Proc. 06 Electrostatics Joint Conerence A lattice Boltzmann method or electric ield-space charge coupled prolems Kang Luo a,, Jian Wu,, Hong-Liang Yi a,*, He-Ping Tan a a. School o Energy Science and Engineering, Harin Institute o Technology, Harin 5000, P. R. China GeoRessources Laoratory, Universitéde Lorraine (ENSG), CNRS, CREGU, F-5450, Vandoeuvre-les-Nancy, France Astract A lattice Boltzmann method (LBM) is developed to solve the electric ield-space charge coupled prolems. Instead o solving the macroscopic current continuity uation and the Poisson s uation, two discrete lattice Boltzmann uations are ormulated and solved to otain the distriutions o charge density and electric potential. The non-uilirium extrapolation scheme is used to treat the oundary conditions with complex geometry. Our technique is veriied with several test cases or which analytical solution and/or numerical results exits. A key eature o this methodology lies in its natural coupling with the LBM or luid low. As a demonstration, the inection induced electroconvection o dielectric liquids in a concentric-cylinder coniguration is considered. I. INTRODUCTION Electrohydrodynamics (EHD) is an interdisciplinary science dealing with the interaction o luid with electric ield []. The complex physics involved in electroconvective phenomena together with some promising applications draw a wide attention to this very active ield. Some representative applications include EHD pumps, electronic cooling, heat transer augmentation, active low control with electric ield, charge inection atomizers, electro-spinning, etc []. In general, EHD lows possess strong nonlinearity, which encourages the use o direct numerical simulation approach to gain deeper insights into such physical phenomena. During the last three decades, the lattice Boltzmann method (LBM), which is a mesoscopic modelling approach, has experienced rapid development and has ecome an estalished alternative or complex lows [3,4]. However, only until recent years, LBM has een introduced into the EHD ield [5], which is in contrast to the act that LBM has long een applied to magnetohydrodynamic (MHD) lows since the early 990s. Indeed EHD and MHD can e viewed as two special suects due to the interaction etween the electromagnetic ield and low motion []. However, as ar as we know, the LBM has not een well estalished or EHD prolems yet. The material presented here represents the irst results o a roader research proect aimed at extending the application o the LBM in simulating EHD lows o single-phase dielectric liquids. In a recent study [6], we have developed a uniied LBM ased on three These two authors contriuted ually to this work. * Corresponding author. Tel. +86 45 864674. Email address: yihongliang@hit.edu.cn (H. L. Yi)
Proc. 06 Electrostatics Joint Conerence consistent lattice Boltzmann uations (LBEs) to calculate the luid low, electrical potential and charge density distriution. The Chapman-Enskog multi-scale analysis has also een perormed to link the mesoscopic LBEs with the macroscopic governing uations. By this way, we provide a solid theoretical asis or our LBM. However, only a simple geometry prolem, i.e., the unipolar inection induced electroconvection in a plate-plate coniguration, was considered to validate the method in [6]. The oective o this study is to extend the LBM to EHD lows with complex geometries. The irst and crucial step is test the easiility o the LBM with the electric ield-space charge coupled prolems with complex geometries. Later a natural coupling with the LB model or low motion can e achieved. The reminder o this paper is organized as ollow. In section II the macroscopic governing uations are descried. Then in section III we present the LBEs or electric potential and charge density uations. In particular, the method or oundary condition treatment is explained in details. Ater that, in section IV the method is veriied with several test cases. As an application, the results with the inection induced electro-convection o dielectric liquids in a concentric-cylinder coniguration are also presented. Finally a conclusion is drawn up in the last section. II. MATHEMATICAL FORMULATION A. Governing uations For electric ield-space charge coupled prolems, the governing uations include the potential and charge conservation uations. In this study, we consider the simplest case with only one charge species, and thus the current continuity uation reads: q J 0, () t where q is the volume charge density and J is the current density. There are three asic transport mechanisms or ree charges in the electric and low ields: drit under the action o the electric ield, convection along with the low ield and charge diusion, J qke qu Dq, () where the vectors E and u are the electric ield and the luid velocity ield; the scalars K and D are the ion moility and the diusion coeicient. For turulence lows, esides the molecular diusion, D also includes an extra contriution due to turulent transport. The space charges are related to the electrical potential y the Poisson s uation, V q/ (3) where V is the electrical potential and is the dielectric permittivity. For vacuum, 0 8.854 0 F/ m. The electric ield is deined as, E V. (4) As shown through Eqns. (-4), the electric ield and charge density distriution are nonlinearly coupled, as the charge distriution inluences the electric ield which in turn, modiies the space charge distriution y the ion drit mechanism. Another characteristic o the electric ield-space charge coupled prolems lies in the typical smallness o the charge diusion coeicient. That is, Eqn. () is a strongly convection-dominating uation.
