LATTICE BOLTZMANN METHOD FOR MICRO CHANNEL AND MICRO ORIFICE FLOWS TAIHO YEOM. Bachelor of Science in Mechanical Engineering.

Transcription

1 LATTICE BOLTZMANN METHOD FOR MICRO CHANNEL AND MICRO ORIFICE FLOWS By TAIHO YEOM Bahelor of Siene in Mehanial Engineering Ajou University Suwon, South Korea 2005 Submitted to the Faulty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE July, 2007

2 LATTICE BOLTZMANN METHOD FOR MICRO CHANNEL AND MICRO ORIFICE FLOWS Thesis Approved: F. W. Chambers K. A. Sallam J. D. Jaob A. Gordon Emslie Dean of the Graduate College ii

3 TABLE OF CONTENTS Chapter Page I. INTRODUCTION Bakground Problem statement and presentation Objetives of researh...8 II. LATTICE BOLTZMANN METHOD Fundamentals of the method Appliation to miro gas flow Boundary Conditions for lattie Boltzmann method...18 III. NUMERICAL APPROACH The LBM ode solution tehnique Sub-routine read_parameters and read_obstales Sub-routine init_density Sub-routine redistribute Sub-routine propagate Sub-routine bounebak Sub-routine relaxation Sub-routine write_results Appliation to the miro hannel Appliation to the miro orifie Analytial solution...38 IV. RERULTS AND DISCUSSION Simulation results of miro hannel flows Simulation results of miro orifie flows...51 iii

4 V. CONCLUSIONS AND RECOMMENDATIONS Conlusions Reommendations...64 REFERENCES...65 APPENDIX A Lattie Boltzmann ode for miro hannel flows...67 APPENDIX B Lattie Boltzmann ode for miro orifie flows...92 iv

5 LIST OF TABLES Table Page I. Classified flow regimes by various ranges of Knudsen number...4 II. The dimensions of the miro hannels...30 III. The dimensions of the miro orifie and miro filters...34 v

6 LIST OF FIGURES Figure Page 1. Two dimensional nine veloity square lattie system (D2Q9 model) Illustrations of on-grid method (left) and mid-grid method (right) Flow hart of the program Miro flows_lbm Diagram of the miro hannel with L/H =100. Cirled area is desribed again in Figure 4 with the magnified diagram Configurations of lattie nodes based on mid-grid model for the upper wall region:, imaginary nodes outside the wall;, fluid nodes Diagram of the miro orifie. The ratio of the open orifie area to the total area (h/h) is 0.6. Cirled area is desribed again in Figure 6 with the magnified diagram Configurations of lattie nodes for the miro orifie flow:, imaginary nodes inside the orifie;, fluid nodes Non-dimensionalized veloity profiles with various values of the aommodation oeffiient (a) for Kn = Non-dimensionalized veloity profiles with various values of the refletion fator (rf) for Kn = Non-dimensionalized veloity profiles of the refletion fator of 0.85 and no-slip boune-bak shemes for Kn = Non-dimensionalized veloity profiles with the no-slip boune-bak and refletion fator boundary onditions for Kn = orresponding to the ontinuum flow regime Non-dimensionalized veloity profiles with the no-slip boune-bak, refletion fator, and aommodation oeffiient boundary onditions for Kn = orresponding to the transitional flow regime...47 vi

7 13. Non-dimensionalized pressure distribution at the enter of the flow along the x-diretion of the miro hannel. The pressure is non-dimensionalized by outlet pressure Non-dimensionalized veloity distribution at the enter of the flow along the x-diretion of the miro hannel. The veloity is non-dimensionalized by inlet veloity Non-linearity of the pressure distribution along the miro hannel Veloity distributions along the flow diretion of the miro orifie at five different Reynolds numbers Density distributions along the flow diretion of the miro orifie at five different Reynolds numbers Pressure distributions along the flow diretion of the miro orifie at five different Reynolds numbers Streamwise u-veloity profiles at various upstream loations of the miro orifie Streamwise v-veloity profile at x-lattie = Streamwise u-veloity profiles at various loations inside the orifie Streamwise u-veloity profiles at the end of the orifie at various Reynolds numbers Veloity vetor field around the miro orifie at Re = Veloity vetor field around the miro orifie at Re = Veloity vetor field around the miro orifie at Re = vii

8 NOMENCLATURE a s aommodation oeffiient partile streaming veloity speed of sound eα α th disrete partile veloity f single partile distribution funtion fα α th disrete partile distribution funtion f F h H k eq Kn l L m P rf equilibrium distribution funtion external fore height of the open orifie area hannel height Boltzmann onstant Knudsen number orifie length hannel length moleular mass pressure refletion fator Re Reynolds number viii

9 t T u U wα time temperature mean thermal veloity of the moleule fluid veloity weighting fator in the LBM x λ position of the partile mean free path µ dynami visosity v ρ ς τ υ veloity of the partile density single relaxation time dimensionless relaxation time kinemati visosity ABBREVIATIONS BE BGK DSMC DVM D2Q9 LBE LBGK Boltzmann equation Bhatnagar, Gross, and Krook Diret Simulation Monte Carlo Disrete Veloity Model two dimensional nine veloities Lattie Boltzmann Equation Lattie Bhatnagar, Gross, and Krook ix

10 LBM LGCA MEMS NS Lattie Boltzmann Method Lattie Gas Cellular Automata Miro-Eletro-Mehanial Systems Navier-Stokes x

11 CHAPTER I INTRODUCTION 1

12 1.1 Bakground Reently, there has been rapid development of Miro-Eletro-Mehanial Systems (MEMS), whih are appliable to a variety of industrial fields and many other sientifi appliations. Many different types of miron-size systems are under development ranging from sensors, atuators, and eletroni iruits to the integrated systems ombining these appliations. These MEMS also ontain many miro-fluidi devies suh as mirohannels, miro-pumps, miro-nozzles, and miro-filters. Due to the advent of MEMS and the miro-fluidi devies inside the systems, the study of miro flows has beome more important than the past. As a result, many researhers have been investigating miro flows experimentally and numerially. Unlike maro flows, fabriating instruments for miro-sale experiments is obviously diffiult work (Arkili, et al. [1]). Therefore as a more effetive and alternative approah, numerial methods have been onsidered by many researhers. In miro gas flow, there are some important effets whih annot be predited in the normal visous flow based on the ontinuum hypothesis (Karniadakis, et al. [2]). The first one is a rarefation effet whih is attributed to the original harateristi of the dilute gas itself. Sine miro flows have dimensions of the order of 0.1 to 10 µm, at this very small sale a fluid partile an travel a relatively long distane and ollide with a boundary before it ollides with another partile (Zea and Chambers [3]). Knudsen number (Kn), whih is the ratio of the mean free path ( λ ) to the harateristi length ( H ) of the system, desribes this effet. As Kn inreases this rarefation effet beomes truly 2

13 signifiant. Another main effet is ompressibility, whih represents that the pressure distribution along the flow diretion is not linear. Basially gas flows are a ompressible flow. In addition, there is a need of the large pressure differene to drive a fluid through a relatively long hannel ompared with maro-sale hannels, so this ompressibility beome important under the large pressure variation between the inlet and outlet (Agrawal and Agrawal [4]). Visous heating and thermal reep (Karniadakis, et al. [2]) are other important effets but the urrent work does not over these effets. Ahmed and Beskok [5] have investigated visous heating as well as ompressibility and rarefation effets in miro gas flow through miro-filters. The final important effet in miro gas flows is the slip veloity at the boundary wall. The present work is mainly foused on obtaining this slip veloity at the wall by applying proper boundary onditions to gain appropriate results for miro hannels and miro filters. Lee and Lin [6] regarded the slip veloity at the boundary wall from the results of LBM simulation as a numerial error produed by the lak of stability of applied boundary onditions. However this slip veloity is obviously an existing phenomenon in the real physis of miro flows. Slip flow already was verified analytially by Karniadakis, et al. [2], who presented a unified flow model for miro flows whih is appliable under the wide Knudsen number range inluding the transitional flow regime. Lee and Lin [6] applied the boundary onditions to generate slip veloity in a physially proper way. Besides, other researhers (Zhang, et al. [7], Tang, et al. [8], Lim, et al. [9]) onduted numerial simulation using various types of boundary onditions to analyze the slip veloity at the wall under various onditions. These papers will be disussed in detail later. 3

