Unified Gas-kinetic Scheme for the Study of Non-equilibrium Flows

Transcription

1 Unified Gas-kinetic Scheme for the Study of Non-equilibrium Flows by WANG, Ruijie A Thesis Submitted to The Hong Kong University of Science and Technology in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Nano Science and Technology Program August 2015, Hong Kong

2 Authorization I hereby declare that I am the sole author of the thesis. I authorize the Hong Kong University of Science and Technology to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the Hong Kong University of Science and Technology to reproduce the thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. WANG, Ruijie 25 August 2015 ii

3 Unified Gas-kinetic Scheme for the Study of Non-equilibrium Flows by WANG, Ruijie This is to certify that I have examined the above PhD thesis and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the thesis examination committee have been made. Prof. Kun Xu, Supervisor Prof. Zikang Tang, Director of Program Nano Science and Technology Program 25 August 2015 iii

4 Acknowledgment Firstly, I would like to express my sincere gratitude to my supervisor Prof. Kun Xu for his patience, encouragement, and immense knowledge during the study. His deep understanding on the research field, innovative ideas, passions in science, and advices exceptionally inspire and enrich me in all the time of the research. I am very glad to have Prof. Kun Xu as my supervisor for my PhD study. I gratefully acknowledge Prof. Xiao-Ping Wang, Prof. Tie-Zheng Qian, Prof. Yi- Kuen Lee, and Prof. Yang Liu for their kindness and readiness to serve on my thesis examination committee. I do appreciate the fruitful discussions, all the support, kindly suggestions, and warm encouragement given to me by Prof. Tie-Zheng Qian, Prof. Zhao-Li Guo, Prof. Quan- Hua Sun, Prof. Graeme Bird, and Dr. Xin-Peng Xu during the period of my postgraduate study. I wish to express my gratitude to my fellow students and friends for their helpful discussions on the research and accompany during the study: Dr. Song-Ze Chen, Dr. Xiao-Dong Ren, Dr. Li-Jun Xuan, Dr. Jun Luo, Dr. Pu-Bing Yu, Liang Pan, Chang Liu, Dr. Sha Liu, Wan-Yang Wang, Dr. Xin-Peng Xu, Jiao-Long Zhang, Yin-Wang, Dr. Tao Lin, Zuo-Gong Yue, Dr. Zong-Long Zhu, Xiao-Ming Liu and many others. I would like to thank the Nano Science and Technology program and the Department of Mathematics for providing me the opportunity and postgraduate studentship to study here, and the research travel grant for providing me the award to attend academic conference. I also wish to express my deep gratitude to my family and my boyfriend for their love, deep caring, continual support, encouragement and understanding during my study. iv

5 Table of Contents Title Page i Authorization Page ii Signature Page iii Acknowledgments iv Table of Contents v List of Figures ix List of Tables xiii Abstract xiv Chapter 1 Introduction Motivation Review of Modeling and Simulation for Gas Flows Objectives and Organization of the Thesis Chapter 2 Numerical Modeling of Single-component Gas Boltzmann Equation and Its Model Equations v

6 2.2 Unified Gas-kinetic Scheme Discretization of Velocity Space Polyatomic Gas Reduced Distribution Functions Calculate Interface Flux Update Cell-averaged Flow Variables Boundary Conditions Numerical Examples Discrete Unified Gas-kinetic Scheme for Compressible Flow Properties of the UGKS and DUGKS Schemes Numerical Examples Conclusion Chapter 3 Numerical Modeling of Multi-component Gas Boltzmann Equation and Its Model Equations Unified Gas-kinetic Scheme Numerical Examples Shock Structure Micro-channel Conclusion vi

7 Chapter 4 Sound-wave Propagation in Monatomic Gas Introduction Methodology Boundary Condition at the Transducer Method for Determining Sound Parameters Numerical Results Behavior at Low Frequencies Behavior at High Frequencies Phase Speed and Attenuation Coefficient Conclusion Chapter 5 Cross-coupling of Mass and Heat Transfer Introduction Mechanism Micro-channel of Planner Surfaces Micro-channel of Ratchet Surfaces Methodology Normalization Data processing Numerical Results vii

8 5.4.1 Cross-coupling for Planner Surfaces Cross-coupling for Ratchet Surfaces Knudsen pump Conclusion Chapter 6 Conclusion and Future Work Conclusion Future Work Bibliography Appendix A Taylor Expansion of Maxwell Distribution Appendix B Moments of Maxwell Distribution viii

9 List of Figures 1.1 Classification of flow regimes based on Knudsen number Flow expansion to vacuum: schematic of the problem Flow expansion to vacuum: pressure contour at rarefaction parameter δ = 100, Flow expansion to vacuum: pressure, temperature, and Mach number along the centerline at rarefaction parameter δ = Normalized coefficients of DUGKS and UGKS as a function of local Knudsen number. Here Kn = τ/ t Sod shock tube: density, velocity, temperature, and heat flux at Kn = Sod shock tube: density, velocity, temperature, and heat flux at Kn = Sod shock tube: density, velocity, temperature, and heat flux at Kn = Shock structure: schematic of the problem Shock structure: density, temperature, heat flux, and shear stress at Ma 1 = Shock structure: density, temperature, heat flux, and shear stress at Ma 1 = ix

10 2.11 Lid-driven cavity: schematic of the problem Lid-driven cavity: flow field at Kn = Lid-driven cavity: velocity profile under different mesh resolutions at Kn = Shock structure in binary gas mixture: number densities and temperatures for Ma 1 = 1.5, mass ratio m B /m A = 0.5, and diameter ratio d B /d A = 1 under diffrent component concentration χ B Shock structure in binary gas mixture: number densities and temperatures for Ma 1 = 1.5, mass ratio m B /m A = 0.25, and diameter ratio d B /d A = 1 under diffrent component concentration χ B Shock structure in binary gas mixture: number densities and temperatures for Ma 1 = 3.0, mass ratio m B /m A = 0.5, and diameter ratio d B /d A = 1 under diffrent component concentration χ B Micro-channel flow of binary gas mixture: schematic of the problem Micro-channel flow of binary gas mixture: particle fluxes due to pressure gradient, temperature gradient, and concentration gradient vs Knudsen number for mass ratio m B /m A = Micro-channel flow of binary gas mixture: particle fluxes due to pressure gradient, temperature gradient, and concentration gradient vs Knudsen number for mass ratio m B /m A = Micro-channel flow of binary gas mixture: particle fluxes due to pressure gradient, temperature gradient, and concentration gradient vs Knudsen number for mass ratio m B /m A = Sound-wave propagation: schematic of the problem x

11 4.2 Sound wave propagation: velocity change on every integer period at the monitor point for frequency ω = Sound wave propagation: A(x) and B(x) for frequency ω = Sound wave propagation: comparision of velocity amplitude of UGKS and the theoretical solutions of R13 at frequency ω = Sound wave propagation: comparison of UGKS and DSMC results using the same boundary condition at Re = γ/ω = 0.5 (ω 3.3) Sound wave propagation: A(x), B(x), and velocity amplitude at frequency ω = Sound wave propagation: location dependent phase speed for frequency ω = Sound wave propagation: phase speed comparison at different frequencies among the results from UGKS, DSMC experimental measurements, and the Navier-Stokes equations Sound wave propagation: attenuation coefficient comparison at different frequencies among the results from UGKS, DSMC experimental measurements, and the Navier-Stokes equations Sound wave propagation: wave speed and attenuation coefficient comparison between UGKS and experimental data of Greenspan Sound wave propagation: wave speed and attenuation coefficient comparison between UGKS and experimental data of Meyer Cross coupling in channel of planner surfaces: schematic of the problem Cross coupling in channel of ratchet surfaces: schematic of the problem. 74 xi

12 5.3 Cross coupling in channel of planner surfaces: ˆL MQ and ˆL QM versus Knudsen number Cross coupling in channel of ratchet surfaces: schematic of the simulation geometry Cross coupling in channel of ratchet surfaces: ˆL MQ and ˆL QM versus Knudsen number Cross coupling in channel of ratchet surfaces: formula fitted and simulated ˆL MQ Cross coupling in channel of ratchet surfaces: temperature contour and streamlines of typical diffusive configuration and diffusive-specular configuration Cross coupling in channel of ratchet surfaces: ˆL MQ as a function of tan α Cross coupling in channel of ratchet surfaces: ˆL MQ as a function of L/H Cross coupling in channel of ratchet surfaces: ˆL MQ as a function of H 1 /H xii

13 List of Tables 2.1 Flow expansion to vacuum: reduced mass flux at L/H = Computational cost comparison of DUGKS and UGKS Sound wave propagation: frequencies and domain lengths Sound wave propagation: wave speed and attenuation coefficient at different frequencies xiii

14 Unified Gas-kinetic Scheme for the Study of Non-equilibrium Flows by WANG, Ruijie Nano Science and Technology Program The Hong Kong University of Science and Technology Abstract There is an increasing demand for multi-scale modeling and simulation of gas flows in various engineering applications, such as the re-entry of space shuttle and heat flow in micro devices. The unified gas-kinetic scheme (UGKS) is a newly developed multi-scale method to study gas flows in all Knudsen regimes from the continuum Navier-stokes solutions to the rarefied non-equilibrium transport. The main objective of this thesis research is to further develop UGKS and apply it to the study of multiple scale transport problems. In this thesis, the UGKS and its simplified variation discrete unified gas-kinetic scheme (DUGKS) are presented and several numerical examples are provided to validate the schemes. UGKS is further constructed for multi-component gas flow and is validated through the simulations of shock structures at different Mach numbers and micro-channel flows driven by small pressure, temperature, and concentration gradients. Then UGKS is used to study the physics of low-speed micro-flows which include the sound-wave propagation and the crosscoupling phenomenon in micro-channel. In the study of sound-wave propagation, the phase speed and attenuation coefficient are extracted from the simulation under a wide range of Knudsen numbers from the continuum flow regime to the free molecular one. The comparison with the experiments shows good agreement in all Knudsen regimes. And the cross-coupling of thermal-osmosis and mechano-caloric effect in slightly non-equilibrium gas is simulated and analyzed for micro-channel with planner and ratchet surfaces. The variation of cross-coupling coefficient as a function of Knudsen number is obtained. At the same time, preliminary optimization for this kind of Knudsen pump is included. xiv

15 Chapter 1 Introduction 1.1 Motivation The theoretical research of continuum and rarefied gas dynamics has a long history. The computational fluid dynamics (CFD) has been greatly developed and successfully applied to many fields in the past decades, such as aerospace, astronautics, microdevices, cars, and other gas machineries. For a long time, the simulation methods of continuum flow and rarefied flow are studied separately. But the demand of multiscale modeling and simulation is arising as a result of fast development of space technologies and micro/nano technologies. The study of multiple scale transport process is still a challenging topic in CFD. In aero-astronautics, the vehicles in outer-space and near-space are either operating in extremely low-density flows or transport between the rarefied and continuum flows. Multi-scale simulation may be needed in both situations. The vehicle in outer-space needs thrusters in the propulsion system to stabilize and control its attitude where the density ratio of the gas inside and outside of the thrusters can be very large. In the spacecraft re-entry passage, the vehicle experiences from the free molecular flow at the edge of atmosphere to the continuum flow region near the ground. The vehicles in near-space may also travel constantly between the low-density and high-density flow regimes. Pure continuum or rarefied gas simulation method is not sufficient and a multi-scale method is therefore preferred. In micro/nano technologies, multi-scale simulation is typically needed in the analysis or design of the micro-electro-mechanical-systems (MEMS). In the thin-film formation 1

16 and MEMS manufacture technologies, such as chemical vapor deposition (CVD), the gas in the decomposition chamber may cover a wide range of rarefaction conditions [1]. And in microfluidic filters, the gas in the filter channel is rarefied whereas the gas outside of the channel is mostly considered as a continuum flow [2]. In micro-nozzles and other vacuum systems, the gas at the entrance can be continuum and the gas at the exit may become rarefied [3, 4]. 1.2 Review of Modeling and Simulation for Gas Flows Under usual circumstances, the flow behavior is a collective effect of particles and the continuum hypothesis is adopted to derive mathematical models for gas flows, such as the Euler equations for inviscid flow and the Navier-stokes equations for viscous flow. The Navier-stokes equations describe the conservation laws of mass, momentum and energy together with the constitutive relations for viscosity (Newton s law) and heat conduction (Fourier s law), and are the fundamentals of the hydrodynamic theory. They are widely studied physically and mathematically, and play an important role in fluid dynamics study. However, the continuum hypothesis breaks down in low-density or small-scale systems, and the flow behavior is clearly influenced or dominated by the individual particle transport. A microscopic description of the gas is therefore needed and this is the subject of kinetic theory. Early works on the kinetic theory can be dated back to 1738, when D. Bernoulli explained the pressure of gas based on the process of particle transport and collision. The important concept of mean free path the average distance traveled by particles between two successive collisions was introduced by R. Clausius. J.C. Maxwell introduced the velocity probability distribution function and derived its form in equilibrium state, i.e. Maxwell distribution. Finally, the solid foundation of kinetic theory was provided by the work of L. Boltzmann. He proposed the dynamic equation for velocity probability distribution function under the effect of particle transport and collision the famous Boltzmann equation. He also introduced the H-theorem and proved that the Maxwell distribution is the only equilibrium distribution for the Boltzmann equation [5]. 2

17 Continuum regime Slip regime Transition regime Free molecular regime Kn Figure 1.1: Classification of flow regimes based on Knudsen number. Due to the complexity of the Boltzmann equation, researchers tried to simplify the equation for theoretical analysis. One typical approach is the BGK model equation proposed by P.L. Bhatnagar, E.P. Gross, and M. Krook [6]. In this model, the complex collision term is replaced by a simple relaxation process of the distribution function from an initial state to the Maxwell equilibrium distribution at the time scale of particle collision time. The original collision term is greatly simplified, whereas many important characters of the Boltzmann equation is still preserved. However, the BGK equation is still a non-linear equation. If the system is slightly disturbed and not far from the equilibrium distribution, the Boltzmann equation or its model equations can be linearized, which provides another popular approach for theoretical research [5, 7]. In the kinetic theory, the degree of rarefaction is usually characterized by the Knudsen number, typically defined as the ratio of the particle mean free path to the characteristic length, Kn = λ/l. (1.1) S. Chapman and D. Enskog expanded the distribution function into a power series of the Knudsen number, recovered the Euler equations as zero-order approximation and the Navier-stokes as first-order approximation of the Boltzmann equation, as well as the formula for transport coefficients [8]. The Chapman-Enskog expansion provides a powerful tool to analyze the kinetic equations in continuum limit. The gas flows can be roughly classified into different flow regimes based on the Knudsen number according to H.S. Tsien. Figure 1.1 shows one of the classifications. Since the flow properties may vary significantly in a flow field, the definition of Knudsen number shall be chosen to characterize the local flow properties [5]. In the continuum regime Kn 0.001, the continuum hypothesis is considered valid and the flow can be well described by the Navier-stokes equations. Many numerical methods are developed for solving the Navier-stokes equations. A typical method is to use the Riemann solvers [9] for inviscid terms, central discretization for viscous 3

