The superscript (B) denotes third-order stress and heat-flux terms in the BGK–Burnett equations. The θ i are given as follows for γ ϭ 1.4: θ 1 ϭ 2.56, θ 2 ϭ 1.36, θ 3 ϭ 0.56, θ 4 ϭ Ϫ0.64, θ 5 ϭ 0.96, θ 6 ϭ 1.6, θ 7 ϭ Ϫ0.4, θ 8 ϭ Ϫ0.24, θ 9 ϭ 1.024, θ 10 ϭ Ϫ0.256, θ 11 ϭ 1.152, θ 12 ϭ 0.16, θ 13 ϭ 2.24, θ 14 ϭ Ϫ0.56, θ 15 ϭ 3.6, θ 16 ϭ 0.6, θ 17 ϭ 1.4, θ 18 ϭ 4.9, θ 19 ϭ 7.04, θ 20 ϭ Ϫ0.16, θ 21 ϭ Ϫ1.76, θ 22 ϭ 4.24, θ 23 ϭ 3.8, and θ 24 ϭ 3.4. Finally, governing Equation (8.1) is nondimensionalized by a reference length and freestream variables and is written in a curvilinear coordinate system ( ξ , η ) by employing a coordinate transformation: τ ϭ t, ξ ϭ ξ (x, y), η ϭ η (x, y) (8.26) 8.4 Wall-Boundary Conditions The no-slip-/no-temperature-jump boundary conditions are employed at the wall when solving the contin- uum Navier–Stokes equations for Kn Ͻ 0.001. In the continuum–transition regimes, the non-slip- boundary conditions are no longer correct. First-order slip/temperature-jump boundary conditions should be applied to both the Navier–Stokes equations and Burnett equations in the range 0.001 Ͻ Kn Ͻ 0.1.The transition regime spans the range 0.01 Ͻ Kn Ͻ 10; the second-order slip/temperature- jump conditions should be used in this regime with the Navier–Stokes as well as the Burnett equations. The Navier–Stokes equations are first-order accurate in Kn, while the Burnett equations are second-order accurate in Kn. Both first- and second-order Maxwell–Smoluchowski slip/temperature-jump boundary conditions are generally employed on the body surface when solving the Burnett equations. The first-order Maxwell–Smoluchowski slip-boundary conditions in Cartesian coordinates are [Smoluchowski, 1898]: U s ϭ Ί s ϩ s (8.27) and T s Ϫ T w ϭ Ί s (8.28) The subscript s denotes the flow variables on the solid surface of the body. First-order Maxwell– Smoluchowski slip-boundary conditions can be derived by considering the momentum and energy-flux balance on the wall surface. The reflection coefficient – σ and the accommodation coefficient α – are assumed to be equal to unity (for complete accommodation) in the calculations presented in this chapter. Beskok’s slip-boundary condition [Beskok et al., 1996] is the second-order extension of the Maxwell’s slip-velocity-boundary condition excluding the thermal creep terms, given as: U s ϭ ΄ s ΅ (8.29) where b is the slip coefficient determined analytically in the slip flow regime and empirically in transi- tional and free molecular regimes. Langmuir’s slip-boundary condition has also been employed in the literature [Myong, 1999]. Langmuir’s slip-boundary condition is based on the theory of adsorption phenomena at the solid wall. Gas molecules do not in general rebound elastically but condense on the surface, being held by the field of force of the surface atoms. These molecules may subsequently evaporate from the surface resulting in some time lag. Slip is the direct result of this time lag. The slip velocity at the wall is given as: U s ϭ (8.30) where β is the adsorption coefficient determined empirically or by theoretical prediction. 1 ᎏ 1 ϩ β p ∂U ᎏ ∂y Kn ᎏ 1 Ϫ bKn 2 Ϫ – σ ᎏ – σ ∂T ᎏ ∂y 1 ᎏ Pr π ᎏ 8RT 2 µ ᎏ ρ 2 γ ᎏ γ ϩ 1 2 Ϫ – α ᎏ – α ∂T ᎏ ∂x µ ᎏ ρ T 3 ᎏ 4 ∂U ᎏ ∂y π ᎏ 8RT 2 µ ᎏ ρ 2 Ϫ – σ ᎏ – σ Burnett Simulations of Flows in Microdevices 8-11 © 2006 by Taylor & Francis Group, LLC In this chapter, these slip boundary conditions are applied and compared to determine their influence on the solution. 