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1. REPORT DATE (DD-MM-YYYY)

17-11-2006

2. REPORT TYPE

Final Report

3. DATES COVERED (From – To)

28 January 2006 - 13-May-10

4. TITLE AND SUBTITLE

-High order numerical schemes for Lattice Boltzmann models: Applications to flows with variable Knudsen

5a. CONTRACT NUMBER

FA8655-05-M-4008

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Dr. Victor Sofonea

5d. PROJECT NUMBER

5d. TASK NUMBER

5e. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

Romanian Academy of Sciences Bd. Mihai Viteazul 24 Timisoara 1900 Romania 8. PERFORMING ORGANIZATION REPORT NUMBER N/A

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EOARD

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SPC 05-4008

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Approved for public release; distribution is unlimited. (approval given by local Public Affairs Office)

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14. ABSTRACT

This report results from a contract tasking Romanian Academy of Sciences as follows: The Grantee will inverigate the development of higher order numerical methods to solve Lattice Boltzmann equations, a system of hyperbolic pde’s obtained from the Boltzmann Equation by discretizing the collision integrals appearing in the latter system.

15. SUBJECT TERMS

EOARD, Finite Difference Methods, Kinetic Theory, Computational Fluid Dynamics (CFD), CFD

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UL

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29

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SURYA SURAMPUDI a. REPORT UNCLAS b. ABSTRACT UNCLAS c. THIS PAGE UNCLAS

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+44 (0)1895 616021

Standard Form 298 (Rev. 8/98)

Romanian Academy – Timis

¸oara Branch

Bd. Mihai Viteazul 24, RO – 300223 Timis¸oara, Romania

High order numerical schemes for lattice Boltzmann models:

Applications to flow with variable Knudsen number

Contract FA8655–05–M–4008

European Office of Aerospace Research and Development Item #0003: Final Report

dr. Victor Sofonea, Principal Investigator Artur Cristea, PhD student

1 Pressure - driven microchannel flow 1

1.1 Introduction . . . 1

1.2 Boundary conditions . . . 2

1.2.1 Walls . . . 2

1.2.2 Inflow/outflow boundaries . . . 4

1.3 Lattice Boltzmann results: short channel . . . 4

1.4 Lattice Boltzmann results: long channel . . . 7 2 Force - driven microchannel flow 13

3 Conclusion 19

1

Pressure - driven microchannel flow

1.1

Introduction

As mentioned in our previous reports [1, 2], Lattice Boltzmann (LB) models [3, 4, 5, 6] are effective for problems where both mesoscopic dynamics and microscopic statistics become important, as in the case of micro - channel flows [7, 8, 9, 10, 11, 12, 13]. In this report, we further investigate the applications of the two-dimensional (2D) thermal finite difference Lattice Boltzmann (FDLB) model introduced in [14]. For convenience, a brief description of this model is provided below.

Lattice Boltzmann (LB) models provide an alternative to particle-based models like Molecular Dynamics (MD) or Direct Simulation Monte Carlo (DSMC), as well as to numerical techniques of Computational Fluid Dynamics (CFD) derived from the principles of continuous media mechanics. The development of Lattice Boltzmann models for the investigation of flow phenomena at non-negligible Knudsen numbers [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] has recently received considerable at-tention due to the increasing interest for mastering gas flow at two extreme scales [7, 8, 9, 10, 11, 12, 13]: high-altitude aerodynamics and micro-electro-mechanical sys-tems (MEMS). Unfortunately, the majority of LB models developed up to date for the investigation of fluid flow at finite Knudsen numbers still refer to the isothermal case.

In the previous reports [1, 2] we approach the non-isothermal gas flow in microchan-nels using the two-dimensional finite difference LB model with multiple speeds of Watari and Tsutahara [14], which allows the correct recovery of mass, momentum and energy equations of a compressible fluid. The model involves a set of 33 nondimensionalized velocities {e00, e_ki}, (k = 1, . . . 4, i = 1, . . . 8) e00 = 0 , eki =  cos π(i − 1) 4 , sin π(i − 1) 4  ck (1)

with ck ∈ {1.0, 1.92, 2.99, 4.49}. The corresponding distribution functions f00= f00(x, t),

fki = fki(x, t), evolve according to the non-dimensionalized equations

∂tfki+ eki· ∇fki = −

1

τ [fki − f

eq

ki] (2)

