MTE 7(1) #6001
Microscale Thermophysical Engineering, 7:51–68, 2003 Copyright © 2003 Taylor & Francis 1089-3954/03 $12.00 + .00 DOI: 10.1080/10893950390150430
PREDICTION OF FLOW AND HEAT TRANSFER CHARACTERISTICS IN MICRO-COUETTE FLOW Hong Xue Mechanical Engineering Department, California State Polytechnic University, Pomona, California
Hongmiao Ji and Chang Shu Department of Mechanical Engineering, National University of Singapore, Singapore
In this study, direct simulation Monte Carlo (DSMC) method, the Navier-Stokes equations, and the Burnett equations are applied to micro-Couette flow to investigate flow and heat transfer characteristics in the slip and transition flow regimes. The unique solutions of the Burnett equations are obtained through determination of additional boundary condition for pressure using the results of DSMC. The effect of rarefaction on velocity, temperature, and pressure distributions is analyzed. The nondimensional parameter PrE characterizing convective heat transfer in Coutte flow increases as flow becomes more rarefied. The prediction of shear stress and heat flux at wall shows that the solution of the Burnett equations is superior than that of the Navier-Stokes equations at relatively high Kn number in the slip flow regime, but the Burnett equations with the slip boundary condition, which is proportional to Kn number, cannot be extended to the transition flow regime.
INTRODUCTION Rarefied gas flows in micron or submicron size at standard atmospheric conditions have found many applications in Micro-Electro-Mechanical-Systems (MEMS) recently. The micro-Couette flow is one of the simplest in shear-driven flow problems; however, no exact solution of the Boltzmann equation has been found so far. The Chapman-Enskog method provides a solution of the Boltzmann equation in which the distribution function f is perturbed by a small amount from the equilibrium Maxwellian form. It is known that the zeroth-order Chapman-Enskog solution reduces the conservation equation to the Eular equations, which describe isentropic flows, and the first-order solution leads to the Navier-Stokes equations, which provides a standard description of viscous flows. From the kinetic theory point of view, the Chapman-Enskog solutions merely reconcile the molecular and continuum approaches for small Knudsen flows. The deviation from the state of gas in continuum flow regime becomes large as Kn number increases from the slip flow regime (10−3 < Kn < 0.1) to the transition flow regime (0.1 < Kn < 10).
Received 13 September 2000; accepted 10 April 2002. Address correspondence to Prof. Hong Xue, Mechanical Engineering Department, California State Polytechnic University, 3801 W. Temple Ave., CA 91768, USA. E-mail:
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NOMENCLATURE c cp E H k Kn L M
√ speed of sound (= γ RT ) specific heat, J/kg · K u2 Eckert number = c (T 1−T ) p
1
2
Pr
distance between two parallel plates, m thermal conductivity, W/m · K Knudsen number Characteristic length scale, m √ Mach number (= u1 / γ RT1 ) c µ Prandtl number = pk
p qi R T u
pressure, Pa heat flux vector, W/m2 gas constant, J/kg · K temperature, K velocity component in the x-direction, m/s
Greek Symbols γ specific heat ratio λ mean free path, m µ dynamic viscosity, kg/m · s ρ density, kg/m3 σij viscous stress tensor accommodation coefficients σu , σT ω exponent in viscous-temperature law Superscripts and subscripts s surface next to the wall w wall 0 reference state 1 plate 1 2 plate 2 ∼ nondimensional variable
Our inquiry starts from the evaluation of the next term in this expansion, which leads to a set of very complicated higher-order continuum equations known as the Burnett equations. Studies on the Burnett solutions to one-dimensional shock wave structure and two-dimensional hypersonic blunt body flows showed that the Burnett equations can provide a more accurate description of a gas flow [1, 2, 3]. Beskok and Karniadakis [4] indicated the possibilities that use of more than two terms of the Chapman-Enskog expansion of the velocity distribution function f could extend the continuum approximation to the transition regime; however, the high-order derivatives present in the Burnett equations would require more boundary conditions than those in the Navier-Stokes equations [5]. Lee [6] confirmed that for the solutions to the Burnett equations to be uniquely determined, an additional boundary condition is needed for each wall, as long as the Knudsen number is not