14 By the Knudsen number (Kn), miro flows an be divided into different flow regimes (Karniadakis, et al. [2]) as an be seen in Table I. For Kn < 0.01 the flow an be regarded as ontinuum flow regime that is governed by ontinuum hypothesis. But as Kn inreases the domination of ontinuum hypothesis for the flow starts to vanish. For 0.01 < Kn < 0.1 the flow is alled slip flow regime and at this flow regime, there exists a slip veloity at the wall boundary. For 0.1 < Kn < 10 the flow beomes transitional regime. These two flow regimes of slip and transitional are typial for gas flows in miro flows. Beyond Kn = 10, the flow is onsidered as free moleular flow regime that should be onsidered by a partile based analysis that solves the equations for motions of moleules. TABLE I. Classified flow regimes by various ranges of Knudsen number Range of Knudsen number Flow regime Kn < 0.01 Continuum flow 0.01 < Kn < 0.1 Slip flow 0.1 < Kn < 10 Transitional flow 10 < Kn Free moleular flow A number of numerial analyses have been performed to investigate miro flows through miro-hannels and miro-filters. Arkili, et al. [1] fabriated a miro-hannel for miro gas flows and demonstrated that a numerial method, whih is based on the ompressible Navier-Stokes equations (NS) with the slip boundary ondition, mathed well with the results from the experiments of fabriated miro-hannel. Chen, et al. [10] also used ompressible NS equations to ondut numerial analysis applying the slip boundary ondition. Another researh group, (Ahmed and Beskok [5], Ahmed and Beskok [11]), simulated gas flows in miro-filters numerially with spetral element method NS equations, whih utilize high-order veloity slip and temperature jump boundary 4

15 onditions. The numerial method that solves the NS equations shows good agreement with the literature in their results. However, they should be inorporated with first-order or higher-order boundary onditions and they are appliable for only limited flow regimes of ontinuum and slip flow (Tang, et al. [12]). Mott, et al. [13] onduted the simulation of miro flows through miro-filters to verify the saling laws, whih represent the pressure drop along the filter, in the ontinuum regime with a Diret Simulation Monte Carlo (DSMC) method. The DSMC is a partile-based numerial simulation and it is basially an effiient solution for the high Knudsen number gas flows of omplex geometries. However, DSMC alulates properties of numerous ells, whih are muh smaller than those of the other numerial methods and ontains at least 20 moleules that should be simulated in the previous step (Karniadakis, et al. [2]). Therefore, DSMC simulations require muh more omputational time for the same size of domain than other numerial methods. The Lattie Boltzmann Method (LBM) is another partile-based method that has been spotlighted reently as an effetive approah for miro flow simulations. In ontrast, the LBM does not alulate properties of eah moleule ontained in the disrete ell. Thus it is a more effiient method than DSMC. In addition, this newly emerged method an be applied to all flow regimes, from the marosale flow to the free moleular flow regime and an be performed with very omplex geometries like porous media. Zea and Chambers [3] onsidered miro gas flows through the miro hannel and miro orifie using LBM. Lee and Lin [6], Zhang, et al. [7], Tang, et al. [8], Lim, et al. [9], Tang, et al. [12] simulated numerial experiments based on LBM for 2D miro hannel gas flows. Jeong, et al. [14] and Agrawal and Agrawal [4] also onduted LBM simulation for 3D miro-hannel gas flows. In addition, Chen and 5

16 Doolen [15] have reviewed a wide range of appliations of the LBM. Important effets of miro flows mentioned previously an be observed well from results of the many researh groups that performed LBM simulations. However, there are some inonsistenies among the results. On that aount, more efforts should be devoted to develop advaned LBM approahes. 6

17 1.2 Problem statement and presentation Zea and Chambers [3] performed 2D LBM simulation for miro hannel and miro orifie flows. The miro orifie was the basi onfiguration of the miro filter. Their simulations yielded ompressibility and density distributions along the hannel and filter that were mathed well with the literature, but did not simulate the slip veloity at the wall of the miro hannel aurately. For the urrent LBM simulation of miro hannels, lattie nodes are used to make the ratio of length to height 100. Implementations are primarily performed in the slip flow regime to examine the slip veloity at the wall. To provide the slip veloity at the wall of the miro hannel and miro filter, the no-slip boune bak, refletion fator (Tang, et al. [8]) and aommodation oeffiients (Zhang, et al. [7]) are applied to the boundary onditions. For the miro orifie simulations, the ratio whih is the ratio of the open orifie area to the total area was seleted as 0.6 to ompare the results with the literature. The shape of the orifie was seleted as a square. Miro orifie simulations are also onduted in the slip flow regime with the proper boundary onditions obtained for the miro hannel simulations. The air was assumed to flow through the miro orifie under isothermal onditions throughout the omputations. 7

18 1.3 Objetives of researh. The objetive of this projet is to investigate important miro gas flow features inluding slip veloity, ompressibility, and rarefation effets for miro hannel and miro orifie flows using the LBM simulation. 8

19 CHAPTER II LATTICE BOLTZMANN METHOD 9

20 2.1 Fundamentals of the method. The Lattie Boltzmann Method (LBM) is a newly emerging method for the numerial solution of visous flow problems, whih is different from the traditional solution with Navier-Stokes equations. The LBM solves the simplified Boltzmann Equation (BE). The original BE an be expressed as: f t f f + v + F x v = Q( f 1, f2) (2.1.1) The term on the left-hand side represents the kineti behavior of a partile following a trajetory influened by the external fore F. The state of this partile an be expressed by the partile distribution funtion f ( x, v, t), the most important variable in the LBM, where x denotes the position of a partile, and v denotes the veloity of the partile. The first term on the left-hand side is the rate of hange of f ( x, v, t) and the seond term is the onvetion of partiles with the veloity v aused by the external fore F (Karniadakis, et al. [2]). The term on the right-hand side is alled the ollision operator that represents the ollisions of partiles. This ollision operator is given by: Q ( f = + 1, f2) R ( 3 S V n f1 f2 f1 f2) dndv (2.1.2) 1 In the above equation, 1 and 2 represent two partiles before the ollision and 1 and 2 represent two partiles after ollision. Also, V indiates the relative veloity of v2 to the 10

21 3 v 1. The integration is alulated in the three-dimensional veloity spae R hemisphere and the + S, whih implies the volume where the partiles are moving after ollisions (Karniadakis, et al. [2]). The ollision term truly makes the BE diffiult to solve analytially and numerially beause of its omplex nonlinear integral term. Thus many approahes to simplify this part have been studied. One of the most widely used methods is the Bhatnagar, Gross, and Krook (BGK) model (Bhatnagar, et al. [16]). The BGK model simplifies the ollision term as: eq f f Q ( f1, f2) = (2.1.3) ς where f f1 and f eq is the equilibrium distribution funtion whih is alled the Maxwell-Boltzmann distribution funtion. ς denotes the single relaxation time whih is assoiated with the distribution funtion f and adjusts the approah rate to the equilibrium state in the loal relaxation proess (Tang, et al. [8]). The equilibrium distribution funtion is expressed by: 2 f m mu eq = exp πkt 2 (2.1.4) 3 / 2 (2 ) kt where k is the Boltzmann onstant, and m, u and T are the moleular mass, mean thermal veloity of moleules and temperature, respetively. The BE with the BGK model an be represented as: 11

22 f t + v f = f ς f eq (2.1.5) The diret origin of the LBM an be found in the Lattie Gas Cellular Automata (LGCA) method whih was developed by Frish, et al. [17]. Some drawbaks of LGCA method led to the LBM by replaing the number of partiles, whih is alled Boolean number, with the single-partile distribution funtion disretized in veloity and time spae. Qian, et al. [18] introdued several sets of lattie systems inluding the two dimensional nine veloities (D2Q9) and three dimensional nineteen veloities (D3Q19). The urrent simulation adopted the D2Q9 square lattie system. The single-partile distribution funtion with the disrete veloity is rewritten as fα ( x, t) = f ( x, eα, t), that is related with the α th veloity in the D2Q9 square lattie system (Figure 1). The disrete veloities in the D2Q9 are given by: e e e 0 α α = 0 = = ( os( α 1) π/ 4, sin( α 1) π/ 4) for α = 1,2,3,4 2( os( α 1) π/ 4, sin( α 1) π/ 4) for α = 5,6,7, 8 (2.1.6) In the above, = x / t is the partile streaming veloity. x and t indiate the onstant lattie spae and the time steps, respetively. Thus fα ( x, t) = f ( x, eα, t) an be interpreted as the distribution funtion with the veloity e α at the position of x at time t. With the disrete distribution funtion, veloity set and the BGK model, the BE beomes: 12