8.5 Linearized Stability Analysis of Burnett Equations Bobylev (1982) showed that the conventional Burnett equations are not stable to small wavelength dis- turbances; hence, the solutions to conventional Burnett equations tend to diverge when the mesh size is made progressively finer. Balakrishnan and Agarwal (1999) performed the linearized stability of one-dimensional original Burnett equations, conventional Burnett equations, augmented Burnett equations, and the BGK–Burnett equations. They considered the response of a uniform gas subjected to small one-dimensional periodic perturbations ρ Ј, uЈ, and TЈfor density, velocity, and temperature respec- tively. Burnett equations were linearized by neglecting products and powers of small perturbations, and a linearized set of equations for small perturbation variables VЈ ϭ [ ρ Ј, uЈ, TЈ] T was obtained. They assumed that the solution is of the form: VЈ ϭ V – e i ω x e φ t (8.31) where φ ϭ α ϩ i β , and α and β denote the attenuation and dispersion coefficients respectively. For stability, α р 0 as the Knudsen number increases. Substitution of Equation (8.31) in the equations for small per- turbation quantities VЈ results in a characteristic equation, |F( φ , ω )| ϭ 0. The trajectory of the roots of this characteristic equation is plotted in a complex plane on which the real axis denotes the attenuation coefficient and the imaginary axis denotes the dispersion coefficient. For stability, the roots must lie to the left of the imaginary axis as the Knudsen number increases. Figures 8.2 to 8.5 show the trajectory of the three roots of the characteristic equations as the Knudsen number increases.The plots show that the Navier–Stokes equa- tions, the augmented Burnett equations, and the BGK–Burnett equations (with γ ϭ 1.667) are stable, but the conventional Burnett equations are unstable. Euler equations are employed to approximate the material derivatives in all three types of Burnett equations. The BGK–Burnett equations, however, become unstable for γ ϭ 1.4. On the other hand, if the material derivatives are approximated using the Navier–Stokes equa- tions, then the conventional, augmented, and BGK–Burnett equations are all stable to small wavelength disturbances. Based on these observations, we have employed the Navier–Stokes equations to approximate the material derivatives in the conventional, augmented, and BGK–Burnett equations presented in Section 8.3. For the detailed analysis behind Figures 8.2 to 8.5, see Balakrishnan and Agarwal (1999). The linearized stability 8-12 MEMS: Introduction and Fundamentals 0 1 −1 −0.5 0.5 Dispersion coefficient () 2 −3 −2 −1 0 1 Attenuation coefficient (␣) Stable ( ␣ ≤ 0 ) Unstable ( ␣ > 0 ) FIGURE 8.2 Characteristic trajectories of the one-dimensional Navier–Stokes equations. © 2006 by Taylor & Francis Group, LLC analysis of conventional, augmented, and super-Burnett equations has also been performed in three dimensions with similar conclusions [Yun and Agarwal, 2000]. 8.6 Numerical Method An explicit finite-difference scheme is employed to solve the governing equations of Section 8.3. The Steger–Warming flux-vector splitting method [Steger and Warming, 1981] is applied to the inviscid-flux terms. The second-order, central-differencing scheme is applied to discretize the stress tensor and heat- flux terms. Converged solutions were obtained with a reduction in residuals of six orders of magnitude. Burnett Simulations of Flows in Microdevices 8-13 0 5 −5 Dispersion coefficient () 10 −15 −10 −5 0 5 Attenuation coefficient ( ␣ ) Stable ( ␣ ≤ 0 ) Unstable ( ␣ > 0 ) FIGURE 8.3 Characteristic trajectories of the one-dimensional augmented Burnett equations ( γ ϭ 1.667); Euler equations are used to express the material derivatives D( )/Dt in terms of spatial derivatives. −40 −35 −30 −25 −20 −15 −5−10 0 −150 −100 −50 0 50 100 150 Attenuation coefficient (␣) Dispersion coefficient () FIGURE 8.4 Characteristic trajectories of the one-dimensional BGK–Burnett equations ( γ ϭ 1.667); Euler equa- tions are used to express the material derivatives D( )/Dt in terms of spatial derivatives. © 2006 by Taylor & Francis Group, LLC All the calculations were performed on a sequence of successively refined grids to assure grid independ- ence of the solutions. 