Issues on Microchannel Boundaries

Microchannel ow has drawn a lot of attentions in recent years, there are several interesting phenomenons in microchannel ow like slip velocity, tem-perature jump and Knudsen minimum [92], simulating microchannel ow and

these phenomenons is a challenging problem. Conventionally, direct simula-tion monte carlo method has been used wildly on this topic and got good results. Recently, LB method is also applied to this topic and got accurate results. In the developing of LB method on microchannel ow, there are two important issues, one is the relations between Knudsen number Kn and the relaxation time τ, the other is the boundary conditions. The detailed descrip-tions covering the rst issue will be discussed in Chapter 5. Here introducing and comparing the state of art boundary conditions for microchannel simula-tions, and the adequate boundary condition is chosen, modied and adopted on semiclassical microchannel ow simulations in Chapter 5. In microchannel simulations, diusive boundary conditions are the most important and worth to introduce in detail. Ansumali et al. disclosed the rst idea of diusive boundary conditions [93]. In their work, derivation of the LB method from the continuous kinetic theory [14][17] is extended to obtain boundary condi-tions. For the model of a diusively reecting moving solid wall, the boundary condition for the discrete set of velocities is derived. This diusive boundary condition is formulated in the continuous kinetic theory and listed in equa-tion (10) of [93] then expanded and discretized on the hermite polynomials.

In 2005, Tang et al. followed the idea and introduced the discrete maxwellian method which use the discrete kinetic theory boundary conditions to get the slip velocity at the solid boundaries [94]. According to the bottom wall, shown in Fig. 4.3, the discrete maxwellian method is described as:

f₂ = f₄^′ f₅ = rKf₅^eq(ρ_w, u_w) + (1− r)f8^′

f₆ = rKf₆^eq(ρ_w, u_w) + (1− r)f7^′ (4.1)

4.3. Boundary Conditions 55

where ρwand uware the density and velocity of wall, and K is a factor dened as: K = _f^{eq f4′+f7′+f8′}

2 (ρw,uw)+f₅^eq(ρw,uw)+f₆^eq(ρw,uw). There is another diusive boundary conditions which reects the sum of incoming distribution function to the wall according to the weight of each distribution function come to the uids. the detail descriptions also referring to the bottom wall of 4.3 is described as:

f₂ = Q× f2^′/(f₂^′ + f₅^′ + f₆^′) f5 = Q× f5^′/(f₂^′ + f₅^′ + f₆^′)

f6 = Q× f6^′/(f₂^′ + f₅^′ + f₆^′) (4.2) where Q = f₄^′+f₇^′+f₈^′ is the sum of incoming distribution function to the wall, f₂^′/(f₂^′+ f₅^′+ f₆^′)is the weight of f2 comes to the uid. The diusive boundary conditions are applied to microchannel ow simulations and got great success.

In 2002, Succi et al. combined bounce back and specular reection methods to capture the gas slip velocity in the microows [95], which is called bounce-back specular reection method here. Referring to the bottom wall of Fig.

4.3. The bounce-back specular reection method is introduced as:

f₂ = f₄^′ f5 = rf₇^′ + (1− r)f8^′

f₆ = rf₈^′ + (1− r)f7^′ (4.3)

f₂ = f₄^′ + rf₇^′ + rf₈^′ f₅ = (1− r)f8^′

f₆ = (1− r)f₇^′ (4.4)

Microchannel ow simulation is used for comparing the inuences of dierent boundary conditions, In these dierent boundary conditions, all of them have a tunable variable r. It can be tuned to match the experimental data. Here we

use classical microchannel ow simulation to validate and compare dierent boundary conditions and the inuences of accommodation coecients. First, we compare the inuences of dierent accommodation coecients on velocity prole based on two dierent boundary conditions are shown in Fig. 4.5.

From the results of this gure, the accommodation coecient r represents the magnitude of the specular, and 1−r means the magnitude of the bounce-back.

That means, the higher r makes slip velocity more obvious. Next, we consider the inuences of dierent boundary conditions on velocity prole based on the same accommodation r = 0.8. We chose two Kn number, Kn = 0.5 and (b) for Kn = 2, and the results are in Fig. 4.6. We found that slip boundary condition and discrete Maxwellian boundary condition both reveals the same velocity prole in low Kn number, but the velocity proles will split in high Kn number. As shown in Fig. 4.6(b), dierent boundary conditions indeed aect the slip velocity on the wall, the slip boundary condition has the maximum slip velocity on the wall. Finally, we consider the velocity prole of dierent Knudsen number based on the same accommodation r = 0.8 for two dierent boundary conditions, as shown in Fig. 4.7(a) for bounce-back specular reection boundary condition and Fig. 4.7(b) for discrete maxwellian boundary condition respectively, we found that dierent Kn number really aects the slip velocity. In high Knudsen number, the velocity slip is obvious.