23 f t α + e α f α = f α ς f eq α (2.1.7) This form of BE, whih is alled the disrete veloity model (DVM) (Yu, et al. [19]), an be solved numerially easily ompared with the original BE, with its omplex nonlinear ollision term and Boolean nature. f f 2 5 f 5 e 6 e 2 e 5 f 3 3 e 3 0 e1 1 f 1 e7 e e f f7 f 4 8 Figure 1. Two dimensional nine veloity square lattie system (D2Q9 model). To solve the DVM (Eq ) numerially, the finite differene sheme an be applied to disretize the DVM. The appliation of the finite differene sheme to the DVM with the time step t and spae step x = eα t gives the formula as: f α eq fα ( x, t) fα ( x, t) ( x + eα t, t + t) fα( x, t) = τ (2.1.8) 13

24 Here τ = ς / t denotes the dimensionless relaxation time that is different from the single relaxation time ς in the BE and x is the position in the disretized physial spae. This equation is the Lattie Boltzmann Equation (LBE) with the BGK model: the so-alled LBGK model. In applying the finite differene sheme, speial aution in setting the range of ς << 1 is needed for low visosity flows (Yu, et al. [19]). In the LBGK model, the equilibrium distribution funtion an be expressed as: f eq α = ρwα 1+ ( eα U ) + ( eα U ) U (2.1.9) where U is the fluid veloity and wα is the weighting fator represented as: w w w α α α = 4/9, = 1/9, = 1/36, for for for α = 0 α = 1,2,3,4 α = 5,6,7,8 (2.1.10) During the alulation of the LBM proess, parameters suh as density and momentum fluxes an be omputed by the following formulas: 8 ρ = f α α= 0 8 ρu = e f α α= 0 8 eq α α= 0 = f α 8 = e f α α= 0 eq α (2.1.11) 14

25 In addition, the pressure is given simply as: P 1 3 = ρ 2 (2.1.12) The inompressible Navier-Stokes (NS) equations an be derived from LBE using the Chapman-Enskog expansion (Sui [20]). The visosity, whih is one of the most important parameters in visous fluid dynamis also an be derived from Eq. (2.1.8) as: 2 υ = ( τ 0.5) s t (2.1.13) where s = / 3. This formula indiates that the dimensionless relaxation time τ must always be larger than 0.5. Also, Qian, et al. [18] notied that τ annot exeed 2 to ensure numerial stability. Lallemand and Luo [21] onsidered the diffiulty of applying the LBGK model to thermal fluid simulations owing to the dependene on τ, whih is not varying parameter in the LBGK model. Beause this fixed relaxation time makes the Prandtl number unity when thermal fluids are simulated. They reommended using the general LBE to overome this drawbak of the LBGK model. 15

26 2.2 Appliation to miro gas flows. To apply the LBM for miro gas flows, speial treatment should be onsidered for the relaxation time ( τ ) beause of its miro-sale and high Knudsen number (Kn) flow orresponding to the slip flow regime. In the LBM simulation of maro-sale flows, the differene of density between nodes does not need to be onsidered. Thus the relaxation time, whih is related to density, is fixed through the alulations. In miro-sale appliations, on the other hand, the differene of density between nodes an affet the result of simulations signifiantly by reason of the invalidity of the ontinuum assumption. As a result, a treatment for the relaxation time is required. The urrent simulation adopted the method of Nie, et al. [22], who proposed the replaement of τ with τ as: 1 τ = ( τ 0.5) (2.2.1) ρ In the above equation, the ρ an be found by summing distribution funtions (Eq ) of eah lattie node. Substituting Eq to Eq gives us the following new expression for the visosity as: 1 2 υ = ( τ 0.5) s t ρ (2.2.2) 16

27 This presents a onstant dynami visosity µ = ρυ. The onstant dynami visosity also implies that urrent simulations onduted under isothermal ondition. Sine the mean free path ( λ ) is linearly proportional to υ, λ an be expressed as: λ = A ρ ( τ 0.5) (2.2.3) Here A = is a onstant determined by Nie, et al. [22] by omparing numerial results with experiment results. Hene, the relation between the Kn and τ is given by: Kn = λ H = A( τ 0.5) ρh (2.2.4) H denotes the height of the flow domain and the relaxation time ( τ ) is assoiated with the visosity (Eq ). Additionally, the mean free path ( λ ), whih is related with Knudsen number (Kn), is linearly proportional to the visosity. Beause of this orrelation between Kn and τ, the treatment of Kn itself an play a role in the handling of τ. Lim, et al. [9] proposed Kn = τ / H that is on aount of the relationship of addition, they linked the loal Kn with Kn o / P λ = τ x. In, where subsript o denotes outlet and P is the pressure distribution of the miro hannel along the flow diretion. Zhang, et al. [7] provided the set of relations of Kn with τ for miro flows along with the lattie model of LBM suh as D2Q9 and D3Q19 that were mentioned previously. For the D2Q9 model, the Kn an be expressed as Kn = M ( τ 0.5) / H, where M = ( 8 / 3π). 17

28 2.3 Boundary onditions for lattie Boltzmann method. Many researhers have been applying speial boundary onditions to obtain the slip veloity at the wall and apture the other features of miro gas flows in the LBM simulations. The boundary onditions inlude the no-slip boune-bak, extrapolation sheme, aommodation oeffiient sheme, and refletion fator sheme, whih will be disussed in this hapter. Sui [20] presented various boundary onditions for the LBM ranging from the omplex boundary ondition to the no-slip boundary ondition regardless of the sale of the flow. He lassified boundary onditions as on-grid and mid-grid for all ases of boundary onditions aording to the positions of the wall boundary. First, the on-grid boundary ondition has the wall boundary at the middle of the node whih is represented by the line in Figure 1. If this node is a bottom node, the area below the line is inside the wall and above the line is the fluid that is neighboring with the wall. However, this on-grid method gives only first-order auray for the boundary alulations due to the one-sided relationship with the fluid node. On the other hand, the mid-grid boundary ondition guarantees a seond order auray. In the mid-grid method, at the upper boundary walls, there is an imaginary layer of nodes whih represents outside the wall above the line in Figure 1. Those two layers of nodes are related to eah other to alulate the properties of the boundary. The on-grid and mid-grid methods are illustrated in Figure 2. 18

29 6 2 5 Upper wall Fluid Figure 2. Illustrations of on-grid method (left) and mid-grid method (right) Sui [20]. Chen, et al. [23] proposed the extrapolation sheme boundary ondition. It is very similar to the traditional finite differene shemes for maro-sale visous flows used to obtain the no-slip boundary ondition. They simply assume that there is an additional layer ( line in Figure 1) of nodes whih follows the same alulation as the real fluid node. They related those nodes with the ondition of f 1 i = f f i i where 1 f i, 0 f i, and 1 f i are the distribution funtions on the layer below the wall (8-4-7 line), the wall layer ( line), and the fluid layer (5-2-6 line). Unlike previous methods (Noble, et al. [24], Maier, et al. [25]], this extrapolation method does not need speial assumptions for the inoming partile distribution funtion (Chen, et al. [23]). This sheme also provides a seond order auray for alulations of boundary. Lim, et al. [9] onduted an LBM simulation for 2D miro-hannel gas flows. The remarkable aspet that they used in their proess was the varying Knudsen number along the hannel with the relation of Kn = Kn / P( ) 0 X. They also applied on-grid type boundary onditions. In their implementation, the line (in Figure 1) is regarded as 19