8.7 Numerical Simulations Numerical simulations have been performed for both the hypersonic flows and microscale flows in the continuum–transition regime. Hypersonic flow calculations include one-dimensional shock structure, two-dimensional and axisymmetric blunt bodies, and a space shuttle re-entry condition. Microscale flows include the subsonic flow and supersonic flow in a microchannel. 8.7.1 Application to Hypersonic Shock Structure The hypersonic shock for argon was computed using the BGK–Burnett equations. The upstream flow conditions were specified and the downstream conditions were determined from the Rankine–Hugoniot relations. For purposes of comparison, the same flow conditions as in Fiscko and Chapman (1988) were used in the computations. The parameters used were T ∞ ϭ 300K, P ∞ ϭ 1.01323 ϫ 10 5 N/m 2 , γ argon ϭ 1.667, µ argon ϭ 22.7 ϫ 10 Ϫ6 kg/sec и m The Navier–Stokes solution was taken as the initial value. This initial Navier–Stokes spatial distribu- tion of variables was imposed on a mesh that encloses the shock. The length of the control volume enclos- ing the shock was chosen to be 1000 ϫ λ ∞ where the mean free path based on the freestream parameters is given by the expression λ ∞ ϭ 16 µ /(5 ρ ∞ ͙ 2π ෆ R ෆ T ෆ ∞ ෆ ).This is the mean free path that would exist in the unshocked region if the gas were composed of hard elastic spheres and had the same viscosity, density, and temperature as the gas being considered. The solution was marched in time until the observed devi- ations were smaller than a preset convergence criterion. A set of computational experiments was carried out to compare the BGK–Burnett solutions with the Burnett solutions of Fiscko and Chapman (1988). Tests were conducted at Mach 20 and Mach 35. In order to test for instabilities to small wavelength disturbances, the grid points were increased from 101 to 501 points. Figures 8.6 and 8.7 show variations of specific entropy across the shock wave. The BGK–Burnett equations 8-14 MEMS: Introduction and Fundamentals −5 0 5 −10 10 Dispersion coefficient () Attenuation coefficient ( ␣ ) Stable ( ␣ ≤ 0 ) Unstable ( ␣ > 0 ) 0 −15 −10 −5 10 15 20 255 FIGURE 8.5 Characteristic trajectories of the one-dimensional conventional Burnett equations; Euler equations are used to express the material derivatives D( )/Dt in terms of spatial derivatives. © 2006 by Taylor & Francis Group, LLC show a positive entropy change throughout the flow field, while the conventional Burnett equations give rise to a negative entropy spike just ahead of the shock as the number of grid points is increased. This spike increases in magnitude until the conventional Burnett equations break down completely. The BGK–Burnett equations did not exhibit any instabilities for the range of grid points considered. Figure 8.8 shows the variation of reciprocal density thickness with Mach number. BGK–Burnett calculations compare well to those of Woods and simplified Woods equations [Reese et al., 1995] and the experimental data of Alsmeyer (1976). Extensive calculations for one-dimensional hypersonic shock structure using various higher order kinetic formulations are given in Balakrishnan (1999). Burnett Simulations of Flows in Microdevices 8-15 Nondimensional specific entropy 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 X Navier−Stokes Burnett (F&C) BGK−Burnett 0 2 4 6 8 10 −2 FIGURE 8.6 Specific entropy variation across a Mach 20 normal shock in a monatomic gas (argon), ∆x/ λ ∞ ϭ 4.0 and γ ϭ 1.667; F & C ϵ Fiscko and Chapman (1988). 