identically zero. Xue et al. [7] compared the solutions of the Navier-Stokes and the Burnett equations on micro-Couette flow. On the other hand, the direct simulation Monte Carlo method (DSMC) has become well established for modeling flows in both slip and transition flow regimes. The DSMC method used traditionally in high-altitude rarefied flows has applied to flows in and around microdevices, including analysis of micro-Couette flows recently [8, 9]; however, the DSMC method is largely constrained by the number of time steps and the resolution required in microflow application, which makes the full three-dimensional simulation impractical because of the large computational resource required [10]. Furthermore, it is also problematic that statistical scatter due to small perturbations increases as flow velocity becomes very subsonic. Encouraged by all of these developments, we intend to predict flow and heat transfer characteristics of micro-Couette flow using both the continuum based approaches and DSMC method. The motivation is two fold. Being able to analyze the characteristics of micro-Couette flow is desirable for many shear-driven flows encountered in engineering applications, such as in micromotors and microbearings. More importantly, based on
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the simple Couette flow study, the development and evaluation of the numerical tools for microflow and heat transfer analysis would help us to tackle growing problems in various MEMS application. MODELING AND METHODS Navier-Stokes and Burnett Equations Flow between two parallel infinite flat plates (Couette flow) is considered. As shown in Figure 1, the space between two infinite parallel plates is separated by a distance H . The lower plate (plate 1) at y = 0 moves with a constant velocity u1 and sets the fluid particles moving in the direction parallel to the plates while the upper plate (plate 2) remains stationary. The flow is considered steady, one-dimensional, and compressible. The governing equations of the Navier-Stokes and the Burnett equations for the Couette flow are cast into nondimensional form. The nondimensional variables are defined as follows [6]: y˜ =
T ρ p u , T˜ = , ρ˜ = , p˜ = , u˜ = √ T0 ρ0 p0 RT0 σij qi µ , q˜i = , µ˜ = , σ˜ ij = ρ0 RT0 µ0 ρ0 (RT0 )3/2
y , H
where the subscript 0 denotes the reference state. Hence, the nondimensional Navier-Stokes equations are expressed as d (µ˜ u˜ ) = 0, d y˜
γ d (−u˜ µ˜ u˜ ) + d y˜ Pr(γ − 1)
d p˜ = 0, d y˜ µ˜ T˜ = 0,
Figure 1. Couette flow coordinate system.
(1) (2) (3)
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where the Pr is Prandtl number and γ is specific heat ratio. For the Burnett equations, the y-momentum equation is different from equation (2) and is written as 2 µ ˜ 1 T˜ T˜ p˜ + Kn20 α6 u˜ 2 + α9 p˜ + α11 2 p˜ 2 + (α12 − 2α9 − 2α11 ) T˜ p˜ p˜ p˜ p˜ p˜ 1 + (α13 + 2α9 + α11 − α12 ) T˜ 2 + (α7 − α9 )T˜ = P0 , T˜
(4)
where P0 is an integration constant and Kn0 is reduced Knudsen number. The coefficients for Maxwellian gas are α6 = −0.667, α7 = 0.667, α9 = −1.333, α11 = 1.333, α12 = −1.333, α13 = 2.0. The superscripts and represent the first and the second derivatives with respect to y, respectively. The details of the derived governing equations can be referred to Xue et al. [7]. The following slip wall boundary conditions are imposed in the calculation. The effects of thermal creep and quadratical variation with Kn are not included: 2 − σu u˜ s − u˜ w = σu
T˜ µ˜ d u˜ π , Kn0 p˜ d y˜ s 2
2γ 2 − σT T˜s − T˜w = σT Pr(γ + 1)
(5)
T˜ µ˜ d T˜ π Kn0 . 2 p˜ d y˜
(6)
s
In the numerical simulation, the accommodation coefficients for u˜ and T˜ are σu − 1 and σT = 1, respectively. The Maxwell molecular is considered, for which the exponent ω in the viscosity-temperature law (µ˜ = T˜ ω ) is 1, Pr = 2/3, and γ = 5/3. The numerical solutions of the Burnett equations in the continuum transitions regime have not been possible for flow with very fine grids, as the Burnett equations are unstable to the disturbances of small wavelength [11]. An accurate numerical method with a possible coarse grid system is desired. Therefore, we selected the generalized differential quadrature (GDQ) method [12, 13] to solve the Burnett equations. The GDQ method discretizes spatial derivatives by a weighted linear sum of all the functional values in the whole domain. The GDQ method can be considered as the highest-order finite difference scheme for a domain with a given mesh. Application of both GDQ and Chebyshev