Here we use the density-dependent relaxation time τ = Λ/n¯c, where Λ is a dimensionless constant, n = n(x, t) is the particle number density and ¯c is the local average speed of fluid particles [27]. The equilibrium distribution functions in eqs. (2)

f_kieq = f_kieq(x, t) = nFkski (3)

are expressed using the series expansion ski = ski(θ, u) up to fourth order with respect to

the Cartesian components uα (α = 1, 2) of the fluid velocity (summation over repeated

Greek indices is understood): ski =  1 − u 2 2θ + u4 8θ2  + 1 θ  1 − u 2 2θ  ekiξuξ + 1 2θ2  1 − u 2 2θ  ekiξekiηuξuη + 1

6θ3 ekiξekiηekiζuξuηuζ +

1

1 PRESSURE - DRIVEN MICROCHANNEL FLOW ₂ The weight factors Fk = Fk(θ) in eq. (3) depend on the local temperature θ = θ(x, t)

and the speeds ck, k = 1, . . . 4. We refer the reader to the literature [1, 2, 14, 27, 28] for

further details on the thermal LB model.

1.2

Boundary conditions

1.2.1 Walls

The thermal LB model, as originally developed by its authors [14], uses the second order upwind finite difference scheme to solve the LB evolution equations (2) in the nodes of a two-dimensional square lattice. The boundary nodes were located on the channel walls and an extrapolation method was used to provide the corresponding values of the distribution functions. This procedure forces the fluid’s temperature and velocity in the boundary nodes to equal the wall’s temperature and velocity. Appropriate boundary conditions [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], many of them based on the concept of diffuse reflection, which dates back to the time of Maxwell and Smoluchowski [29, 30, 31, 32], were already introduced in the isothermal LB models to capture the fluid velocity slip near the walls when the Knudsen number is no longer negligible.

To reduce numerical errors, the MCD flux limiter finite difference scheme [33, 34, 35] is used in our simulations. A first implementation of diffuse reflection boundary conditions in the thermal LB model of Watari and Tsutahara relies on the redistribution (thermalization) of the particle distribution functions in the ghost nodes outside the walls [27]. Although successful in capturing the temperature jump and slip velocity in Couette flow, this implementation of the diffuse reflection boundary conditions does not allow the fluid density to vary along the channel walls, as happens in the case of pressure-driven gas flow. To overcome this drawback, in the previous report [2] we managed the distribution functions to mix (redistribute) themselves in wall nodes instead of ghost nodes. More precisely, the distribution functions whose corresponding velocities point to the wall in the normal direction mix separately from those distribution functions with corresponding velocities orientated along the diagonals of the square lattice. For convenience, we refer to the left wall of the vertical channel in Figure 1, where the black squares (j = 1/2, l) and (j = 1/2, l + 1/2), l = 0, 1, . . ., denote the mixing points on the wall. The corresponding (unknown) values of the particle number density are n1w/2,l+1/2

and n1w/2,l+1, respectively. Although the mixing points may have different temperatures,

in this paper we restrict ourselves to the case when the channel wall temperature is constant (θ1w/2,l = θ

1/2,l+1/2

w = θw, for all l).

In the finite difference LB model we need appropriate procedures to compute the values of the distribution functions f_k20,l, f_k80,l+1, f_k10,l+1 (l = 0, 1, . . .) defined in the ghost nodes outside the left wall of velocity uw [27]. The requirement that the distribution

2 l=2 l=1 l=0 j=1/2 j=3/2 j=5/2 j=7/2 j=4 j=3 8 j=2 j=1 j=0 1 l=7/2 l=5/2 l=3/2 l=1/2 l=4 0 Y X 4 5 6 7 l=3 3

Figure 1: Implementation of diffuse reflection boundary conditions in the lattice nodes near the lower left end of a vertical channel (the dashed line marks the outflow bound-ary): • - bulk nodes, ◦ - boundary nodes, 2 - ghost nodes,  - wall nodes where the distribution functions f_kijl follow the Maxwellian distribution.

