A Coupled DSMC/Navier-Stokes Method for Multiscale
Analysis of Gas Flow in Microfluidic Systems
Ozgur Aktas, Umberto Ravaioli, N. Aluru
Beckman Institute for Advanced Science and Technology,
University of Illinois at Urbana-Champaign
405 N. Mathews Avenue, Urbana, Illinois, 61801
Abstract. A multiscale method that couples direct simulation Monte Carlo (DSMC) method with Navier-Stokes
equations is presented. The multiscale method is based on the Schwarz coupling of the DSMC and Navier-Stokes
subdomains. Dirichlet boundary conditions are used at the coupling interfaces. The Navier-Stokes equations are solved
using a scattered point based finite cloud method. Data interpolation between Navier-Stokes and DSMC subdomains is
also achieved using a scattered point interpolation scheme. With the present method, multiscale problems that exhibit
compressibility in the continuum subdomains can be simulated. As an example, a microfilter is simulated using the
multiscale method, and it is shown that the interface can be positioned in a region exhibiting velocity, pressure, and
temperature gradients. As compared to a multiscale analysis of the same problem using DSMC/Stokes coupling [8], the
present method achieves a larger reduction in computation time. Detailed comparison of the multiscale solution with the
DSMC results is presented.
INTRODUCTION
Simulation of gas flow through microfluidic devices often necessitates a multiscale approach for achieving
computational efficiency. The rarefaction of flow in the small dimensions of the microfluidic devices can be
simulated by direct simulation Monte Carlo method [1]. However, the complete system often involves regions with
larger dimensions in which the flow is not rarefied and a DSMC simulation is expensive [2]. Thus, the system to be
investigated exhibits a large range of length scale and can be solved naturally and efficiently using multiscale
methods. An approach to multiscale simulation of such systems is to use domain decomposition: In this approach,
the simulation domain is divided into subdomains each of which is simulated using the appropriate model. The
solution in the subdomains can be coupled by enforcing various boundary conditions [3].
For multiscale simulation of gas flow in microfilters, the most general model for the continuum subdomains is
the compressible Navier-Stokes equations. The achievement of proper DSMC/Navier-Stokes coupling provides two
main advantages: First, by using the methods developed for DSMC/Navier-Stokes coupling, multiscale flows where
the continuum flow exhibits compressibility or non-isothermal effects can be addressed. Second advantage is that,
even in cases where part of the continuum region is incompressible and a multiscale analysis by DSMC/Stokes
coupling is possible, the use of compressible Navier-Stokes equations increases the region that can be simulated by
continuum methods and makes it possible to improve the speed-up that can be obtained. Thus, DSMC/Navier-Stokes
coupling enables multiscale simulation for the complete range of problems encountered in multiscale simulation of
gas flow in microfludic devices. Furthermore, the problems that can be addressed include not only gas flows
simulated by Navier-Stokes equations, but also coupling of other equation systems that are similar in form to
Navier-Stokes equations (such as Poisson-Nernst-Planck equations) with atomistic models.
Previous work on coupling of DSMC with Navier-Stokes equations has focused on high-speed flows [4-6].
However, in microfluidic devices the flow velocity is low and thus the estimation of fluxes across the boundary is
expensive. For this reason, previous approaches to DSMC/Navier-Stokes coupling that enforce flux boundary
conditions at the coupling interfaces are not well suited for multiscale simulation of microfluidic devices. For low
speed flows, the application of Dirichlet boundary conditions at the coupling interfaces is more appropriate [7]. In
our previous work [8], the coupling of DSMC with Stokes equations using Schwarz coupling and Dirichlet-Dirichlet
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
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boundary conditions was investigated and it was demonstrated that using Schwarz coupling the computational time
required for the simulation of the microfluidic problem can be reduced. Also, the convergence properties of the
coupled method were investigated in detail. In this paper the application of the Schwarz method is extended for
coupling of DSMC with Navier-Stokes equations. As in reference [8], the multiscale method presented in this paper
uses a scattered point finite cloud method (FCM) [9, 10] for the solution of the Navier-Stokes equations in the
continuum regions. FCM uses a fixed kernel technique for the construction of interpolation functions and a
collocation technique for the discretization of the governing equations. A fixed kernel technique is also used to
construct a scattered point interpolation scheme between continuum and DSMC regions.
