01

-4 -0.02 -0.03 -0.02 -001 0

X

0.02

>-0.01 a =0.02693 III! �

blade3 0 >- 0

-2

-0.01

-4

-0.02

Figure 5 The three different blades considered in the simulations.

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Blade 2 and 3 are much sharper than the blade l. The different numerical domains have been meshed with the meshing software Pointwise, excepted for the blade 1 mesh which has been generated according iso-potential and iso­electric field lines (see figure 2). Is should be pointed out that the quality of the mesh is superior when following iso­potential and iso-electric field lines. This has a non negligible impact on the convergence rate of the numerical code especially in terms of CPU time consuming. Indeed the computational cells generated by Pointwise are more skewed than in the case of the blade 1 mesh. The computation is then more demanding.

Mesh of blade 3

Zoom around the tip

0.0008

0.0006 r---____ ____

0.0002

>- 0

-0.0002

-0.0004

-0.0008 �-:::>-<----

x

Figure 6 Detailed on the blade 3 mesh.

B. injections laws

Several injection laws glvmg the charge density on the surface of the blade have been considered in this paper.

The fust one can be described as: qo(s) = C (6)

The second injection law is inspired from the paper of Vasquez [8] and Suh [12]. The charge injected in the bulk as boundary condition is dependent on the magnitude of the local electric field on the

injector Einj and is expressed as:

C qo(s) = 112 2bE(s)injK1(2bE(s)inj)

(7)

K/ is the modified Bessel function of the second kind and

order one, b is a dimensionless parameter depending on the charge of electron, on the Boltzmann constant, on the absolute temperature and on the potential on the injector. In

our computations b = 0.2

The third law is defmed as:

(8)

Where Ec is a kind of electric threshold which determines the

injecting zone, typically Ec = 0.6Emax where 0.6 has been

chosen arbitrarily. Each time the curvilinear coordinate along the blade s is not comprised between [So I , S02] then q 0 (s) = 0

IV. RESULTS

A. Flow structure

In a previous study [19] the authors have shown that such EHD flow was characterized by the development of a plume similar to a jet from the blade towards the vertical electrode. The flow is steady depicting two steady vortices until the Reynolds number is below a crititical value found to be within the interval [1000, 1100] (see figure 7). But this critical value is dependent on the injection law and on the blade shape too. The range [1000, 1100] has been found for a autonomous law qo(s) = C and for the hyperbolic blade 1. For the second

injection law our computations show that the transition occurs sooner and the critical Reynolds number belongs rather to the range [700-800] always for the blade 1. As we shall see it later it is due to the fact than the second injection law injects a greater amount of charge in the bulk and even more when the electrode is sharper.

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Figure 7 Steady stream function at Re=200

Above this critical transition value the jet starts to be unsteady with a characteristic flapping behaviour often encountered in impinging jets. When the Reynolds is still increased the flow turns to become turbulent but as we are in 2D we should rather say chaotic. On figure 8 we can observe three instantaneous snapshots of the unsteady vorticity field at Re=lOOOO. This highlights the formation of small eddies due to the development of the Kelvin Helmoltz instability characteristic of jet flows. We can observe the development of a vortex shedding as in a typical Von Karman street.

Figure 8 Three instantaneous contourmap of the vorticity field at Re=lOOOO.

B. Effect a/the injection law.

In this numerical experiment we performed some computations with the same blade, but using the three injection laws that we have implemented. The first law is autonomous and does not depend on the local electric field. The two others are non autonomous and are related to the electric field computed at the blade surface and follow the expressions given in section III B. First we examine the distribution of the charge density on the blade 1 surface. On figure 9 we can observe that the injection law 2 is the one who injects the most, while the injection laws 1 and 3 are of the same order.

35 --- Injection law 1 --- Injection law 2

30 --- Injection law 3

25

20

"

15

7\ 10

5

0 50 100 150 200 250 300

J

FIgure 9 Charge denSIty dlstnbutlon on the blade surface.