0 2 4 6 8 10 Non-dimensional specific entropy 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 X Navier−Stokes Burnett (F&C) BGK−Burnett FIGURE 8.7 Specific entropy variation across a Mach 35 normal shock in a monatomic gas (argon), ∆x/ λ ∞ ϭ 4.0 and γ ϭ 1.667; F & C ϵ Fiscko and Chapman (1988). © 2006 by Taylor & Francis Group, LLC 8.7.2 Application to Two-Dimensional Hypersonic Blunt Body Flow The two-dimensional augmented Burnett code was employed to compute the hypersonic flow over a cylindrical leading edge with a nose radius of 0.02 m in the continuum–transition regime. The grid sys- tem (50 ϫ 82 mesh) used in the computations is shown in Figure 8.9. The results were compared with those of Zhong (1991). The flow conditions for this case are as follows: M ∞ ϭ 10, Kn ∞ ϭ 0.1, Re ∞ ϭ 167.9, P ∞ ϭ 2.3881 N/m 2 , T ∞ ϭ 208.4 K, T w ϭ 1000.0 K The viscosity is calculated by Sutherland’s law, µ ϭ c 1 T 1.5 /(T ϩ c 2 ). The coefficients c 1 and c 2 for air are 1.458 ϫ 10 Ϫ6 kg/(sec m K 1/2 ) and 110.4K, respectively. Other constants used in this computation for air are γ ϭ 1.4, Pr ϭ 0.72 and R ϭ 287.04 m 2 /(sec 2 K). The comparisons of density, velocity, and temperature distributions along the stagnation streamline are shown in Figures 8.10, 8.11, and 8.12 respectively. The results agree well with those of Zhong (1991) for both the Navier–Stokes and the augmented Burnett computations. Because the flow is in the continuum– transition regime (Kn ϭ 0.1), the Navier–Stokes equations become inaccurate, and the differences between the Navier–Stokes and the augmented Burnett solutions are obvious. In particular, the difference between the Navier–Stokes and Burnett solutions for the temperature distribution is significant across the shock. Te mperature and Mach number contours of the Navier–Stokes solutions and the augmented Burnett solutions are compared in Figures 8.13 and 8.14 respectively. The shock structure of the augmented Burnett solutions agrees well with that of Zhong (1991). The shock layer of the augmented Burnett solutions is thicker, and the shock location starts upstream of that of the Navier–Stokes solutions. However, because the local Knudsen number decreases and the flow tends toward equilibrium as it approaches the wall surface, the differences between the Navier–Stokes and augmented Burnett solutions become negligible near the 8-16 MEMS: Introduction and Fundamentals 0.5 0.4 0.3 0.2 0.1 0.0 1 2 3 4 5 6 7 8 91011 M Navier−Stokes Navier−Stokes Woods Simplified Woods BGK−Burnett Reciprocal density thickness FIGURE 8.8 Plot showing the variation of reciprocal density thickness with Mach number, obtained with the Navier–Stokes, Woods and Simplified Woods [Reese et al., 1995], and BGK–Burnett equations for a monatomic gas (argon). Experimental data were obtained from Alsmeyer (1976). (Reprinted with permission from Alsmeyer, H. (1976) “Density Profiles in Argon and Nitrogen Shock Waves Measured by the Absorption of an Electron Beam,” J. Fluid Mech. 74, pp. 497–513.) © 2006 by Taylor & Francis Group, LLC wall, especially near the stagnation point. Thus, the Maxwell–Smoluchowski slip boundary conditions can be applied for both the Navier–Stokes and the augmented Burnett calculations for the hypersonic blunt body. 8.7.3 Application to Axisymmetric Hypersonic Blunt Body Flow The results of the axisymmetric augmented Burnett computations are compared with the DSMC results obtained by Vogenitz and Takara (1971) for the axisymmetric hemispherical nose. The computed results are also compared with Zhong and Furumoto’s (1995) axisymmetric augmented Burnett solutions. The flow conditions for this case are M ∞ ϭ 10, Kn ∞ ϭ 0.1 ϭ 1.0, ϭ 0.029 T 0 is the stagnation temperature. The gas is assumed to be a monatomic gas with a hard-sphere model. The viscosity coefficient is calculated by the power law µ ϭ µ r (T/T r ) 0.5 . The reference viscosity µ r and the reference temperature T r used in this case are 2.2695 ϫ 10 Ϫ5 kg/(sec m) and 300 K, respectively. Other constants used in this computation are γ ϭ 1.67 and Pr ϭ 0.67. T w ᎏ T 0 T w ᎏ T ∞ Burnett Simulations of Flows in Microdevices 8-17 FIGURE 8.9 Two-dimensional computational grid (50 ϫ 82 mesh) around a blunt body, r n ϭ 0.02 m. © 2006 by Taylor & Francis Group, LLC The comparisons of