3D Unstructured Couette Flow Simulation in Transition
Region
HsinChih Frank Liu
Department of Information Management, Shih Chien University, Kaohsiung, Taiwan
Abstract. Couette Flow in transition region is simulated in this research by using 3D Unstructured DSMC method
(UDSMC in [1]). A couple of the boundary conditions are developed and applied in the computational domain including
periodic condition, specular reflection, movingly diffusive reflection, and fixedly diffusive reflection. After applying
these boundary conditions, the Couette flow with an infinite length in the real case can be simulated in a cubical domain.
In order to extend the techniques to arbitrary geometries in the fields of rarefied gas flow and micro flow, we use
tetrahedral meshes and develop unstructured DSMC procedures in this research. In each time step, the trajectory of each
sample particle will be considered to be divided into multiple stages to cover all of the possibilities that particles interact
with different types of boundary conditions. The physical data including the 3D results of velocity, density, and
temperature are demonstrated in this research.
INTRODUCTION
Plane Couette flow is an essential fluid dynamic problem and its results are widely used as a benchmark for
verification on the simulation field [4][5]. Plane Couette flow is driven by one moving surface on the top and the
other stationary surface on the bottom. Both of them interact with particles by diffusive reflections in an infinite
length. We choose Direct Simulation Monte Carlo (DSMC) method to study the problem, since it has been a
popular way to simulate the flow in a broad Knudsen number range [2][3]. In this research, we construct two
specular surfaces and two periodic boundaries, together with two diffusive surfaces, to form a 3D domain to perform
Couette flow in the transition region. We also expect the computational procedures and algorithms developed here
are capable for arbitrary geometry problems such as Couette flow with riblets that can be dealt with by unstructured
meshes and periodic condition. Our simulation results [Fig. 4] show a similar shape to the one-dimensional
simulation results of Couette flow in [4]. The velocity gradient is nearly uniformly distributed from the moving wall
to the stationary wall. Moreover, both temperature and density perform parabolic shapes of distribution.
PERIODIC AND MULTIPLE-REFLECTION PROCEDURES
Multi-reflection process consists of diffusive reflections and specular reflections. In our combined periodic and
multiple-reflection process, the multi-reflection process will be considered first [Fig. 1] for particles. Once these
particles move out of the cubical domain, the periodic process will be undertaken. After the periodic process is
completed, particles might encounter another multi-reflection process in the other side of the domain. Thus, multireflection process and periodic process contribute into one recurrence stage [Fig. 1].
When a particle goes through a periodic process, the position of the particle will be shifted to the other side of the
domain. Zn is defined as the coordinates of the nth particle. Zn ∈ R. L is the width of the domain. When the
periodic process occurs, the downstream’s particles will be shifted to the upstream, Zn = Zn + L; the upstream’s
particles will be shifted to the downstream, Zn = Zn - L.
When particles go across periodic boundaries, the original properties of the particles will be remained as
deterministic properties in one time step. Once the process is completed, the indices of the particles will be
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
Start
Move particles along their
trajectories in each time step.
Computing Multi-reflection process
with solid boundaries as they occur
Perform Periodic Process
Particles move out of the
Domain?
Yes
No
Stop
FIGURE 1. Flow Chart of Multi-reflection process and Periodic process.
Periodic
Periodic
Periodic
Periodic
Specular
Reflection
(a)
Specular
Reflection
M oving
Wall
(b)
Stationary
Wall
(c)
FIGURE 2. Periodic condition, specular reflection, and diffusive reflection (moving wall and stationary wall) used in the
simulation domain. The upper particle (green color) hits the top wall and experiences reflection process first. The lower particle
(blue color) goes through periodic boundary first. (a) A top view of the domain. (b) A front view of the domain. (c) A side view
of the domain.
refreshed for the computations in next stages.
In Figure 2, two possible motions of particles are illustrated. The upper particle hits the moving wall and invokes
a diffusive reflection process. Then the particle is assigned with a new velocity for moving out of the domain and
undertakes a periodic process; after that, its deterministic physical properties are remained and the particle keeps
moving until hitting a specular wall. The lower particle moves out of the domain and proceeds to a periodic process.
In this process, its deterministic velocity is kept and leads the particle to hit the bottom wall and encounter a
diffusive reflection process.
The procedures demonstrated in Figure 1 are applicable for arbitrary geometry cases where particles might move
across a single periodic boundary back and forth, as the reflection occurs on the other side of the domain and
bounces particle back [1].