18 terms, and Runge-kutta method for time integration. The turbulent flow is either simulated by the Reynold-averaged Navier-stokes (RANS) equations with turbulence models, Large-eddy Simulation (LES), or Direct Numerical Simulation (DNS) [10]. Since the Navier-stokes equations can be derived form the Boltzmann equation, a Navier-stokes solver can be developed from the kinetic theory as well, such as the Lattice Boltzmann method (LBM) [11] and the Gas-kinetic scheme (GKS) [12]. In the slip regime Kn 0.1, the non-equilibrium phenomena appear near the boundary. The tangential velocity of the gas near the solid surface is non-zero relative to the surface and the temperature of the gas near the surface is not equal to the surface temperature. The phenomena are called velocity slip and temperature jump, and their expressions can be derived from the kinetic theory to different order of approximations [5, 13]. Usually the Navier-stokes equations are considered valid in the bulk region of the flow in this regime, and slip boundary conditions are supplied for the simulation. Higher-order macroscopic equations can be derived from the kinetic theory. D. Burnett derived the second-order approximations of stress tensor and heat flux from Chapman-Enskog expansion [5, 14]. H. Grad derived the extended macroscopic equations from Hermite expansion for 13 macroscopic quantities density, velocity, pressure, stress tensor, and heat flux, namely Grad 13 moments equations [5, 15, 16]. In recent years, H. Struchtrup proposed the Regularized 13 moments equations (R13) based on the Chapman-Enskog expansion and Grad s method [16]. Y. Sone also developed another asymptotic theory to analysis the non-equilibrium phenomena under small Knudsen numbers [17]. In the transition regime 0.1 Kn 10 and free molecular regime Kn 10, the non-equilibrium phenomena become significant in the whole system, and even the high-order macroscopic equations can hardly go beyond Kn 1. For the free molecular regime, the Boltzmann equation can be greatly simplified due to the absence of collision term, and many problems can be solved theoretically. For the transition regime, the particle collision still plays an important role and is very difficult for theoretical analysis. The direct numerical simulation becomes a dominant tool in this regime. The numerical simulation methods for rarefied gas can be classified into two types. The first type is the particle-based method to directly simulate the physical behavior 4

19 of gas particles. Another type is the numerical methods based on partial differential equations (PDEs), such as the Boltzmann equation or its model equations. The Direct Simulation Monte Carlo (DSMC) method proposed by G.A. Bird [18] is a famous particle-based method. DSMC use simulation particles to represent real particles, and decompose the particle behavior during a time step into free transport stage and collision stage. The free transport stage is deterministic and the collision stage is Monte Carol. Since DSMC is particle-based, it s not restricted by the assumptions in the Boltzmann equation, such as two-body collisions. It s relatively easy to implement physical models for complex flow situations, such as chemical reactions and radiations. DSMC is also very efficient for high speed rarefied flow and is widely applied to astronautics. But due to the decomposition, the time step should be smaller than the collision time and the mesh size should be smaller than the mean free path. This limits the wide application of DSMC in small Knudsen number flows. In low-speed micro flows, DSMC may experience significant statistical errors. The discretization of the full Boltzmann equation consists of a large part of the PDEbased methods. The key difficulty is the computation of the multiple integrals in the collision term. Different approaches are proposed to address this problem, including the discrete velocity method (DVM) [19, 20] and spectral method [21, 22, 23]. The numerical method can be simplified and the computational cost can be reduced if model equations are employed, instead of the full Boltzmann equation. This includes the discrete ordinate method (DOM) [24, 25] and the unified gas-kinetic scheme (UGKS) [26, 27]. But the collision term in model equations may not be able to capture all the delicate properties of the gas flow in transitional regime in some cases. The PDEbased methods are generally free of statistical errors, but not as flexible as DSMC due to the intrinsic modeling underlying the PDEs. The PDE-based methods require discretization of the particle velocity space. The full Boltzmann equation has to be solved in a six-dimensional space (three for location and three for particle velocity) instead of three spatial degrees of freedom in the macroscopic equations. For model equations, reduced distribution functions [28] can be introduced to lower the required simulation dimensions for one-dimensional and two-dimensional problems. In small Knudsen numbers and low-speed flows, where the non-equilibrium phenomena are weak, the discrete velocity points can be reduced. However, in highly non-equilibrium flows, numerous velocity points are required to capture the irregular distribution functions. 5

20 Thus, the PDE-based methods are usually not as efficient as DSMC in hypersonic flow simulations (typically highly non-equilibrium). In order to relieve this problem, adaptive algorithms are proposed [29, 30]. Most PDE-based methods, except UGKS, use a similar strategy as that in DSMC method which decomposes the particle motion into free transport and collision, where the flux is computed from a free transport process and the collision is only handled as a source term. Consequently, they experience the same limitation on time step and mesh size as the DSMC method. Although both continuum regime and transitional/free molecular regime have relatively mature numerical methods for their study, the numerical method which covers all flow regimes is still highly demanded. To develop such a scheme is challenging. A traditional approach for the multi-scale problem is the hybrid method. In most hybrid method, the physical computational domain is decomposed into different regions according to the local degree of rarefaction or non-equilibrium status, where each region only covers one flow regime, and different methods are used in different regions [31, 32, 33, 34]. The main difficulty of this approach is how to determine the criteria for the decomposition and how to exchange the data between different methods. Other methods including decomposition of the particle velocity space into fast particles solved by kinetic equations and slow particles solved by hydrodynamic equations [35], or simultaneously solving the kinetic equations to provide transport coefficients for the hydrodynamic equations [36]. Unified gas-kinetic scheme [26, 27] is another approach developed in recent years aiming to solve the multi-scale problems. In comparison with the hybrid approach, the UGKS uses a single method for all flow regimes without any decomposition in physical or velocity space. It is based on the evolution solution of the BGK-type model equation, but different from any other PDE-based method, the free transport and collision are not treated separately due to the evolution solution. Thanks to the coupling of transport and collision, the time step and mesh size are not limited by the particle collision time and mean free path, and the flow physics from free molecular flow to continuum flow can be recovered automatically in different flow regimes. These features enable UGKS to be an efficient multi-scale method in the simulation of low-speed micro-flows [37]. 6

21 1.3 Objectives and Organization of the Thesis The main objectives of the current thesis research are, Validate unified gas-kinetic scheme for the simulation of non-equilibrium flows in various flow regimes; Study the non-equilibrium phenomena in low-speed micro-flows in various flow regimes; Further develop the scheme for a wider applicable area. The thesis is organized as follows: Chapter 2 starts with a brief introduction of the Boltzmann equation and its model equations for single-component gas. Then the unified gas-kinetic scheme and discrete unified gas-kinetic scheme (DUGKS) for single-component gas are presented. Some numerical examples are provided to validate the schemes and the simulation codes. Chapter 3 presents with a brief introduction of the Boltzmann equation and its model equations for multi-component gas. Then the UGKS for multi-component gas is constructed. Several test cases are simulated to validate the scheme, including shock structures at different Mach numbers and micro-channel flows driven by small pressure, temperature, and component concentration gradients. Chapter 4 covers the study of the sound-wave propagation in monatomic gas under a wide range of sound wave frequencies. The sound propagating parameters are extracted and compared with different methods, including experimental measurements from continuum to free molecular flow. This provides a solid validation for the UGKS to study non-equilibrium flow in different flow regimes. Chapter 5 studies the cross-coupling of thermal-osmosis and mechano-caloric effect in slightly non-equilibrium gas for micro-channels with planner surfaces and ratchet surfaces. The variation of cross-coupling coefficients as a function of Knudsen number is obtained. And a preliminary optimization for the Knudsen pump is also included. Chapter 6 summarizes current work and discusses further research directions. 7

22 Chapter 2 Numerical Modeling of Single-component Gas 2.1 Boltzmann Equation and Its Model Equations The Boltzmann equation for single-component monatomic gas without chemical reactions and external forces is [5] f t + u f x = Q( f), Q( f) = 4π 0 ( f f 1 f f 1)u r σdωdu 1, (2.1) where f = f(x, t, u) is the velocity distribution function, x = (x 1, x 2, x 3 ) = (x, y, z) is the coordinates, u = (u 1, u 2, u 3 ) = (u, v, w) is the particle velocity, u r is the relative velocity, σ is the differential collision cross section, Ω is the solid angle, and du = du 1 du 2 du 3. The right hand side is the collision term describing the change of f due to two-body collision between particles, where f is the distribution function of postcollision velocity. The equilibrium distribution function is unique, and is a Maxwellian, ( ) m 3/2 g = ρ exp( m ) 2πk B T 2k B T (u U)2, (2.2) where ρ is density, T is temperature, U = (U 1, U 2, U 3 ) = (U, V, W) is the macroscopic velocity, m is molecular mass, and k B is the Boltzmann constant. Macroscopic quantities can be obtained by taking the corresponding moments of f. Specifically, ρ W = ρu = ψ f du, (2.3) ρe 8

23 P ij = c i c j f du, (2.4) p = 1 3 P ii, (2.5) q i = 1 2 c i c j c j f du, (2.6) where W are the conservative flow variables, E is the total energy density, ψ = (1, u, u 2 /2) T are the collisional invariants, c = u U is the peculiar velocity, P is the stress tensor, p is pressure, and q is heat flux. Due to the conservation of mass, momentum, and energy during collisions, Q( f) satisfies the compatibility condition, ψq( f)du = 0. (2.7) The Chapman-Enskog expansion of the Boltzmann equation gives the Euler equations as zero-order approximation and the Navier-stokes as first-order approximation, as well as the expressions for transport coefficients [5, 8]. Bhatnagar, Gross, and Krook [6] proposed a simplified model equation, i.e. equation. It has the following general form, BGK f t + u f x = Q( f), Q( f) = f+ f τ, (2.8) or d f dt = Q( f), (2.9) where f + is the post-collision distribution function and τ is relaxation time. The BGK equation maintains most important characters of the Boltzmann equation, such as the H-theorem, conservations, etc. Integrating Eq.(2.9) along the characteristic line and assuming τ is a local constant, an analytic solution can be constructed [38], f(x, t, u) = 1 τ t t n f+ (x, t, u)e (t t )/τ dt + e (t tn )/τ f 0 (x u(t t n ), u), (2.10) where x = x u(t t ) and f 0 is the distribution function at t = t n. 9

24 In the original BGK equation, f + = g. The Chapman-Enskog expansion to the firstorder gives the same Navier-stokes equations, except the different transport coefficients. Prandtl number from the BGK equation is Pr BGK = c p µ BGK κ BGK = 1, (2.11) where c p is the specific heat at constant pressure, µ is the viscosity coefficient, and κ is the heat conduction coefficient. However, the correct value should be Pr = 2/3 for monatomic gas. Various improved BGK-type model equations are proposed to give a realistic Prandtl number, such as BGK-Shakhov model [39], ES-BGK model [40], and the general model [41]. The idea of BGK-Shakhov model and ES-BGK model can also be applied to diatomic gas, such as the Rykov model [42] and the polyatomic ES-BGK model [43]. In the BGK-Shakhov model, f + is a third-order Hermite polynomial, where the coefficients are determined by requiring the first 13 moments equations coincide with that of Boltzmann equation of pseudo-maxwell molecules [39]. It takes the following form, ( f + = g+ g + = g [1+(1 Pr)c i q i c i c i The relaxation time τ is equal to the collision time, ) ] m m k B T 5 5pk B T ). (2.12) τ = µ p. (2.13) In the ES-BGK model, f + is chosen to maximize the entropy while satisfies the moments of f up to second-order [40]. It takes the following form, f + 1 = ρ ( 2πT exp 1 ) 2 c T 1 c, (2.14) where T is a tensor, And the relaxation time is T = ( ) ( 1 kb Pr m Tδ ij+ 1 1 ) Pij Pr ρ. (2.15) τ = 1 Pr µ p. (2.16) In the general model proposed by Chen et al. [41], f + is a combination of BGK- Shakhov and ES-BGK model, and τ takes the same form as that in Eq.(2.16). f + = g + Shakhov + f+ ES, (2.17) 10

25 where Pr in g + Shakhov is replaced by a coefficient C Shak, Pr in f + ES and τ is replaced by a coefficient C ES. The dynamic viscosity coefficient µ can be computed from the Sutherland s law, µ = µ ref ( T T ref ) 3/2 ( Tref + T s T+T s ), (2.18) or other molecular models, such as hard sphere or variable hard sphere, ( ) T β µ = µ ref, (2.19) T ref where µ ref and T ref are the reference viscosity and temperature, and T s is Sutherland temperature. Here β is the temperature dependency index. For example, β is equal to 1/2 for hard sphere model. The mean free path is defined as [5] λ = c/(u r σ T n), (2.20) where c is the mean thermal speed, u r is relative velocity, σ T is the collision cross section, and n is number density. For variable soft sphere in equilibrium [5], λ = 4β 1(7 2β 2 )(5 2β 2 ) 5(β 1 + 1)(β 1 + 2) ( m ) 1/2 µ 2πk B T ρ, (2.21) where β 1, β 2 are two coefficients, and β 2 is equal to the temperature dependency index in Eq.(2.19). For hard sphere or variable hard sphere molecule, β 1 = 1. It can be found that λρ is a function of temperature only, and is constant for hard sphere. Since the definition of λ may take different form in the literature, we will present the definition of λ for each simulation. 2.2 Unified Gas-kinetic Scheme Gas-kinetic scheme was originally proposed by Xu for the simulation of continuum flow [38, 44, 12], and then extended to a multi-scale method for all Knudsen numbers monatomic flow [26, 27], and also diatomic gas [45]. An important idea of the unified gas-kinetic scheme is to model the gas flows physically in the discrete simulation space, instead of direct discretization of the partial differential equations [46]. In the 11

26 finite-volume framework, the evolution of distribution function and the conservative variables are f n+1 = f n 1 V t n+1 t n V u n f dsdt+ 1 V t n+1 t n Q( f)dvdt, (2.22) and W n+1 = W n 1 V t n+1 t n V ψu n f dudsdt, (2.23) where V is the volume of the cell, V is the cell interface, s is the area, and n is the outward unit normal. The modeling of f at the interface and Q( f) inside the cell depend on the spatial and time scales to identify the flow evolution. Different models can be used for the interface distribution function and the collision term [47]. Other physical considerations can also be embedded [48]. In UGKS, the interface distribution function is modeled by the BGK-type equation and is calculated from the analytic solution Eq.(2.10), instead of an upwind scheme. As shown in Eq.(2.10), the solution consists of a hydrodynamic part and a kinetic part. The hydrodynamic part is the integration of f +, which include the contribution from collisions. And the kinetic part is the transport of initial condition f 0, which recovers the contribution from free transport. The contribution from the two parts to the final distribution function is dynamically determined by the ratio of relaxation time to the numerical time step. If the relaxation time is much smaller than time step, the hydrodynamic part dominants. With appropriate modeling, the hydrodynamic part can recover the Navier-Stokes distribution function with second-order accuracy. If the relaxation time is much larger than the time step, the kinetic part dominants. With well prepared initial condition, the physics of free transport can be recovered with second-order accuracy. This property has exactly the asymptotic preserving property [49]. In addition, the time step and cell size are not limited to the collision time and mean free path due to the coupling of collision and free transport in the evaluation of interface flux. In UGKS, the distribution function and conservative variables are updated simultaneously. This seems unnecessary at first glance since macroscopic quantities are moments of the distribution function. The particle velocity space is continuous in the domain of (, ). However, it s discretized and truncated in numerical simulation. And the moments of the distribution function is obtained through numerical quadrature. Consequently, the compatibility condition in Eq.(2.7) can t be accurately 12

