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Nonlinear Filtering for Low-Velocity Gaseous Microflows Carolyn R. Kaplan and Elaine S. Oran Laboratory for Computational Physics and Fluid Dynamics Naval Research Laboratory, Washington, DC Abstract. Gaseous flows in microfluidic devices are often characterized by relatively high Knudsen numbers. For such flows, the continuum approximation is not valid, and Direct Simulation Monte Carlo (DSMC) is an appropriate solution. However, for low-velocity flows, where the fluid velocity is much smaller than the mean molecular velocity, large statistical fluctuations in the solution mean that the features of the flow may be obscured by noise in the solution. In this paper, we evaluate a method for numerically simulating low velocity microflows, in which we use a high-order, nonlinear monotone convection algorithm as a filter to extract the solution from the noisy DSMC calculation. We test the filtering method on a high-velocity microchannel flow problem, for which we know the correct DSMC solution, and show that the filtered and correct DSMC solutions are qualitatively similar and follow the same trends. We then apply the filter to a low-velocity microchannel flow and compare the results with an analytical continuum solution using slip-wall boundary conditions. Results indicate that the filtering operation successfully removes the large statistical fluctuations, reduces computational time, and makes it feasible to do low-velocity microflow calculations. INTRODUCTION Many practical MEMS devices involve gas flows in micron-sized systems at very low velocities. These are characterized by relatively high Knudsen numbers, Kn=A/L, where A, is the mean free path and L is a characteristic length. For a gas in a microdevice at 1 atm and 298K, Kn~0.15-0.20, which means that the continuum approximation does not apply. One possible method for computing the properties of such flows is DSMC, which is a statistical method in which the motion and interactions of a number of simulated particles are used to modify their positions and velocities [1,2]. However, DSMC is not practical for very low-velocity flows with high number densities. That is, at 1 atm and 298 K, the mean collision time is ~10~10 s, but the mean particle transit time for a slow flow through a microchannel is on the order of 10"2 s. Therefore, it would take an enormous number of timesteps (10s) for the particles to make one complete pass through the entire computational domain [3] and it usually requires several complete transits to obtain an equilibrium solution. An additional problem occurs when the flow velocity, Vfl0w5 is much smaller than the mean molecular thermal velocity [2]. For a gas at 1 atm and 298 K, the most probable thermal speed, Vth, is on the order of 100,000 cm/s. If Vfiow = 20 cm/s, the statistical noise is 104 greater than the solution and features of the flow are completely lost in the noise. In principle, the correct result should emerge as the ensemble or time average converges, because the statistical fluctuations decrease with the number of samples. However, too many timesteps (~108) are required to resolve such a low-velocity flow. Because of these problems, DSMC calculations of gaseous microflows have been conducted only for highvelocity flows [4-8], while the low-velocity calculations have been made with continuum methods incorporating slip boundary conditions [9-13]. Most recently, a low-speed gaseous microchannel calculation was demonstrated by implementation of an "Information Preservation" (IP) method [14], in which the statistical scatter is reduced by preserving and updating the physical signal using a macroscopic point of view. The IP part of the calculation has no effect on the DSMC part; that is, particles redistribute their translational velocities and internal energies as usual in the DSMC calculation. But, during the IP step, the preserved information is redistributed by different rules. That is, all particles in a cell share a common acceleration, which is calculated from the net pressure force acting on a cell and the total particle mass in a cell. That common acceleration is used to change the preserved velocity of each particle in the cell. This preservation method is based on some theoretical simplifications, including the assumption that the flow is isothermal, and the assumption that when two particles collide, there is no internal energy CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis 2001 American Institute of Physics 0-7354-0025-3 472 redistribution and their preserved translational velocities are divided equally. In this paper, we evaluate another method that uses a nonlinear filtering technique to extract a solution from the noisy DSMC calculation. NUMERICAL