On figure lO we can observe the effect of these different injection laws on the flow itself. We have displayed the contourmap of the charge density distribution in the early stages of the simulation. All the three figures are displayed at the same dimensionless time t=0.25. The Reynolds number

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has been set to 500 for the three cases and the same blade (blade 1) is considered too. Only the injection laws differ. First we can observe that the TVD numerical scheme used to solve the charge density transport equation is capable to keep the sharp gradients without smearing the solution. We observe

0.05

the formation of an EHD plume. The charge density is Law 3 >-

convected in the bulk and adopts the shape of a mushroom which is slightly different according the considered injection law. It is also visible that the amount of charge injected in the bulk is also very different from one law to another. We can

-0.05

see that injection law 2 is the most injecting one. We also notice that for the same time, the injected charge has reached the abscise x=0.0808 with the second law while for the two other cases law 1 and law 3 it is a bit less: 0.0572 and 0.056

0.1 -0.05 o 0.05

x

respectively This clearly indicates that these injecting laws Figure 10 Charge density distribution for the three different laws. have a real effect on the dynamic of the flow itself.

0.1

0.05

>-

Law 1

-0.05

-0.05 o

0.05

Law 2 >-

- 0.05

-0.05 o

0.05 0 . 1

x

0.05 0.1

x

C. Effect a/the blade shape.

On figure 11 we have displayed the evolution of the E(S)inj qo (s) along the blade for the 3 different blades

the second injection law (equation 7).

E

o c:

60

50

40

I 30 W

20

10

I njection law 2

--- blade 1 --- blade 2 _ . . _ . . _ . . blade 3

-" -"" --';;. ,

150 200

(a) Ma nitude of the Electric field alon the blade surface

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I njection Law 2 150

--- blade 1 --- blade 2 _ . . _ . . _ . . blade 3

100

a

50

As expected and due to the sharpness of blade 2 and 3 it is observed that the electric field magnitude is higher in these two cases compare to the blade 1 case. As a consequence blades 2 and 3 inject more charge than the blade 1. It appears that the blade 2 is the most injecting one due to his thinness

On the next figure we have displayed the charge density distribution obtained with the second injection law but for the three different blades. As previously all the results are displayed at the same time which is here t=0.03 and for the same Reynolds number Re=500. It is noticeable that at this time only small amount of charge is effectively injected in the bulk from blade 1, although for blade 2 and blade 3 the characteristic EHD plume arises in a mushroom shape. When comparing blade 2 and blade 3 it is worth noting that although the behaviours are very similar the dynamic is a bit enhanced in the case of blade 2. Indeed we have noticed that the maximum amount of charge injected, which can be seen in the middle of the plumes, was higher in the case of blade 2 compared to blade 3. This is coherent with the distribution of electric field on the surface blade which is higher for the second injecting law (see figure lla).

blade 1

blade 2

blade 3

-0.04

0.01

>- 0

-0.01

-0.02

-0 03 -0.02

0.01

>- 0

-0.01

-0.02

-0.03 -0.02

-0.02

X

-0.01

X

-0.01

X

0 0.02

0 0.01

o 0.01

Figure 12 Charge density distribution for the three blades.

V. CONCLUSION

In this paper numerical investigations for the study of an EHD plume induced by sharp blade electrode in a dielectric liquid have been undertaken. Several different injection laws autonomous and non-autonomous ones as well as different blade shapes have been considered. Different flow regimes according the value of the Reynolds number have been emphasized. The critical Reynolds number for which the transition between steady and unsteady regimes occurs depends on the injection law and on the blade shape. In the

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case of an hyperbolic blade and for autonomous injection the critical Reynolds number is determined in the range [lOOO, 1100] while for a non-autonomous injection laws, the critical Reynolds number is lower and belongs to the range [700-800]. It has been shown than for a Reynold number lOOOO the flow turns to be turbulent and behaves as jet flow with the development of a Kelvin-Helmoltz instability. The dynamic of the EHD plumes is significantly dependent on the blade shape as well as on the injection law.

