Collisionless gas flows. I. Inside arbitrary enclosures - NMSU

PHYSICS OF FLUIDS 20, 067105 共2008兲

Collisionless gas flows. I. Inside arbitrary enclosures Chunpei Caia兲 and Danny D. Liub兲 ZONA Technology Inc., Scottsdale, Arizona 85258, USA

共Received 30 January 2008; accepted 22 May 2008; published online 27 June 2008兲 This study concentrates on steady collisionless gas flows inside arbitrary enclosures, which are either convex or concave, two-dimensional or three-dimensional, formed by several plates maintained at different temperatures. If the molecular reflections on these plates are completely diffuse, then at the final steady flow stage for any point inside the enclosures, the velocity distribution function 共VDF兲 is completely determined, and macroscopic properties such as density, velocities, temperature, and heat flux can be exactly determined by integrating the VDF with different moments. The result from this study leads to many exact solutions for internal collisionless gas flow and thermal fields, such as those inside vacuum packaged micro-/nanoelectromechanical system devices. These solutions are applicable as base solutions to solve flows with a much smaller Knudsen number. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2947585兴 I. INTRODUCTION

This study concentrates on collisionless gas flow fields, such as those inside vacuum packaged micro-/nanoelectromechanical system 共MEMS/NEMS兲 devices. As well known, most MEMS devices have to be packaged in vacuum chambers before usage in order to obtain stable performances.1,2 Usually, a package consists of a hot chip on one plate and several other plates maintained at lower temperatures. Inside the enclosure formed by those plates, there is internal rarefied gas flow transferring heat from the hot chip to the cold plates. The dimensions of the MEMS devices can be very small and the vacuum chambers can maintain at very low pressures. Especially nowadays, due to the continuous development of MEMS/NEMS devices with smaller dimension, and lower background pressure in vacuum chambers, it is very reasonable to believe that gas flows inside vacuum packaged MEMS/NEMS devices can be highly rarefied. A collisionless flow state provides extreme heat transfer rate and may be very helpful for design purpose. In the literature, there are very rare analytical results about the internal collisionless gas flow and temperature fields inside enclosures formed by several plates maintained at different temperatures. Previously, we reported a study of such free molecular gas flows inside vacuum packaged MEMS devices.3 In that study, we proposed a four-node model which was based on the fact that on the same straight plates maintained at the same temperature, the distribution functions at different locations must be the same due to completely diffuse reflections. However, for enclosures of concave shapes with curvatures, it is impossible to construct a similar “N-node” model; and for more practical 3D enclosures, it is rather complex to construct a similar N-node model. Hence, our previous work is not general enough for real enclosures formed by multiple plates with complex gea兲

Electronic mail: [email protected]. Electronic mail: [email protected].

b兲

1070-6631/2008/20共6兲/067105/8/$23.00

ometries, and the major goal of this short paper is to extend the applicability of the previous speculations3 from convex enclosures to general enclosures of arbitrary shapes. II. SPECULATIONS

Here we extend the speculations obtained in our previous study3 to the following new formats. Speculations. Inside an arbitrary shaped enclosure formed by different plates which are maintained at different temperatures, no matter whether this enclosure is two dimensional or three dimensional, convex or concave, as long as the molecular reflections at the plates are completely diffuse and the plates are stationary, then at the final steady state 共i兲 the internal flow field has zero macroscopic velocity everywhere; 共ii兲 if the group of particles, reflected at and traveling away from the ith plate which is maintained at Ti, has a number density ni, then ni冑Ti is a constant for all plates; 共iii兲 for any point inside the enclosure, the velocity distribution function at that point consists of several Maxwellian distribution functions; each of these functions is valid inside a specific solid angle subtended by the point and different plate; 共iv兲 the pressure field inside the enclosure is not constant; 共v兲 for flows with multiple species, the above speculations are applicable separately for each species as well. These speculations have been discussed previously for a convex enclosure.3 The same arguments in our previous work3 are still applicable here for speculations 共i兲, 共iii兲, and 共iv兲; while the validity of 共v兲 is evident because the flows are collisionless. We do not repeat them here to keep this paper concise. Instead we concentrate on the second speculation. First we revisit the definition of a completely diffuse reflection. No matter how fast molecules travel toward a plate before their collisions, after the diffuse reflections on a plate those molecules can travel away along any direction in front of the plate, with an equal probability inside a solid angle of ␲ for a two-dimensional case 共2␲ for a 3D case兲 and with a thermal speed characterized by the plate temperature Ti,