METHOD DSMC is a direct particle simulation method based on kinetic theory. The fundamental idea is to track a large number of statistically representative particles. The particles' motion and interactions are then used to modify their positions, velocities, or chemical reactions. Conservation of mass, momentum, and energy is enforced. The primary approximation of DSMC is to uncouple the molecular motions and the intermolecular collisions over small time intervals. Particle motions are modeled deterministically, while the collisions are treated statistically. Modeling the collisions is always a three-dimensional calculation, even when the computational grid dimensions are reduced to account for symmetries in physical space. Statistical errors are inherently introduced into DSMC calculations because each simulated molecule represents a large number of actual molecules. Increasing the number of simulated molecules reduces the statistical error, but increases the computational work required for the calculation. The core of the DSMC algorithm consists of four primary processes: move the particles, index and crossreference the particles, simulate collisions, and sample the flow field. These procedures are uncoupled during each timestep. Of primary importance is the selection of a timestep that is less than the local mean collision time and cell size that is less than the local mean free path. The first process, moving simulated molecules, enforces the boundary conditions and samples macroscopic properties along solid surfaces. Modeling molecule-surface interactions requires applying the conservation laws to individual molecules instead of using the velocity distribution function. The second process involves indexing and tracking the particles. Accurate and fast indexing and tracking are key to practical DSMC applications for large-scale processing. The next step is simulating collisions between randomly selected pairs of molecules within each cell. This step is a probabilistic process. There are several different collisional modeling techniques that have been used in DSMC codes over the years. In this work, we use the notime-counter technique in conjunction with the sub-cell technique [1]. The sub-cell method calculates local collision rates based on the individual cells, but restricts possible collision pairs to sub-cells. This procedure improves accuracy by ensuring that collisions occur only between near neighbors. The final process is sampling the macroscopic flow properties. The spatial coordinates and velocity components of molecules in a particular cell are used to calculate macroscopic quantities at the geometric center of the cell. The other steps of the DSMC procedure do not depend on the sampling process. The DSMC technique is explicit and time-marching. All of the microchannel flow calculations presented here are steady flows. Each computation proceeds until a steady flow is established, and the desired result is a time average of all values calculated after reaching the steady state. RESULTS Test Problem The methods for treating low-velocity flows are evaluated by comparing a series of two-dimensional DSMC calculations for helium at 1 atm and 298 K flowing through a 1 x 12 |im2 channel. Calculations were done for inflow velocities ranging from 200 to 80,000 cm/s. For helium at 298 K, Vth =110,000 cm/s. Therefore, we expect that the statistical fluctuations would significantly affect the solution for the cases in which the flow velocities are 10,000 cm/s or less. In all cases, the pressure at the inflow and outflow boundaries are set to 1 atm; however, the simulations show that the pressure builds up at the inlet, especially for the higher velocity cases, due to the highly viscous flow conditions. For all of the calculations shown here, the grid consisted of 150x30 cells, and approximately 100,000 simulated Variable Hard Sphere (VHS) molecules were included in the calculation. Tests in which the grid resolution and number of simulated molecules were doubled (i.e., 300x60 cells and 200,000 particles) are not shown as they resulted in no appreciable differences in the solution. Figure 1 shows contours of velocity, pressure, temperature, and density for the 60,000 cm/s case. Although the inflow velocity at the boundary is maintained at 60,000 cm/s, the velocity at the inlet is reduced to approximately 10,000 cm/s while the pressure builds up to over 2 atm. The velocity contours show the boundary layer which forms immediately near the inlet boundary. The temperature remains almost constant, except for a slight increase at the 473 Velocity, cm/s 1{4 Pressure, atm !