Proc. 06 Electrostatics Joint Conerence 3 For conventional methods ased on the partial dierential uations (PDEs), special discretization schemes or methods (such as the particle in cell method, the method o characteristic, the lux corrected transport and total variation diminishing schemes and so on) are unusually ruired to solve this uation to otain low numerical diusion and oscillation-ree solutions [7,8]. B. Boundary conditions For the test cases presented in Section 4, there are two types o oundary conditions with the two independent variales q and V : Dirichelt and Neumann conditions. The electric potentials on the electrodes are either speciied with a given value or set to e zero to represent the grounded case: Velectrode Vapplied or 0. Depending on the physical prolems modelled, the value o charge densities on the electrodes can either e prescried or show a zero gradient: qelectrode qapplied or qelectrode n 0, with n denotes the normal to the electrode surace. In addition, on the symmetry planes, all dependent variales are assumed to have a zero gradient in the direction normal the plane (i.e., Neumann condition). III. 3. THE LATTICE BOLTZMANN METHOD The LBM is a mesoscopic method and it is most ruently used to compute solutions o the Navier-Stokes uations. However, the idea o mesoscopic modelling can also e applied to other macroscopic systems. In this study, a LBM approach is ormulated or the Poisson s uation and the current continuity uation. The asic idea o LBM is to solve a set o discrete uations or the mesocopic distriution unctions in a domain discretized y the Cartesian grid. Then the macroscopic quantities (i.e., the luid density and velocity, electrical potential and charge density) can e determined rom their corresponding distriution unctions [9]. 6 5 3 0 (a) () Fig.. (a) Lattice Boltzmann discretization o a domain containing a circle and () the DD9 model. Since the discrete uations are irst order PDEs, they are much easily to solve than the macroscopic governing uations. In addition, the solution procedure o these discrete uations can e vividly understood y the collision-streaming process o some pseudo particles. An example o the discretization grid or the domain containing a circular is shown in Fig. a. For D prolems, the DQ9 velocity discretization scheme is commonly used. In other words, during one time step, the pseudo particles can either stay at the original place or stream to the eight neighoring locations in certain directions; see Fig.. For DQ9 model, the nine velocity vectors are given y 7 4 8
Proc. 06 Electrostatics Joint Conerence 4 (0, 0) 0 c c(cos ( ),sin ( ) ) 4, (5) c(cos ( ),sin ( ) ) 5 8 4 4 where the streaming speed c is deined as c x / t, x and t eing the size o lattice cell and the lattice time step, respectively. The weight unction or the th velocity direction is given as 4 / 9 0 / 9 4. (6) / 36 5 8 Based on the aove DQ9 model, we have developed two LBEs or electrical potential and the charge density in [6]. In addition, the two LBEs are coupled to another LBE with a ody orce or low motion. In the ollowing, we irst make a rie description o the LBEs. Then we ocus on the treatment o curved oundary. A. Lattice Boltzmann method or electric potential and electric ield The LBM is a method that intrinsically deals with paraolic uations with temporal terms. However, the Poisson s uation is an elliptical uation, thus it is necessary to introduce an artiicial time dependent term into Eq. (3) [0]: V q V. (7) t A nonzero coeicient γ (γ > 0) is also introduced to control the evolution speed. Note that a steady solution o Eqn. (7) will also satisy Eqn. (3) or any γ. The value o γ will aect the numerical staility and also the computational cost o the solution procedure. Its optimal value depends on the speciic prolems under consideration. For the test cases presented later, an optimal value 0.3 is chosen ased on some preliminary tests. The LBE or Eqn. (7) can e ormulated as g ( x c, ) (, ) (, ) t t t g x t g t g (, t) ts x x (8) where g is the distriution unction o electric potential and its uilirium distriution g is given y g ( x, t) V. The source term is deined as S q/. The relaxation time in Eq. (8) is computed rom, 3. (9) c t The electric potential is related to its corresponding distriution unctions y V g. (0) The electric ield can also e directly determined rom the distriution unctions [],