1 PRESSURE - DRIVEN MICROCHANNEL FLOW ₄ reads (k = 1, . . . 4): f_k20,l+ f_k21,l+1 Fk(θw)sk2(θw, uw) = f 0,l+1 k8 + f 1,l k8 Fk(θw)sk8(θw, uw) = 2n1/2,l+1/2 w (5a) f_k10,l+1+ f_k11,l+1 Fk(θw)sk1(θw, uw) = 2n1/2,l+1 w (5b)

Eqs. (5), together with the requirements that there is no mass flux perpendicular to the wall in the mixing nodes:

X k ck h f_k41,l+ f_k61,l+1i = X k ck h f_k20,l + f_k80,l+1i (6a) X k ckf 1,l+1 k5 = X k ckf 0,l+1 k1 (6b)

may be solved to get the values of the distribution functions in the ghost nodes (0, l) and (0, l + 1) after each time step. Similar relations may be used to compute the values of the corresponding distribution functions in the ghost nodes outside the right wall of the channel.

1.2.2 Inflow/outflow boundaries

The wall temperature and velocity are set to θw = 1 and uw = 0, while the pressure on

the inflow and outflow boundaries are pin and pout, respectively. At both inflow/outflow

boundaries, the x and y components of the fluid velocity are linearly extrapolated along the y direction to get the corresponding values in the ghost nodes. The linear extrapo-lation is used also to define the temperature values in the ghost nodes below the outflow boundary (Figure 1), while the temperature in the ghost nodes above the inflow bound-ary is set equal to the wall temperature θw, as done by other authors [36, 37, 38]. The

values of the fluid density in the inflow/outflow ghost nodes are calculated according to the non-dimensionalized form of the ideal gas equation of state p = nθ. Once the values of n, θ and u are known in the inflow/outflow ghost nodes, the values of the dis-tribution functions in these nodes are set to the corresponding values of the equibrium distribution functions, which may be calculated according to eq. (3).

1.3

Lattice Boltzmann results: short channel

Figure 2 shows the stationary profiles of pressure, density, temperature and y component of the fluid velocity u in the stream direction down the central line x = L/2 of a short channel with L = 0.2, h = 0.6, pout = 5 × 107 and P = pin/pout = 3. These profiles

were recovered after 500,000 iterations with time step δt = 10−4

on a square lattice with 21 × 63 nodes and spacing δs = L/21. We choose Λ = 106

to achieve the initial value Knout = 0.1 of the Knudsen number Kn = Λ/nL [27] on the outflow boundary.

5e+07 6e+07 7e+07 8e+07 9e+07 1e+08 1.1e+08 1.2e+08 1.3e+08 1.4e+08 1.5e+08 0 0.1 0.2 0.3 0.4 0.5 0.6 pressure p y coordinate 7e+07 8e+07 9e+07 1e+08 1.1e+08 1.2e+08 1.3e+08 1.4e+08 1.5e+08 0 0.1 0.2 0.3 0.4 0.5 0.6

particle number density n

y coordinate

Figure 2: Stationary profiles (pressure, density, temperature and y component of the fluid velocity) in the stream direction y down the central line x = L/2 of a short channel (L = 0.2, h = 3L, Knout = 0.1).

1 PRESSURE - DRIVEN MICROCHANNEL FLOW ₆ 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.1 0.2 0.3 0.4 0.5 0.6 temperature θ y coordinate -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 velocity u y y coordinate

Figure 2: (continued) Stationary profiles (pressure, density, temperature and y compo-nent of the fluid velocity) in the stream direction y down the central line x = L/2 of a short channel (L = 0.2, h = 3L, Knout = 0.1).

[36, 37, 39, 40], as well as to the BGK-Burnett solution of microchannel flow in the slip regime [38]. For this purpose, the profiles in Figure 2 should be rescaled taking into account that we used the reference values p0 = 108, n0 = 108, c_s =

√

2 and T0 = 1

for pressure, density, sound velocity and temperature, while the corresponding reference values in [36, 37, 38] are p0 = 6.05 × 10

−4

, n0 = 1.21 × 10

−3

, cs = 0.91 and T0 = 1,

respectively. A main characteristic of the gas flow in the short channel, which is well captured by both LB and DSMC simulations, is the large temperature drop near the ouflow boundary.