27 satisfied. The independent updating of the conservative variables can ensure the conservation. Moreover, the independent updating of the conservative variables allows an implicitly or semi-implicitly discretized collision term to be evaluated explicitly. It will be shown in later sections. The general steps of UGKS are as follows: Step 1: Initialization of the flow field. The flow field can be initialized by assuming equilibrium distribution or other specific form. Step 2: Determination of the time step. Similar to the traditional CFD, the time step is calculated from the Courant-Friedrichs-Lewy (CFL) condition. For example, where s i is the projected area in x i direction. Step 3: V t = CFL 3 i=1 max(u, (2.24) i)s i Reconstruction. The derivatives of the distribution function at each particle velocity and conservative flow variables in each cell are constructed and constrained by a slope limiter. In this thesis, the van Leer limiter is used for all the simulations, where s r and s l are the slopes. Step 4: s = (sign(s r )+sign(s l )) s r s l s r + s l, (2.25) Calculating interface flux. The flux of distribution function and conservative variables are evaluated from the analytic solution of BGK-type equation. Step 5: Updating cell-averaged flow variables. The conservative variables are first updated, then followed by the distribution function. Step 6: Checking the output condition. If not satisfied, go back to step Discretization of Velocity Space Physically, particles may take any velocity in range (, ). However, the particle velocity space needs to be discretized and truncated by a finite number of velocity points in the simulation. The moments of the distribution function are then calculated from numerical quadrature. 13

28 The criteria of the discretization is to get reasonable accurate moments especially the low order ones from numerical quadrature, and to maintain the number of velocity points as small as possible at the same time. In continuum flow and slightly disturbed rarefied flow, the distribution function is not far away from Maxwellian. Gaussian quadrature is quite accurate and efficient under such a condition. And the truncation range can be roughly estimated as (U i 4 k B T/m, U i + 4 k B T/m), i = 1, 2, 3. If the flow stays in strong non-equilibrium state, the distribution function may be quite different from a Maxwellian. Newton-cotes integration can be used in such a situation with increased velocity points. The truncation range can be the same as the above for low-speed flows. But in hypersonic flows, the truncation range should be enlarged. In this thesis, a specially designed Gaussian Hermite quadrature [50] and the compound Boole rule are used. And the same discretization is applied to all cells. A more efficient way for velocity space discretization is to use the adaptive algorithms [29, 30], where the discrete velocity space can be different in each cell and is dynamically adjusted Polyatomic Gas The formulas presented in section.2.1 only apply to monatomic gas, but most of the time we would like to simulate polyatomic gas, especially the diatomic one (e.g. the air). The implementation of UGKS for diatomic gas can be found in reference [45], but here a simpler method is chosen without distinguishing the temperatures of different types of motion. For polyatomic gas molecule, the degree of freedom other than the translational ones might be considered as the internal degree of freedom. For example, the internal degree of freedom in diatomic gas without vibrational excitation is the two rotations. Denote the internal degree of freedom as ξ = (ξ 1,..., ξ M ), where M is the total number of internal degree of freedom. The distribution function is now also a function of ξ and becomes f = f(x, t, u, ξ). becomes And specifically, the Maxwell distribution function ( ) m (3+M)/2 g = ρ exp( m ( ) ) (u U) 2 + ξ 2πk B T 2k B T i ξ i. (2.26) If the BGK-Shakhov model is chosen, its formula keeps unchanged since it is derived 14

29 for monatomic gas thus only applies to translational degree of freedom. The macroscopic variables are and W = P ij = ψ f dudξ, (2.27) c i c j f dudξ, (2.28) q i = 1 2 c i (c j c j + ξ k ξ k ) f dudξ, (2.29) where the collisional invariants are ψ = (1, u,(u 2 + ξ i ξ i )/2), and dξ = dξ 1,...dξ M Reduced Distribution Functions The particle velocity space has 3 + M dimensions. However, by introducing the reduced distribution functions [28], the number of dimensions requiring discretization can be reduced. Suppose the simulation problem is in N-dimensions, the other 3 N dimensions can be considered as internal degree of freedom and are denoted by ξi (i = 1,..., 3 N), then a pair of reduced distribution functions are h = f dξ, b = (ξi ξ i + ξ j ξ j ) f dξ, (2.30) where dξ = dξ1...dξ 3 N dξ 1...dξ M. Then Eq.(2.8) becomes a pair of equations, h t b t where ũ = (u 1,..., u N ) and x = (x 1,..., x N ). h + ũ x = h+ h, τ b + ũ x = b+ b, τ (2.31) The Maxwell distribution becomes g h = g b = ( ) m N/2 gdξ = ρ exp( m ) 2πk B T 2k B T c i c i, ( ) (ξi p ξ i + ξ j ξ j )gdξ = (3+ M N) g ρ h, (2.32) where Ũ = (U 1,..., U N ) and c = ũ Ũ. 15

30 For BGK-Shakhov model, f + becomes [ ( h + =g h 1+(1 Pr) c i q i c i c i m k B T 2 N ) ] m 5pk B T ), b + m =g b + g h (1 Pr) c i q i 5pk B T [ c i c i (3+ M N)+ p ( N 2 N(3+ M) 2M) ], ρ (2.33) where q = (q i,..., q N ). found in appendix B. Some useful results of the moments of Maxwellian can be The macroscopic variables are hdũ W = ũ i hdũ, (2.34) 1 2 (ũ iũ i h+b)dũ and P ij = c i c j hdũ, (2.35) q i = 1 2 c i ( c j c j h+b)dũ. (2.36) For simplicity, the formulas presented in latter sections are for monatomic gas with the original f Calculate Interface Flux The distribution function at the interface is described by Eq.(2.10). And the flux can be calculated with proper approximation of f + and f 0. Suppose the interface is located at x 0 with a local coordinate (e 1, e 2, e 3 ) and e 1 is the outward unit normal n. In UGKS, the initial condition f 0 is assumed to be linearly distributed inside each cell 16

31 and is discontinuous across the interface, ( f 0 (x, u) = + ( f L 0 (x 0)+ x f L 0 x f R 0 (x 0)+ x f R 0 x ) ) (1 H[ x n]) H[ x n], (2.37) where f0 L and f 0 R are the initial conditions at the left and right hand side cell of the interface and x = x x 0. H[x] is the Heaviside step function, 0, x < 0, H[x] = (2.38) 1, x 0. The post-collision distribution function f + is approximated by a first order Taylor expansion at the interface. It is assumed to be continuous at the interface, but has different normal derivatives at the left and right sides of the interface. For simplicity, the derivatives of f + is actually replaced by the derivatives of Maxwellian distribution, f + (x, t,u) = f + 0 (x 0) [ ] +g 0 (x 0 ) (1 H[ x])a L x+ H[ x]a R x+bȳ+c z+ A(t t n ), (2.39) where f + 0 and g 0 are the distributions at t = t n and x = x n, ȳ = x e 2, z = x e 3. Substituting Eq.(2.37) and Eq.(2.39) into the analytic solution Eq.(2.10), the distribution function at the interface is ( ) f(x 0, t, u) = 1 e (t tn )/τ f + 0 (x 0) ( )( ) + (t t n + τ)e (t tn )/τ τ a L H[ū]+a R (1 H[ū]) ūg 0 (x 0 ) ( ) + (t t n + τ)e (t tn )/τ τ (b v+c w)g 0 (x 0 ) ( ) + t t n + τ(e (t tn )/τ 1) Ag 0 (x 0 ) +e (t tn )/τ +e (t tn )/τ ( ( f L 0 (x 0) (t t n )u f L 0 x f R 0 (x 0) (t t n )u f R 0 x 17 ) ) H[ū] (1 H[ū]), (2.40)

32 where ū = u n, v = u e 2, w = u e 3. Since the interface distribution function in Eq.(2.40) contains the information of time, there is no need to use Runge-Kutta time stepping method. The integration of Eq.(2.40) over t already has second-order accuracy in time, t n+1 t n f(x 0, t, u)dt = +τ +τ ( ) τe t/τ + t τ f + 0 (x 0) ( )( ) e t/τ ( t+2τ) t+2τ a L H[ū]+a R (1 H[ū]) ūg 0 (x 0 ) ( ) e t/τ ( t+2τ) t+2τ (b v+c w)g 0 (x 0 ) ( + τ 2 e t/τ + t 2 /2 τ t+τ 2) Ag 0 (x 0 ) +τ +τ ( 1 e t/τ)( ) f0 L(x 0)H[ū]+ f0 R(x 0)(1 H[ū]) ( ) f e t/τ L ( t+τ) τ u ( 0 x H[ū]+ f 0 R x (1 H[ū]) ), (2.41) where t = t n+1 t n. Note that when τ/ t approaching infinity, the time related coefficients in Eq.(2.41) have limiting values. However, the simulation program may not able to correctly reproduce the limits and large numerical errors may occur. In practice, the coefficients are expanded at τ/ t if τ/ t excess some threshold value. In current simulation, the threshold is chosen as Now expressions are needed for the coefficients a L, a R, b, c, A, relaxation time τ, f + 0 (x 0), and g 0 (x 0 ) in Eq.(2.41). Here τ, f + 0 (x 0), and g 0 (x 0 ) are fully determined by the macroscopic quantities at (x 0, t n ), if τ = τ 0 (x 0 ) is chosen. The required macroscopic quantities are obtained by taking moments of f(x 0, t n, u), for example, ( ) W 0 (x 0 ) = ψ f0 L(x 0)H[ū]+ f0 R(x 0)(1 H[ū]) du. (2.42) The coefficients a L, a R, b, c, A are related to the spatial and time derivatives of g 0, for example, a L,R = 1 ( ) ( ) g0 W L,R 0, g 0 (x 0 ) W 0 (x 0 ) x x=x 0 (2.43) A = 1 ( )( ) g0 W(x0 ). g 0 (x 0 ) W 0 (x 0 ) t t=t n (2.44) 18

33 And the coefficients are functions of particle velocities in the form of a = a i ψ i, where ψ i are the collisional invariants. Taking A as an example, a i are ( 2 ρe ( + U t i U i 3p ) ρ ρ t 2U i a 5 = ρ 3p 2 a i+1 = 1 p a 1 = 1 ρ ( ρui t ρu i t ), (2.45) ) ρ U i U t i a 5 (i = 1, 2, 3), (2.46) ρ t U ia i ( U i U i + 3p ) a 5, (2.47) ρ where the macroscopic quantities are those at(x 0, t n ). Detailed derivations of Eq.(2.45) - Eq.(2.47) are given in appendix A. Derivatives of conservative variables are still needed to fully determine a L, a R, b, c, A. For example, the derivatives with respect to x are ( ) W L,R 0 = W L,R 0 (x 0 ) W 0 (x L,R ), (2.48) x (x 0 x L,R ) n x=x 0 where x L,R are the coordinate of the left and right cell centers. The derivative with respect to time is determined by the conservative moment requirements on the first order Chapman-Enskog expansion ψ(g t (x 0, t n )+u g x (x 0, t n ))du = 0 [29], ( ) W(x0 ) ( ) = (a L H[ū]+a R (1 H[ū]))ū+b v+c w g 0 (x 0 )ψdu. (2.49) t t=t n Now all the variables in Eq.(2.40) are known. The flux of distribution function and conservative variables across the interface from t n to t n+1 are t n+1 t n V t n+1 t n V u n f dsdt = ψu n f dudsdt = N N t n+1 F i = s i u n f(x 0, t, u)dt, (2.50) i=1 i=1 t n N N t n+1 F i = s i i=1 i=1 t n ψu n f(x 0, t, u)dudt, (2.51) where N is the number of interfaces of a cell. The moments of g over the particle velocity can be calculated analytically instead of numerical quadrature [26], and some useful results are given in appendix B. The procedure of flux evaluation can be summarized as follows, Step 1: Preparing the initial conditions f L,R 0 (x 0 ) and f L,R 0 / x. 19

34 Step 2: Calculating W 0 (x 0 ) from Eq.(2.42) and the corresponding τ = τ 0 (x 0 ), f + 0 (x 0) and g 0 (x 0 ). Step 3: Calculating the spatial derivatives of W 0 from Eq.(2.48), and then the coefficients a L,R, b, c from Eq.(2.45) - Eq.(2.47). Step 4: Calculating the time derivative of W from Eq.(2.49) and coefficient A. Step 5: Calculating the interface flux by using Eq.(2.41) Update Cell-averaged Flow Variables For BGK-type model equations, the collision term can be discretized by the trapezoidal rule. Then Eq.(2.22) and Eq.(2.23) become ( ) f n+1 = f n 1 N V F i + t Q n + f+(n+1) f n+1 2 τ i=1 n+1, (2.52) W n+1 = W n 1 V N F i. (2.53) i=1 In the simulation, the conservative variables are first updated by Eq.(2.53), then f +(n+1) and τ n+1 are known. Finally, the semi-implicit Eq.(2.52) is updated explicitly, ( f n+1 = 1+ t ) [ ( )] 1 2τ n+1 f n 1 N V F i + t Q n + f+(n+1) 2 τ i=1 n+1. (2.54) Boundary Conditions Solid surface Due to the existence of velocity slip and temperature jump in rarefied flow, kinetic boundary conditions should be used, such as the Maxwell boundary condition and the Cercignani-Lampis-Lord (CLL) model [5]. In Maxwell boundary condition, there are two types of interaction between the particles and the solid surface diffusive reflection and specular reflection. 20

35 For diffusive reflection, the distribution function of reflected particles is a Maxwellian determined by the density, velocity, and temperature on the surface. Typically, the velocity and temperature of the surface are given. And the density is calculated from the requirement of no particles penetrating the surface, t n+1 t ūg w dudt+ ū f in dudt = 0, (2.55) ū 0 t n where g w is the Maxwellian at the surface and f in is the distribution function of incoming particles. The surface is assumed to be located at the left hand side. Although the moments of g w can be calculated analytically, numerical integration is used in the simulation to minimize the error in macroscopic conservation. ū<0 For specular reflection, the particles are reflected with unchanged tangential velocity but opposite normal velocity. The distribution function of the reflected particles is f r (ū) = f in ( ū). (2.56) For a surface not parallel to the coordinate system, velocity of the reflected particles may not fall in the discrete velocity points, and interpolation is generally needed. But the conservation has to be maintained during the interpolation. In this thesis, the surface is always parallel to the coordinate system for the specular reflection cases. Finally, the distribution function at the solid boundary is a combination of the two streaming flows, f = (βg w +(1 β) f r ) H[ū]+ f in (1 H[ū]), (2.57) where β is the thermal accommodation coefficient in [0, 1]. The incoming f in can be calculated from different methods. A simple method is to extrapolate the distribution function from interior region to the surface, and use it as f in. A more complex method is to use the extrapolated distribution function as an initial condition, and based on the method described in section to obtain a time accurate distribution function at the interface as f in. When calculating f in through this method, the derivatives of f and W on both sides can be assumed to be equal. Mirror symmetry The implementation is the same as solid surface with specular reflection. 21