30 the wall boundary. At the upper wall, the distribution funtions of f 8, f4 and f 7 treated as inoming partiles from the boundary node to the fluid node and at the lower wall, the distribution funtions of f 5, f2 and f 6 are are treated as inoming partiles from the fluid node to the boundary node. Thus these distribution funtions of inoming partiles from boundary nodes are unknown properties whih need to be determined by boundary ondition treatments. In order to update unknown distribution funtions as well as apture the slip veloity at the wall, they used a kind of a speular refletion. Considering the upper wall, f 8, f4 and f7 are unknown distribution funtions that needed to be speified. The speular boundary onditions used in this implementation are given as : f ( x, y, t) = 8 f ( x, y, t) = 4 f ( x, y, t) = 7 f ( x, y, t) 5 f ( x, y, t) 2 f ( x, y, t) 6 (2.3.1) All these distribution funtions are alulated at the same time step. At the inlet and outlet boundary onditions, equilibrium funtions ( f eq ) were adopted to fill blanks of unknown distribution funtions. Another speial feature they applied for the boundary ondition was an extrapolation sheme. They approximate the density and veloity of the boundary node by a seond-order polynomial extrapolation. After the approximation, the unknown distribution funtions of inoming partiles from the boundary node are adjusted by equilibrium funtions under the ondition that all partiles are going to be in 20

31 the state of equilibrium following the equilibrium funtion in their loal relaxation proess. Zhang, et al. [7] applied the aommodation oeffiient (a) to the speular refletion boundary ondition based on mid-grid type method to gain a slip veloity in the LBM simulation of gas flows through the miro-hannel. The aommodation oeffiient weighs the portion between the speular refletion and diffusive refletion. In the speular refletion, the inoming partiles to the wall are refleted as light is refleted from a mirror after the ollision. However, in the diffusive refletion, the inoming partiles are refleted diffusively without a partiular angle of refletion. Sine the proess of the propagation of distribution funtions is taking plae horizontally within only boundary nodes without the vertial relationship with neighboring fluid nodes this sheme is slightly different from the traditional mid-grid method. Formulas of boundary onditions at the upper wall whih employ the aommodation oeffiient (a) are expressed by: f ( x, y, t) = (1 a) f 8 f ( x, y, t) = (1 a) f 7 f ( x, y, t) = af ( x x, y, t t) ( x + x, y, t t) ( x x, y, t t) + af 6 ( x + x, y, t t) + f ( x, y, t t) 2 (2.3.2) When the aommodation oeffiient (a) is 1, the formulas represent diffusive refletion. In the ase that the aommodation oeffiient (a) is 0, the formulas represent speular refletion, whih is desribed above in the approah of (Lim, et al. [9]). However, Zhang, et al. [7] take the value of f 5, f2 and f 6 from the orresponding distribution funtion at 21

32 the previous time step while Lim, et al. [9] take values of f 5, f2 and f6 from distribution funtions at the same time step as used for f 8, f4 and f 7. Tang, et al. [8] simulated 2D miro-hannel gas flows by LBM. They foused on the onept that both no-slip boune-bak and speular refletion might not be able to explain the momentum exhange and frition of the real wall reasonably and physially in the miro gas flows. Thus they proposed the refletion oeffiient ( r ) that weighs the portion of the ontribution of the no-slip boune-bak and speular refletion respetively. These boundary onditions for the upper wall, whih are based on the ongrid method, are: f ( x, y, t) = r 8 f ( x, y, t) = r 7 f ( x, y, t) = 4 f ( x, y, t) + (1 r ) f 2 6 f ( x, y, t) + (1 r ) f 5 f ( x, y, t) 5 6 ( x, y, t) ( x, y, t) (2.3.3) They verified boundary onditions with the refletion oeffiient ( r ) by alulating the sums of momentum along the x and y diretions. When the sum of y-momentum of the partiles after the ollision step beomes 0 ( = f f + f f + f f 0), this M y = indiates there is no vertial momentum exhange of partiles after the ollision step. Thus v-veloities at the wall beomes 0 regardless of the value of refletion oeffiient ( r ). Whereas, along the sum of x-momentum of partiles after the ollision, the wall an be regarded as an extremely rough wall at r = 0. 5 and as an absolutely smooth wall at r = 0. The sum of x-momentum for the refleted partiles from the bottom wall after the 22

33 ollision an be alulated as M = f f = 1 2r )( f ). Thus when r = 0. 5, x re 5 6 ( 8 f7 the sum of momentum beomes 0. This means that the partiles refleted bak diffusively from the wall without a ertain angle of refletion. Tang, et al. [8] regarded this state of wall as an extremely rough wall that partiles are refleted perfetly diffusively. See Tang, et al. [8] for more details. In their simulations, when r = 0. 7 the slip veloity at the wall was predited well. Sui [26] investigated the slip boundary ondition with the refletion oeffiient ( r ) and the slip oeffiient ( s ) whih is very similar to that used by Tang, et al. [8] for 3D mirohannel flows. This method, however, is based on the mid-grid sheme: f ( x, y, t) = rf 8 7 f ( x, y, t) = f ( x, y, t) = rf 5 ( x + x, y y, t) + sf ( x x, y y, t) + sf f ( x, y y, t) 5 6 ( x x, y y, t) ( x + x, y y, t) (2.3.4) where r is the refletion oeffiient and s = 1 r is the slip oeffiient. Results from this LBM simulation show that when r goes down below 0.01 slip flows surely take plae. Thus he set this value as a ritial refletion oeffiient. In these simulations, he primarily foused on the range of 0 r

34 CHAPTER III NUMERICAL APPROACH 24

35 3.1 The LBM ode solution tehnique This hapter illustrates the omputational proedure of the program Miro flows_lbm for urrent miro flow simulations. Zea and Chambers [3] have adopted and modified the program anb developed by Bernsdorf [27] at the C&C Researh Laboratories, NEC Europe to perform the miro flow simulations. The program Miro flows_lbm is based on the program anb and the boundary onditions have been modified from the Zea and Chambers [3] s program. The program Miro flows_lbm is written in FORTRAN 77. The program is omposed with three initializing sub-routines and five main sub-routines. To get perfetly onverged results, iteration steps were onduted for miro hannel and miro orifie flow simulations Sub-routine read_parameters and read_obstales In the sub-routines read_parameters and read_obstales, the program reads the initializing parameters and geometry information from the input files anb.par and anb.obs respetively. The program Miro flows_lbm starts its proess with these two sub-routines. The onfiguration file anb.par onsists of four lines and eah line indiates the number of iterations, the density per link, the aeleration term, and relaxation time in order. Among these parameters, the density and the relaxation time are related to the Knudsen number. Thus the density and relaxation time were arranged to hange the Kn. The obstale file anb.obs has the geometry information of the flows with the two rows of data. The first row represents the x-oordinate of the obstale nodes, whih are interpreted as the obstale or wall boundary. The seond row represents the y- oordinate of the obstale nodes. 25

36 3.1.2 Sub-routine init_density In the program, the values of partile distribution funtions fα are stored in the ell node ( α, x, y ). α indiates the nine partiles in the disrete lattie nodes by the two dimensional nine veloity (D2Q9) model (Figure 1). The initial values of partile distribution funtions fα are imposed to the ell node ( α, x, y ) in the sub-routine init_density. The initial values are deided from the equilibrium distribution funtions eq f α (Eq ) with zero initial veloity and it beomes the produt of the density and weighting fators (Eq ) of eah partiles Sub-routine redistribute The flows implemented in the urrent simulations are basially pressure-driven flows. Thus the speial inlet boundary ondition is used to impose the pressure differene between the inlet and outlet of the hannel in the sub-routine redistribute. The parameter ael from the input file anb.par is multiplied to the weighting fator. Thus the produt of the density, ael, and weighting fators is used to inrease or derease the values of the partile distribution funtions of the inlet lattie nodes. At the beginning of every iteration, the program ompares the magnitude between the produt term and f 3,6,7. If the values of f3,6, 7 are greater than the produt of the density, ael, and weighting fators the produt is added to the f1,5, 8 and subtrated from f 3,6, 7. This addition and subtration gives an aeleration to the flow from the inlet to the outlet. However, This redistribution method does not allow diret ontrol of the Reynolds number at the inlet of the hannel. 26