> } f 1 T f 1r 1r ( 1 I J ' ! . V V i V J 0,8 Temperature, K Density, g/cm3 |LLm \ 2.80-04 | 2.20-04 I 1 1.80-04 1.20-04 | 12|Lim FIGURE 1. Contours of selected variables for 60,000 cm/s case. inlet and slight decrease at the outlet. For the lower velocity cases (not shown here), the pressure build up at the inlet is much smaller, and the entire microchannel remains at constant temperature. Figure 2 shows a comparison between the DSMC solution and an analytical solution to the Navier-Stokes equations for the velocity and pressure along the centerline longitudinal axis for this 60000 cm/s case. The analytical solution, described in detail by Arkilic [15], is obtained by neglecting the nonlinear inertial term in the Navier-Stokes equations, and is represented by 2-a 2-a 4 a (1) (2) where P(x) is the dimensionless pressure distribution, Kn0 is the Knudsen number at the outlet, a is the accommodation coefficient (assumed to be 0.85), Pf is the inlet to outlet pressure ratio, x is the streamwise distance from the inlet, L is the channel length, U(x) is the streamwise velocity, [i is the dynamic viscosity, y is the vertical distance from the centerline, H is the channel height, and A is the mean free path. This solution incorporates a slip boundary condition at the walls: (3) wall The accuracy of the analytical solution is zero order, and is only valid when the Reynolds number is very small. Although 60,000 cm/s is a high velocity flow, the Reynolds number for this case is only six, due to the very small length scale. The DSMC and analytical solutions for the pressure distribution along the centerline are in good agreement and follow the same trends, as shown in Fig. 2. The velocity profiles are in reasonable agreement, except near the outlet, where the local Kn is highest (Kn>0.2), and where slip flow conditions (assumed in the analytical solution) may no longer be valid. 474 35000 -S2. 25000 ' DSMC Analytical NS 15000 > 10000 8 10 12 14 DSMC . Analytical NS I- 4 6 8 10 Distance, microns FIGURE 2. Comparison between the DSMC and analytical Navier-Stokes solution for the 60,000 cm/s case shows that both solutions follow the same trends. Quantities shown are values along the longitudinal centerline axis. Nonlinear Filtering The filter that we consider is Flux-Corrected Transport (FCT) [16], which is a high-order, nonlinear, monotone, positivity-preserving convection algorithm. FCT solves the generalized continuity equations and is frequently used to convect sharp discontinuities with little numerical diffusion. Monotone algorithms add numerical diffusion to assure positivity and accuracy, which can be mutually exclusive requirements in the solution of flows with steep local gradients. These algorithms add the most diffusion in areas of steep gradients to ensure positivity, and add the minimum amount of diffusion (required for stability) in smooth regions where monotonicity is not threatened, and then use an anti-diffusion step as a correction to remove the strong diffusion. However, the anti-diffusion stage can reintroduce negative values or nonphysical overshoots. FCT limits the antidiffusion flux to ensure positivity and stability, so that the antidiffusion stage does not generate new nonphysical maxima or minima in the solution. This flux-limiter works as a nonlinear filter, and is discussed in more detail by Boris et al. [17]. As a test case, we chose the high-velocity flow (60,000 cm/s) for which we knew the late-time converged DSMC solution. We compare the converged DSMC solution with an FCT-filtered solution, where the filtering operation was performed on an early-time noisy DSMC result. Figure 3-a shows the starting point for the filtering operation, and Figs. 3-b through 3-d show the results of the filtering operation after one, 100 and 10,000 passes, respectively, through the FCT filter. Comparison between the late time-converged DSMC solution (shown in Fig. 3e) and the filtered solution (Fig. 3-d) shows that the filtered solution is not as smooth as the converged DSMC solution, but the two solutions are qualitatively similar. Continued filtering does not further change the filtered solution. Figure 4 shows a quantitative comparison of velocity, pressure and temperature along the centerline 475 (a) Starting point for filtering operation (early time DSMC solution) (b) 1 pass through filter (c) 100 passes through filter (d) 10000 passes through filter 10000 MOOOO (e) Converged DSMC solution FIGURE 3. Comparison of velocity (cm/s) images between converged DSMC solution and filtered solution. Images in (a) through (d) show the filtered solution at various stages. Image (a) shows the starting point for the filtering operation (a very early time DSMC solution), while images (b), (c) and (d) show the solution after one, 100, and 10000 passes through the filter, respectively. Image in (e) shows the late-time converged DSMC solution (after 