pseudospectral methods provides the same weighting coefficients for the first derivative when the grid points are chosen as the roots of the Nth order Chebyshev polynomial for the both methods. Equations (1)–(6) were solved numerically by the GDQ method. For the unique determination of the Burnett equations, the pressure at walls must be prescribed accordingly. P1 = P2 = Pb is assumed. The accurate determination of Pb will be described in the next section. Thirteen grid points were used along the y direction based on a grid refinement study. The convergence criterion for the simulation was based on the residual value for the continuity equation. The criterion ensured that the maximum residual −4 |Resn+1 ij |max ≤ 10 .
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DSMC Method The DSMC method is used simultaneously for the numerical simulation of microCouette flow. A DSMC code is developed based on the method proposed by Bird [14]. DSMC is a particle-based numerical modeling technique. It computes the trajectories of a larger number of particles and calculates macroscopic quantities by sampling particle properties. The procedure is based on a scheme composed of the following steps: it cycles the movement of molecules, indexes molecules into cells, selects collision pair, and calculates postcollision properties. The flow field and surface quantities are sampled repeatedly, starting from the initial data. In this study, the variable hard sphere (VHS) model and argon gas were used in all simulations. It is assumed that the surface is diffuse and fully accommodating. In the calculation, each cell contains two collision subcells, and each subcell employs an initial number of 20 to 200 molecules. The number density ranges from 1020 to 1024 , which ensures that a certain number of statistical collisions can be computed by the no-time-counter (NTC) method. The time step is chosen at about 10−9 to 10−6 so that a typical molecule moves about one third of the cell dimension at each time step, which is also much less than the mean collision time. Different ranges of Kn numbers from the slip to transition flow regimes were tested. The variation of Kn number was achieved by varying the distance between the two plates. The total number of samples collected were from 1.5 × 104 to 8.0 × 106 , depending on different Kn numbers.
RESULTS AND DISCUSSIONS The profiles of velocity, temperature, and pressure predicted by the DSMC are shown in Figure 2. In Figure 2(a), the velocity profiles are plotted at the different Kn numbers. The gradient of the velocity profiles increases as the flow becomes more rarefied. This means that the velocity slip between the molecule velocity near the plate and the plate velocity increases with Kn number. When Kn = 0.01, the slip velocity is just 0.015, and when Kn = 1.0, it increases to 0.281. Furthermore, at Kn = 8.0, it becomes 0.407. On the other hand, when Kn = 0.01, the velocity distribution is linear, but when Kn > 0.1, the linear velocity profiles no longer exist. This is due to the increase of the thickness of the Knudsen layer in the transition flow regime. The Knudsen layer is a very thin layer (about one to a few mean free paths) next to the wall. When Kn number is small, the change of velocity in the Knudsen layer is negligible in the entire velocity profile; however, in the transition flow regime, the thickness of the Knudsen layer is comparable to the distance between the two plates in the Couette flow. The bending of the velocity profiles becomes pronounced. It is interesting to note that the bending occurs most significantly at Kn number around 1. With further increase in Kn number, the flow will enter the free molecular regime, where the shear stress approaches a constant. Therefore, the linearity of the velocity profiles resumes gradually after Kn number exceeds 1. It can be predicted that the slip velocity at the upper and lower plates would reach the maximum when Kn number approaches infinite. Figure 2(b) illustrates the temperature distributions at the different Kn number. As Kn number increases, temperature jump at the upper and the lower plates becomes more significant, but the curvature of temperature profiles reduces. In addition, similar to the velocity, the temperature jump at the plates will reach maximum, as Kn number approaches infinite, and the temperature
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Figure 2. Numerical solution by DSMC: (a) velocity u; (b) temperature T ; (c) pressure p.