1.4

Lattice Boltzmann results: long channel

Figure 3 shows the stationary profiles of temperature and pressure in the stream direc-tion down the central line x = L/2 of a long channel (21 × 1050 nodes) with L = 0.2, h = 10. These profiles were recovered after 2,000,000 iterations, while the other simu-lation paramenters remained the same as in the case of the short channel. Unlike the short channel case, the temperature variation along the long channel is less than 2%. The nonlinear presure profile along the central line, due to the competition between the compressibility and rarefection effects [7], is in good agreement with the analytical solution of Arkilic, Schmidt and Breuer [38, 41]:

p(y) pout = −6Knout +  (6Knout+ P) 2 − (P − 1)(P + 1 + 12Knout) L − y L 2 (7) The cross-stream profiles in Figure 4 demonstrate the capability of the thermal LB model to capture also the non-vanishing velocity component normal to the walls, due to the non-uniform fluid density in the cross-stream direction [41]. As expected, the y component of the fluid velocity (whose cross-stream profile is parabolic), as well as the slip velocity at the walls (Figure 5) increase when approaching the outflow boundary, where pressure drops faster [7, 25, 41].

We investigated also the capability of the thermal LB model with diffuse reflection boundary conditions to capture the minimum of the volumic flow rate as a function of the Knudsen number. We follow [7] and express the non-dimensionalized volumic flow rate in the middle of the channel (y = L/2) as

Q(y) = ¯ p Z L 0 n(x, y)uy(x, y)dx −(dp/dy)¯nL2cR ' constant × Z L 0 n(x, y)uy(x, y)dx pin− pout (8) where ¯p and ¯_{n are the mean fluid pressure and density, dp/dy ' (p}in− pout)/h and we

took advantage of the fact that the temperature is quite constant in the channel. To achieve various values of Knout, we changed the value of pout while preserving L = 0.2,

h = 10, P = 3 and θw = 1. In Figure 6 we compare our thermal LB results with those

obtained using the CLL model of Cercignani, Lampis and Lorenzani [42], as well as with the analytical solutions in the zero- and infinite-Kn limits, where s = 1.01615 is the slip

1 PRESSURE - DRIVEN MICROCHANNEL FLOW ₈ 0.984 0.986 0.988 0.99 0.992 0.994 0.996 0.998 1 1.002 0 1 2 3 4 5 6 7 8 9 10 temperature θ y coordinate 4e+07 6e+07 8e+07 1e+08 1.2e+08 1.4e+08 1.6e+08 0 1 2 3 4 5 6 7 8 9 10 pressure p y coordinate LB simulation Arkilic et al., Eq.(7) linear

Figure 3: Stationary profiles (temperature and pressure) in the stream direction y down the central line x = L/2 of a long channel (L = 0.2, h = 50L, pin = 3pout, Knout = 0.1).

0.9988

0.999

0.9992

0.9994

0.9996

0.9998

1

0

0.05

0.1

0.15

0.2

temperature

θ

x coordinate

Figure 4: Stationary profiles (temperature, pressure, density and x component of the fluid velocity) in the cross - stream direction x of a long channel (L = 0.2, h = 50L, y = L/2, pin= 3pout, Knout= 0.1).

1 PRESSURE - DRIVEN MICROCHANNEL FLOW ₁₀

1.078e+08

1.0782e+08

1.0784e+08

1.0786e+08

1.0788e+08

1.079e+08

1.0792e+08

1.0794e+08

0

0.05

0.1

0.15

0.2

particle number density n

x coordinate

-2e-05

-1.5e-05

-1e-05

-5e-06

0

5e-06

1e-05

1.5e-05

2e-05

0

0.05

0.1

0.15

0.2

velocity u

x coordinate

Figure 4: (continued) Stationary profiles (temperature, pressure, density and x compo-nent of the fluid velocity) in the cross - stream direction x of a long channel (L = 0.2, h = 50L, y = L/2, pin= 3pout, Knout= 0.1).

0 0.05 0.1 0.15 0.2 0 2 4 6 8 10 -0.16 -0.12 -0.08 -0.04 0 velocity u_y x y velocity u_y

Figure 5: Stationary profile of the y omponent of the fluid velocity in the long channel. coefficient [43]: Q0 = 1/(6Kn) + s + (2s 2 − 1)Kn (9a) Q∞ ∼ π −1/2_{log(Kn) (9b)}

For Kn < 1.0, our thermal LB results fit well to the CLL model and the minimum of the flow rate is situated around Kn = 0.625. For larger values of Kn, the thermal LB results are situated between the analytical results predicted by eq. (9b) multiplied by a factor of 4, and those derived using the CLL model. This behaviour is very similar to the results of Toschi and Succi, recovered using an isothermal LB model for force-driven Poiseuille flow [20].