37 3.1.4 Sub-routine propagate The values of partile distribution funtions stored in the ell node ( α, x, y ) propagate to the neighboring lattie nodes and stored in temporary ell n_hlp. Between the outlet and inlet and between the upper wall nodes and bottom wall nodes, the periodi boundary onditions (Sui [20]) are used Sub-routine bounebak In the sub-routine bounebak, the boundary onditions for the solid wall or obstales are inorporated. The propagated partile distribution funtions from the fluid nodes to the wall nodes are repositioned rotating to the opposite diretion. Then the partile distribution funtions from the fluid nodes are sent bak to the fluid nodes at the next propagation step. This method illustrates the mehanism that inoming partiles to the wall are bouned bak into the fluid. You an see more details of the boundary onditions in the setions 3.2 and 3.3 for the urrent miro flow simulations Sub-routine relaxation In this sub-routine, the partile distribution funtions after the relaxation proess are alulated by the Lattie Boltzmann Equation (Eq 2.1.8). First, the program alulates the integrated loal density and the veloity omponents using Eq Then the equilibrium distribution funtions are omputed by Eq Finally, the partile distribution funtions of the equilibrium state, whih will be used as the initial values of the distribution funtions at the new iteration step, an be alulated by Eq For the relaxation time of the Eq 2.1.8, new relaxation time model for the miro flows (Nie, et al. [22]) is adopted. Only fluid nodes are onsidered in the sub-routine relaxation. 27

38 3.1.7 Sub-routine write_results In this sub-routine, the veloity and pressure are alulated by Eq and Eq after the last iteration step is ompleted. We an find the results of u-veloity, v-veloity and pressure from the output files ux_array.dat, vy_array.dat, and pressure.dat respetively. We an open these output files with Mirosoft Exel program. Figure 3 shows the flow hart of the program. read_parameters and read_obstales Initialization init_density Initial onditions of partile distribution funtions redistribute Density redistribution for the inlet and outlet boundary onditions propagate The values of partile distribution funtions propagate to the neighboring lattie nodes Bounebak The partile distribution funtions from the fluid nodes are sent bak to the fluid nodes relaxation The partile distribution funtions of the equilibrium state, whih will be used as the initial values of the distribution funtions at the new iteration step, are alulated by LBE write_results the veloity and pressure are alulated after the last iteration step is ompleted Figure 3. Flow hart of the program Miro flows_lbm. 28

39 3.2 Appliation to the miro hannel In the LBM simulation of miro flows, the most important onern is obtaining a orret slip veloity at the wall. This an be done by modeling suitable boundary onditions at the wall that satisfy physial phenomena in miro flows. In general, for LBM simulations, the diffusively sattering partile model has been widely used for very small Kn orresponding to the ontinuum flow regime (Zhang, et al. [7]). This model is based on the interpretation that refleted partiles from the wall after the ollision are relaxed along the Maxwell-Boltzmann distribution funtion. However, in the miro flow LBM simulation, partiles are not always refleted diffusively satisfying the Maxwell- Boltzmann distribution funtion (Zhang, et al. [7]). Thus more general boundary onditions are required for the LBM of miro flows. The present LBM simulation adopts both the no-slip boune-bak boundary ondition and slip boune-bak boundary ondition to investigate miro hannel flows. Figure 4 illustrates a diagram of the miro hannel used in urrent implementations. The ratio of length to height of the hannel is maintained as 100 through omputations using 21 lattie nodes in the y-diretion and 2100 lattie nodes in the x-diretion. Tang, et al. [8] onduted the miro hannel LBM simulations varying the ratio of length to height from 10 to However, in their simulations, lattie nodes in the y-diretion seemed to be fixed as 10 throughout omputations. Nie, et al. [22] simulated the miro hannel flows with 10 lattie nodes for the height maintaining the ratio as 100. Lim, et al. [9] also used 10 and 21 lattie nodes in hannel height with the fixed ratio 10. On the other hand, Lee and Lin [6] applied relatively finer grids of between 10 and 320 lattie nodes. However, there are still few 29

40 results of LBM miro flow simulations using finer grids of more than 20. Thus the simulations implemented by finer grids are highly enouraged in the future work, but the oarse grid is judged adequate for the present work. The dimensions of miro hannels used in present study and in the literatures are arranged in Table II. TABLE II. The dimensions of the miro hannels Length (Number of nodes) Height (Number of nodes) L/H Present study Tang, et al. [8] 12 μm ~ 1200 μm 1.2 μm 10 ~ 1000 Nie, et al. [22] Lim, et al. [9] 100, , Lee and Lin [6] N/A 10 ~ ~ 160 The dashed lines in Figure 4 and Figure 5 indiate the real physial wall of the miro hannel. There are also imaginary nodes outside the wall to play a role of taking bak refleted partiles from the wall into fluid nodes. This sheme is based on the mid-grid method (Sui [20]). The height of the hannel an be presented as H = ( NY 2) y on the mid-grid method and H = ( NY 1) y on the on-grid method. In the LBM of the square lattie node, used in the urrent projet, the grid sale is based on the relation of x = y = t =1 (Yu, et al. [19]). Therefore H beomes 19 on aount of N = 21. In the no-slip boune bak boundary ondition of the urrent omputational study, inoming partiles to the wall are bouned away faing an inoming diretion of partiles. The unknown distribution funtions that need to be determined from the boundary ondition are f 7, f4 and f 8 of the wall nodes. Y 30

41 Physial wall H = 21 lattie nodes Y X L = 2100 lattie nodes Figure 4. Diagram of the miro hannel with L/H = 100. Cirled area is desribed again in Figure 4 with the magnified diagram. Upper wall N Y N Y 1 N Y 2 N X = Figure 5. Configurations of lattie nodes based on mid-grid model (Sui [20]) for the upper wall region:, imaginary nodes outside the wall;, fluid nodes. In Figure 5, the distribution funtions of inoming partiles, f 5, f 2 and f 6, to the wall are stored in f 7, f4 and f8 of the wall node (5, N Y ) respetively. Then they are propagated to adjaent fluid nodes at the next time step. This no-slip boune-bak boundary ondition for the upper wall an be expressed as: 31

42 f ( x, y, t) = 4 f ( x, y, t) = 7 f ( x, y, t) = 8 f ( x, y y, t t) 2 f ( x x, y y, t t) 5 f ( x + x, y y, t t) 6 (3.2.1) For the slip boundary ondition, the refletion fator (Tang, et al. [8]) and the aommodation oeffiient (Zhang, et al. [7]), whih are disussed in the literature review setion of setion 2.1, are applied to get more aurate results. The refletion fator ( r f ) (Tang, et al. [8]) weighs the proportion of the ontribution of no-slip bounebak and speular shemes respetively as mentioned in the previous hapter. The unknown distribution funtions of the slip boundary ondition with node are found as: r f of the upper wall f ( x, y, t) = 4 f ( x, y, t) = r 7 f ( x, y, t) = r 8 f ( x, y y, t t) f f 2 f ( x x, y y, t t) + (1 r 5 f ( x + x, y y, t t) + (1 r 6 f f ) f ) f 6 5 ( x + x, y y, t t) ( x x, y y, t t) (3.2.2) The aommodation oeffiient ( a ) (Zhang, et al. [7]) is inorporated to determine the portion of diffusive refletion and speular refletion. As a approahes 1 the boundary ondition beomes diffusive refletion and as a approahes 0 the boundary ondition beomes speular refletion. On the other hand, unlike the refletion fator and no-slip boune-bak shemes, unknown distribution funtions f7 and f 8 of the wall node are not influened by distribution funtions f5 and f 6, whih are oming from the opposite diretion and having the nature of no-slip. The slip boune-bak boundary ondition with a for the upper wall an be represented by: 32

43 33 ),, ( ) (1 ),, ( ),, ( ) (1 ),, ( ),, ( ),, ( ),, ( ),, ( t t y y x x f a t y x f t t y y x x f a t y x f t t y y x f t t y y x x af t t y y x x af t y x f = + = = (3.2.3)