36 Inflow and outflow In continuum flow, the inflow/outflow boundary conditions are usually determined based on the characteristics or Riemann invariants. For example, the pressure inlet and outlet boundary conditions for internal flow can be 1. Inlet: extrapolating the normal velocity from the interior region. Pressure, temperature, and the tangential velocity are specifically given. 2. Outlet: only the pressure is specified, other quantities are extrapolated from interior region. In micro-channel flows, the inlet/outlet boundary conditions are determined in the same way as in continuum flows, unless otherwise stated. For external flows, especially the hypersonic ones, a semi-empirical boundary condition can be used to take into account the rarefied effects [29], W b = βw +(1 β)w R, (2.58) where W are the specified far-field macroscopic quantities, W R are the macroscopic quantities constructed by Riemann invariants, and β is a coefficient in[0, 1]. A possible choice for the coefficient is β = exp( 1/Kn) [29], where Kn is the global Knudsen number Numerical Examples In vacuum technology, the conductance C cond of a duct between the vacuum system and the vacuum pump is an important design property, which is defined as [4] C cond = J M / p, (2.59) where J M is the mass flux and p is the pressure difference of the inlet and outlet. In this section, we will present the simulation of gas flow expansion to vacuum through a short channel. Figure 2.1 shows the schematic of the problem. A short channel of length L and height H is connected to two identical reservoirs of length L R and height H R. The left reservoir contains equilibrium gas with pressure p 1 and temperature T 0, and the right 22

37 Diffusive reflection T 0 p 1,T 0 H R /2 p 2 = 0 L R H/2 L Symmetry Figure 2.1: Flow expansion to vacuum: schematic of the problem. reservoir is vacuum. The solid lines are solid surface with fully diffusive reflection maintained at T 0, the dashed lines are inlet/outlet boundaries, and the dash-dotted line is the symmetry line. At the inlet, the gas is maintained at p 1, T 0. At the outlet, the interface distribution function is chosen as f = f in H[ū], (2.60) since there are no particles entering the channel. The Knudsen number is defined as Kn = λ ( ) L c H = u r σ T n L 1 H, (2.61) where λ L is the mean free path at the left reservoir. In the literature, the rarefaction parameter δ is more frequently used instead of Knudsen number [51], From Eq.(2.21), their relation for hard sphere molecule is δ = p 1H m. (2.62) µ 1 2k B T 0 δ = π Kn. (2.63) The mass flux can be calculated analytically in some limiting cases. If the length of the channel is infinitely small (a slit), or the particles are specularly reflected in the free molecular limit, the mass flux only consists of particles entering the channel from the left reservoir, J 0 M = H 0 m ug L du = p 1 H. (2.64) 2πk B T 0 If the particles are diffusively reflected in the free molecular limit, the mass flux can be calculated as [52] J 0 M = β L RH 0 m ug L du = β L R p 1 H, (2.65) 2πk B T 0 23

38 Table 2.1: Flow expansion to vacuum: reduced mass flux at L/H = 1. δ J M (current) J M (DSMC [53]) Difference % % % % % % % % % % % where β L R is the transmission probability. It is the probability of a particle entering the channel from the left and going to the right. In the literature, a reduced mass flux is usually used to characterize the channel [51], J M = J M/J 0 M. (2.66) The conductance in Eq.(2.59) now becomes C cond = J M H m 2πk B T 0. (2.67) Since the Knudsen number is changed by choosing different p 1, where H and T 0 are kept constant, the conductance is proportional to the reduced mass flux. In the simulation, the channel geometry is set to be L/H = 1 and the reservoir size is chosen as L R = H R = 20H. The gas is assumed to be hard-sphere argon, and the Shakhov model is chosen. The transmission probability at the free molecular limit has theoretic solution, which gives β L R = for L/H = 1 [52]. Then from Eq.(2.65) and Eq.(2.66), the corresponding reduced mass flux is J M = β L R = Table.2.1 shows the reduced mass flux at various rarefaction parameters, compared 24

39 (a) Pressure contour at δ = 100. (b) Pressure contour at δ = 1. Figure 2.2: Flow expansion to vacuum: pressure contour at rarefaction parameter δ = 100, 1. Pressure, temperature, and Mach number Figure 2.3: Flow expansion to vacuum: pressure, temperature, and Mach number along the centerline at rarefaction parameter δ =

40 with reference [53]. Good agreement is found for all rarefaction parameters with a maximum difference of 1.6%, and the theoretic value at δ = 0 is reproduced correctly. Current results are slightly different from that in reference [4], since the mesh here is non-uniform and more refined near the channel in current simulation. Figure 2.2 shows the pressure contour around the channel at δ = 100, 1, and Figure 2.3 shows the pressure, temperature, and Mach number along the centerline at δ = Discrete Unified Gas-kinetic Scheme for Compressible Flow The discrete unified gas-kinetic scheme (DUGKS) is a simplified variation of the UGKS method proposed by Z.L. Guo et al [54, 55]. The key difference between DUGKS and UGKS is the way to get the cell interface gas distribution function, where DUGKS discrete Eq.(2.9) into the form f(x, t) f(x u(t t n ), t n ) = t tn (Q(x, t)+q(x u(t t n ), t n )), (2.68) 2 instead of integrating Eq.(2.9) as that in UGKS to use the analytic solution. Its influence on the interface flux will be discussed in detail. In DUGKS, the evolution of distribution function and conservative variables are discretized in the same way as in Eq.(2.52) and Eq.(2.53) for BGK-type equation. The evolution equation of conservative variables is not necessary for the procedure given in reference [54, 55]. But here we present a slightly different procedure which needs to update the conservative variables for exactly the same reason as in the UGKS. The fluxes are N F i = t n+1 s i (u n) f(x 0, t n+1/2, u), (2.69) i=1 i=1 N F i = t n+1 s i i=1 i=1 ψ(u n) f(x 0, t n+1/2, u)du, (2.70) where t n+1 = t n+1 t n. And the interface distribution function at t = t n+1/2 is calculated from Eq.(2.68), f(x 0, t n+1/2 ) f 0 (x 0 u t n+1 /2) = tn+1 4 ( ) Q(x 0, t n+1/2 )+Q 0 (x 0 u t n+1 /2), (2.71) 26

41 where f 0 and Q 0 are evaluated at t = t n. Instead of f, a new variable f = f + t Q, (2.72) 4 is stored and updated in the simulation. Since f is also a function of t, it is not a state variable anymore as f. It should be careful to use the correct time step when performing the transformation. In terms of f, the evolution of the distribution function becomes f n+1 = β n+1 1 (1 β n 2 ) f n + β n+1 1 β n 2 f+(n) +(1 β n+1 1 ) f +(n+1) β n+1 1 and the calculation of interface distribution function in Eq.(2.71) becomes 1 V N F i, (2.73) i=1 f(x 0, t n+1/2 ) = (1 β n+1/2 3 ) f 0 (x 0 u t n+1 /2)+ β n+1/2 3 f + (x 0, t n+1/2 ), (2.74) the macroscopic flow variables are where the coefficients are W = P ij = 1 ( c 1 β i c j f du β4 4 q i = 1 ( c 1 β i c j c j f du β4 4 β n+1 1 = 4τn+1 t n+1 2(2τ n+1 + t n+1 ), ψ f du, (2.75) ) c i c j f + du, (2.76) ) c i c j c j f + du, (2.77) β n 2 = 2 tn+1 t n 4τ n t n, β n+1/2 3 = β 4 = t 4τ. t n+1 4τ n+1/2 + t n+1, At the cell interface, the initial condition in Eq.(2.74) is approximated by ( ) f 0 (x 0 u t n+1 /2) = + ( f 0 (x L )+(x 0 x L u t n+1 /2) f L 0 x f 0 (x R )+(x 0 x R u t n+1 /2) f R 0 x ) H[u n] (1 H[u n]), (2.78) (2.79) 27

42 where x L, x R are the coordinates of the left and right cell center. And the post-collision term in Eq.(2.74) is calculated from the compatibility condition, ψ f + (x 0, t n+1/2 )du = ψ f 0 (x 0 u t n+1 /2)du. (2.80) When updating the cell-averaged flow variables, the conservative variables are first updated, then f is updated by Eq.(2.73). The implementation of boundary conditions is the same as in UGKS, except f is transformed to f. In numerical implementation, the limiting values of the time related coefficients should be recovered correctly as an AP scheme. The procedure of DUGKS can be summarized as follows: Step 1: Initializing the flow field f and W. Step 2: Determining the time step from CFL condition. Step 3: Reconstructing the derivatives of f. Step 4: Calculating interface flux. 1. Prepare the initial condition from Eq.(2.79) 2. Calculate f + (x 0, t n+1/2 ) from Eq.(2.80) 3. Calculate the flux with Eq.(2.74) Step 5: The conservative variables are first updated, then f is updated by Eq.(2.73). Step 6: Checking the output condition. If not satisfied, go back to the step Properties of the UGKS and DUGKS Schemes It is clear that the procedure to update the cell-averaged flow variables are almost the same in DUGKS and UGKS, thus there is no much difference in the computational cost. Now comparing the steps needed to evaluate the flux, 1. Both DUGKS and UGKS need to prepare an initial condition f 0 or f 0, and calculate conservative variables W and the corresponding g. 2. DUGKS don t need to calculate a L,R, b, c, A and the moments of g 0, due to its transformation. 28

43 3. DUGKS has fewer terms in the expression of interface distribution function thus needs less computer operations, DUGKS : f(t n+1/2 ) = a 1 f0 + a 2 g(t n+1/2 ), UGKS : f(t) = (a 1 + a 2 au+a 3 A)g 0 + a 4 f 0 + a 5 f 0 x. (2.81) From the comparison, it is expected that less computational efforts are required in DUGKS. For low speed isothermal flows, g can be further expanded in terms of Mach number, and less velocity points are needed as in LBM [54]. The procedure presented in [54, 55] requires two distribution functions to be stored in a cell, which further reduces the computational effort, but increases the memory consumption and communication time in parallel computation. With a large number of velocity points, the memory and the communication will become a bottleneck. Therefore, one distribution function is stored in this thesis. The simplification of flux evaluation in DUGKS heavily relies on the newly introduced variable f and the property ψ f du = 0, so it might be difficult to apply DUGKS to some specific model equations. Now we compare the difference of interface flux calculated by UGKS and DUGKS. It can be shown that the interface fluxes of both methods share the similar structure. For simplicity, f + = g, continuous reconstruction, and constant t and τ are assumed in the following analysis. In UGKS, the interface distribution function integrated along particle trajectory is given in Eq.(2.41). Noting that (a u)g 0 (x 0 ) = u g 0 x, Ag 0 = ( g t ) t=t n, g t(t n ) u g 0 x, (2.82) the interface distribution function integrated by time in Eq.(2.41) can be casted into t n+1 f(x 0, t)dt = a 1 g 0 (x 0 )+a 2 (u gx)+a 0 3 f 0 (x 0 )+a 4 (u fx), 0 (2.83) t n where a 1 =τe t/τ + t τ, a 2 = τe t/τ ( t+τ) t 2 /2+τ 2, a 3 =τ(1 e t/τ ), a 4 = τe t/τ ( t+τ) τ 2. (2.84) 29

44 In DUGKS, Eq.(2.79) can be rewritten as ( ( ) f 0 (x 0 u t/2) = (1 β 4 ) f 0 (x 0 )+ β 5 (u fx) )+ 0 β 4 g 0 (x 0 )+ β 5 (u gx) 0, (2.85) where β 5 = t/2. (2.86) From Eq.(2.80), ψg(x 0,t n+1/2 )du = ψg 0 (x 0 )du+(1 β 4 )β 5 ψg 0 (x 0 )du+(1 β 4 )β 5 ψ(u f 0 x)du+ β 4 β 5 ψ(u g 0 x)du+ β 4 β 5 ψ(u g 0 x)du ψ(u g 0 x)du (2.87) ( ) ψ g 0 (x 0 )+ β 5 (u gx) 0 du, then g(x 0, t n+1/2 ) g 0 (x 0 )+ β 5 (u g 0 x). (2.88) Now the interface distribution in Eq.(2.74) integrated by time becomes t n+1 t n f(x 0, t)dt = t f(x 0, t n+1/2 ) =a 1 g 0 (x 0 )+a 2 (u g 0 x)+a 3 f 0 (x 0 )+a 4 (u f 0 x), (2.89) where a 1 = 2 t2 t+4τ, a 2 = t3 t+4τ, a 3 = t(4τ t), a 4 = t2 ( t 4τ) t+4τ 2( t+4τ). (2.90) It is now clear that the interface fluxes in UGKS and DUGKS share the same structure, but with different coefficients given in Eq.(2.84) and Eq.(2.90). For UGKS, the coefficients contain exponential functions. For DUGKS, the coefficients are polynomials. Here continuous reconstruction is assumed for simplicity, discontinuous reconstruction has the similar result. From the comparison, it is expected that DUGKS share similar properties as UGKS, such as second-order asymptotic preserving in the free molecular limit and the Navierstokes limit. 30

45 Normalized coefficients Figure 2.4: Normalized coefficients of DUGKS and UGKS as a function of local Knudsen number. Here Kn = τ/ t. Figure 2.4 shows the normalized coefficients of DUGKS and UGKS as a function of local Knudsen number. Here Kn = τ/ t. When τ/ t =, the flux from both methods is t n+1 t n f(x 0, t)dt = t f 0 (x 0 ) t2 2 (u f 0 x). (2.91) When τ t, we expand the coefficients at τ and retain only the leading orders, the fluxes are UGKS : DUGKS : t 2 2τ g 0(x 0 ) t3 3τ (u g0 x)+ t f 0 (x 0 ) t2 2 (u f 0 x), (2.92) t 2 2τ g 0(x 0 ) t3 4τ (u g0 x)+ t f 0 (x 0 ) t2 2 (u f 0 x). (2.93) Here only the leading term of a 2 is different. This is reasonable, since UGKS uses analytic integration, so t 0 t 2 dt = t 3 /3. And DUGKS uses the mid-point rule, so t 0 t 2 dt t 3 /4. 31

46 When τ < t, the differences in the coefficients seem large. But due to the symmetry of the coefficients around zero axis in Figure 2.4 and f g for small Knudsen number, the differences can cancel each other mostly, so the effective differences are still small. For example, when τ/ t = 0, the flux from both methods is t n+1 t n f(x 0, t)dt = tg 0 (x 0 ) t2 2 (u g0 x), (2.94) since f = g at the continuum limit. The main reason for the differences in the coefficients between UGKS and DUGKS are due to the transformation used in DUGKS, which combines the equilibrium and non-equilibrium gas distribution functions because of the BGK-type relaxation models. Detailed analysis of the AP property can be found in references [54, 55] Numerical Examples Sod shock tube In this section, the standard Sod shock tube is tested from free molecular limit to the continuum limit. For a computational domain in x [0, L], the initial condition is x L/2 : ρ L, U L = 0, p L, x > L/2 : ρ R = 0.125ρ L, U R = 0, p R = 0.1p L. (2.95) The left and right boundaries are maintained at its initial condition. The gas is assumed to be air with hard sphere intermolecular interaction, then Pr = 0.72 and the internal degree of freedom is 2. The reference Knudsen number is defined based on the left initial state, Kn = λ L L = 1 ( ) 16 m 1/2 µ L. (2.96) L 5 2πk B T L ρ L Since λρ is constant for hard sphere molecule, the Knudsen number of the right initial state is Kn R = 8Kn. The simulation is performed with Shakhov model for three different Knudsen numbers, Kn = 10, 10 3, 10 5, ranging from free molecular flow to continuum one. The results are compared with that of UGKS. In the simulation, the physical space is discretized into 100 cells, and the velocity space is discretized into 201 points in the range 32