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will be uniform along the y-axis. The pressure profiles are depicted in Figure 2(c). The pressure distribution from the Navier-Stokes equations is uniform due to cut-off of the high-order terms of the stress tensor, but the DSMC predicted a pressure jump at the wall. For the case of Kn = 0.1 to 4, which typically represents the pressure distribution in the transition flow regime, pressure gradients near the wall are identified due to the effect of the Knudsen layer. As mentioned in the previous section, for the solution of the Burnett equations to be uniquely determined, an additional boundary condition is needed as long as Kn = 0. Although proposals were made to restrict the solution procedure to the extrapolation strategy [11, 15], Lee [6] showed that the solutions were not uniquely determined, as the degrees of freedom still remain in the initial data set of the iterative procedure, which strongly affects the final solutions. Figure 3 shows the results of DSMC and the Burnett equations at Kn = 0.0, 0.01, 0.1 and M = 1. The solutions of the Burnett equations were obtained under the assumption of Pb = 1.0. Obviously there are significant discrepancies between the results by DSMC and the Burnett equations in the pressure profiles. This implies that a modification on the boundary condition for pressure is necessary as long as Kn = 0. Using the results of DSMC at 0.001 ≤ Kn ≤ 0.1, the boundary condition for pressure was regressed. The corrected boundary condition for pressure is a function of Kn number and can be written as Pb = −
Kn2 + a log Kn + b, 2
(7)
where a = 1/72 and b = 1.068 are two constants. The results of the Burnett equations with corrected boundary conditions for pressure are compared with those of DSMC again in Figure 4, where the results of velocity, temperature, and pressure profiles fit the solutions of the DSMC well. To examine the effects of Mach numbers on the boundary condition of the pressure, simulations on the Burnett equations were also carried out at two different Mach numbers, M = 0.5 and M = 3. The results of the Burnett equations and DSMC at M = 3 are shown in Figure 5. It is seen that the predicted pressure profiles of the Burnett equations are consistent with that of the DSMC using the same equation (7). This indicates that Mach number has little effect on the boundary condition of pressure. From Figures 4 and 5, it is also noticed that the velocity and temperature distributions are little influenced by the boundary condition of the pressure due to the decoupled nature of equation (4) with equations (1) and (3). The effect of rarefaction on heating at different temperature ratio is illustrated in Figure 6. It is observed that when the temperature ratio is small, the deviation of the case Kn = 0.01 and Kn = 0.1 from the cases Kn = 0 is relatively small. The effect of rarefaction on the heating is insignificant. The temperature jump becomes distinguished at higher Kn numbers only when an extreme temperature difference exists between the two plates as shown in Figure 6(c). In classical heat transfer, the direction of heat flow at lower plate (ie, whether the heat transfer is into the fluid or the wall for T1 > T2 ) depends on the magnitude of the nondimensional parameter Pr · E [16], where Pr is Prandtl number and E is Eckert number. For Pr · E = 2, there is no heat transfer at the lower plate. From Figure 7, it is interesting to see that the nondimensional parameter Pr · E is deviated from the value 2 as the rarefaction increases. Subsequently, the value of Pr · E, at which the heat flow is set to be zero at the lower plate, becomes 2.008 for Kn = 0.01, 2.011 for Kn = 0.05,
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Figure 3. Comparison of the Burnett and DSMC solution with Pb = 1.0: (a) velocity u; (b) temperature T ; (c) pressure p.
FLOW AND HEAT TRANSFER IN MICRO-COUETTE FLOW
Figure 4. Comparison with Burnett and DSMC solution with corrected boundary condition for pressure at M = 1: (a) velocity u; (b) temperature T ; (c) pressure p.
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Figure 5. Comparison of Burnett and DSMC solution with corrected boundary condition for pressure at M = 3: (a) velocity u; (b) temperature T ; (c) pressure p.
