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
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f
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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
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DSMC
Analytical NS
15000
> 10000
8
10
12
14
DSMC
.
Analytical NS I-
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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
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<D
DSMC (converged)
Filtered DSMC
2.0 104
1.5 104
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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
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10
12
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300
DSMC
Filtered DSMC
299
^
Js
I
298
29V
296
295
4
6
8
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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. Computing
resources were supplied by the DoD High Performance Computing Modernization Program.
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