MULTIPLE-REFLECTION PROCESS
We develop a computational sequence suitable for both specular reflections and diffusive reflections in multiplereflection process. Whenever a particle starts moving and a reflection occurs, the intersection point between the
particle and the surface will be found, and the reflected velocity will be calculated. In this research, solid boundary
is considered as an area covered by triangles that consist of one single side of the tetrahedrons adjacent to the solid
boundary. On the other hand, particle’s motion is thought of as a straight line. Therefore, we can treat the multiplereflection process as solving a computational geometry problem.
The normal vector of triangles is x0. Each triangle has a set of edge vectors, x3. The trajectories of the particle’s
movement will be treated as a line segment with a set of edge vectors, x2. We use the following steps to compute the
intersection point and the reflected physical properties:
(1) Use linear operator L, L: R3 → R2. R2 is spanned by standard bases.
(2) Compute L(x0) = Ax0 and find the matrix A.
(3) Applying the rotational (orthogonal) matrix A on x2.
(4) Compute x3 = {Ax2
∩ R2} and rule out the particles with impossibilities of reflection.
(5) Apply matrix A on x1 to get a set of vectors, x4, where x4
∈ R2.
( y − y 4) 
 ( x 3 − x 4)
3i
i
 , i = 1~3. When (i+1)>3, assign x 3
(6) Compute Bi = det 
= x 3 and
i +1
1
(
−
)
(
− )
y
x
x
 3i + 1 4
3i + 1 y 4 
= y . When B1, B2, and B3 have the same sign, the index will be established.
y3
31
i +1
(7) Go through either diffusive or specular reflection process and compute reflected properties.
(8) After calculating reflected velocity and forming a new line segment, rotating it back to its original space by
applying the matrix AT.
(9) Go back to step (1) if another reflection process might occur.
GROUP ESTABLISHMENT
In each time step, resulting from the molecular collisions or surface reflections, once the particle moves out from
the original cell to a new cell, a new index between the particle and the new cell will be established. In this research,
we form a group by a number of cells to reduce the computing time for indexing and locating particles. Groups can
be formed by either triangular cells in 2D [Fig. 3] or tetrahedral cells in 3D. We will describe the 3D group forming
in the next section.
Group can have multiple layers formed by cells. The size of a group, which is equal to the number of a group’s
layers, is required to be large enough to cover all of particle’s trajectories and is determined by the time step. More
layers will be established for each group in a bigger time step. Once the accuracy is concerned in DSMC method by
limiting the length of the time step, an upper bound will be enforced on the number of layers in each group.
(a)
(b)
FIGURE 3. Group forming illustrated by 2D triangular cells. Black color represents the original cell. (a) One layer group
without contacting the boundary. (b) One layer group along with a boundary.
DEFINITION AND FORMULATION
k
Different types of groups are built for different purposes in the simulation stages. G n is the group of the nth cell
k
k
with k layers; when particles collide with other particles, cells in group G n will be checked. B n is the group
k
k
formed by solid-boundary cells only. A n is the group formed by B n group’s cells and is built for every multiplek
k
reflection process. D n is the group formed by the cells in inlet-outlet boundaries only. C n is the group formed by
k
k
D n group’s cells; when a particle goes across a periodic boundary, cells in group C n will be searched for new
indexing.
k
E n is formed by the cells in Ω E and is built specifically for the combined periodic and multiple-
reflection process.
1
Definition 1. G n = {m: ∃ i, j = 1, 2, 3, and 4, such that Xm(j) = Xn(i)}, n
k
Definition 2. G n =
UG
m∈
1
m,
n ∈Ω.
k −1
Gn
k
Definition 3. B n = {m: (m ∈ G kn ) ∩ (m ∈ Ωb )}, n ∈ Ωb .
k
Definition 4. A n =
UG
m∈
q
Bn
k
m
,n
∈ Ωb .
∈Ω.
k
Definition 5. D n = {m: (m
k
Definition 6. C n =
UG
D
m∈
k
Definition 7. E n =
k
m
, n ∈ Ωo .
q
n
U
m∈
∈ G kn ) ∩ (m ∈ Ωo )}, n ∈ Ωo .
k
Bm .
ΩE
n is the cell number and k is the layer number. Xn(i) is defined as the coordinates of the nth cell’s edge. Xn(i) ∈
R, edge number i = 1, 2, 3, and 4. Ω represents the entire computational domain. Ωo represents the domain where
the cells locate at inlet-outlet boundaries. Ωb represents the domain where the cells locate at solid boundaries.