FLOW AND HEAT TRANSFER IN MICRO-COUETTE FLOW
Figure 6. Effects of rarefaction on heating at different temperature ratios: (a) T1 /T2 = 1; (b) T1 /T2 = 2; (c) T1 /T2 = 20.
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Figure 7. Nondimensional temperature distribution as a function of Kn number and PrE (zero heat flux at lower plate).
and 2.1505 for Kn = 0.1. This implies that to keep zero heat flux at lower plate, the moving velocity of the plate must increase as the flow becomes more rarefied. To examine the performance of the Burnett equations at the wall, the relationship between the nondimensional shear stress, heat flux, and the Kn number at M = 1 and M = 3 are shown in Figures 8 and 9. The results of the Navier-Stokes equations and the Burnett equations were obtained with velocity slip and temperature jump conditions at different Kn numbers. At M = 3, the solution of the six-moment method by Liu and Lees [17] and the solution of direct simulation by Nanbu [9] were available for comparison. The nondimensional shear stress and heat flux predicted by DSMC are very close to the results of Nanbu [9]. It is clear that the Navier-Stokes equations are only accurate in the region of Kn < 0.04 for shear stress and even lower Kn number region for heat flux. Similar to the solution of the six-moment method, the results of the Burnett equations are close to the solution of the direct simulation in the entire slip flow regime up to about Kn = 0.1; however, we were not able to obtain solutions for the Burnett equations with the first-order slip boundary conditions after Kn number reaches 0.18. The reason is due to the inability of the continuum approaches with slip boundary conditions in predicting flow in the transition flow regime, where the physical changes take place in the Knudsen layer. The error of using the slip velocity to approximate the real velocity at the wall in the transition flow regime would increase significantly as the Knudsen layer becomes comparably thick to the distance between the two plates. As indicated in Figure 8, the gradient of the shear stress becomes nonlinear after the Knudsen number reaches 0.04
63 Figure 8. Variation of shear stress with Kn number: (a) M = 3.
64 Figure 8. Variation of shear stress with Kn number: (b) M = 1.
65 Figure 9. Variation of heat flux with Kn number: (a) M = 3.
66 Figure 9. Variation of heat flux with Kn number: (b) M = 1.
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and it approaches asymptotically to a constant, the free molecular limit, as Kn → ∞. Hence, the assumption on the Kn number as a linear coefficient of the slip velocity and temperature jump in equations (5) and (6) is only valid for the Knudsen number less than 0.04. Although the Burnett equations can improve the prediction of the NavierStokes equations in the region from Kn = 0.04 to about Kn = 0.1, it is impossible to further extend the Burnett equations to the entire transition flow regime. In fact, even if the second-order slip boundary conditions developed by Schamberg [18] are used, the Burnett equations still fail as flow enters the transition flow regime. In our previous study of microchannel flows [19], a hyperbolic tangent function of Kn number was proposed in the power series of the velocity distribution function and the slip boundary condition. The result works well in isothermal channel flow; however, the hyperbolic tangent function only fits the curve of the shear stress. It fails to follow the trend of heat flux curve shown in Figure 9. CONCLUSIONS Predictions of flow and heat transfer characteristics in micro-Couette flow using both the continuum-based approaches and the DSMC method have been carried out. The DSMC method has not only proven to be a valuable tool for investigating the behavior of microflows in the transition flow regime, but also has been benchmarked to determine the boundary condition for pressure in the Burnett equations. As an efficient mathematical tool, the continuum-based approaches are still useful, particularly for computing lowspeed, three-dimensional microflow, to which the current DSMC method is constrained by the requirement of extremely fine time step and resolution, as well as the problem of statistical scatter [10]. The results show that the Burnett equations are superior than that Navier-Stokes equations at relatively high Kn numbers in the slip flow regime. The Burnett equations can predict wall shear stress and heat flux well up to Kn = 0.1 but fails to extend the solutions further in the transition flow regime. The effect of rarefaction on heating is insignificant unless an extremely large temperature difference exists between the two plates. The nondimensional parameter PrE characterizing zero convective heat transfer is no longer a constant in micro-Couette flow. It increases as the flow becomes more rarefied. The failure of the Burnett equations to obtain solutions at upper Kn number limit indicates that the challenge on simulation of the low-speed micro flows in the transition flow regime is still in front of us. Improving the efficiency of the DSMC method is definitely necessary. On the other hand, taking more radical steps to find workable approximations for the systematic expansion methods is also attractive. REFERENCES 1. G. C. Pham-Van-Diep, D. A. Erwin, and E. P. Muntz, Testing continuum descriptions of low Mach number shock structures, J. Fluid Mech. vol. 232, pp. 403–413, 1991. 2. F. E. Lumpkin III, Accuracy of the Burnett equations for hypersonic real gas flows, Journal of thermophysics and Heat Transfer, vol. 6, no. 3, pp. 419–425, 1992. 3. X. L. Zhong, On numerical solutions of Burnett equations for hypersonic flow past 2-D circular blunt leading edges in continuum transition regime, AIAA 42 Fluid Dynamic Conf., AIAA 933092, Orlando, 1993.