density and temperature distributions along the stagnation streamline among the current axisymmetric augmented Burnett solutions, the axisymmetric augmented Burnett solutions of Zhong and Furumoto, and the DSMC results are shown in Figures 8.15 and 8.16,respectively. The corre- sponding Navier–Stokes solutions are also compared in these figures. The axisymmetric augmented Burnett solutions agree well with Zhong and Furumoto’s axisymmetric augmented Burnett solutions in both density and temperature. The density distributions for both the Navier–Stokes and augmented 8-18 MEMS: Introduction and Fundamentals 0 5 10 15 25 20 Nondimensional density (/ ∞ ) −0.25 x/r n Navier−Stokes Navier−Stokes (Zhong) Augmented Burnett Augmented Burnett (Zhong) −1 0 −0.75 −0.5 FIGURE 8.10 Density distributions along stagnation streamline for blunt body flow: air, M ∞ ϭ 10, and Kn ∞ ϭ 0.1. 0 2500 3000 2000 1500 1000 500 Velocity (m/sec) Navier−Stokes Navier−Stokes (Zhong) Augmented Burnett Augmented Burnett (Zhong) x/r n −1 0 −0.75 −0.5 −0.25 FIGURE 8.11 Velocity distributions along stagnation streamline for blunt body flow: air, M ∞ ϭ 10, and Kn ∞ ϭ 0.1. © 2006 by Taylor & Francis Group, LLC Burnett equations show little differences from the DSMC results. The temperature distributions, however, show that the DSMC method predicts a thicker shock than the augmented Burnett equations. The max- imum temperature of the DSMC results is slightly higher than those of the augmented Burnett solutions. However, the augmented Burnett solutions show much closer agreement with the DSMC results than the Navier–Stokes solutions. Burnett Simulations of Flows in Microdevices 8-19 0 5000 4000 3000 2000 1000 Temperature (K) x/r n −1 0 −0.75 −0.5 −0.25 Navier−Stokes Navier−Stokes (Zhong) Augmented Burnett Augmented Burnett (Zhong) FIGURE 8.12 Temperature distributions along stagnation streamline for blunt body flow: air, M ∞ ϭ 10, and Kn ∞ ϭ 0.1. Augmented Burnett Navier−Stokes FIGURE 8.13 Comparison of temperature contours for blunt body flow: air, M ∞ ϭ 10, and Kn ∞ ϭ 0.1. © 2006 by Taylor & Francis Group, LLC Overall, the axisymmetric augmented Burnett solutions presented here agree well with Zhong and Furumoto’s (1995) axisymmetric augmented Burnett solutions and describe the shock structure closer to the DSMC results than the Navier–Stokes solutions. As another application to the hypersonic blunt body, the augmented Burnett equations are applied to compute the hypersonic flow field at re-entry condition encountered by the nose region of the space shuttle. 8-20 MEMS: Introduction and Fundamentals Navier−Stokes Augmented Burnett FIGURE 8.14 Comparison of Mach number contours for blunt body flow: air, M ∞ ϭ 10, and Kn ∞ ϭ 0.1. 0 Navier−Stokes 50 40 30 20 10 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 Augmented Burnett Augmented Burnett (Zhong and Furumoto) DSMC (Vogenitz and Takara) x/r n / ∞ FIGURE 8.15 Density distributions along stagnation streamline for a hemispherical nose: hard-sphere gas, M ∞ ϭ 10 and Kn ∞ ϭ 0.1. © 2006 by Taylor & Francis Group, LLC [...]... dynamics on a finite-difference, finite-volume, or finite-element mesh, the LBM derives its basis from the kinetic theory that models the microscopic behavior of gases The fundamental idea behind LBM is to construct the simplified kinetic models that capture the essential physics of microscopic behavior so 9-1 © 2006 by Taylor & Francis Group, LLC 9-2 MEMS: Introduction and Fundamentals that the macroscopic... unit time There are two types of moving particles: the particles that move along the axis with velocity|ei| ϭ 1, i ϭ 1, 2, 3, 4 and the particles that move along the diagonals with velocity|ei| ϭ ͙ෆ, i ϭ 5, 6, 7, 8 Also, there are rest particles with speed zero at each node 2 The occupation of these three types of particles is described by the single particle distribution function fi where the subscript... equations: helium, M∞ ϭ 5 and Kn∞ ϭ 0 .7 condition predicts lower slip velocity near the entrance and higher slip velocity near the exit As the figures show, there is very little difference between the Navier–Stokes solutions and the augmented Burnett solutions at the entrance, but as the local Knudsen number increases toward the exit of the channel, the difference between the Navier–Stokes solutions and the. .. increases as expected 8 .7. 