47 UGKS DUGKS UGKS DUGKS UGKS DUGKS UGKS DUGKS Figure 2.5: Sod shock tube: density, velocity, temperature, and heat flux at Kn = 10. of [ 10 2k B T L /m, 10 2k B T L /m] with Newton-Cotes quadratures. The CFL number in both methods is 0.5 and the output time is t = 0.15(L m/(2k B T)). The simulation results are presented in non-dimensional form where ˆρ = ρ ρ L, Û = ˆT = T T L, ˆq = U (2k B T L /m) 1/2, q ρ L (2k B T L /m) 3/2. (2.97) Figure 2.5 shows the density, velocity, temperature and heat flux at Kn = 10. The flow is in free molecular regime and the flow field is well resolved. Figure 2.6 shows the density, velocity, temperature, and heat flux at Kn = The flow is in slip regime and discontinuities begin to occur in the flow field. This is because the mean free path becomes smaller compared to the cell size, so the flow structure such as shock is not fully resolved. The scheme is gradually behaving like a shock-capturing scheme 33

48 1 0.9 UGKS DUGKS UGKS DUGKS UGKS DUGKS UGKS DUGKS Figure 2.6: Sod shock tube: density, velocity, temperature, and heat flux at Kn = as the Knudsen number reduces. Figure 2.7 shows the density, velocity, temperature, and heat flux at Kn = The flow is in continuum regime and the flow field clearly shows a rarefaction wave, a contact discontinuity, and a shock as that in a typical continuum solution. The scheme is now a shock capturing scheme. It can be found that the solutions of DUGKS and UGKS are almost indistinguishable in all three figures. Shock structure Figure 2.8 shows the schematic of the shock structure problem. The upstream gas p 1, T 1, Ma 1 and downstream gas p 2, T 2, Ma 2 satisfy the Rankine-Hugoniot condition 34

49 1 0.9 UGKS DUGKS UGKS DUGKS UGKS DUGKS UGKS DUGKS Figure 2.7: Sod shock tube: density, velocity, temperature, and heat flux at Kn = and form a normal shock, Ma 2 1 Ma 2 = (γ 1)+2 2γMa 2 (2.98) 1 (γ 1), ρ 2 = (γ+1)ma2 1 ρ 1 (γ 1)Ma 2 (2.99) 1 + 2, T 2 = (2+(γ 1)Ma2 1 )(2γMa2 1 γ+1) T 1 Ma 2, (2.100) 1 (γ+1)2 where Ma is the Mach number and γ is the ratio of specific heat. The gas is assumed to be hard sphere argon gas, so µ T 0.5, Pr = 2/3, and γ = 5/3. The mean free path is defined by Eq.(2.21). The simulation is performed with Shakhov model for Ma = 3, 8. The physical space is in the range of x [ 25λ 1, 25λ 1 ] and discretized into 100 cells, where λ 1 is the up- 35

50 p 1,T 1,Ma 1 p 2,T 2,Ma 2 Figure 2.8: Shock structure: schematic of the problem. Normalized density and temperature Normalized heat flux and shear stress Figure 2.9: Shock structure: density, temperature, heat flux, and shear stress at Ma 1 = 3.0. stream mean free path. The velocity space is in the range of[ 15 2k B T 1 /m, 15 2k B T 1 /m] and discretized into 101 points. The CFL number in all cases is The origin of the figures is determined by requiring ρ(0) = (ρ 1 + ρ 2 )/2. And the simulation results are presented in non-dimensional form, ˆρ = ρ ρ 1 ρ 2 ρ 1, ˆT = T T 1 T 2 T 1, ˆq = q ρ 1 (2k B T 1 /m) 3/2, ˆτ xx = P xx p 2p 1, (2.101) where τ xx is the shear stress. Figure 2.9 shows the normalized density, temperature, heat flux, and shear stress at Ma 1 = 3.0. The density and temperature increase up to the downstream value in the thin layer of the shock, and the intensive temperature increment induces a significant heat flux within the shock. Figure 2.10 shows the normalized density, temperature, heat flux, and shear stress at Ma 1 = 8.0. When shock is stronger, the heat flux becomes even higher. And it can be found that the simulation results of DUGKS match very well with UGKS in both figures. 36

51 Normalized density and temperature Normalized heat flux and shear stress Figure 2.10: Shock structure: density, temperature, heat flux, and shear stress at Ma 1 = 8.0. Lid-driven cavity T 0,U w T 0 ρ 0,T 0 T 0 L T 0 L Figure 2.11: Lid-driven cavity: schematic of the problem. Figure 2.11 shows the schematic of the lid-driven cavity problem. The gas initially at rest with ρ 0, T 0 is confined by a rectangular container. Solid surfaces on the left, right, and bottom are stationary with constant temperature T 0. The solid surface on the top is moving with velocity U w and keeps a constant temperature T 0. The gas is assumed to be argon with VHS model such that µ = µ 0 (T/T 0 ) The Knudsen number is defined as Kn = λ 0 L, (2.102) 37

52 where λ 0 is given by Eq.(2.21). Simulations are performed with Shakhov model for Ma = U w /(γk B T 0 /m) 1/2 = 0.15 and Kn = 0.1. In the simulation, the physical space is discretized into mesh points for Kn = 0.1. The particle velocity space is discretized into mesh points with Gaussian quadrature. The CFL number is 0.5. Figure 2.12 shows the flow field at Kn = 0.1. The velocity magnitude is almost symmetric and a large vortex is formed in the container. The heat flux is flowing from the low temperature region to the high temperature region due to the non-equilibrium effect. The horizontal velocity along the vertical centerline and the vertical velocity along the horizontal centerline are also plotted and compared with the results of UGKS. Excellent agreements are found. In reference [27], UGKS is compared with DOM under different mesh resolutions, and it turns out that UGKS is not very sensitive to it while DOM deteriorates quickly when reducing the mesh points. Since DUGKS also coupled the free transport and collisions during the flux evaluation, we would like to check whether this is still true for DUGKS. Figure 2.13 shows the velocity profile along the centerline under three mesh resolutions 61 61, 31 31, and It can be found that even with meshes, the velocity profiles are still well captured. Computational cost In this section, the computational cost of DUGKS and UGKS are compared by using a one-dimensional shock structure problem. The codes of both methods are compiled with parallelization turned off and all the parameters are set equal. To minimize the fluctuation of the execution time, each case is repeated 5 times to obtain an average value and 4000 iterations are performed for each simulation to ensure an execution time > 10s. Table 2.2 shows the execution time of DUGKS and UGKS for different velocity points. It can be found that DUGKS is faster in general. And the difference tends to increase to some constant value around 20% when increasing the velocity points. 38

53 UGKS DUGKS (a) Temperature field and heat flux in DUGKS. (b) Vertical velocity along the horizontal centerline UGKS DUGKS (c) Velocity magnitude and streamlines in DUGKS. (d) Horizontal velocity along the vertical centerline. Figure 2.12: Lid-driven cavity: flow field at Kn = 0.1. Table 2.2: Computational cost comparison of DUGKS and UGKS. Velocity points Execution time (s) DUGKS UGKS Difference % % % % % 39

54 X61 (DUGKS) 31X31 (DUGKS) 11X11 (DUGKS) X61 (DUGKS) 31X31 (DUGKS) X11 (DUGKS) (a) Horizontal velocity along the vertical centerline under different mesh resolutions. (b) Vertical velocity along the horizontal centerline under different mesh resolutions. Figure 2.13: Lid-driven cavity: velocity profile under different mesh resolutions at Kn = Conclusion In this chapter, a brief introduction of the Boltzmann equation and its model equations for single-component gas are given. Then the unified gas-kinetic scheme is presented in detail. The simulation of flow expansion to vacuum through a short channel is provided as a test case to validate the scheme and the simulation code. The reduced flow rate matches with the reference solution of DSMC very well in different flow conditions with a maximum difference of 1.6%. At the same time, the discrete unified gas-kinetic scheme for compressible flow is presented. The differences between UGKS and DUGKS are analysed. And it can be concluded that DUGKS preserves many main properties of UGKS, such as second-order asymptotic preserving while reducing the computational cost. Several numerical examples are provided to validate DUGKS, including the sod shock tube, shock structure, and lid-driven cavity. The solutions from UGKS and DUGKS are compared and excellent agreement is found, which is consistent with the analysis. The comparison of computational cost between DUGKS and UGKS shows that DUGKS is approximately 10% 20% faster than that of UGKS. 40

55 Chapter 3 Numerical Modeling of Multi-component Gas 3.1 Boltzmann Equation and Its Model Equations For multi-component gas, the Boltzmann equation for component α is [56] where Q α ( f, f) = f α t + u α f α x α = Q α ( f, f), (3.1) N N Q αr ( f α, f r ) = r=1 r=1 ( f α f r f α f r )u αr σ αr dωdu r. (3.2) Term Q αα ( f α, f α ) is called self-collision term, and Q αr ( f α, f r ), α = r is called crosscollision term. In equilibrium, all components shall have Maxwell distributions with the same velocity and temperature. And macroscopic quantities of individual component α are W α = ψ f α du, (3.3) P α = c α c α f α du, (3.4) q α = 1 2 p α = 1 3 Pα ii, (3.5) c α c 2 α f α du, (3.6) where c α = u U α. Due to the momentum and energy exchange between components, there are source 41

56 terms in the momentum and energy equations of individual component, ρ α U α + ρ αu α U α + P α t x x = uq α ( f, f)du, (3.7) ρ α E α t + ρ αe α U α x + P α U α x + q α x = 1 2 u2 Q α ( f, f)du. (3.8) The non-zero source terms uq α ( f, f)du and 1/2u 2 Q α ( f, f)du are called exchange relations [57]. Similar to the single-component situation, a BGK-type model can be constructed for the simplification of analysis and simulation. Existing BGK-type gas mixture models can be classified into two categories. One is the multiple-bgk-operator model and the other is the single-bgk-operator model. In the multiple-bgk-operator models, every collision operator Q αr is approximated by a BGK operator. The original model is proposed by Gross and Krook [58] for binary mixture, f α t + u f α x = f+ αα f α + f+ αr f α, (3.9) τ αα τ αr where the relaxation time τ αr has property n α /τ αr = n r /τ rα and n is the number density. The post-collision term is, ( ) 3/2 f αr + mα = ρ α exp( m ) α (u U αr ) 2. (3.10) 2πk B T αr 2k B T αr If r = α, T αr = T αα and U αr = U αα are the temperature and velocity of component α. If r = α, T αr and U αr are the mixture temperature and velocity of component α and r, and are usually defined by requiring the exchange relations in coincidence with that of the Boltzmann equation of Maxwell molecule. Several authors [59, 60, 61, 62] proposed a modified form of the model, which linearize f + αr in terms of f + αα or vice versa or combined. In the single-bgk-operator models, only one global collision operator is used for each component to take account of both self-collision and cross-collisions. One typical model is proposed by Andries, Aoki, and Perthame (AAP model) [57], where f + α f α t + u f α x = f+ α f α τ α, (3.11) ( ) 3/2 mα = ρ α exp( 2πk B Tα m ) α 2k B Tα (u U α) 2. (3.12) 42

57 The parameters Tα and U α are chosen to recover the exchange relations for Maxwell molecule. Another typical model is the Ellipsoidal model for gas-mixture [63, 64]. Suppose a binary mixture with components α, r, the main idea of the model is to impose an additional constraint, for example, 1 ρ α uq α du 1 uq r du = η(u α U r ), (3.13) ρ r to allow the velocity of each component to relax to its equilibrium value at different rate, then maximize the entropy to determine the form of f +. Most existing models are derived by assuming Maxwell molecules, thus the cross coupling of Dufour effect and Soret effect is missing [59]. In this thesis, the single- BGK-operator models are preferred since 1. The multiple-bgk-operator models generally don t satisfy the in-differentiability principle [60, 57], which requires the model to fall back to the single component BGK equation when all components are equal. But the single-bgk-operator models usually do satisfy it. 2. If there are more than two components in the mixture, the multiple-bgk-operator models are more complex. In multi-component gas, there are three types of transport coefficients: viscosity, heat conduction, and mass diffusion. Since the AAP model is derived based on the original BGK, only one transport coefficient can be recovered correctly. While the Ellipsoidal model has additional free parameter η that enables it to recover the viscosity and diffusion coefficients. In this thesis, the AAP model is considered, the formulation for Ellipsoidal model will be future work. In the AAP model described by Eq.(3.11) and Eq.(3.12), the parameters Tα and U α are connected to the macroscopic properties of individual components via [57] ρ r U N α =U α + τ α 2 θ αr (U r U α ), (3.14) m r=1 α + m r 3 2 k BT α = 3 2 k BT α m α 2 (U α U α ) 2 ρ r N +τ α 4m α (m r=1 α + m r ) 2 θ αr ( 3 2 k BT r 3 2 k BT α + m ) r 2 (U r U α ) 2, (3.15) 43

58 where θ is the interaction coefficient between particles. The collision time is determined by 1 = β τ α N r=1 θ αr ρ r m r, (3.16) where β is either 1 or chosen to coincide with the τ of single-component gas when all components are equal. In this thesis, β = 1 is used for in the simulations. Different type of molecules can be approximated by the choice of θ αr, for example [65], 4 ( π 2kB T α + 2k ) 1/2 ( ) BT r dα + d 2 r Hard sphere 3 m θ αr = α m r 2 ( ) aαr (m α + m r ) 1/ π Maxwell, m α m r (3.17) where d α, d r are the diameters of molecule and a αr is the constant of proportionality in the intermolecular force law. 3.2 Unified Gas-kinetic Scheme Similar to the single component formulation, the evolution of macroscopic variables and distribution function for component α are and W n+1 α = W n α 1 V i ( fα n+1 = 1+ t ) [ 1 2τα n+1 fα n 1 V i F i α+ t τ α (W (n) α W n α), (3.18) F i α+ t 2 where W α = (ρ α, ρ α U α, ρ α E α) are the moments of ψ f + α du. ( Q n α + f+(n+1) α τα n+1 )], (3.19) Eq.(3.15) and Eq.(3.16) show that W α and τ α are determined by the macroscopic quantities of all components. Once (W 1,..., W N ) for all the components are known, the calculation (W 1,..., W N ) (W1,..., W N ) and (W 1,..., W N ) (τ1,..., τ N ) can be done from Eq.(3.15) and Eq.(3.16). Noting that there is a source term in Eq.(3.18) that is evaluated explicitly. Theoretically, summation of the source term in Eq.(3.18) for all components is zero. But numerical errors may influence this conservation. Further improvements may be made to remove the explicit evaluation and ensure the conservation of mass, momentum, and energy for mixture properties. 44