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4. A. Beskok and G. E. Karniadakis, A Model for Flows in Channels, Pipes, and Ducts at Micro and Nano Scales, Microscale Thermophysical Engineering, vol. 3, pp. 43–77, 1999. 5. S. A. Schaaf and P. L. Chambre, Flow of Rarefied Gases, vol. 8, Princeton Aeronautical Paperbacks, Princeton Univ. Press, NJ, 1961. 6. C. J. Lee, Unique determination of solutions to the Burnett equations, AIAA Journal, vol. 32, no. 5, pp. 985–990, 1994. 7. H. Xue, H. M. Ji, and C. Shu, Analysis of micro-Couette flow using the Burnett equations, Int. J. of Heat Mass Transfer, vol. 44, pp. 4139–4146, 2001. 8. K. Nanbu, Analysis of the Couette flow by means of the new direct-simulation method, Journal of the Physical Society of Japan, vol. 52, no. 5, pp. 1602–1608, 1983. 9. L. S. Pan, G. R. Liu, and K. Y. Lam, Determination of slip coefficient for rarefied gas flows using direct simulation Monte Carlo, J. Micromech. Microeng., vol. 9, pp. 89–96, 1999. 10. E. S. Oran, C. K. Oh, and B. Z. Cybyk, Direct simulation Monte Carlo: Recent advances and applications, Annu. Rev. Fluid Mech., vol. 30, pp. 403–441, 1998. 11. X. L. Zhong, R. W. MacCormack, and D. R. Chapman, Stabilization of the Burnett equations and application to hypersonic flows, AIAA Journal, vol. 31, no. 6, pp. 1036–1043, 1993. 12. C. Shu, B. C. Khoo, and K. S. Yeo, Numerical solutions of incompressible Navier-Stokes equations by generalized differential quadrature, Finite Elements in Analysis and Design, vol. 18, pp. 83–97, 1994. 13. C. Shu, Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation, Ph.D. Thesis, University of Glasgow, UK, 1991. 14. G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flow, Oxford University Press, London, 1994. 15. X. L. Zhong, R. W. MacCormack, and D. R. Chapman, Evaluation of slip boundary conditions for the Burnett equations with application to hypersonic leading edge flow, Proceedings of the 4th International Symposium on Computational Fluid Dynamics, pp. 1360–1043, 1991. 16. M. N. Ozisik, Heat Transfer, McGraw-Hill, New York, 1985. 17. C. Y. Liu and L. Lees, Kinetic theory description of plane compressible Couette flow, Rarefied Gas Dynamics (ed. L. Talbot), Academic Press, New York, pp. 391–428, 1961. 18. R. Schamberg, The fundamental differential equations and the boundary conditions for high speed slip-flow, and their application to several specific problems, Ph.D. Thesis, California Institute of Technology, 1947. 19. H. Xue and Q. Fan, A new analytic solution of the Navier-Stokes equations for microchannel flows, Microscale Thermophysical Engineering, vol. 4, no. 2, pp. 125–143, 2000.