20, 067105-1

© 2008 American Institute of Physics

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067105-2

Phys. Fluids 20, 067105 共2008兲

C. Cai and D. D. Liu

f i共u, v,w兲 = ni

冉 冊 冉 1 2␲RTi

3/2

exp −

冊

u2 + v2 + w2 , 2RTi

共1兲

where R is the gas constant. Then from a point on the ith plate, along any direction ni⬘ or any ray from this point, the molecular flux for the outflow is ˙i= m

冕

+⬁

0

u ⬘n i

冑2␲RTi

冉

exp −

冊

u ⬘2 du⬘ = ni冑RTi/共2␲兲, 2RTi

共2兲

where u⬘ is the thermal velocity along the ni⬘ direction. The probability distribution function follows Eq. 共1兲. Because the reflection is completely diffuse, molecules travel along any ray from the point on the plate with the same probability, hence the above relation is valid for any direction ni⬘. Equation 共2兲 indicates that the mass flux is related with the wall temperature and the number density for the group of molecules only. Secondly, if this ray interacts the jth plate with a different temperature T j, then with a steady state of zero flux ˙ i=m ˙ j or ni冑Ti = n j冑T j. Whether everywhere, we must have m the enclosure is convex or concave does not change the result: for a concave domain, the ray originating from one point can reflect at different plates several times, and finally when it reaches the other side of the concave domain, the flux on the other side of the concave part must satisfy the flux balance relation as well. For example, in Fig. 1, along ˙ 4=m ˙1 ˙ 4=m the ray starting from point F, we must have m ˙ 2 since the ray reflects on the plates with different tem=m peratures. Hence, for all the plates maintained at different

fP =

冦

冋 冋 冋 冋

册 册 册 册

− 共u2 + v2 + w2兲 n1 exp , u/v 苸 ⬔A1OB1 共2␲RT1兲3/2 2RT1 − 共u2 + v2 + w2兲 n2 , u/v 苸 ⬔B1OC1 3/2 exp 共2␲RT2兲 共2RT2兲 − 共u2 + v2 + w2兲 n3 exp , u/v 苸 ⬔C1OE1 共2␲RT3兲3/2 2RT3 共u2 + v2 + w2兲 n4 , u/v 苸 ⬔E1OA1 , 3/2 exp − 共2␲RT4兲 2RT4

where A1O in the right phase plot is parallel to line PA in the left domain; E1O in the right phase plot is parallel to PE in the left domain. In the right phase plot, the region inside solid angle E1OA1 is formed by the molecules in the left domain which travel to point P and within solid angle ⬔APE. Correspondingly, the molecules traveling toward point P have velocities within different solid angles subtended by point P and specific plate ends. Narasimha originally adopted this solid angle approach to study an effusion flow problem.4 From the classical gaskinetic theory, if at one point P,

FIG. 1. An arbitrary two-dimensional enclosure with plates maintained at four temperatures, and the velocity phase space of point P.

temperatures, the second speculation must hold. At a steady state, molecules reorganize their number densities only according to the plate temperatures, and local curvatures have no effects on the number density at all. This explanation is more general and simpler than the four-node model approach in our previous study.3 The third speculation is illustrated by Fig. 1 as well, which is an arbitrary two-dimensional enclosure formed by four plates maintained at different temperatures. On the boundary there are five arbitrary points, A, B, C, D, and E, and they divide the boundary into four regions of different temperatures. The velocity phase for point P consists of four pieces of Maxwellian distribution functions characterized by four temperatures and four number densities,