Ω E represents the domain where the cells locate at solid boundaries and domain edges.
SIMULATION RESULTS
The simulation results are shown in Figure 4 and Figure 5. The condition settings are described as following:
Moving wall velocity is 300 m/s. The global Knudsen number is 0.1. The initial free stream velocity is 300 m/s.
Wall temperature is 273 K for both sides. 4206 tetrahedral cells are used in this computing. The VHS molecular
model is employed and the gas is Nitrogen. The computational domain expands from –1.0 to 1.0 in x-axis, from -1.0
to 1.0 in y-axis, and from –0.2 to 0.2 in z-axis. The moving wall condition is assigned on x-z plane in y = 1.0. The
stationary wall condition is assigned on x-z plane in y = -1.0. The specular conditions are assigned on x-y planes in
z = 0.2 and z = -0.2. The periodic conditions are assigned on y-z planes in x = 1.0 and x = -1.0.
Figure 4 demonstrates 4 plots: mesh architecture and boundary conditions, the profile of velocity-U distribution,
the profile of temperature distribution, and the profile of density distribution. Due to the effect of specular
conditions, the physical values along the x-z plane in each plot are without variation as expected. Velocity
component U is shown with nearly linear distribution. Both temperature and density have parabolic distributions.
These characteristics match the results with one-dimensional Couette flow simulation in [4]. Velocity slip and
temperature jump, proportional to Knudsen number (or mean free path) as well as the gradient of physical
properties, has a larger scale compared to [4] which is in a smaller Knudsen range. Figure 5 demonstrates the
profiles of velocity-V and velocity-W. As expected, both plots show the value with small deviation around zero in
the whole domain.
U
Moving Wall
U = 300
244.622
1
Periodic
Periodic
172.996
1
0.5
0.5
Y
101.37
0
Y
0
-0.5
-0.5
1
29.7442
0.5
1
0
-1
-0.2
Z
0
-0.5
0.2
0
-1
0.5
-0.2
0
Z
X
-0.5
X
-1
0.2
-1
Velocity - U
Specular
Stationary Wall
Y
X
D
Z
OT
287.959
3.03664E-07
1
1
2.97691E-07
282.279
0.5
0.5
Y
0
Y
278.221
0
2.91717E-07
-0.5
-0.5
1
1
0.5
0
-1
-0.2
Z
0
-0.5
0.2
X
-1
Temperature
0.5
274.164
0
-1
-0.2
Z
0
-0.5
0.2
X
2.85744E-07
-1
Density
FIGURE 4. Couette flow in 3D unstructured simulation domain. The upper-left plot shows the mesh architecture and
boundary condition setting. The upper-right plot shows the profiles of velocity-U with a nearly linear distribution. The lower-left
plot shows the temperature distribution. The lower-right plot shows the density distribution. The distributions of temperature
and density perform in parabolic shapes.
V
7.27402
1
3.57248
0.5
Y
0
-0.129062
-0.5
1
0.5
0
-1
-0.2
0
Z
-0.5
0.2
X
-1
-3.8306
Velocity - V
Y
W
X
2.68635
Z
1
0.5
0.60391
Y
0
-1.47853
-0.5
1
0.5
0
-1
-0.2
Z
0
-0.5
0.2
X
-1
-3.56097
Velocity - W
FIGURE 5. The profiles of Velocity-V and Velocity-W for Couette flow in 3D unstructured simulation domain.
REFERENCES
1. Liu, H. F., “2D & 3D Unstructured Simulations and Coupling Techniques for Micro-geometries and Rarefied Gas Flow”,
Brown University, PhD, 2000
2. Liu, H. F., Gatsonis N., Beskok A., Karniadakis G. E., “Simulation Models for Rarefied Flow Past a Sphere in a Pipe”, 21st
International Symposium on Rarefied Gas Dynamics, 1998.
3. Liu, H. F., Gastsonis N., Beskok A., Karniadakis G. E., “Flow Past a Micro-Sphere in a Pipe: Effects of Rarefaction”, ASME
Micro-Fluidics Symposium, 1998.
4. Bird G.A., “Molecular Gas Dynamics and the Direct Simulation of Gas Flows”, Clarendon Press, Oxford, 1994.
5. Cercignani, C., “Rarefied Gas Dynamics”, Cambridge University Press, 2000.