6 Supersonic Flow in a Microchannel The Navier–Stokes equations and the augmented Burnett equations are applied to compute the supersonic flow in a microchannel The geometry and grid of the microchannel are shown in Figure 8.29 As the flow enters the channel, the tangential velocity component to the wall is retained, while the velocity component normal to the wall is neglected... used in the calculation for air are γ ϭ 1.4, Pr ϭ 0 .72 , and R ϭ 2 87. 04 m2/(sec2 K) Figure 8.21 shows the density contours of the Navier–Stokes solution with the first-order Maxwell– Smoluchowski slip-boundary conditions These contours using the continuum approach agree well with those of Sun et al (2000) using the information preservation (IP) particle method At these Mach and Knudsen numbers, the contours... identical velocity profiles These velocity profiles agree well with the velocity profiles from the micro-flow calculation by Beskok and Karniadakis (1999) Nondimensional mass flow rates along the microchannel are shown in Figure 8.25 All the mass flow rates from both equations and both slip-boundary conditions are about 0 .76 and almost constant along the channel, as should be the case This mass flow rate... for Moment Models of Dilute Gases, Ph.D thesis, University of Michigan Burnett, D (1935) The Distribution of Velocities and Mean Motion in a Slight Non-Uniform Gas,” Proc London Math Soc 39, pp 385–430 Chapman, S., and Cowling, T.G (1 970 ) The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, New York © 2006 by Taylor & Francis Group, LLC 8-3 4 MEMS: Introduction and Fundamentals. .. Pressure-Driven Slip Flow in a Microchannel without and with Magnetic Field 9.6.1 Analytical and Numerical Solutions We consider the pressure-driven MHD slip flow in a long constant area microchannel as shown in Figure 9.2 subjected to a constant magnetic field B0 in y-direction and a constant electric field E0 in the – z-direction Let the bar “ ” over a flow quantity denote the average value at the exit... 1.0E+ 07 5.0E+06 0.0E+00 0.0 1.0 2.0 x/ymax Mach number 5.0 4.0 3.0 2.0 1.0 0.0 0.0 1.0 2.0 x/ymax FIGURE 8.31 Comparisons of density, temperature, pressure, and Mach number profiles along the centerline of the channel: helium, M∞ ϭ 5 and Kn∞ ϭ 0 .7 Maxwell–Smoluchowski slip-boundary conditions are employed at the rest of the wall boundaries The channel height and length are 2.4 and 12 µm, respectively The. .. significant differences These flow property contours also agree well with the DSMC solutions obtained by Oh et al (19 97) Figure 8.31 compares the density, temperature, pressure, and Mach number profiles along the centerline of the channel using the Navier–Stokes, augmented Burnett, and DSMC formulations [Oh et al., 19 97] The profiles generally agree well with the DSMC results The temperature and Mach . from the DSMC results. The temperature distributions, however, show that the DSMC method predicts a thicker shock than the augmented Burnett equations. The max- imum temperature of the DSMC results. 19 97] . The profiles generally agree well with the DSMC results. The temperature and Mach number profiles especially show very close agreement with the DSMC 8-3 0 MEMS: Introduction and Fundamentals 0.0. boundaries in the region 0 Յ x Յ 1 m. The first-order 8-2 8 MEMS: Introduction and Fundamentals y/y max 0.0 0.5 1.0 0.0 1.0 2.0 3.0 4.0 5.0 x/y max FIGURE 8.29 Microchannel geometry and grid. y/y max 1.0 0.5 0.0 0.0 1.0 2.0 3.0