冧

共3兲

the velocity distribution function is known, then with higher order moments, the macroscopic properties, such as density, temperature, and gaskinetic heat flux can be obtained as

n共x,y兲 =

冕

T共x,y兲 =

1 3Rn共x,y兲

f P共x,y兲dudvdw,

共4兲

冕

共5兲

共u2 + v2 + w2兲f P共x,y兲dudvdw,

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Phys. Fluids 20, 067105 共2008兲

Collisionless gas flows. I. Inside arbitrary enclosures

1

0

420

400

0 38

1.02

6 1.04

0

0

1.0

50

48

460

0.8

440

92 0.

1. 08

0

52

4

0.96 0.98

1

0.9

0.8

1

0.9

1.1

36

067105-3

0.6

Y/H

Y/H

0.6

0.4

0.4

0.2

0.2

0

0

0.2

0.4

X/L

0.6

0.8

0

1

FIG. 2. Contours of normalized number density for the first test case; solid line: analytical; dashed line: DSMC.

q˙i共x,y兲 =

冕冋

册

1 ci ⑀ + m共u2 + v2 + w2兲 f P共x,y兲dudvdw. 2

0

0.2

0.4

X/L

0.6

0.8

1

FIG. 3. Contours of temperature 共kelvin兲 for the first test case; solid line: analytical; dashed line: DSMC.

T共x,y兲 =

1 兵␣共x,y兲TLnL + 关2␲ − ␣共x,y兲兴TRnR其, 2␲n共x,y兲

共6兲

共8兲

Equation 共5兲 utilizes the first speculation that macroscopic velocity equals to zero everywhere inside such an enclosure; ⑀ is the gas internal energy and ci is a thermal velocity component. As we have pointed out in our previous paper,3 those internal collisionless flows represent a challenge to the direct simulation Monte Carlo 共DSMC兲5 method because of the absolute zero macroscopic velocity; due to the nonuniform pressure distributions inside an enclosure, they present a challenge to the current information preservation6–9 共IP兲 schemes as well.

␣共x , y兲 = arccos关x / 冑x2 + 共y − L / 2兲2兴 + arccos where 2 2 冑 关x / x + 共y + L / 2兲 兴, it is the solid angle subtended by point P共x , y兲 and points Q1共L / 2 , 0兲, Q2共L / 2 , L兲. The next step is to determine nL and nR. Suppose there is no leakage from the enclosure, then the mass of argon gas inside the enclosure must be a constant value before and after the packaging process. Integrating Eq. 共7兲 over the whole square domain, we obtain

III. EXAMPLES AND NUMERICAL VALIDATIONS

To demonstrate the extended speculations, we present several test cases, with their analytical and simulation results. For the simulation cases in this paper, we use argon gas and the initial number density is set to n0 = 1 ⫻ 1020 m−3. A. An enclosure by four plates, two temperatures

The first case is a square enclosure formed by four plates with a length L = 1 m. The left half of the enclosure, x / L ⬍ 0.5, is maintained at TL = 300 K, while the right half is maintained at TR = 600 K. Figure 2 illustrates the domain configuration, which is similar to that investigated by Aoki,10,11 but we concentrate on the collisionless flow situation only. We set the coordinate origin at the bottom left corner of the enclosure. The velocity distribution for any point P共x , y兲 consists of two pieces of Maxwellian distribution functions. The density and temperature fields inside the enclosure are n共x,y兲 = 兵␣共x,y兲nL + 关2␲ − ␣共x,y兲兴nR其/共2␲兲,