Proc. 06 Electrostatics Joint Conerence 5 B. Lattice Boltzmann method or charge density E c g. () c t s Inspired y the method proposed in [], the ollowing LBE is used to solve the current continuity uation, h ( x c t, t t) h ( x, t) h (, t) h (, t) x x, () where h is the distriution unction or charge density and its uilirium distriution is given as ( ) ( ) ( K ) K cs K c E u h (, t) q c E u E u x 4. (3) cs cs The relaxation time q in Eq. () is deined as 3D q c t. (4) The charge density is related to its corresponding distriution unctions y q h. (5) In [6] we have perormed the Chapman-Enskog analysis to prove that the LBEs (8) and () can recover to the macroscopic Eqns. (7) and () with second-order accuracy. C. Boundary condition treatment The treatment o mesoscopic oundary conditions is also a key issue in the application o LBM. In the two-step collision-streaming implementation style, eore the streaming step some distriution unctions rom the oundary nodes or outside o the domain are unknown, which is ruired to e supplemented with the given macroscopic oundary conditions. For the treatment o straight line oundaries with simple geometries, please reer to [6]. Here we ocus on the curved oundary treatment. q x x e e x w x Fig.. Illustration o the non-uilirium extrapolation scheme or the treatment o a curved oundary. As shown in Fig., a part o the curved wall separates the whole region into two parts. The lattice node on low region is denoted as x and that on the inner side o oundary is
Proc. 06 Electrostatics Joint Conerence 6 denoted as x. The small circles on the curved oundary, x w, denote the intersection position o the solid wall with other lattice links. The raction is deined as x xw x x, 0. The post-collision distriution unctions g ( x, t) and h ( x, t) rom the node x to a the node x are unknown. Their values are determined y idea o non-uilirium extrapolation scheme [3]. Taking g ( x, t) as the example, the unknown distriution unction can e separated into an uilirium component and a non-uilirium one, n g ( x, t) g ( x, t) g ( x, t). (6) The uilirium part can e calculated as, g ( x, t) w V, (7) where V is determined y the ollowing extrapolation [3], Vw ( ) V / 0.75 V. (8) Vw ( ) V ( ) Vw ( ) V / ( ) 0.75. For the Dirichlet condition, the value o V w is directly known. For the Neumann condition, its value should e determined y extrapolation with the known values o outer V 4 V V / 3 is used [4], and the nodes. In this study a simple second order scheme w more accurate schemes can e ound in [5]. n The next task is to determine the non-uilirium component g ( x, t). We consider the ollowing second-order approximation [3], n g ( x, t) g ( x, t) g ( x, t) 0.75. (9) n g (, t) g (, t) g (, t) ( ) g (, t) g (, t) x x x x x 0.75 Note that or this component, there is no dierence etween Dirichlet and Neumann conditions. D. Calculation algorithm The initialization is done y setting all distriution unctions with the uilirium values computed with the given macroscopic initial conditions. Then a successive iteration in time is perormed. At each time step, the LBE (8) is irst solved, and then the electric ield is otained with Eqn. (). Ater that, the LBE () is solved with the latest electric ield to otain the charge density distriution. I we also consider the low motion, the LBE with a ody orce model should then e solved to otain the low velocity ield, which will e used to determine the charge density distriution at the next time step. IV. NUMERICAL TESTS In [6], our LBM or the coupled Poisson s uation and the charge transport uation has een veriied with the hydrostatic regime o the inection induced electroconvection prolem in a plate-plate coniguration. For this simple geometry case, our LBM or the