59 The integral solution for each component remains unchanged, so the interface distribution function integrated along particle trajectory for component α is t n+1 ) f α (x 0, t, u)dt = (τ α e t/τ α + t τ α f + t n α (x 0, t n ) +τ α ( e t/τ α ( t+2τ α ) t+2τ α )(a L H[ū]+a R (1 H[ū]) +τ α ( e t/τ α ( t+2τ α ) t+2τ α )(b v+c w) f + α (x 0, t n ) ( ) + τα 2 e t/τ α + t 2 /2 τ α t+τα 2 A f α + (x 0, t n ) )( ) +τ α (1 e t/τ α fα(x L 0, t n )H[ū]+ fα R (x 0, t n )(1 H[ū]) +τ α ( e t/τ α ( t+τ α ) τ α ) u ( f L α (t n ) x ) ū f α + (x 0, t n ) H[ū]+ f α R (t n ) ) (1 H[ū]). x (3.20) The preparation of initial conditions f L,R α component formulation. (t n ), fα L,R (t n )/ x is the same as the single f + α (x 0, t n ) and τ α = τ α (x 0, t n ) can be fully determined by the macroscopic quantities of all components. For individual component α, W α (x 0, t n ) are calculated from W α (x 0, t n ) = ( ) ψ fα(x L 0, t n )H[ū]+ fα R (x 0, t n )(1 H[ū]) du. (3.21) After the macroscopic quantities for all components are known, the transformation (W 1 (x 0, t n ),..., W N (x 0, t n )) (W 1 (x 0, t n ),..., W N (x 0, t n )) are calculated from by Eq.(3.15), thus f + α (x 0, t n ) is fully determined. And (W 1 (x 0, t n ),..., W N (x 0, t n )) (τ 1,..., τ N ) are calculated from Eq.(3.16). The coefficients a L,R, b, c, A are calculated from the derivatives of W α(x 0 ). The approximation of the spatial derivative is the same as that in the single component formulation, for example, ( W (L,R) ) α (t n ) = W (L,R) α (x 0, t n ) W α(x L,R, t n ). (3.22) x (x 0 x L,R ) n x=x 0 And the time derivative is calculated from ( W ) α (x 0 ) ( ) = (a L H[ū]+a R (1 H[ū]))ū+b v+c w f α + (x 0, t n )ψdu. (3.23) t t=t n 45

60 3.3 Numerical Examples Shock Structure Consider a binary gas mixture with components A and B of mass m A, m B and diameter d A, d B that forms a normal shock. The upstream component concentrations, number densities, velocity, and temperature are denoted by χ A,B 1, n A,B 1, U 1, T 1, and the corresponding downstream ones are χ A,B 2, n A,B 2, U 2, T 2, where χ A,B = n A,B /(n A + n B ). The Mach number of the shock is then defined by U Ma =, (3.24) (γk B T/m) 1/2 where m = m A χ A + m B χ B. For each component, the Rankine-Hugoniot condition holds, so the upstream and downstream conditions are related through Ma 2 1 Ma 2 = (γ 1)+2 2γMa 2 (3.25) 1 (γ 1), χ A,B 2 = χ A,B 1, (3.26) n A 2 n A 1 = nb 2 n B 1 = (γ+1)ma2 1 (γ 1)Ma 2 (3.27) 1 + 2, U 2 = Ma2 1 (γ 1)+2 U 1 Ma 2 1 (γ+1), (3.28) T 2 = (2+(γ 1)Ma2 1 )(2γMa2 1 γ+1) T 1 Ma 2. (3.29) 1 (γ+1)2 The gas is assumed to be hard sphere argon gas, and the reference mean free path is defined by λ = 1 2πd 2 A n 1. (3.30) In the simulation, the physical space is in the domain x [ 25λ, 25λ ], which is discretized by 100 cells. The velocity space range is [ 8 2k B T 1 /m, 8 2k B T 1 /m], which is discretized by 101 points. The CFL number in all cases is The origin of the figures is determined by requiring n(0) = (n 1 + n 2 )/2. And the simulation results are presented in non-dimensional form, ˆn A,B = na,b n A,B 1 n A,B 2 n A,B 1, ˆT A,B = TA,B T 1 T 2 T 1. (3.31) 46

61 A (UGKS) B (UGKS) A (Boltzmann) B (Boltzmann) A (UGKS) B (UGKS) A (Boltzmann) B (Boltzmann) Figure 3.1: Shock structure in binary gas mixture: number densities n A,B and temperatures T A,B for Ma 1 = 1.5, mass ratio m B /m A = 0.5, and diameter ratio d B /d A = 1 under different χ1 B, and the reference solutions [66]. The solid lines are profiles of A component, and the dashed lines are profiles of B component from UGKS. The square symbols are profiles of A component, and triangle symbols are profiles of B component from the reference. The hat will be dropped in the figures for simplicity. The number densities and temperatures of each component under different Mach numbers and concentrations are shown in Figure 3.1, Figure 3.2, and Figure 3.3. And the simulation results are compared with the Boltzmann solution [66]. As shown in Figure 3.1 and Figure 3.2 for Ma = 1.5, the solutions from current scheme show good agreement with the reference in both number density and temperature profiles under different mass ratios. In Figure 3.2 for Ma = 1.5, m B /m A = 0.25, and χ B 1 = 0.1, the temperature profile of light component B slightly deviates from the reference. For Ma = 3.0 shown in Figure 3.3, the number density profile is still good, while the temperature profiles of UGKS arises earlier, especially the light component B. It is also observed in the single component simulation [67] but is severer in the multicomponent case. The deviation of temperature profile before the shock is partially due to the incorrect transport coefficient produced by the model. Since the light component is more likely to be influenced for the same amount of momentum and energy transfer, it s more sensitive to the transport coefficient. 47

62 A (UGKS) B (UGKS) A (Boltzmann) B (Boltzmann) A (UGKS) B (UGKS) A (Boltzmann) B (Boltzmann) Figure 3.2: Shock structure in binary gas mixture: number densities n A,B and temperatures T A,B for Ma 1 = 1.5, mass ratio m B /m A = 0.25, and diameter ratio d B /d A = 1 under different χ1 B, and the reference solutions [66]. The solid lines are profiles of A component, and the dashed lines are profiles of B component from UGKS. The square symbols are profiles of A component, and triangle symbols are profiles of B component from the reference. A (UGKS) B (UGKS) A (Boltzmann) B (Boltzmann) A (UGKS) B (UGKS) A (Boltzmann) B (Boltzmann) Figure 3.3: Shock structure in binary gas mixture: number densities n A,B and temperatures T A,B for Ma 1 = 3.0, mass ratio m B /m A = 0.5, and diameter ratio d B /d A = 1 under different χ1 B, and the reference solutions [66]. The solid lines are profiles of A component, and the dashed lines are profiles of B component from UGKS. The square symbols are profiles of A component, and triangle symbols are profiles of B component from the reference. 48

63 H/2 H/2 p 0 (1+C p x/h) T 0 (1+C T x/h) χ A 0 +C χ x/h Figure 3.4: Micro-channel flow of binary gas mixture: schematic of the problem Micro-channel Figure 3.4 shows the schematic of the problem. A long channel is formed by two parallel plates in x direction and has height H in y direction. A binary gas mixture with components A and B of mass m A, m B and diameter d A, d B in the channel has uniform pressure gradient, temperature gradient, or concentration gradient in x direction, i.e. p = p 0 (1+C p x/h), T = T 0 (1+C T x/h) or χ A = χ A 0 + C χx/h. The plates are assumed to be fully diffusive and have temperature gradient T = T 0 (1+C T x/h). The inlet and outlet are imposed with pressure boundary conditions based on characteristics as described in section The gas is assumed to be hard sphere gas, and the Knudsen number is defined by Kn = λ 0 H, λ 0 = 1/( 2πn 0 d 2 A ), (3.32) where n 0 is the number density at the inlet. In the simulation, the channel with pressure gradient, temperature gradient, or concentration gradient are considered separately. If C T and C χ are zero, the non-dimensional particle flux of each component due to pressure gradient is defined by [68] Mp A,B = 1 1/2 U A,B d(y/h). (3.33) C p 1/2 2kB T 0 /m A M A,B T and M A,B χ follow similar formula. In our simulation, the channel has a length/height ratio equal to 40 and the gradients C p, C T, C χ are kept very small. The simulation results are compared with the work of Kosuge [68], where the McCormack model [69] for linearized Boltzmann equation is used under the assumption of small C p, C T, and C χ. Figure 3.5, Figure 3.6 and Figure 3.7 show the particle fluxes due to pressure gradient, temperature gradient, and concentration gradient vs Knudsen number under 49

64 A (current) B (current) A (linearized Boltzmann) B (linearized Boltzmann) A (current) B (current) A (linearized Boltzmann) B (linearized Boltzmann) (a) M α p (b) M α T (current) (current) (linearized Boltzmann) (linearized Boltzmann) (c) χ A 0 MA χ and χ B 0 MB χ Figure 3.5: Micro-channel flow of binary gas mixture: particle fluxes due to pressure gradient, temperature gradient, and concentration gradient vs Knudsen number for m B /m A = 2, d B /d A = 1, and χ A 0 = 0.5. The square symbols are profiles of A component, and triangle symbols are profiles of B component from UGKS simulation. The solid lines are profiles of A component, and dashed lines are profiles of B component from the reference [68]. 50

65 A (current) B (current) A (linearized Boltzmann) B (linearized Boltzmann) A (current) B (current) A (linearized Boltzmann) B (linearized Boltzmann) (a) M α p (b) M α T (current) (current) (linearized Boltzmann) (linearized Boltzmann) (c) M α T Figure 3.6: Micro-channel flow of binary gas mixture: particle fluxes due to pressure gradient, temperature gradient, and concentration gradient vs Knudsen number for m B /m A = 4, d B /d A = 1, and χ A 0 = 0.5. The square symbols are profiles of A component, and triangle symbols are profiles of B component from UGKS simulation. The solid lines are profiles of A component, and dashed lines are profiles of B component from the reference [68]. 51

66 A (current) B (current) A (linearized Boltzmann) B (linearized Boltzmann) A (current) B (current) A (linearized Boltzmann) B (linearized Boltzmann) (a) M α p (b) M α T (current) (current) (linearized Boltzmann) (linearized Boltzmann) (c) M α T Figure 3.7: Micro-channel flow of binary gas mixture: particle fluxes due to pressure gradient, temperature gradient, and concentration gradient vs Knudsen number for m B /m A = 10, d B /d A = 1, and χ A 0 = 0.5. The square symbols are profiles of A component, and triangle symbols are profiles of B component from UGKS simulation. The solid lines are profiles of A component, and dashed lines are profiles of B component from the reference [68]. 52

67 three different molecular mass ratios m B /m A = 2, 4, 10. From the figures of particle fluxes due to pressure gradient, it can be seen that the overall agreement with the reference solution is good for all mass ratios. Around Kn 1, M p takes a minimum value. This is called Knudsen minimum which is well known for pressure-driven Poiseuille flow in rarefied gas. The figures of particle fluxes due to temperature gradient show that the profile of heavy component B matches with the reference solution quite well, but the profile of light component A deviates in the transition regime. One reason for the disparity in temperature induced flow rate may come from the discrepancy in transport coefficients from different models, and the light component is more sensitive to it. Other factors may also have some impact on M T. Since it is calculated from thermal-creep flow in the open channel, the pressure at the inlet and outlet may not exactly keep its prescribed value. From the figures of particle fluxes due to concentration gradient, the profiles of both components have good agreement with the reference solution for all mass ratios. The particle flux of A component is in the negative direction due to the increasing of concentration and the particle flux of B component is in the positive direction due to the decreasing of concentration. In the profile of M p, M T, and M χ, the values of A, B components have the largest difference at the free molecular limit, then the difference reduces as the Knudsen number decreases and finally becomes zero at the continuum limit. This can be explained from the fact that M A,B p,t UA,B as shown in Eq.(3.33). In the high Knudsen number case, U A and U B may have large difference due the insufficient collisions. In the continuum limit, U A and U B tend to have the same value due to intensive collisions. 3.4 Conclusion In this chapter, a brief introduction of the Boltzmann equation for multi-component gas is given and the model equations for multi-component gas are briefly reviewed and compared. Then a unified gas-kinetic scheme for multi-component gas based on a single-bgk-operator model the AAP model is constructed. In order to validate current scheme, simulations are performed for the shock structure problems under different Mach numbers and component concentrations, and the micro-channel flow driven by small pressure gradient, temperature gradient, and con- 53

68 centration gradient under different molecular mass ratios. Good agreement with the reference solution is obtained at moderate Knudsen numbers and mass ratios. There are some deviations in the temperature profile of shock structure at high Mach numbers, and in the particle fluxes due to the temperature gradient in micro-channels, especially the light component. It may be caused by the transport coefficients, which are not fully accurate from the current kinetic model, except in case of Maxwell molecules [57]. And the light component is more sensitive to it. However, theoretically different kinetic models, other than the current AAP, can be employed in the UGKS as well. Future improvements may include the implicit evaluation of the source term in macroscopic equations, the enforcement of the conservation of macroscopic equations for mixtures properties, and the recovering of accurate transport coefficients, such as the adoption of the ellipsoidal models [64, 63]. 54

69 Chapter 4 Sound-wave Propagation in Monatomic Gas 4.1 Introduction In continuum flow regime, the sound propagation in gas can be described by the Navier-Stokes equations. However, as Knudsen number increases to the transition regime, the sound wave parameters, i.e., phase speed and attenuation coefficient, deviate from the classical prediction. Most existing hydrodynamic equations fail to describe the ultrasound propagation since the period of the sound wave propagation becomes comparable with the particle collision time. In order to investigate the high frequency sound wave propagation, many researchers turned attention to the kinetic equations by means of theories based on the expansion of Boltzmann equation. Wang Chang and Uhlenbeck [70] utilized the Super-Burnett equations, which were then extended by Pekeris et al. [71] up to 483 moments. However, the success of these theories cannot be extended to high Knudsen number flow regime. A remarkable success that performs well for a wide range of Knudsen numbers is the work of Sirovich and Thurber [72], and also Buckner and Ferziger [73]. Sirvoich and Thurber used Gross-Jackson model and analyzed the dispersion relation, where Buckner and Ferziger solved the half-space problem by means of elementary solutions, with diffusely-reflecting boundary. Besides the Gross-Jackson model, another popular kinetic model used for the study of sound wave is the BGK model. Thomas and Siewert [74] and Loyalka and Cheng [75] adopted the BGK model and solved the problem in half-space together with diffusely-reflecting boundary. Their results agreed with each other. Another successful method in simulating ultrasound wave propagation is the DSMC method [76]. In this thesis, the simulation is performed by UGKS and the solutions are compared with experimental results of Greenspan [77] 55

70 transducer (diffusive) U = U 0 cos(ωt),t = T 0 ρ 0,T 0 receiver (specular) L Figure 4.1: Sound-wave propagation: schematic of the problem. and Meyer and Sessler [78], and the DSMC results of Hadjiconstantinou and Garcia [76]. Figure 4.1 shows the schematic of the simulation geometry. The monatomic gas initially at rest with ρ 0, T 0 is enclosed between two solid surfaces separated by a distance L. The left surface is the transducer which is imposed by a periodical velocity U(t) = U 0 cos ωt, and the particles are diffusely reflected from the surface. On the other hand, the right surface is a stationary receiver and the particles are specularly reflected, which leads to total reflection of the propagating waves. The flow field is assumed to be one-dimensional. There are two relevant Knudsen numbers for this problem, one is defined as the ratio of mean free path λ to the domain length L, and the other is the ratio of wave frequency ω to particle collision frequency 1/τ, Kn L = λ L, Kn ω = ωτ, (4.1) where λ is the particle mean free path, L is the domain length, ω is the angular frequency of wave. 4.2 Methodology Boundary Condition at the Transducer In our simulation, the boundary treatment at the transducer is different from the Maxwellian reservoir method used by Hadjiconstantinou and Garcia in their DSMC simulation [76]. Following Loyalka and Cheng and others [79, 75], the Maxwellian 56