共7兲

n0 = 0.5nL + 0.5nR ,

共9兲

where n0 is the initial or average density inside the enclosure and 0.5 is an average value of ␣共x , y兲 / 共2␲兲 over the whole domain. From the second speculation, we obtain nL冑TL = nR冑TR. Then, nL = 2冑2n0 / 共冑2 + 1兲 and nR = 2n0 / 共冑2 + 1兲 are completely determined. A DSMC simulation is performed with a specific package named MONACO.12 The collision function in MONACO is turned off, and a true collisionless flow is guaranteed. Because the flow is collisionless, hence the requirements on the simulation parameters are significantly relaxed. We utilize 100⫻ 100 cells along the X- and Y-directions in the simulation domain and about a half million particles in this simulation. Figures 2–4 show the normalized number density, by n0, the translational temperature, and normalized pressure distributions in the enclosure, respectively. These distributions are not uniform, and the numerical simulation results agree with the analytical results very well. In fact, the smooth contours from the analytical results are circular lines passing the two ending points Q1共L / 2 , 0兲 and Q2共L / 2 , L兲. Along the centerline Q1Q2, the temperature distribution is T共L / 2 , y兲 = 冑TLTR = 424.2 K, while the normalized density has a constant value of 1.0.

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067105-4

Phys. Fluids 20, 067105 共2008兲

C. Cai and D. D. Liu

1

80

Exact, no collision DSMC, no collision

1. 55 0.8

1.4

1.35

Y/H

1.45

0.6

q, W/m2

1.5

1.3

40

0.4

0

-40

0.2

0

0

0.2

0.4

X/L

0.6

0.8

-80 -1

1

FIG. 4. Contours of normalized pressure for the first test case; solid line: analytical; dashed line: DSMC.

The gaskinetic heat flux vector is very complex to compute, and we first illustrate the gaskinetic thermal heat flux over the bottom plate as an example. In this study, we consider gaskinetic thermal heat flux only and neglect other formats of heat transfer, such as radiation. Take one point P共x , 0兲 on the left side of the bottom plate, 0 ⬍ x ⬍ L / 2, if we denote ⬔Q1 PQ2 = ␣1 = arctan关L / 共L / 2 − x兲兴 which is an angle less than 90°, then Fig. 5 illustrates the velocity phase for point P共x , y兲 on the bottom plate. The I and II quadrants represent the particles reflected at the bottom plate, and the corresponding Maxwellian distribution function is characterized by number density nL and temperature TL. The III and IV quadrants which represent the particles traveling toward P共x , 0兲, inside a solid angle range of ⍀ = 关␲ , ␲ + ␣1兴, are the particles traveling from the right plates maintained at temperature TR = 600 K, and the Maxwellian distribution function is characterized by number density nR and temperature TR; the rest of the velocity phase represents the particles traveling from the left plates maintained at temperature TL = 300 K, and the Maxwellian distribution function is the same as that for quadrants I and II. Then the gaskinetic thermal heat flux over the bottom plate at point P共x , 0兲 , 0 ⬍ x ⬍ L / 2, is

0

1

s/L

2

FIG. 6. Heat flux at the left, bottom, and right plates; solid line: analytical; symbol: DSMC 共only one of every six symbols is shown兲. n0 = 1 ⫻ 1020 m−3, TL = 300 K, TR = 600 K.

qn,P共x,0兲 =

冕

1 nR共␤R/␲兲3/2 m共u2 + v2 + w2兲 2 ⍀

⫻exp关− ␤R共u2 + v2 + w2兲兴dudvdw +

冕

1 nL共␤L/␲兲3/2 m共u2 + v2 + w2兲 2 2␲−⍀

⫻exp关− ␤L共u2 + v2 + w2兲兴dudvdw =

m 关1 − cos共␣1兲兴 4

冋冑

nL

␲␤L3/2

−

nR

冑␲␤R3/2

册

共10兲

,

where ␤R = 1 / 共2RTR兲, ␤L = 1 / 共2RTL兲, and 0 ⬍ x ⬍ L / 2. After a similar derivation, we obtain the formula for the heat flux over the right half of the bottom plate, and the formula is asymmetric to the value on the left part. We obtain the thermal heat flux over the left vertical plate as qn,P共0,y兲 =

m 关sin共␣3兲 + sin共␣2兲兴 4

冋冑

nR

␲␤R3/2

−

nL

冑␲␤L3/2

册

, 共11兲

where ␣2 = arctan关2共L − y兲 / L兴 and ␣3 = arctan共2y / L兲. Figure 6 shows the gaskinetic thermal heat flux results over the left, bottom, and right plates. The X-axis represents the normalized distance by L from the origin point 共0, 0兲. E Ttop P F L Tbot A

FIG. 5. Velocity phase for a point on the left side of the bottom plate.