Proc. 06 Electrostatics Joint Conerence 7 potential and charge density shows second order accuracy in space, which is consistent with the accuracy order predicted y the Chapman-Enskog analysis. Here three test cases with a more complex geometry are considered to veriy the proposed method. The irst two cases only solve the potential uation: one without any space charge and the other one with constant space charge. In the third case, the electric ield and space charge distriution are ully coupled with each other. Solution domain a O V V, q q 0 0 V V q/ y 0 Fig. 3. Modal geometry, solution domain and oundary conditions or the concentric cylinder coniguration. A concentric cylinder coniguration is shown in Fig. 3. The geometry o this prolem is deined y the radii o the inner and outer cylinders, a and. A high DC voltage (V 0 > 0) is applied to the inner electrode while the outer electrode is grounded (V = 0). For the ollowing case A, there is no ree charge rom the electrode and the charge density in the solution domain is set to e zero or a constant value. For case B, charges are generated at the inner electrode and then enter into the solution domain, thus Eqns. (-4) are ruired to e solved simultaneously. A. Poisson s uation with a wire-cylinder coniguration For the concentric cylinder coniguration, the analytical solutions o Eq. (3) are availale or oth zero and constant space charge cases [6]: q c q c ln r ln V ( r) ( r ) V0 ( a ), (0) 40 40 ln a ln Where r is radial distance rom the inner cylinder center. For no space charge qc 0 3 and constant space charge q 0μC/m, with the geometry a =.77 mm, = 03.mm, c the applied voltage V 0 = 50 KV and 0 8.854 0 F/m, the analytical and numerical results are shown in Fig. 4. A very good agreement or oth cases is readily seen. The numerical results are otained with a 30 30 lattice nodes. In this case, since the radius ratio etween the inner and outer cylinders is airly small and at least 0 lattice cells are ruired to represent the inner cylinder, a large numer o lattice nodes is ruired. However, the LBM shows its advantage o using Cartesian grids, easy oundary condition treatment and computational eiciency. To reduce the numer o lattice nodes, the non-uniorm LBM [7] or the multi-lock technique [8] may e considered in the uture study.
Electric potential V (KV) Proc. 06 Electrostatics Joint Conerence 8 50 40 30 LBM Analytical LBM Analytical } } Space charge q = 0 Space charge q = 0C/m 3 0 0 0 0 0 40 60 80 00 Distance rom inner electrode r (mm) Fig. 4. Comparison etween numerical and analytical solutions with the zero or constant space charge cases. B. Hydrodynamic solutions etween two concentric cylinder In the second case, we consider a uniorm distriution o constant charge density q 0 enters into the solution domain rom the inner cylinder. Such a process can represent the charge inection at the dielectric liquid/electrode interace and the corona discharge o air close to the electrode. Here we consider the ormer case with dielectric liquids. The Coulom orce due to the electric ield exerting on the inected space charges tends to destailize the system and to induce low motion (named the annular electroconvection). Because o the symmetrical characteristic o the coniguration and the uniorm inection, the prolem is characterized y a linear instaility iurcation. In other words, low motion arises only when the Coulom orce is suiciently strong and overcomes the damping action o the viscous orce. Otherwise, there is no low motion and system is at the hydrostatic state. The linear staility analysis with this prolem was analyzed in [9]. In a recent study [0], we have perormed a numerical study with this prolem with FVM. The purpose here is to veriy our LBM with the hydrodynamic regime solution. Taking (-a), V 0 and εv 0 /a as the scales or length, electric potential and charge density, Eqs. () and (3) can e transormed into the ollowing dimensionless orm (or clarity, the same symols are used or dimensionless variales), q ( qe) D q () t V q () where D is the dimensionless diusion coeicient. The non-dimensional oundary conditions or this prolem are: Vinector, qinector C or inner cylinder and Vcollector 0, q n 0 or outer cylinder. The inection strength parameter C is deined C q ( a) / V. The analytical solution o hydrostatic state with the case o D = 0 is: as 0 0 / q() r Ae r B A / e e and Er() r r Be r, (3) where A e and B e are two constants depending on the ratio etween the radii o cylinders a/ and the inection strength C. For C = 0 and various value o, the values o A e