44 3.3 Appliation to the miro orifie. The refletion fator ( r f ) is also applied to the boundary ondition of the miro orifie flow beause it shows exellent results in the miro hannel simulations. Figure 6 presents a shemati of the miro orifie, whih simulates an array of miro filter fibers. The fators suh as geometry and dimension of the hole and the edge-shape of the filter an affet the result of the filtration. In the urrent simulation, the two dimensional square orifie is used. Also, north and south boundaries of the hannel part are periodi sides that are neighbored with other arrays of miro filter fiber. The ratio of the open orifie area to the total area (h/h) is seleted as 0.6 with 2 μm height in order to ompare results with the work of Ahmed and Beskok [5]. In addition, 20 lattie nodes in the y-diretion and 340 lattie nodes in the x-diretion are used setting the ratio of length to height as 17. Ahmed and Beskok [5] used various dimensions for their miro filters. The dimensions of the miro orifie and miro filters used in urrent study and in the literature are speified in Table III. TABLE III. The dimensions of the miro orifie and miro filters H (μm) h (μm) l (μm) L/H Present study Ahmed and Beskok [5]

45 Periodi sides l = 8 lattie nodes l In-flow h = 12 lattie nodes Out-flow H = 20 lattie nodes Y X 340 lattie nodes Figure 6. Diagram of the miro orifie. the ratio of the open orifie area to the total area (h/h) is 0.6. Cirled area is desribed again in Figure 6 with the magnified diagram. Vertial wall 8 Periodi side Horizontal wall Figure 7. Configurations of lattie nodes for the miro orifie flow:, imaginary nodes inside the orifie;, fluid nodes. Figure 7 is the magnified part from the irled area of Figure 6 showing the onfiguration of lattie nodes and the relationship between the distribution funtions of the nodes. Sine the flow diretion varies along the position of the orifie wall, rearrangement of the 35

46 distribution funtions is required in order to apply the refletion fator ( r f ) to the boundary ondition of the miro orifie. For instane, in the ase of vertial wall nodes of the orifie, Figure 7, the flow diretion is top to bottom along the orifie wall. Thus the unknown distribution funtions that need to be determined are f 3, f6 and f 7. These are different from the ase of horizontal north wall of miro hannel. The boundary ondition with r f for vertial wall nodes of the miro orifie an be expressed as: f ( x, y, t) = f ( x x, y, t t) 1 f ( x, y, t) = r f ( x, y, t) = r f f f ( x x, y + y, t t) + (1 r 8 f ( x x, y y, t t) + (1 r 5 f f ) f ) f 5 8 ( x x, y y, t t) ( x x, y + y, t t) (3.3.1) In the above formulas, unknown distribution funtions f 3, f6 and f 7 distribution funtions f 1, f5 and f 8 of neighboring fluid nodes. are influened by For the orner wall node of the orifie, more speial treatments are needed beause the orner wall nodes are affeted by both vertial diretion flow and horizontal diretion flow. For example, the distribution funtion f 7 of the orner wall node an be orrelated with f 8 (from vertial flow), f 6 (from horizontal flow) and f 5 (as a no-slip boune-bak omponent). Other distribution funtions f6 and f 8 of the orner wall node also an be handled the idential way. The boundary ondition for the horizontal wall node does not neessitate any more speial treatment, as it is the same as the miro hannel horizontal 36

47 37 wall nodes. The boundary ondition with f r of the orner wall node for the miro orifie is presented as: ),, ( ) 0.5(1 ),, ( ) 0.5(1 ),, ( ),, ( ),, ( ) 0.5(1 ),, ( ) 0.5(1 ),, ( ),, ( ),, ( ) 0.5(1 ),, ( ) 0.5(1 ),, ( ),, ( ),, ( ),, ( ),, ( ),, ( t t y y x x f r t t y y x x f r t t y y x x f r t y x f t t y y x x f r t t y y x x f r t t y y x x f r t y x f t t y y x x f r t t y y x x f r t t y y x x f r t y x f t t y y x f t y x f t t y x x f t y x f f f f f f f f f f = = = = = (3.3.2)

48 3.4 Analytial solution. Karniadakis, et al. [2] developed the unified flow model for miro flows based on the ompressible Navier-Stokes (NS) equation onsidering veloity saling and rarefation effets. This unified model is valid for the wide Knudsen number ( Kn ) regime from the ontinuum to the transitional flow regime and independent of the gas type. In addition, the unified flow model has shown exellent agreement with results of the Diret Simulation Monte Carlo (DSMC) method and solutions of the linearized Boltzmann equation (Karniadakis, et al. [2]). The unified flow model was also validated in miro filter flows at various Kn between and (Ahmed and Beskok [5]). Besides, the experimental results for miro flow are not available in the literature for miro hannel flows and miro orifie flows under onditions mathing the urrent simulation (Zea and Chambers [3]). On this aount, the unified flow model of (Karniadakis, et al. [2]) is the best analytial solution for the present study at this time. Their nondimensionalized veloity profile ( U ) for the miro hannel flow, whih is a funtion only of the y to the hannel height and the Knudsen number ( Kn ), is: U ( y, Kn) 2 ( y / h) + ( y / h) + Kn /(1 bkn) (1/ 6) + Kn /(1 bkn) (3.4.1) When b = 0 in the above equation the unified model orresponds to the first order auray boundary ondition. When b = 1, as used in urrent implementations, the model has seond order auray for a wide range of Kn. The unified flow model for 38

49 miro hannels also provides the pressure distribution (Karniadakis, et al. [2]). The pressure distribution funtion is expressed as: ~ P 2 2 σ 1+ 2(6 + α) σ v v ~ Kn ( P 1) o 2 σ + 2(6b + α) σ v v ~ 2 P bkno x Kno loge( ) = B(1 ) 1 bkn L o (3.4.2) Where ~ P ( x) = P( x) / P, whih represents the pressure at x oordinate normalized with o the exit pressure. B is a onstant with a value that makes ~ P (0) / = P i Po. The i P and P o denote the pressure at inlet and outlet respetively. The onstant σv is the tangential momentum aommodation oeffiient that represents tangential momentum exhange of partiles with the wall. When the surfae is rough with the harateristi length sale of moleules, the partiles refleted from the wall diffusively and σv beomes lose to 1 (Lee and Lin [6]). The real engineering ases suh as air or CO2 on mahined brass or shella have σ v = 1(Lee and Lin [6]). Thus σv is seleted as 1, whih represents diffuse refletion, for the urrent simulation. σ is expressed as σ = m m ) /( m m ). Here v v ( i r i w m i, mr and mw are the tangential momentum of inoming, refleted partiles and the wall respetively (Karniadakis, et al. [2]). mw is 0 for the stationary wall. The tangential momentum aommodation oeffiient σv and the aommodation oeffiient ( a ) of Zhang, et al. [7] for the boundary onditions both at similarly and weigh the proportion of diffusive refletion and speular refletion. However they are not diretly related. The 39

50 onstant α has the value of 2.2 that was determined for nitrogen in a finite-length hannel with the ratio L/h = 20 (Karniadakis, et al. [2]). 40

51 CHAPTER IV RESULTS AND DISCUSSION 41

52 4.1 Simulation results of miro hannel flows. In the LBM simulations of miro hannel flows, three different Knudsen numbers (Kn) of , and 0.194, whih indiate the ontinuum, slip and transitional flow regimes respetively, were used. For boundary onditions at miro hannel walls, the noslip boune-bak, aommodation oeffiient ( a ) (Zhang, et al. [7]) and refletion fator ( r f ) (Tang, et al. [8]) were applied to obtain a slip veloity at the boundary wall. Veloities from the result are non-dimensionalized by the mean veloity with the y oordinate non-dimensionalized by the height of hannel. Results at the loation x/l = 0.9 are presented to show the fully developed veloity profile of the flow. Veloity profiles are analyzed by omparing with the analytial unified flow result of Karniadakis, et al. [2] whih is disussed in setion 4.3. First of all, simulations were onduted to aquire proper slip veloity at the wall in the slip flow regime. Thus for Kn = , the aommodation oeffiient and refletion fator were inorporated into the boune-bak boundary ondition. Figure 8 shows the results of the aommodation oeffiient sheme. Overall, the aommodation oeffiient sheme does not provide very good results. The profiles do not agree well with the analytial result. When a = 0. 3 the veloity profile does not have a paraboli shape espeially around the wall area and it has a huge deviation in the enter of the flow. As the aommodation oeffiient inreases, the veloity profile approahes the paraboli shape and the slip veloity dereases. When a = 0. 99, whih indiates 42