This paper is aimed mainly towards demonstrating successful coupling between Navier-Stokes equations and
DSMC. Microfilters are used as a prototypical test case since they exhibit clear multiscale behavior. In addition,
some of the important details of the coupling of the Navier-Stokes equations and DSMC method are discussed.
DSMC/NAVIER-STOKES COUPLING
The multiscale approach discussed in this paper uses an overlapped Schwarz method with Dirichlet-Dirichlet
type boundary conditions for solving the steady-state flow problems encountered in microfluidic filters. It is
assumed that the validity regions for continuum and atomistic models have already been identified. The continuum
region, where continuum models hold good, is simulated by Navier-Stokes equations using the Finite Cloud method
[10], and the atomistic region, where continuum models fail, is simulated by DSMC [11].
A Schwarz technique is employed to solve the coupled DSMC/Navier-Stokes problem on overlapping
subdomains. In Figure 1, a sample simulation domain, which is divided into two overlapping subdomains, is shown.
The alternating Schwarz method for the overlapping subdomains shown in Figure 1 can be summarized as follows:
Begin : n = 0; u 2( 0 )
Γ1
= initial condition
Repeat{ n = n + 1
Solve Lu1( n ) = f1 on Ω1 with u1( n ) = u 2( n −1) on Γ1
Solve Lu 2( n ) = f 2 on Ω 2 with u 2( n ) = u1( n −1) on Γ2
} until convergence
where n is the iteration number, ui(n) is the solution in domain Ωi at iteration n, L is the partial differential operator
describing the governing equations and fi are forcing functions of position in domain Ωi. In the alternating Schwarz
method the subdomains are overlapped and Dirichlet type boundary conditions are employed on the boundaries Γ1
and Γ2 of both subdomains.
Ω1
Ω2
Γ2
Γ1
Figure 1: Decomposition of a sample geometry into two overlapping subdomains.
Coupled Approach
A high-level description of the coupled DSMC/Navier-Stokes approach is shown in Algorithm 1. Given an
arbitrary initial state and an initial set of boundary conditions along the coupling interfaces, a Schwarz technique is
implemented to find a self-consistent solution to the Navier-Stokes and the DSMC subdomains. At each Schwarz
iteration, the updated boundary conditions at the coupling interfaces are calculated using scattered point
interpolation as described in [8]. After the convergence of the coupling iterations, additional coupling iterations
between Navier-Stokes and DSMC subdomains are performed and the intermediate DSMC results are saved to a file
for a post-processing step. The final results in the DSMC subdomain are obtained as an average over all of the
collected samples. As a last step, the final results from the DSMC subdomains are interpolated on to the boundary
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Coupling iteration
for each DSMC subdomain Di do
simulate Nstep DSMC time steps in Di
end for
interpolate DSMC to Navier-Stokes
reset accumulators
for each Navier-Stokes subdomain Si do
solve Navier-Stokes equation in Si
end for
interpolate Navier-Stokes to DSMC
Main loop
initialize DSMC particle states
initialize DSMC boundary conditions
while coupling iterations not converged do
coupling iteration
check for convergence
end while
start saving accumulators
while statistical noise > limit do
coupling iteration
check for convergence
end while
compute averages from saved accumulators
interpolate DSMC to Navier-Stokes
for each Navier-Stokes subdomain Si do
solve Navier-Stokes equation in Si
end for
Algorithm 1: Description of Navier-Stokes/DSMC coupling
DSMC
Stokes
Stokes
C
A
dov
Di
G
So
p1
Si
hc
dext
dMC
B
hf
Do
p2
y
dext
H
D
lin
E
dov
dMC
x
F
lc
l out
Figure 2: The geometry of the microfilter device. Also shown in the figure are the Navier-Stokes and DSMC subdomains
and the overlap between the two subdomains.
nodes of the continuum subdomains, and the governing equations are solved to find the solution in the continuum
subdomains.
The geometry of the microfilter for coupled simulation is shown in Figure 2. The variables interpolated between
subdomains for coupling is specified in Table 1. The coupling of the pressure and velocity is performed in the same
manner as the DSMC/Stokes coupling [8]. For the coupling of temperature, several alternatives were tried. It was
found that coupling the temperature in the same way as velocity gives the best results. That is, the temperature
estimated from DSMC is interpolated to the Navier-Stokes subdomains; and after the Navier-Stokes solution, the
temperature from within the continuum subdomain is interpolated back to the DSMC boundary cells. Unlike
velocity coupling, in the absence of overlap, the temperature does not converge. For this reason, all tests of
DSMC/Navier-Stokes coupling uses non-zero overlap.