71 distribution at the transducer is ( ) m 3/2 g w = ρ w exp( m ) ((u U 0 cos ωt) 2 + v 2 + w 2 ), 2πk B T 0 2k B T 0 u > 0, (4.2) The density ρ w at the transducer is determined by t n+1 t n+1 (u U 0 cos ωt)g w dudt+ (u U 0 cos ωt) f in dudt = 0, (4.3) t n u>0 t n where f in is the distribution function of particles impinging on the transducer. The fluxes are then F = S t n+1 u 0 u(g w H[u]+ f in (1 H[u]))dt, F = t n ψf du. (4.4) The effect of different boundary treatment on the solution will be discussed later Method for Determining Sound Parameters In the experiment, one measures the pressure signal at the receiver and assumes that the pressure is a single damped wave of the form p(x, t) = A exp(i(ωt kx+ ϕ) αx), (4.5) where A is amplitude, k is wave number, α is attenuation coefficient, and ϕ is phase shift. Linear fits are performed in logarithm plots of amplitude and phase over a range of distances between the transducer and receiver in order to determine the parameters in the above equation. In our simulation, the flow variables in the whole domain is obtained in each computation, which enables us to extract the sound parameters without changing the domain length. Similar to the method used by Hadjiconstantinou and Garcia [76], the wave can be expressed as U = A(x) cos ωt+b(x) sin ωt, (4.6) and a least-square method is used to extract A(x) and B(x) from the numerical solution after the initial transients (approximately after 60 periods), which are given by A(x j ) = i sin 2 ωt i i U(x j, t i ) cos ωt i i sin ωt i cos ωt i i U(x j, t i ) sin ωt i i cos 2 ωt i i sin 2 ωt i ( i sin ωt i cos ωt i ) 2, B(x j ) = i cos 2 ωt i i U(x j, t i ) sin ωt i i sin ωt i cos ωt i i U(x j, t i ) cos ωt i i cos 2 ωt i i sin 2 ωt i ( i sin ωt i cos ωt i ) 2, (4.7) 57

72 with x j being the x coordinate of cell center and t i being the time to do the sampling. Then the amplitude can be calculated by A(x) 2 + B(x) 2. If we further assume the wave propagating in the positive direction as U m exp(i(ωt kx+ ϕ) αx), (4.8) and the reflected wave as U m exp(i(ωt+k(x 2L)+ ϕ)+α(x 2L)), (4.9) the superposition leads to U =U m exp( αx) cos(ωt kx+ ϕ) U m exp(α(x 2L)) cos(ωt+k(x 2L)+ ϕ). (4.10) Combining Eq.4.6 with Eq.4.10 gives, A(x) = U m exp( αx) cos(kx ϕ) U m exp(α(x 2L)) cos(k(x 2L)+ ϕ), (4.11) and B(x) = U m exp( αx) sin(kx ϕ)+u m exp(α(x 2L)) sin(k(x 2L)+ ϕ). (4.12) In the above formulas, U m, ϕ, α, k are unknowns, and are obtained by parameter estimation using the Nelder-Mead simplex method, which is available in most mathematical softwares. The formula used by Hadjiconstantinou and Garcia is a little different from Eq They simplified the expression under the condition L = (7/4)l, where l is the wavelength. In low frequencies, the estimation of wave parameters are based on the amplitude A(x) 2 + B(x) 2 = U m 4e 2Lα (cosh(2α(x L)) cos(2k(l x))). (4.13) However, in high frequencies, the reflected wave is very weak in comparison with incoming one and the amplitude approximately takes the following form, U m exp( αx). (4.14) The information of phase speed is lost in the expression of the above amplitude, and the estimation of wave parameters for high frequency wave is directly based on both A(x) and B(x) in Eq.4.11 and Eq According to the analysis in [80], the wave 58

73 is composed of several modes in low frequencies, instead of one mode described by Eq Within all transport modes, the so-called acoustic mode dominates the transport, and the other modes get damped quickly. By excluding the region near to the transducer, the formula fitted result using Eq.4.13 is actually the acoustic mode in low frequencies (shown in next section). In high frequencies, however, the sound parameters show an increasing dependence on the location, which was observed in other numerical computations [76, 75, 74] and was analyzed in free-molecular limit [81]. Under this condition, Eq.4.5 and Eq.4.10 are not applicable for high-frequency waves in the whole domain. Since all experimental measurements do not include the information about the region where the sound parameters are measured, we determine the numerical region for estimating wave parameters by gradually excluding the region near the transducer until the best fit for the rest of the domain is obtained. In latter sections, A(x), B(x), and the amplitude calculated directly from the sampling by Eq.(4.7) are referred as the simulated results, A(x), B(x), and the amplitude obtained by Eq.(4.11), Eq.(4.12), and Eq.(4.13) after the parameter estimation are referred as the formula fitted results. 4.3 Numerical Results The simulation is performed with Shakhov model and the gas is assumed to be hard sphere monatomic. The results are presented in non-dimensional form, ˆt = t τ 0, û = where L = µ 0 (k B T 0 /m) 1/2 /p 0. u x, ˆx =, (4.15) (k B T 0 /m) 1/2 L For hard-sphere molecule, the particle mean free path is λ 0 = ( 2/π)(8/5)L, and the two relevant Knudsen numbers are Kn L = λ 0 L = π 5 ˆL, Kn ω = ωτ 0 = ˆω. (4.16) The dimensionless form of classical sound speed at rest state is Ĉ 0 = C 0 /(k B T 0 /m) 1/2 = γ. For monatomic gases, Ĉ0 = 5/3. In latter sections, the hat will be dropped for simplicity. 59

74 Table 4.1: Sound wave propagation: frequencies and domain lengths. ω L ω L We have performed simulation for a wide range of frequencies from ω = 0.08 to ω = 32. Based on Eq.4.16, the corresponding Knudsen number Kn ω ranges from 0.08 to 32. The domain length L is chosen to be no more than a few wave lengths, which is approximately L (7/4)l. The wave frequencies and domain lengths are listed in Table 4.1. Based on Eq.4.16, the Knudsen number Kn L changes from to In order to avoid nonlinear effect, such as shock formation in the wave propagation, the requirement ρ 0 U 0 C 0 ωµ 0 1, (4.17) should be met. The requirement can be written as U 0 ω/ γ. For frequencies ω 0.25, we use U 0 = 0.005; for 0.4 ω 2.5, we use U 0 = 0.01; and for even higher frequencies, we use U 0 = To capture the wave profile accurately, we use 140 cells in most cases, which is approximately 80 cells per wavelength. For ω = 8.0, 70 cells are used. For extremely high frequencies ω = 16.0, 32.0, only 35 cells are used. requirement, The time step is determined by the CFL condition, and it also satisfies the t < 1 (2π/ω), (4.18) 60 in order to accurately capture the time evolution of the wave profile. The time to start sampling is determined by setting a monitor point, where the velocities at each moment of integer period, i.e., at time t = N(2π/ω) with N = 1, 2, 3,..., are recorded. When the changing of velocities becomes substantially small, we start the sampling. Figure 4.2 shows the velocity change at the monitor point for ω = 0.1. To extract A(x) and B(x), we start sampling from 100 periods to 110 periods, with 100 time samples in each period. Figure 4.3 shows A(x) and B(x) for ω =

75 ω = Velocity Period Figure 4.2: Sound wave propagation: velocity change on every integer period at the monitor point for frequency ω = Behavior at Low Frequencies In low frequencies, the estimation of wave parameters is based on the amplitude definition in Eq Figure 4.4 shows the amplitude obtained by direct sampling from Eq.(4.7) (simulated results), and the amplitude obtained by Eq.(4.11) and Eq.(4.12) after parameter estimation (formula fitted results) at ω = 0.1 from the UGKS simulation. In order to validate our result, the re-scaled analytic solution of Regularized 13-moments (R13) equation [80] is also included for comparison. To have a real comparison with the R13 result, the same isothermal wall boundary condition as R13 is used in our simulation. Since ω = 0.1 is a relative low frequency, the R13 result should be reliable even though it does not work properly for high frequency wave. The above comparison confirms that by excluding the region near to the transducer, the fitted amplitude using Eq.4.13 recovers the acoustic mode Behavior at High Frequencies Figure 4.5 shows A(x) and B(x) for Re = γ/ω = 0.5 (ω 3.3) using the UGKS and the DSMC data, with the same boundary condition (Maxwellian reservoir method). 61

76 A(x) B(x) A(x),B(x) X Figure 4.3: Sound wave propagation: A(x) and B(x) for frequency ω = ω = 0.1 Velocity Amplitude R13 R13-acoustic mode UGKS-simulation UGKS-fitting X Figure 4.4: Sound wave propagation: simulated amplitude (from direct sampling) and formula fitted amplitude (from the assumed wave form after parameter estimation) for UGKS results and the theoretical solutions of R13 [80] at ω =

77 A(x)-DSMC B(x)-DSMC A(x)-UGKS B(x)-UGKS A(x),B(x) X Figure 4.5: Sound wave propagation: comparison of UGKS and DSMC results using the same boundary condition at Re = γ/ω = 0.5 (ω 3.3). The results are almost the same from two different methods, and the estimated wave parameters only have a slight difference from that obtained by the boundary treatment in Eq.4.2 and Eq.4.3. Since Kn ω = ω 3.3 is a pretty large Knudsen number, the perfect match between UGKS and DSMC solution confirms the accuracy of the UGKS method in capturing non-equilibrium flow. In high frequencies, the estimation of wave parameters are based on both A(x) and B(x) presented in Eq.4.11 and Eq.4.12, and the region used for fitting is determined by gradually excluding the region near the transducer until the best fit for the rest domain is obtained. Figure 4.6 shows one of the fitted result of A(x), B(x) at ω = 5 by matching the numerical solutions with the analytical ones in Eq.4.11 and Eq.4.12, and the corresponding velocity amplitude. The location dependent behavior of phase speed and attenuation coefficient in high frequencies can be observed by changing the region used for fitting. Figure 4.7 shows the fitted phase speed starting from different locations x min for ω = 5. The fitted result in Figure 4.6 is based on the sampling point starting from x min In high frequency case, the point-wise definition of wave parameters used by Schotter [82], Garcia and Siewert [79], and Sharipov [81], may be another choice for their evaluation. 63

78 Figure 4.6: Sound wave propagation: simulated results (from direct sampling) and formula fitted results (from the assumed wave form after parameter estimation) at ω = Phase Speed and Attenuation Coefficient The extracted phase speed and attenuation coefficient are listed in Table 4.2. The comparison among the experiments, DSMC [76], Navier-Stokes solutions, and the UGKS results are presented in Figure 4.8 and Figure 4.9, respectively. It is obvious that the UGKS results have a good agreement with the experimental data. Although the boundary treatment and numerical method used in our simulation are different from that in DSMC calculation, the main difference in the UGKS and the DSMC results does not come from them, but the region used for fitting the numerical solution by the formula. The DSMC simulation fixed the starting location for fitting at x min = λ or x min = 0.5λ. If the same fitting location is used, it is expected to obtain similar results for both UGKS and DSMC methods. The UGKS results are also compared with the original experimental data presented by Greenspan [77] and Meyer [78] in Figure 4.10 and Figure It can be seen that the current results have good agreement with the experimental data in general, especially in the continuum regime and free-molecular regime. In regime 2 ω 4, the phase speed have a slightly overshot. This overshot is also observed in other computations [73, 76, 75, 74]. 64

79 k/k X min Figure 4.7: Sound wave propagation: location dependent phase speed for frequency ω = 5. Table 4.2: Sound wave propagation: wave speed and attenuation coefficient at different frequencies. ω k 6.160E E E E E-01 α 3.310E E E E E-02 ω k 5.929E E E E E+00 α 2.287E E E E E-01 ω k 2.028E E E E+01 α 1.109E E E E+00 65

80 k/ k Re= γ/ ω Navier-Stokes DSMC(x min = λ) DSMC(x min = 0.5λ) Experiment(Greenspan) Experiment(Meyer) UGKS Figure 4.8: Sound wave propagation: phase speed comparison at different frequencies among the results from UGKS, DSMC [76], experimental measurements, and Navier-Stokes equations / k Navier-Stokes DSMC(x min = λ) DSMC(x min = 0.5λ) Experiment(Greenspan) Experiment(Meyer) UGKS Re= γ/ ω Figure 4.9: Sound wave propagation: attenuation coefficient comparison at different frequencies among the results from UGKS, DSMC [76], experimental measurements, and Navier-Stokes equations. 66

81 10 0 (a). Phase speed k/ k 0 Navier-Stokes Experiment(Greenspan) UGKS r = 1/ ω (b). Attenuation coeff cient / k Navier-Stokes Experiment(Greenspan) UGKS r = 1/ ω Figure 4.10: Sound wave propagation: wave speed and attenuation coefficient comparison between UGKS and experimental data of Greenspan [77]. 67

82 (a). Phase speed k/ k Navier-Stokes Experiment(Meyer) UGKS r = 1/ ω (b). Attenuation coeff cient / k Navier-Stokes Experiment(Meyer) UGKS r = 1/ ω Figure 4.11: Sound wave propagation: wave speed and attenuation coefficient comparison between UGKS and experimental data of Meyer [78]. 68

83 4.4 Conclusion In this chapter, the sound wave propagation in monatomic gases is simulated with hard-sphere molecule for the whole Knudsen regime, and the phase speed and attenuation coefficient are obtained. The good agreement between the UGKS results and the experimental data is another validation of the UGKS method in capturing the physical solutions for non-equilibrium flows. There are several differences between the UGKS and the DSMC method for the sound wave simulation. First, the boundary treatment of the transducer in our simulation is different with the Maxwellian reservoir method used by the DSMC method [76]. However, as shown in this paper, different boundary treatments only have marginal effect on the evaluation of phase speed and attenuation coefficient. Second, the hardsphere molecule is used in our simulation, which corresponds to µ T. We can easily extend the UGKS method to simulate any viscosity law, such as the Sutherland s law. For the DSMC method, it is not straightforward to incorporate any general viscosity laws, except the hard-sphere and variable hard-sphere models. By contrast, the viscosity coefficient in UGKS can be directly implemented through the determination of the local particle collision time. The wave propagation in high frequencies/high Knudsen number flow is quite different from the classical sound wave propagating results. With the dependence of phase speed and attenuation coefficient on the location for their evaluation, the wave behavior deviates from its classical form, and cannot be described by Eq.4.5 and Eq A point-wise definition [79, 82, 81] of phase speed and attenuation coefficient may be another choice for their evaluation. 69

84 Chapter 5 Cross-coupling of Mass and Heat Transfer 5.1 Introduction Onsager s reciprocal relations for linear irreversible processes [83, 84] play an important role in the theory of non-equilibrium thermodynamics. In a thermodynamic process, the entropy production rate can be expressed by N ds dt = J i X i, (5.1) i=1 where S is entropy, J i are thermodynamic fluxes, and X i are the conjugate thermodynamic forces. For small deviation away from the equilibrium, we have the linear response expressed by J i = N L ij X j, (5.2) j=1 where L ij are the kinetic coefficients. Onsager s reciprocal relation states that as a consequence of microscopic reversibility. L ij = L ji, (5.3) Starting from the Gibbs equation, the thermodynamics fluxes and forces can be identified for gas flows, and constitutive equations can be derived. In multi-component gas, the Onsager reciprocal relation shows up in the diffusion coefficients between two components and the coupling of diffusion-thermal and thermal-diffusion coefficient for each component. The cross effects of thermal-diffusion and diffusion-thermal are also called the Soret effect and Dufour effect [56]. In single-component gas, the Onsager reciprocal relation shows up in the cross coupling of thermal-osmosis and 70