TL

B

R θ

C

TR

D

FIG. 7. Illustration for the second problem.

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Phys. Fluids 20, 067105 共2008兲

Collisionless gas flows. I. Inside arbitrary enclosures

5

0 X, m

0.5

0

0.0001

52

-1

460

0

-1.5

50

0.2 0.25 0.3 0.4

0

0.1

48

0. 15

0.0

067105-5

420

380 -0.5

0 X, m

0.5

1

1.5

-1.5

-1

-0.5

1

1.5

FIG. 8. Contours of solid angles ␣共x , y兲 / 共2␲兲.

FIG. 10. Contours of translational temperature 共kelvin兲 for the second test case; solid line: analytical; dashed line: DSMC.

The spans 共−1, 0兲, 共0, 1兲, and 共1, 2兲 represent the left, bottom, and right plates, respectively. The solid line represents analytical results corresponding to Eqs. 共10兲 and 共11兲, while the symbols are the DSMC simulation results, to clearly show the results, Fig. 6 only plots one of every six symbols. As we can see, the DSMC simulation results agree perfectly well with the theoretical results.

共1兲 From an internal point P共x , y兲, shown in Fig. 7, determine the two tangent points on the bottom semicylinder, L and R. If these two points are below line AB or line CD, then set the corresponding point, L or R to B or C. The solid angle ␤共x , y兲 is determined as ⬔LPR. 共2兲 At the left side, connect A to point P共x , y兲, then ⬔APL is ␣共x , y兲. If segment AP intersects with the bottom cylinder surface, then ␣共x , y兲 = 0, which means point P共x , y兲 is invisible from point A, and the left plate has a zero contribution to the density and temperature values at point P共x , y兲. Figure 8 shows the solid angle distribution ␣共x , y兲 / 共2␲兲. 共3兲 Repeat the above process for the right hand side plate CD, and ␥共x , y兲 is determined. 共4兲 ␨共x , y兲 is determined as 2␲ − ␣共x , y兲 − ␤共x , y兲 − ␥共x , y兲.

B. A concave enclosure

The second test case demonstrates the second speculation’s applicability to a concave enclosure, shown as Fig. 7. It has four plates, one top semicylinder, one bottom semicylinder, and they are connected by a left flat plate and a right flat plate. These four plates are maintained at different temperatures. After simple derivations, we obtain n共x,y兲 =

1 关␣共x,y兲nL + ␤共x,y兲nB + ␥共x,y兲nR + ␨共x,y兲nT兴, 2␲ 共12兲

T共x,y兲 =

␣共x,y兲 ␤共x,y兲 ␥共x,y兲 n LT L + n BT B + n RT R 2␲n 2␲n 2␲n +

␨共x,y兲 n TT T , 2␲n

共13兲

where subscripts L,B,R,T represent the left, bottom, right, and top plates; ␣共x , y兲, ␤共x , y兲, ␥共x , y兲, and ␨共x , y兲 represent specific solid angles subtended by point P共x , y兲 and different plates, respectively. The major steps to evaluate the four solid angles are summarized as follows.