53 almost a diffusive refletion, the veloity profile still has deviations in the enter and wall area of the flow Kn = U/Umean a = 0.3 a = 0.5 a = 0.7 a = 0.9 a = 0.99 Karniadakis et al.[2] y/h Figure 8. Non-dimensionalized veloity profiles with various values of the aommodation oeffiient (a) for Kn = A diffusive refletion implies natural movement of partiles after ollisions obeying the Maxwell-Boltzmann distribution funtion. For diffusive phenomena there should be enough partile distribution funtions to represent real physial phenomena with large numbers of partiles. However, in the LBM, only nine partile distribution funtions represent movements of all partiles due to the disretized veloity model suh as D2Q9. Thus the differenes of the aommodation oeffiient sheme from the analytial result an be attributed in part to this trait of the LBM. The results of the refletion fator boundary ondition are presented in Figure 9. Veloity profiles for the refletion fator sheme show similar patterns as the aommodation 43

54 oeffiient sheme as the refletion fator inreases. When r = 0. 3, there are large deviations in the enter and wall area of the flow. As the refletion fator inreases, the veloity profile approahes the result of Karniadakis, et al. [2]. In the ase of r = 0. 7, deviations of the maximum veloity and slip veloity are approximately 2.38% and 38% eah. In the ase of r = 0. 85, the veloity profile is in very good agreement with the f analytial result showing approximately only 0.3% and 2.34% deviations in the maximum veloity and slip veloity. f f Kn = U/Umean rf = 0.3 rf = 0.5 rf = 0.7 rf = 0.8 rf = 0.85 rf = 0.9 Karniadakis et al.[2] y/h Figure 9. Non-dimensionalized veloity profiles with various values of the refletion fator for Kn = Figure 10 illustrates the ase of r = f again omparing with the no-slip boune-bak sheme under the Kn = It is shown that the veloity profile applied with no-slip boune-bak boundary ondition also provides slip veloity at the wall in the slip flow regime. The no-slip boune-bak boundary ondition originated for the ase of no-slip 44

55 veloity at the wall. However, the veloity profile of the no-slip boune-bak shows approximately 35.65% underestimated slip veloity in the wall. In the maximum veloity, the profile overestimates the veloity approximately 2.52%. The urrent simulations were not onduted to determine the relation between the Kn and r f through the entire slip flow regime hanging both parameters. Sine inreasing r f shows a trend of making slip veloity derease in the ase of Kn = , One may speulate that the optimum refletion fator may derease with inreasing Kn in the slip flow regime Kn = U/Umean boune bak rf = 0.85 Karniadakis et al.[2] y/h Figure 10. Non-dimensionalized veloity profiles of the refletion fator of 0.85 and no-slip bounebak shemes for Kn = Additionally, simulations were performed with Kn = and Kn = These simulations were used to assess the apability of the LBM using the boundary onditions applied to slip flow for the ontinuum and transitional flow regimes. Figure 11 shows the omparison of veloity profiles between the no-slip boune bak and the refletion fator 45

56 of 0.85 for Kn = , orresponding to the ontinuum flow regime. These two boundary ondition shemes yield onsistent veloity profiles in the ontinuum flow regime, although there are small deviations from the analytial result around the wall area inluding a slip veloity at the wall. In the very low Knudsen number flow, the slip veloity should be very small. However, the urrent simulations alulate notieably large slip veloity at the wall in the ontinuum flow regime Kn = U/Umean boune bak rf = 0.85 Karniadakis et al.[2] y/h Figure 11. Non-dimensionalized veloity profiles with the no-slip boune-bak and refletion fator boundary onditions for Kn = orresponding to the ontinuum flow regime. The omparison of results between the no-slip boune bak, refletion fator and aommodation oeffiient boundary for Kn = are presented in Figure 12. The results of urrent LBM simulations for the transitional flow regime are not well predited. This disrepany may be attributed to the single relaxation time model used in the urrent simulations. The single relaxation time model represents only one ollision step for partiles until they approah the equilibrium state after ollisions. In the low Knudsen 46

57 number flow, there are more partiles in the same area ompared with the high Knudsen number flow. Thus the spae that partiles an move before they ollide with other partiles is relatively small in the low Knudsen number flow. So the partiles an relax with small relaxation time and just one ollision step. On the other hand, in the high Knudsen number flows, due to the rarefied effet, several partile ollision steps an be reasonably expeted before approahing the equilibrium state. Hene more sophistiated relaxation time models should be developed to improve the performane of the LBM simulation for high Knudsen number flows Kn = U/Umean rf = 0.85 rf = 0.5 a = 0.3 a = 0.7 Karniadakis et al.[2] y/h Figure 12. Non-dimensionalized veloity profiles with the no-slip boune-bak, refletion fator and aommodation oeffiient boundary onditions for Kn = orresponding to the transitional flow regime. One of the purposes of this projet is to onfirm the ompressibility effet for miro flows. This an be performed by investigating the pressure distribution along the flow diretion of the miro hannel. Sine the refletion fator sheme showed proper veloity 47

58 profile that agreed well with the literature, the pressure distribution for the ase of rf = 0.85 and Kn = was investigated. The dereasing pressure distribution along the flow diretion, whih is non-dimensionalized by outlet pressure, is shown in Figure 13. The x oordinate is normalized by the length of the miro hannel. Due to dereasing pressure and density, enterline veloities inrease along the miro hannel Kn = rf = P/Po P/Po x/l Figure 13. Non-dimensionalized pressure distribution at the enter of the flow along the x-diretion of the miro hannel. The pressure is non-dimensionalized by outlet pressure. Figure 14 shows inreasing veloity along the hannel whih is a typial feature of miro hannel flow (Lim, et al. [9]). Similar results an be found in many papers (Agrawal and Agrawal [4], Lee and Lin [6], Lim, et al. [9], Jeong, et al. [14]). The miro flows implemented in the urrent projet are pressure driven flows. To impose the pressure differene between the inlet and outlet the density redistribution method is used for the inlet boundary ondition, whih inreases or dereases the density of the inlet aording 48

59 to the densities of the inlet nodes at the end of eah iteration time step. However the density redistribution method does not seem to provide aurate results at the inlet and outlet of the miro hannel. We an see small jumped veloities in the inlet and outlet of the veloity profile in Figure Kn = rf = 0.85 U/Uinlet U/Uinlet x/l Figure 14. Non-dimensionalized veloity distribution at the enter of the flow along the x-diretion of the miro hannel. The veloity is non-dimensionalized by inlet veloity. In order to ensure the ompressibility effet is simulated aurately, the differene between the non-linear pressure distribution of miro hannel flows and the orresponding linear pressure distribution of the inompressible flow an be onsidered. This is the traditional omparison used in the literature. This non-linearity of the pressure distribution along the hannel is presented in Figure 15. Here, Pl = P + ( P P )(1 x) is o i o the linear pressure distribution whih is a funtion of only x oordinate. Pi and P o 49

60 indiate inlet and outlet pressure respetively. The LBM simulation differenes from the linear pressure distribution are in exellent onsisteny with the analytial result of Karniadakis, et al. [2] in the figure. Another feature of these results is the loation where the maximum pressure deviation is indiated. The maximum point is shifted toward the outlet at approximately x/l = In the results of Lee and Lin [6], Lim, et al. [9], the non-linearity dereases when the Knudsen number inreases. This means that the ompressibility effets derease in more rarefied gas flows Kn = (P-Pl)/Po LBM Rf = 0.85 Karniadakis et al. [2] x/l Figure 15. Non-linearity of the pressure distribution along the miro hannel. 50