85 mechano-caloric effect of rarefied gas in system not far away from equilibrium. This provides an interesting case and was studied by various authors. Groot and Mazur [85] as well as Waldmann [86] studied the coupling phenomena in parallel planner channel in both free molecular and slip regime (0.001 Kn 0.1). Loyalka [87, 88] and Sharipov [89] analyzed the cross effect by means of the linearized Boltzmann method and obtained the coupling coefficients numerically. Although the theoretical analysis of Loyalka [87] is valid for capillary of arbitrary shape, most of the work, especially the numerical calculations [88, 52], were devoted to capillary of planner surfaces or circular cross sections. The thermal-osmosis effect attracts much more attention since it can be used to design pumping devices without any moving part, i.e. the Knudsen pump [17]. Despite the proposed Knudsen pump [17, 90], capillary with ratchet surfaces has the potential for another possible configuration [91]. The driving mechanism of such system has been analyzed by Wüger [91] as well as Hardt et al. [92], and the mass and momentum transfer is studied by Donkov et al. [93]. In this chapter, we will study the cross coupling phenomena for a long capillary of both planner and ratchet surfaces by unified gas-kinetic scheme. The mechanism of cross coupling for both cases are presented. The coupling coefficients for planner surfaces are numerically calculated and compared with the literature. And the coupling coefficients for ratchet surfaces are presented and analyzed. A preliminary geometry optimization for use as Knudsen pump is also included. 5.2 Mechanism Micro-channel of Planner Surfaces The schematic of cross coupling in channel of planner surfaces is shown in Figure 5.1. A long channel is confined by two parallel solid plates of distance H and connected to two reservoirs. The left reservoir is maintained at pressure p 0 p/2 and temperature T 0 T/2. The right reservoir is maintained at p 0 + p/2 and T 0 + T/2. p < 0 and T > 0 are used in the simulation, where p/p 0 1 and T/T 0 1 to ensure linear response. Typically, a mass flux to the right is generated by the pressure gradient and a heat flux to the left is generated by the temperature gradient. For 71

86 p 0 p/2 T 0 T/2 H heat flow mass flow p 0 + p/2 T 0 + T/2 ( p < 0) ( T > 0) L Figure 5.1: Cross coupling in channel of planner surfaces: schematic of the problem. rarefied gas, p also contributes to the heat flux and T contributes to the mass flux. These cross-coupling effects are called mechano-caloric effect and thermo-osmosis effect. For single-component gas, the entropy production can be expressed by [86] ( ) ds 1 ( dt = J E + J M ν ), (5.4) T T where ν is the chemical potential per unit mass, J E and J M are the energy flux and mass flux from the left reservoir to right reservoir, and means the quantity at the right reservoir minus the quantity at the left reservoir. Here ν and J E can be written as ν = h Ts, (5.5) J E = J Q + hj M, (5.6) where s and h are the entropy and enthalpy per unit mass and J Q is the heat flux. Together with the Gibbs-Duhem equation dν = sdt+dp/ρ, (5.7) Eq.(5.4) becomes ds dt = 1 ρt J Mdp 1 T 2 J QdT. (5.8) From Eq.(5.8), the thermodynamic forces and fluxes are connected as J M = L MM L MQ T 1 0 ρ0 1 p, (5.9) J Q L QM L QQ T0 2 T and L MQ = L QM, (5.10) 72

87 due to Onsager reciprocal relation. Note that J Q is measured at the outlet for a long channel [94]. The detailed mechanism may vary under different configurations and rarefactions. For free molecular regime and specular reflection on plates, the molecules travel ballistically from on side to the other. And the distribution function at any point can be treated as the combination of two half-space Maxwellians from the two reservoirs. The kinetic coefficients can be calculated analytically in such case, as given by Waldmann [86], L MM L QM L MQ L QQ = 1 4 Hρ 0T 0 8kB T 0 mπ ρ 0/p 0 1/2. (5.11) 1/2 9p 0 /4ρ 0 If the temperature gradient is imposed on the plates and particles are diffusive reflected, the mass flux due to temperature gradient is generated by thermal creep on the plates [85, 17, 5]. The kinetic coefficients for such system is calculated by several authors with different methods [52]. Assuming the length to height ratio of the channel is fixed and noting that ρλ = constant, µ T 1/2 for hard-sphere molecule, the average velocity induced by thermal creep can be estimated from the Maxwell slip boundary condition [5], U µ 0 ρ 0 T 0 T T T0 Kn, (5.12) where Kn = λ 0 /H. In later sections, we will show that L MQ = L QM holds for this case and is an increasing function of Kn Micro-channel of Ratchet Surfaces The mechanism in channel of ratchet surfaces is a little complex. Consider a long channel consists of repeating structure similar to the reference [93] as shown in Figure 5.2, where the two ends are connected to two reservoirs with p 0 p/2 ( p < 0), T 0 and p 0 + p/2, T 0 respectively. The upper wall (solid line colored in blue) are diffusive and maintained at T 0 + T/2( T > 0), the lower inclined walls (solid lines colored in blue) are diffusive and maintained at T 0 T/2. And the lower horizontal and vertical walls (dot-dashed lines colored in green) are specular. Typically, a mass flux pointing to the right is generated by the pressure gradient and a heat flux pointing to the bottom is generated by the temperature gradient. In rarefied gas, p also 73

88 p 0 p/2 T 0 H T 0 + T/2 ( T > 0) mass p 0 + p/2 T 0 ( p < 0) H 0 heat α H 1 T 0 T/2 L Figure 5.2: Cross coupling in channel of ratchet surfaces: schematic of the problem. The upper wall and the lower inclined walls (solid lines colored in blue) are diffusive. The lower horizontal and vertical walls (dot-dashed lines colored in green) are specular. contributes to the heat flux and T contributes to the mass flux, where p /p 0 1 and T /T 0 1. The thermodynamic forces and fluxes are connected as J M = L MM L MQ T 1 0 ρ0 1 p, (5.13) J Q L QM L QQ T0 2 T where J Q is the normal heat flux at the solid surfaces. And due to Onsager reciprocal relation. L MQ = L QM, (5.14) Since the upper wall and the lower inclined walls are maintained at different temperatures, the isothermal lines near the tip (indicated by a dashed circle) are sharply curved and thermal-edge flow is induced at the tip from top to bottom [17, 92]. The non-parallel isothermal lines along the inclined walls also induce a thermal-stress slip flow in the direction opposite to the thermal-edge flow, but it is not significant if the Knudsen number is small [17, 92]. It s possible to have a rough estimation of the induced velocity at the tip [17, 91]. If all the walls are assumed diffusive, the temperature gradient along the inclined wall near the tip is approximated by [91], π T = 2 ( α ) H π/(2π α) T (2π α) 2 cos, (5.15) 2 λ 1 π/(2π α) 0 where λ 0 is the mean free path at p 0, T 0. Recalling ρλ = constant and µ T 1/2 for hard-sphere molecule, the induced velocity in slip regime can be estimated from the 74

89 formula for slip boundary condition [5], U µ 0 T ρ 0 T T 1 ( α ) ( ) 0 T0 (2π α) 2 cos λ π/(2π α) 0. (5.16) 2 H Since 0 α 2π and Kn = λ 0 /H < 1 in the slip regime, the induced velocity is 1. Proportional to T. 2. Decreasing function of T Decreasing function of α, since smaller α means sharper edges. 4. Increasing function of Kn, due to the stronger non-equilibrium effect. However, U will decrease if Knudsen number excess certain values since the thermalinduced flows are typically strongest in the lower transition regime [17]. Under current configurations, the average velocity can be expressed by where C 1, C 2 are some constants for a specific geometry. U C 1 T T0 Kn C 2, (5.17) 5.3 Methodology Normalization The kinetic coefficients are calculated and presented in non-dimensional form as follows ( ) 2kB ˆL MM = L MM, mρ 0 C 0 H ( ) 2k ˆL MQ = B L MQ mρ 0 C0 3H, ( ) 2k ˆL QM = B L QM mρ 0 C0 3H, ( ) 2k ˆL QQ = B L QQ mρ 0 C0 5H, (5.18) where ρ 0, C 0 = (2k B T 0 /m) 1/2 are the reference state and H is the height of the channel to define Knudsen number Kn = λ 0 /H. λ 0 is the mean free path at reference state. 75

90 Since the density variation is small in the simulation, the mass flux can be expressed by Further assume that there is no pressure difference, then J M ρ 0 UH. (5.19) ( mρ0 C0 ρ 0 UH = ˆL 3H ) MQ 2k B T T0 2, (5.20) or U ˆL MQ T T0. (5.21) Comparing Eq.(5.21) with Eq.(5.12) and Eq.(5.17), ˆL MQ is assumed to have the form ˆL MQ = C 1 Kn C 2, (5.22) where C 1, C 2 are some constants for a specific geometry and can be obtained by data fitting of the simulated solution Data processing Suppose a system has n kinds of driven forces, which generates n kinds of fluxes. The forces and fluxes are related through J 1 L L 1n F 1... = or J = LF, (5.23) L n1... L nn F n J n where J and F are directly extracted from the simulation data, and the coefficients L are unknowns to solve. For a single simulation, there are n equations and n n unknowns, so n sets of different simulation data are needed to solve L, L = provided that the matrix of F is invertible. [J 1... J n ][ F 1... F n ] 1, (5.24) For the cross coupling considered here, this requires the simulations to be performed twice with different p and T for a single system (same geometry and Knudsen number) to determine all the kinetic coefficients. 76

91 Figure 5.3: Cross coupling in channel of planner surfaces: ˆL MQ and ˆL QM versus Knudsen number. The reference is the S-model solution based on variational method by Chernyak et al. [95]. 5.4 Numerical Results Cross-coupling for Planner Surfaces Before proceeding to the ratchet surfaces, the kinetic coefficients are calculated for channel with planner surfaces and compared with those in literature. The schematic of the simulation geometry is shown in Figure 5.1. The solid surfaces are assumed to be diffusive and have linearly distributed temperature from T 0 T/2 to T 0 + T/2. The gas is assumed to be hard-sphere and monatomic so that Pr = 2/3, µ T 0.5. The Knudsen number is defined as Kn = λ 0 /H and the mean free path is defined by Eq.(2.21). The simulation is performed with Shakhov model. p and T are kept small enough so that the response of fluxes to forces is linear and the length/height ratio of the channel is taken to be 20 in order to reduce the influence of inlet and outlet. When extracting the coefficients, the pressure and temperature difference are measured from 77

92 Figure 5.4: Cross coupling in channel of ratchet surfaces: schematic of the simulation geometry. the inlet and outlet, the mass flux J M is averaged over the cross section at the inlet and outlet, and the heat flux J Q is measured along the cross section in the middle of the channel. Figure 5.3 shows the normalized off-diagonal coefficients ˆL MQ, ˆL QM versus Knudsen numbers. The coefficients are quite close to each other with a maximum difference of 5%, and also have good agreement with the S-model solution based on variational method by Chernyak et al. [95]. The normalized coefficients are zero at Kn = 0 since there is no thermal-induced flow in the continuum limit and the heat flux follows Fourier s law. Then the normalized coefficients increases as the Knudsen number becomes larger. The profiles are almost linear which means ˆL MQ = ˆL QM log(kn). And this agrees with the conclusion obtained from linearized Boltzmann equation for two-dimensional infinitely long channel [7, 52] Cross-coupling for Ratchet Surfaces In the simulation, the channel is consisted by seven repeating blocks as shown in Figure 5.2 where each one has L/H 0 = 1, H = H 1, and α = 45. And two parallel sections with specular walls of length L are attached at two ends. The channel is then connected to two reservoirs. The schematic of the whole system is shown in Figure 5.4. The parallel specular wall sections are introduced to reduce the influence of inlet/outlet to the ratchet sections and also simplifies the extraction of mass flux. The gas is still assumed to be hard-sphere monatomic so that Pr = 2/3 and µ T 0.5. The Knudsen number is defined as Kn = λ 0 /H and the mean free path is defined by Eq.(2.21). When extracting the coefficients, the pressure difference is measured from the inlet and outlet, the mass flux J M is averaged over the cross sections at the inlet and outlet, and J Q is integrated along all the inclined walls. 78

93 Figure 5.5: Cross coupling in channel of ratchet surfaces: ˆL MQ and ˆL QM versus Knudsen number. Figure 5.5 shows the normalized off-diagonal coefficients ˆL MQ, ˆL QM versus Knudsen number. It can be seen that the two coefficients are quite close to each other with a maximum difference of 2%, which indicates the relation L MQ = L QM is well satisfied. The off-diagonal coefficients are zero at Kn = 0 since there is no thermal-induced flow in continuum limit and the heat flux follows the Fourier s law. As the Knudsen number increases, the rarefied effects occurs at the sharp edge of the ratchet and leads to an increasing of the normalized off-diagonal coefficients. Both ˆL MQ and ˆL QM exhibits a maximum at Kn 0.28, which means the maximum average velocity is achieved if T and T 0 are the same according to Eq.(5.21). This is consistent with the observations in reference [93] for a similar ratchet geometry with periodic boundary condition. The cross coupling effect arises from the thermal induced flow on the rough surface, especially the sharp edge. As the Knudsen number becomes higher than a certain value, the roughness can hardly be seen by the particles, thus the coefficient gradually decreases. Now we perform the data fitting of the simulated cross coupling coefficients by Eq.(5.22). Since the formula is valid in the slip regime, two additional simulations 79

94 Figure 5.6: Cross coupling in channel of ratchet surfaces: formula fitted and simulated ˆL MQ. are performed to obtain ˆL MQ at Kn = 0.02 and Kn = 0.04, and only the first four points are taken into consideration. The fitted coefficient is then ˆL MQ = Kn (5.25) Inserting α = π/4 into Eq.(5.16), one finds Ū Kn (5.26) This is very close to the fitted parameter in Eq.(5.25). Figure 5.6 shows the fitted coefficients and the simulated coefficients as a function of Knudsen number. 5.5 Knudsen pump Wüger [91] and Donkov et al. [93] proposed the capillary with ratchet surfaces as another possible configuration for Knudsen pump. In this section, a preliminary optimization is provided for this purpose. For simplicity and accuracy, only one block as shown in Figure 5.2 is used and the inlet/outlet is replaced by periodic boundary 80

95 (a) Temperature contour and streamlines of diffusive configuration where α = 0. (b) Temperature contour and streamlines of a typical diffusive-specular configuration. Figure 5.7: Cross coupling in channel of ratchet surfaces: temperature contour and streamlines of typical diffusive configuration and diffusive-specular configuration. condition. The upper wall is still diffusive, but the lower walls have two different configurations: All the lower walls are diffusive (will be referred as diffusive configuration). The vertical and horizontal lower walls are diffusive, and the inclined wall is specular (will be referred as diffusive-specular configuration). This configuration is the same as reference [93]. Figure 5.8 shows the ˆL MQ as a function of tan α for L/H = 2, H 1 /H = 1, and Kn = The left figure is for diffusive configuration and the right figure is for diffusive-specular configuration. In the diffusive configuration, thermal-edge flow arises on both sides of the sharp edge and diminishes each other, thus the induced velocity is reduced [92]. If α = 0, there will be zero net mass flow in the x direction, and two identical vortices are formed between the needles as shown in Figure 5.7. The optimized value of α occurs around tan α In the diffusive-specular configuration, thermal-edge flow occurs only on the diffusive surfaces, thus the induced velocity is much higher than that in the diffusive configuration. The temperature contour and streamlines of a typical diffusive-specular configuration is shown in Figure 5.7, and is very close to the result in reference [93]. And ˆL MQ is an increasing function of tan α in a diffusive-specular configuration. 81