By utilizing the mass conservation law and with a simple computer code, we can numerically integrate out the density distributions with Eq. 共12兲 over the whole concave domain: n0 = 0.1045nL + 0.1554nB + 0.1045nR + 0.6395nT. Further use speculation II, nL冑TL = nB冑TB = nR冑TR = nT冑TT, the complete relations among the number densities are uniquely determined. Integrating the velocity distribution function with different moments leads to density and temperature fields. A DSMC simulation is performed, the bottom and outer cylinder radii are set to R1 = 0.5 m and R2 = 1.5 m, respectively, mesh size of 100⫻ 50 along the circular and radical directions. In this simulation, we utilize 2 ⫻ 106 particles, TL = 200 K, TB = 300 K, TR = 400 K, and TT = 600 K. Figures 9 and 10 show satisfactory matches between the numerical and analytical results of normalized number density and temperature, respectively.

0.98 1

-1.5

-1

-0.5

0 X, m

0.5

94 0.

0.96

1.04 1.08 1.14

1

1.5

FIG. 9. Contours of normalized number density for the second test case; solid line: analytical; dashed line: DSMC.

A

B

O

θ

P α C C’

D

FIG. 11. Illustration of the computation relations for the thermal heat flux at point P.

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067105-6

Phys. Fluids 20, 067105 共2008兲

C. Cai and D. D. Liu

bottom cylinder temperature TB and the number density nB, and ⬔vOC⬘ = ␪. Region ⍀2 is within ⬔C⬘OD⬘ = ␣ and is formed by those particles traveling from the right flat plate, and the Maxwellian PDF is characterized by the right flat plate temperature TR and number density nR. The rest of the domain, ⍀3, represents the particles traveling from the top cylinder surface, and the velocity distribution function is governed by a Maxwellian distribution function, characterized by the top plate temperature TT and number density nT. Then the gaskinetic thermal heat flux at point P共x , y兲 is computed as

qn,P共␪兲 = FIG. 12. Velocity phase for a point P, shown in Fig. 11, on the right side of the bottom cylinder surface.

qn,P共␪兲 =

冦

mnB

+

mnR共cos ␣ − 1兲

2冑

−

+ , 2冑␲␤B3/2 2冑␲␤T3/2

−

+ 2冑␲␤B3/2

␲␤B3/2

mnB

mnB

4␤R3/2

冑␲

+

+

4␤L3/2

冑␲

冕

1 vnnR共␤R/␲兲3/2 m共u2 + v2 + w2兲 2 ⍀2

⫻exp共− ␤R共u2 + v2 + w2兲兲dudvdw +

冕

mnT共cos ␣ + 1兲 4␤T3/2冑␲

mnT共1 + cos ␨兲 4␤T3/2

冑␲

where in the last expression, ␨ = arccos共R2 sin ␪ / 冑R21 + R22 + 2R1R2 cos ␪兲. Figure 13 shows the steady state of gaskinetic thermal heat flux across the bottom cylinder surface. The solid line is the analytical results from Eq. 共15兲, and the circular symbols are results from the DSMC simulation. To clearly illustrate

共14兲

where ␤i = 1 / 共2RTi兲 and vn is the velocity along the cylinder normal direction at point P. A simple evaluation of the above integration is to rotate the velocity phase clockwise by an angle of ␪, then integrate the above integration with u instead of vn. The heat flux for other locations on the bottom cylinder plate can be evaluated similarly. For the final thermal heat flux result, if we denote ␪0 = arccos共R1 / R2兲, then