61 4.2 Simulation results of miro orifie flows. The main purpose in onduting simulations of gas flows through the miro orifie, whih simulates an array of miro filter fibers, is to investigate the ation of flows passing the miro filter. 2 μm for hannel height and 1.2 μm for the height of open orifie area are used to keep the ratio of the open orifie are to the total area as 0.6. The length from the inlet to the outlet of the hannel is 34 μm. The shemati view of the miro orifie is illustrated in Figure 6. The simulations are performed at atmospheri onditions, with air assumed to flow through the miro orifie. The temperature of the miro orifie and surrounding areas is kept at 298 K, so the simulation has isothermal onditions throughout the omputations. The refletion fator boundary ondition, whih shows exellent agreement with the analytial result in the simulations of miro hannel flows, is applied to the wall of orifies in the limit of slip flow regime. The simulations are onduted for Kn = and at five different Reynolds numbers. In the simulations of miro orifie flows, the Knudsen number is alulated using the orifie open area height. Figures 16, 17 and 18 present veloity, density and pressure variations along the miro orifie at five different Reynolds numbers with Kn = The orifie is loated at the x-lattie nodes between 87 and 94. The dimension of veloity is m/s. In the urrent LBM ode, partile streaming veloity ( ) in the Eq is set as 1, whih is different from the real partile streaming veloity under the atmospheri onditions. The real partile streaming veloity an be alulated as = 3s where s is the speed of sound under the simulation ondition. The veloity of the flow is deided by Eq Sine the veloity 51

62 has the same ratio to the partile streaming veloity ( and e α have same dimensions from Eq ) the real veloity of the flows an be alulated by multiplying the results of the LBM by 3 s. Veloity (m/s) Kn = Re = 2.15 Re = 4.23 Re = 6.16 Re = 7.91 Re = x-lattie Figure 16. Veloity distributions along the flow diretion of the miro orifie at five different Reynolds numbers. In Figure 16, as the flow approahes the orifie it aelerates and hits its maximum veloity. However maximum veloities our at different loations for the different Reynolds numbers. For example, when the Re = the flow reahes its maximum veloity at x-lattie = 94 while the maximum veloity ours at x-lattie = 91 when the Re = This means that the flow at high Reynolds number aelerates for a somewhat longer time ompared with the low Reynolds number flow. This phenomenon, whih owes to the ompressibility effet, an be more severe for the high Reynolds number flow than for the low Reynolds number flow. After the flow reahes its maximum 52

63 veloity, it begins to slow down and reahes onstant veloity again. Another indiator of the ompressibility effet an be found here. There are deviations between the inoming onstant veloity to the orifie and the outgoing onstant veloity from the orifie for the different Reynolds numbers. At Re = the outgoing veloity is inreased about 35.11% ompared to the inoming veloity. However at Re = 2.15 the flow shows only approximately a 5% differene between inoming and outgoing veloities. Thus it is onfirmed that different densities resulting from the ompressibility effet produe greater hanges at high Reynolds number. Similar behavior also an be observed in the results of Ahmed and Beskok [5]. The veloity distributions are orrelated very well with the density and pressure distributions in Figures 17 and 18. The density distribution is a diret evidene of the ompressibility effet (Ahmed and Beskok [5]). Density (Kg/m3) Kn = Re = 2.15 Re = 4.23 Re = 6.16 Re = 7.91 Re = x-lattie Figure 17. Density distributions along the flow diretion of the miro orifie at five different Reynolds numbers. 53

64 Figure 17 shows that the largest density-drop take plae at the highest Reynolds number of The densities maintain nominally onstant values until the flows ome to the orifie, and densities drop to minimum values after the orifie. However the loation where the minimum density takes plae does not orrespond to the loation where the maximum veloity is aptured. At lower Reynolds numbers of 2.15, 4.23 and 6.16, after densities reah their minimum they remain as onstant without upturning regions. On the other hand, at the higher Reynolds number of 7.91 and 10.82, there are small upturning regions in whih densities inrease, reahing onstant values after a short distane. This is unexpeted behavior whih is different than the results of Ahmed and Beskok [5]. This feature requires additional study. 190 Pressure (KPa) Kn = Re = 2.15 Re = 4.23 Re = 6.16 Re = 7.91 Re = x-lattie Figure 18. Pressure distributions along the flow diretion of the miro orifie at five different Reynolds numbers. 54

65 The pressure distributions in Figure 18 show the same pattern as the density distributions. The largest pressure drop ours at the highest Reynolds number. In addition, there are upturning regions of the pressure right after the orifie for two higher Reynolds number ases. However unlike the density distribution, these upturning regions of the pressure also an be found in the results of Ahmed and Beskok [5]. For more detailed analysis, upstream veloities of the orifie and veloities inside the orifie along the y diretion are investigated as Ahmed and Beskok [5] did. Simulation results at Re = 6.16 are seleted to be investigated. Figure 19 presents streamwise u- veloity profiles at various loations upstream from the orifie. Again, the miro orifie region is between x-lattie 87 and Kn = Re = 6.16 u-veloity (m/s) x-lattie = 50 x-lattie = 78 x-lattie = 82 x-lattie = y/h Figure 19. Streamwise u-veloity profiles at various upstream loations of the miro orifie. 55

66 The orifie open area is between y/h = 0.2 and y/h = 0.8. At x-lattie = 50, whih is far away from the orifie, the flow has a uniform streamwise veloity. As the flow approahes the orifie, the profile begins to develop a paraboli shape. The enter of the flow starts to aelerate and wall area of the flow starts to deelerate due to the blokage of the orifie. At x-lattie = 86, whih is just before the inlet of the orifie, u-veloities around the wall area have almost vanished due to the orifie. The streamwise v-veloity profile at x-lattie = 86 is presented in Figure 20. Aording to this v-veloity profile from just before the orifie, the flow starts to aelerate to the y-diretion following the vertial wall of the orifie as expeted. However v-veloity starts to derease after the end of the vertial wall and beomes zero at the enter of the orifie where the flow is aelerated to the x-diretion Kn = Re = 6.16 x-lattie = v-veloity (m/s) y/h Figure 20. Streamwise v-veloity profile at x-lattie =

67 Figure 21 shows streamwise u-veloity profiles at various loations inside the orifie. The interesting thing is that the flows already have developed the paraboli shape before entering the orifie. The flows at very low Reynolds numbers implemented in the present study have relatively little inertia ompared with the maro sale visous flows so the flows reat quikly to the surroundings. This helps explain the paraboli shape of the flow before entering the orifie. Thus there are only slight hanges in the shapes of the veloity profiles along the various loations inside the orifie. In addition, as disussed previously, the flow reahes its maximum veloity before the end of the orifie at the x- lattie = Kn = Re = 6.16 u-veloity (m/s) x-lattie = 87 x-lattie = 88 x-lattie = 90 x-lattie = 92 x-lattie = y/h Figure 21. Streamwise u-veloity profiles at various loations inside the orifie. Figure 22 presents u-veloity profiles at the end of the orifie under various Reynolds numbers. It is shown that as the Reynolds number inreases, veloities at all the streamwise loations from the wall to the enter of the flow also inrease. Overall 57

68 behavior of the flows an be observed from Figures 23, 24 and 25. The figures illustrate the veloity vetor fields around the orifie from x-lattie = 70 to x-lattie = 120. The x- oordinate is normalized by the length between x-lattie = 70 and x-lattie = 120. The y- oordinate also is normalized by the hannel height. When the Re = 7.91 the veloity vetor field shows reasonably expeted behavior of the flows. As the uniform flows approah the orifie the flows are aelerated to the enter of the orifie. Then the flows pass away from the orifie and beome uniform flows again after the orifie. 300 u-veloity (m/s) Kn = x-lattie = 94 Re = 2.15 Re = 4.23 Re = 6.16 Re = 7.91 Re = y/h Figure 22. Streamwise u-veloity profiles at the end of the orifie at various Reynolds numbers. These veloity vetor fields do not indiate the strength of the veloities, as all the arrows in the vetor fields have the same length. Figures 24 and 25 show interesting features right after the orifie. When the Re = and Re = the behaviors of the flows before the orifie are almost same with the ase of Re = However, there are separation areas at the downstream orner of the orifie. 58

69 Figure 23. Veloity vetor field around the miro orifie at Re = Figure 24. Veloity vetor field around the miro orifie at Re =

70 Figure 25. Veloity vetor field around the miro orifie at Re = The separation area seems to start to appear at the Re = When the Re = the separation area is larger than the ase of Re = These behaviors of the flow need to be studied more in future work. The limited number of lattie points in this region should be noted. 60