ACKNOWLEDGMENT

The authors would like to gratefully acknowledge Professor Adamiak from Western University of Ontario, Canada for the blade 1 mesh he kindly provided us for our computations.

This work was partially funded by the French Government program "Investissements d' Avenir" (LABEX INTERACTIFS, reference ANR-II-LABX-0017-01

REFERENCES

[I] F. M. 1. McCluskey and A T. Perez, The electrohydrodynamic plume between a line source of ions and a flat plate, IEEE Trans. Electr. Insulation 2(27), (1992) 334.

[2] B. Malraison, P. Atten and A. T. Perez, Panaches charges resultant de I'injection d'ions dans un liquide isolant par une lame ou une pointe placee en face d'un plan, J. Phys. III France 4, (1994) 75.

[3] A T. Perez, P. A Vazquez, and A Castellanos, Dynamics and linear stability of charged jets in dielectric liquids, IEEE Trans. Industry App\. 31 (1995) 761.

[4] P. A Vazquez, A T. Perez, and A Castellanos Thermal and electrohydro- dynamic plumes. A comparative study, Phys. Fluids 8, (8) (1995) 2091-2096.

[5] P. A. Vazquez, A Castellanos, A.T. Perez and R. Chicon, Numerical modelling of EHD flows due to injectors of finite size, Conference on Electrical Insulation and Dielectric Phenomena, (2000).

[6] P. A. Vazquez, E. Chacon Vera, A Castellanos and T. Chacon Rebollo, Finite element-particle method calculation of EHD plumes, Annual Report Conference on Electrical Insulation and Dielectric Phenomena, (2002) 208-211.

[7] P. A. Vazquez, E. Chacon Vera, A Castellanos and T. Chacon Rebollo, Finite element-particle method calculation of EHD plumes, Annual Report Conference on Electrical Insulation and Dielectric Phenomena, (2003) 706-709.

[8] P. A. Vazquez, C. Soria and A Castellanos, Numerical simulation of two-dimensional EHD plumes mixing finite element and particle methods, Annual Report Conference on Electrical Insulation and Dielectric Phenomena, (2004) 122-125.

[9] A Perez , P. Traore, D. Koulova-Nenova and H. Romat, Numerical study of an electrohydrodynamic plume between a blade injector and a flat plate, IEEE Transactions on Dielectrics and Electrical Insulation, 16, (2009) 448-455.

[10] A Denat,B.Gosse, J.P. Gosse, Ion injections in hydrocarbons, Journal of Electrostatics, 7, (1979), pp 205-225.

[II] A Castellanos, (ed) Electrohydrodynamics, (New-York) Springer, (1998).

[12] Y.K. Suh, Modeling and simulation of ion transport in dielectric liquids- Fundamentals and Review. IEEE Transactions on Dielectrics and Electrical Insulation, Vol 19, No 3; (2012) 831848.

[13] Patankar S.V., Numerical Heat Transfer and Fluid Flow, Stockholm, Washington, DC, (1980)

[14] P. Traore , Y. Ahipo, C. Louste, "A robust and efficient finite volume scheme for the discretization of diffusive flux on extremely skewed meshes in complex geometries." Journal of Computational Physics, 228, pp. 5148-5159,2009.

[15] S.F. Davis, TVD finite difference schemes and artificial viscosity, ICASE Report 84-20, NASA CR-I72373, NASA Langley Research Center (1984).

[16] Gaskel P. H and Lau AK.C, Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm, International Journal for Numerical Methods in Fluids, 8, (1988) 617-641.

[17] P.Traore and A Perez, Two-dimensional numerical analysis of electroconvection in a dielectric liquid subjected to strong unipolar injection, Physics of Fluid 24, (2012) 037102.

[18] J. Wu, P. Traore, C. Louste An efficient finite volume method for electric field-space charge coupled problems, Journal of Electrostatics, (2013), pp 319-325.

[19] J. Wu, P. Traore" C.Louste, D.Koulova, H. Romat, Direct numerical simulation of electrohydrodynamic plumes generated by a hyperbolic blade electrode, Journal of Electrostatic, 3, (2013), pp 326-331

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