, 0 ⬍ ␪ ⬍ ␪0

␪0 ⬍ ␪ ⬍ ␲ − ␪0 +

1 vnnT共␤T/␲兲3/2 m共u2 + v2 + w2兲 2 ⍀3

⫻exp共− ␤T共u2 + v2 + w2兲兲dudvdw,

mnT

mnL共1 − cos ␨兲

1 vnnB共␤B/␲兲3/2 m共u2 + v2 + w2兲 2 ⍀1

⫻exp共− ␤B共u2 + v2 + w2兲兲dudvdw

For a heat flux computation, the process is very complex. Here we only provide the theoretical and DSMC simulation results for the gaskinetic thermal heat flux at the bottom cylinder surface. For each point P共x , y兲 on the bottom cylinder surface, the velocity phase consists of several parts: half of the space is composed of particles reflected on the bottom cylinder surface and they are traveling away from point P共x , y兲; while the other half space is composed of those particles traveling toward point P共x , y兲. For example, Fig. 11 shows one point P on the right side of the bottom cylinder surface. If we denote ⬔POC = ␪, and ⬔DPC⬘ = ␣, then we have ␣ = arcsin关共R2 cos ␪ − R1兲 / 冑R22 + R21 − 2R1R2 cos ␪兴. Figure 12 shows the corresponding velocity space for point P共x , y兲, on the right side of the bottom cylinder surface. In this figure, region ⍀1 represents the reflected particles and the Maxwellian distribution function is characterized by the

−

冕

,

␪ ⬎ ␲ − ␪0 ,

冧

共15兲

the results, Fig. 15 only show one every four symbols. It is very evident that the simulation results agree with the analytical results well. Based on the heat flux across the bottom circular surface, we can strongly believe that the general speculations proposed in this paper are valid for concave enclosures as well.

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067105-7

Phys. Fluids 20, 067105 共2008兲

Collisionless gas flows. I. Inside arbitrary enclosures

1

120

360 100

3 80

340

0.8

80

300

0.6 Y, M

q, W/m2

320

60

0.4 DSMC, collisionless Analytical, collisionless

40

320

0.2

340 20

0

30

60

90 θ

120

150

180

0

360

0

0.2

0.4

X, M

0.6

0.8

1

FIG. 13. Heat flux at the bottom cylinder surface; solid line: analytical; symbol: DSMC 共only one of every four symbols is shown兲, n0 = 1 ⫻ 1020 m−3, TL = 200 K, TR = 400 K, TB = 300 K, TT = 600 K.

FIG. 15. Contours of translational temperature 共kelvin兲 for the 3D test case, for Z = 0.5 m plane; solid line: analytical; dashed line: DSMC.

C. A 3D enclosure with six plates

IV. CONCLUSION

As the final example, we show a simple unit cube formed by six identical plates. The boundary conditions are T共z = 0兲 = 100 K, T共z = 1兲 = 200 K, T共x = 0兲 = 300 K, T共x = 1兲 = 400 K, T共y = 0兲 = 500 K, and T共y = 1兲 = 600 K. Similar steps as the first test case are followed to compute the analytical results of the normalized number density and translational temperature. A DSMC simulation is performed with about 620 000 tetrahedral cells and 9 ⫻ 106 particles. Figures 14 and 15 show the contours at the z = 0.5 m plane, the analytical and DSMC simulation results of normalized number density and translational temperature. The comparisons yield very good matches for this 3D case. We can believe that the speculations are valid for 3D enclosures as well.

In this supplemental note to our previous study,3 we have reported extensions of several speculations regarding steady internal collisionless gas flows inside arbitrary enclosures formed by several completely diffuse plates maintained at different temperatures. The molecules inside the enclosures reorganize themselves according to the plate temperatures. At any point inside the enclosures, the velocity distribution functions consist of several pieces of accurately determined Maxwellian distribution functions which are characterized by plate temperatures and number densities. The number densities are completely determined by the plate temperatures. Macroscopic properties, such as density, temperature, and heat flux inside the enclosures can be completely determined. We did not include any properties related to higher order moments with the velocity distribution function, but we can expect very satisfactory matches. Those speculations are expected to work well for enclosures which are not single connected, as well. The speculations can lead to many exact solutions for collisionless internal gas flows inside enclosures formed by plates maintained at different temperatures. The solutions can serve as base solutions to study less rarefied flow problems.10,13 They are good benchmarks to develop the information preservation method as well.

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ACKNOWLEDGMENTS

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The authors would like to acknowledge some partial supports from ZONA Internal Research and Development funding. 2 0.90.9 0.8

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FIG. 14. Contours of normalized number density for the 3D test case, for Z = 0.5 m plane; solid line: analytical; dashed line: DSMC.

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