q Proc. 06 Electrostatics Joint Conerence 9 and B e can e ound in Tale o [0]. On Fig. 5 we have displayed the proiles o charge density and the electric ield strength in the radial direction versus the modiied distance r* = (r a) or three radius ratios 0., 0.3 and 0.5. For all cases, the computational domain is discretized with 30 30 lattice nodes. A very good agreement etween our numerical solutions and the analytical ones is otained. In particular, the sharp variation o charge density in the region close to the inner electrode is accurately captured; see Fig.5 (a). On Fig. 6 we have presented the charge density iso-contours, and no unphysical-oscillation is oserved. 0 8 LBM results Analytical results.4. = 0.5, 0.3, 0. 6 = 0.5, 0.3, 0. E r.0 0.8 4 0.6 0 0.0 0. 0.4 0.6 0.8.0 0.4 0. LBM results Analytical results 0.0 0. 0.4 0.6 0.8.0 r* r* (a) () Fig. 5. Comparison etween numerical and analytical solutions o the hydrostatic regime with three radius ratios : (a) charge density, and () the electric ield in the radial direction. (a) () (c) Fig. 6. Iso-contours o the numerical solutions o charge density with three radius ratios : (a) =0.5, () = 0.3 and (c) = 0.. C. Annular electroconvection etween two concentric cylinders An attractive advantage o the proposed LBM or electric ield-space charge coupled prolems is its natural coupling with the LBM or low motion. An illustration example is provided in this susection. In [], we have coupled the LBMs or energy uation and low motion ased on the split-orcing model [3] and investigated the natural convection heat transer prolem with concentric and eccentric cylinders. The same coupling idea is adopted in [6]. We consider the annular electroconvection induced y unipolar inection rom the inner electrode. This prolem has een careully investigated y the FVM in [0,
Proc. 06 Electrostatics Joint Conerence 0 4]. When the Coulom orce is suiciently strong, the hydrostatic state descried y Eqns. (3) is no longer stale and annular electroconvection arises. In the dimensionless orm, the Coulom orce is controlled y the electric Rayleigh numer T and the ion moility is expressed through the moility numer M. As shown in Fig. 7, or the set o parameters (C = 0, T = 0, M = 0, D = 0-4 and = 0.5), the low shows a steady motion with 8 pairs o counter-rotating vortices, which is the same numer as predicted y FVM [0]. The results are computed with 40 40 lattice nodes. We have also compared the amplitude o the luid velocities, and the maximum dierence with the solutions otained y FVM and LBM is less than %. In Fig. 7a, we oserve eight regions, which are almost ree o charges. This kind o charge void region is a very characteristic eature o Coulom-driven lows with symmetrically placed electrodes. Our LBM have accurately captured the transition rom the charge covered region to the charge void region. (a) () Fig. 7. (a) Charge density distriution and () stream unction or an inection induced annular electro-convection case. Parameters: C = 0, T = 0, M = 0, D = 0-4 and = 0.5. V. CONCLUSION In this paper, we present a lattice Boltzmann model to solve the electric ield-space charge coupled prolems in complex geometries. Instead o solving the macroscopic uations, two consistent lattice Boltzmann uations are ormulated or charge density and electric potential. The curved oundaries are treated y a non-uilirium extrapolation idea. An attractive advantage o the proposed LBM lies in its direct coupling with the LBM or low motion. Three test cases with availale analytical solutions were used to veriy our method. The good agreement etween numerical and analytical results demonstrates that LBM is a promising alternative or electric ield-space charge coupled prolems and EHD lows in simple and complex geometries. In a uture work, we plan to extend the physical model to take into account multi-species ions and non-isothermal ield.
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