In addition to the scheme described in Table 1, other possibilities for the coupling of temperature were also
investigated: It was observed that the Navier-Stokes solution does not converge if the temperature is not specified at
the interfaces with the atomistic model. Also, it was observed that when the temperature at the boundary of the
DSMC subdomain is updated by extrapolating the value from the neighboring cells, the method becomes unreliable,
with the temperature solution differing significantly from the DSMC solution in some cases.
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Table 1: A summary of boundary conditions on various surfaces of the microfilter geometry.
Surface
A
B
C,E
D,F
Si,So
Di,Do
G,H
Pressure
1.3 atm
1.0 atm
∂P/∂y = 0
∂P/∂y = 0
–
P = NS solution
–
x-velocity
–
–
0
0
vx = DSMC estimate
vx = NS solution
diffusive
y-velocity
0
0
∂vy/∂y = 0
∂vy/∂y = 0
vy = DSMC estimate
vy = NS solution
diffusive
Temperature
300 K
300 K
∂T/∂y = 0
∂T/∂y = 0
T = DSMC estimate
T = NS solution
300 K
SIMULATION RESULTS
For the filter shown in Figure 2, hf = 5 µm, lc = 1 µm, hc = 0.8 µm, lin = 6 µm, and lout = 8 µm is used. Figure 2
also shows the decomposition of the filter geometry into Navier-Stokes and DSMC subdomains. The extension of
the DSMC subdomain on each side of the channel is denoted by dext. The overlap between DSMC and the NavierStokes subdomains is denoted by dov. The overlap distance is measured from the center of the DSMC estimation
cells to the continuum nodes, i.e., the generation cells are not counted in the overlap, as these cells do not have valid
data that can be used. An identical overlap distance, dov, is used for both the input and the output regions. For the
initial state and boundary conditions vx=0 m/s, vy=0 m/s, T=300 K was used. The initial state and the boundary
conditions for pressure were set to P=1.3 atm at the high-pressure side, P=1.0 atm at the low-pressure side, and
P=1.15 atm within the channel. The boundary conditions imposed on various surfaces of the microfilter geometry
are listed in Table 1. For all the simulations, a DSMC time step of 10 ps was used. For the coupled DSMC/NavierStokes analysis, a total of 80e3 DSMC iterations were performed to make certain the coupling procedure has
converged, and the averages were collected for at least 2 µs. For the DSMC simulations, a transient of 1.5 µs was
simulated and averages were collected for 1 µs. In the DSMC subdomain, the simulated fluid was nitrogen with the
internal degrees of freedom ignored. The parameters for the fluid in the Navier-Stokes subdomain were set to
correspond to the parameters of the fluid simulated in the DSMC subdomain.
The filter geometry shown in Figure 2 was simulated with the coupled method using dMC = 0.4 µm, and dov =
0.17 µm. The results for pressure, x-velocity, and temperature are compared with the DSMC solution in Figures 3
and 4. From these figures a very good agreement of the coupled and DSMC results is observed. The speed-up
obtained by the coupled approach is defined as the ratio of the CPU time used by the DSMC simulation to the CPU
time used by the coupled approach. For the example given here, the speed-up was measured to be 8.26, showing that
significant speed-up can be obtained by DSMC/Navier-Stokes coupling. The speed-up obtained for the same
problem with DSMC/Stokes coupling was 2.17. Thus, using DSMC/Navier-Stokes coupling the speed-up is
increased.
The plots in the cross-sectional direction (y-direction) are provided in Figures 6-7 to show the degree of
agreement obtained in a direction perpendicular to the main flow direction. The results are plotted along the
interface of the DSMC and Navier-Stokes subdomains on the low-pressure side (x=7.4 µm). From these results it is
seen that within the noise levels all results show good agreement with DSMC results. In Figure 5 two plots of
temperature are provided along lines separated by a single DSMC cell distance to clarify that the perceived trends in
the DSMC data are only short-distance correlations of noise. From the plot of x-velocity and y-velocity, shown in
Figures 6, a very good agreement of DSMC and coupled results is observed. In analyzing Figure 7, attention needs
to be paid to the scale of the vertical axis. The pressure difference encountered in the solution is 28.5e3 Pa, as seen
from Figure 3, whereas the difference in the results for pressure, shown in Figure 7, is only about 20 Pa. The
comparison of these two figures indicates a very high level of accuracy for coupled pressure.