10000 timesteps). longitudinal axis for the late-time converged DSMC and FCT-filtered solutions for this high-velocity case. As shown, the two solutions are comparable and follow similar trends. Some of the differences between the late time converged DSMC solution and the filtered solution can be attributed to boundary effects. That is, when using FCT as a filter, the fluid is convected at zero velocity, and the boundary conditions (at the walls and even at the inflow and outflow) are zero gradient conditions. These boundary conditions are used because our intent is to use FCT as a filter, and not to simulate fluid convection. Further study is needed to determine the appropriate way to treat boundaries when using FCT as a high-frequency filter. The time required to obtain a converged DSMC solution (Fig. 3-e) is approximately six hours on one processor of an Origin 2000. However, the time required to obtain the filtered solution (Fig. 3-d) is approximately five minutes to obtain the noisy early-time DSMC calculation (shown in Fig. 3-a) and an additional five minutes of FCT filtering. We have demonstrated the filtering technique for a high-velocity flow, because we know the correct DSMC solution for this case. The real benefit of the filtering method is to allow us to simulate a low-velocity flow, where the statistical noise overpowers the signal to the degree to which it prevents the flow solution from evolving. For the 200 cm/s case, Figure 5 shows a comparison between the DSMC solution (after 60,000 timesteps), the filtered solution and an analytical Navier-Stokes solution. As shown, the DSMC solution is still very noisy even after 60,000 timesteps. This figure shows that the filtered DSMC and the analytical Navier-Stokes solutions follow the same trends. Because the velocity is so low, the pressure drop in this short microchannel is relatively small, as shown in this figure. The temperature throughout the microchannel is essentially isothermal. Use of the filtering technique was essential in getting the solution for the 200 cm/s case. The noisy DSMC solution shown in Fig. 5 ran for several weeks on one processor of an Origin 2000, while the FCT filtering required only five minutes of CPU time. 476 3.5 104 3.0 104 2.5 104 o ."o1? o <D DSMC (converged) Filtered DSMC 2.0 104 1.5 104 1.0 104 5.0 103 0.0 10° 2.5 DSMC (converged) Filtered DSMC 2.0 1.5 £ 1.0 0.5 360 DSMC (converged) Filtered DSMC 340 320 300 280 260 4 6 8 Distance (microns) 10 12 14 FIGURE 4. Comparison between known DSMC solution and filtered solution (where the filtering starts from a noisy earlytime DSMC solution) for the 60,000 cm/s case. 477 140 DSMC Filtered DSMC Analytical N-S 120 — 100 I & 80 | 60 13 20 0 1.008 10 12 14 10 12 14 1.006 -£ 1.004 l' 1.002 £ 1.000 0.998 0.996 300 DSMC Filtered DSMC 299 ^ Js I 298 29V 296 295 4 6 8 Distance (microns) 10 12 14 FIGURE 5. Comparison between the noisy late-time DSMC solution, the filtered solution and the analytical solution of the Navier-Stokes equations for the 200 cm/s case. 478 SUMMARY In this paper, we have discussed a promising approach for the removal of statistical noise from DSMC solutions of low-velocity microflows. One of the more advantageous aspects of filtering is that it does not affect the DSMC calculation as it is running; rather the filtering is done as a post-processing operation. The filtering calculations presented here show the cost-savings (in computational time) of the technique and also demonstrate that it is feasible to do a low-speed microflow calculation. The quality of the filter used to remove statistical noise is important. When applying FCT, the filtered solution eventually stops changing with time even though the filtering continues. This is in contrast to using a more diffusive type of filter (such as an arithmetic smoothing function) in which the filtered solution continues to change with time and eventually goes to the wrong solution. Another potential method to reduce statistical noise is to combine a DSMC and Navier-Stokes (NS) solution. By using both methods interactively, the output from one method can provide the input conditions for the other method. In this sense, the DSMC calculation may be viewed as providing the physically correct boundary conditions and transport coefficients in high-Kn flow regimes to the NS computation. Previous work on a highspeed microchannel flow problem [18] showed that using NS as a filter for the DSMC calculation could eliminate some of the noise and reduce the computational time. However, the filtering could not be applied until a near-steady solution had been attained. ACKNOWLEDGMENTS This project is supported by the DARPA Design for Mixed Technology Integration Program. 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