We should notice that there are still many other boundary conditions for the slip velocity boundaries not discussed here, for example, [96] introduced a modied LB model with a stochastic relaxation mechanism mimicking virtual wall collisions between free streaming particles and solid walls. Virtual wall collisions model combined with bounce back method or diusive boundary conditions [93] were compared to show that this model can help the latter two methods work in high Kn number ow.

4.3. Boundary Conditions 57

Fluid Node Solid Node

Real Boundary Intersection Point Real Boundary

t a

f

_ȟ G

x 

x

f

x

b

x

w

Figure 4.1: Layout of the curved wall boundary in the regularly spaced lattices.

1 2

4

5

8 3 0

7 6

Figure 4.2: Periodic boundary conditions.

1 2

3

4 6 5

7 8

0

xbb

xb

xf

Figure 4.3: Layout of dierent boundary conditions in semiclassical LB method.

4.3. Boundary Conditions 59

Wall

A C B

D E

F

(a) 1

q 2

q AC AB

L R

Wall

A D C B

E F

(b) 1

qt 2

q AC AB

L R

Figure 4.4: Bounce back boundary conditions (a) q < 1/2, (b) q ≥ 1/2.

y/H

Vx/U0

0.2 0.4 0.6 0.8

1.06 1.08 1.1 1.12

BSR_accom=0.4 BSR_accom=0.6 BSR_accom=0.9 Kn=2

(a)

y/H

Vx/U0

0.2 0.4 0.6 0.8

1.08 1.09 1.1 1.11 1.12

DM_accom=0.4 DM_accom=0.6 DM_accom=0.9 Kn=2

(b)

Figure 4.5: The inuences of dierent accommodation coecients on velocity prole based on two dierent boundary conditions. (a) Bounce-back specular reection boundary condition, (b) Discrete maxwellian boundary condition.

4.3. Boundary Conditions 61

y/H

Vx/U0

0.2 0.4 0.6 0.8

0.7 0.8 0.9 1 1.1 1.2 1.3

SlipBC BounceBack BSR

DM (a)

Kn=0.5

y/H

Vx/U0

0.2 0.4 0.6 0.8

1.06 1.08 1.1 1.12 1.14

SlipBC BounceBack BSR

DM Kn=2 (b)

Figure 4.6: The inuences of dierent boundary conditions on velocity prole based on the same accommodation r = 0.8. (a) Kn=0.5, (b) Kn=2.

y/H

Vx/U0

0.2 0.4 0.6 0.8

0.8 1 1.2

Kn=2 Kn=1 Kn=0.9 Kn=0.6 Kn=0.3 (a)

Accommodation Numbe=0.8

y/H

Vx/U0

0.2 0.4 0.6 0.8

0.9 1 1.1 1.2

Kn=2 Kn=1 Kn=0.9 Kn=0.6 Kn=0.3 (b)

Accommodation Numbe=0.8

Figure 4.7: The velocity prole of dierent Knudsen number based on the same accommodation coecient r = 0.8 for two dierent boundary condi-tions. (a) Bounce-back specular reection boundary condition, (b) Discrete maxwellian boundary condition.

Chapter 5

Numerical Results

Contents

5.1 Overview . . . 63 5.2 One Dimensional Shock Tube . . . 64 5.3 Two Dimensional Microchannel Flow. . . 65 5.4 Two Dimensional Flow over Cylinder . . . 69 5.5 Two Dimensional Natural Convection and

Rayleigh-Benard Convection Flow . . . 71 5.6 Three Dimensional Lid Driven Cavity Flow . . . 73

5.1 Overview

In this chapter, Some numerical examples for testing theory and illustrating the present semiclassical LB method are reported. For validation and compari-son purposes, the present single relaxation time semiclassical LB method is ap-plied to one-dimensional quantum gas ows in a shock tube, two-dimensional quantum gas ow over cylinder, two-dimensional quantum gas microchannel

ow and three-dimensional quantum gas ow lid-driven cavity. Moreover, a new thermal LB method presented in Appendix 2 is validated by two dimen-sional natural convection ow and two-dimendimen-sional Rayleigh-Benard thermal

convection.

在文檔中 半古典晶格波滋曼方法 (頁 69-80)