In order to further verify the results, the total temperature, shown in Figure 7 was integrated along the line at
which the results are plotted x=7.4 µm). The results of the integration from the DSMC simulation, the DSMC
subdomain of the coupled simulation, and the Navier-Stokes subdomain are 7.5158e4, 7.5159e4, and 7.5271e4 K,
respectively. The results agree to within 0.1%, and demonstrate the successful coupling achieved by the multiscale
method.
From Figure 8 it is seen that the overlap between subdomains does not significantly affect the convergence of
velocity. However, it must be noted that, the DSMC/Navier-Stokes coupling by the present method did not converge
when the overlap was reduced to zero. This convergence behaviour needs to be further investigated.
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5
1.35
x 10
100
coupled
DSMC
DSMC
coupled
DSMC
90
1.3
80
1.25
Navier−Stokes
subdomain
x−velocity (m/s)
pressure (Pa)
70
1.2
Navier−Stokes
subdomain
1.15
60
50
Navier−Stokes
subdomain
Navier−Stokes
subdomain
40
1.1
30
1.05
DSMC
20
1
0
0.5
1
1.5
position (m)
10
0
0.5
1
1.5
position (m)
−5
x 10
−5
x 10
Figure 3: The DSMC and coupled results for pressure and x-velocity plotted along the midline. The vertical lines denote the
boundaries of the Navier-Stokes subdomains.
301
coupled
DSMC
DSMC
300
temperature (K)
299
298
Navier−Stokes
subdomain
Navier−Stokes
subdomain
297
296
295
294
0
0.5
1
1.5
position (m)
−5
x 10
Figure 4: The DSMC and coupled results for temperature plotted along the midline. The vertical lines denote the boundaries of
the Navier-Stokes subdomains.
302
301
coupled
DSMC
301.5
coupled
DSMC
300.5
300.5
temperature (K)
temperature (K)
301
300
299.5
299
300
299.5
299
298.5
298.5
298
297.5
−3
−2
−1
0
position (m)
1
2
3
−6
x 10
298
−3
−2
−1
0
position (m)
1
2
3
−6
x 10
Figure 5: The DSMC and coupled results for temperature. The first plot is made along the interface of the Navier-Stokes
subdomain at the low-pressure side. The second plot is made along a line displaced by one DSMC cell size from the interface of
Navier-Stokes subdomain.
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80
20
coupled
DSMC
70
60
10
50
y−velocity (m/s)
x−velocity (m/s)
coupled
DSMC
15
40
30
20
5
0
−5
−10
10
−15
0
−10
−3
−2
−1
0
position (m)
1
2
3
−20
−3
−2
−1
−6
x 10
0
position (m)
1
2
3
−6
x 10
Figure 6: The DSMC and coupled results for x-velocity and y-velocity plotted along the interface of the Navier-Stokes/DSMC
subdomains on the low-pressure side.
5
1.035
x 10
coupled
DSMC
1.03
pressure (Pa)
1.025
1.02
1.015
1.01
1.005
1
−3
−2
−1
0
position (m)
1
2
3
−6
x 10
Figure 7: The DSMC and coupled results for pressure and energy plotted along the interface of the Navier-Stokes/DSMC
subdomains on the low-pressure side.
45
overlap = 0.9 µm
overlap = 2.9 µm
40
Absolute error (m/s)
35
30
25
20
15
10
5
0
0
10
20
30
40
Number of coupling iterations
50
60
Figure 8: Comparison of convergence of velocity boundary conditions transferred from the Navier-Stokes subdomain on the
low-pressure side of the filter.
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CONCLUSIONS
The DSMC/Navier-Stokes coupling using Schwarz method was demonstrated. Details of the coupling process
for proper coupling of temperature were investigated, and a method that yields agreement with DSMC results is
presented. Good agreement of the coupled results and the achievement of increased computational timesavings as
compared to DSMC/Stokes coupling are shown. The coupled simulation is able to achieve agreement with the
DSMC results in the presence of large gradients at the coupling interface.
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