1 PRESSURE - DRIVEN MICROCHANNEL FLOW ₁₂

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5

3

3.5

mass flux Q

Knudsen number Kn

thermal LB results

CLL results, Ref.[14]

Eq.(9a)

Eq.(9b)

Figure 6: Normalized mass flux Q as function of Knudsen number Kn: comparison of thermal LB simulation results with calculated values in the CLL model [42], as well as with analytical solutions, Eqs.(9)

2

Force - driven microchannel flow

Let us consider the LB evolution equation with a constant external force Fγ = maγ

acting on the fluid particles of mass m [20, 23, 33, 34] ∂tfki+ eki· ∇fki+ aγ(ekiγ− uγ)fkieq = −

1

τ [fki− f

eq

ki] (10)

The boundary conditions on the left/right walls of the channel are the same as in Section 1.2.1 and periodic boundary conditions are considered in the vertical direction. The acceleration vector is orientated along the channel (ax = 0, ay = a).

Simulations were done on a 500 × 3 lattice with spacing δs =. We fixed the non-dimensionalized acceleration value a = 1.0 and the time step δt = 1.0 × 10−5

. Two values of the mean fluid density were considered: ¯_{n = 1.0 × 10}7

and ¯_{n = 5.0 × 10}7

. The corresponding values of the Knudsen number are Kn = 0.5 and Kn = 0.1, respectively. In the initial state (iter = 0), the fluid is at rest and begins to flow under the action of the external force. Figures 7 and 8 show the evolution of the cross-stream density, temperature, velocity and pressure profiles for both values of ¯n mentioned above. Although the velocity profile is always parabolic, a striking feature is the occurence of the bimodal temperature profile, as well as the concave pressure profile. These features were observed also by other authors [36, 37, 38] using Direct Simulation Monte Carlo (DSMC) or BGK-Burnett solvers.

As done in the case of the pressure - driven flow, in the force - driven case we checked also the existence of the minimum of the volumic flow rate as a function of the Knudsen number. The normalized volumic flow rate is shown in Figure 9 as function of the Knudsen number. Unlike the pressure - driven case, the simulation results in the force - driven case fit well to the analytical formula, Eq. (9a), even at large values of the Knudsen number.

2 FORCE - DRIVEN MICROCHANNEL FLOW ₁₄

density temperature velocity pressure

9.99e+06 9.992e+06 9.994e+06 9.996e+06 9.998e+06 1e+07 1.0002e+07 1.0004e+07 1.0006e+07 1.0008e+07 1.001e+07 0 0.05 0.1 0.15 0.2 0.99 0.995 1 1.005 1.01 0 0.05 0.1 0.15 0.2 -1 -0.5 0 0.5 1 0 0.05 0.1 0.15 0.2 9.99e+06 9.992e+06 9.994e+06 9.996e+06 9.998e+06 1e+07 1.0002e+07 1.0004e+07 1.0006e+07 1.0008e+07 1.001e+07 0 0.05 0.1 0.15 0.2 iter = 0,000 9.997e+06 9.998e+06 9.999e+06 1e+07 1.0001e+07 1.0002e+07 1.0003e+07 1.0004e+07 1.0005e+07 0 0.05 0.1 0.15 0.2 1.00025 1.0003 1.00035 1.0004 1.00045 1.0005 1.00055 1.0006 1.00065 0 0.05 0.1 0.15 0.2 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0 0.05 0.1 0.15 0.2 1.0001e+07 1.0002e+07 1.0003e+07 1.0004e+07 1.0005e+07 1.0006e+07 1.0007e+07 1.0008e+07 1.0009e+07 0 0.05 0.1 0.15 0.2 iter = 10,000 9.994e+06 9.996e+06 9.998e+06 1e+07 1.0002e+07 1.0004e+07 1.0006e+07 1.0008e+07 1.001e+07 0 0.05 0.1 0.15 0.2 1.0024 1.00245 1.0025 1.00255 1.0026 1.00265 1.0027 0 0.05 0.1 0.15 0.2 0.115 0.12 0.125 0.13 0.135 0.14 0.145 0.15 0.155 0 0.05 0.1 0.15 0.2 1.0018e+07 1.002e+07 1.0022e+07 1.0024e+07 1.0026e+07 1.0028e+07 1.003e+07 1.0032e+07 1.0034e+07 1.0036e+07 0 0.05 0.1 0.15 0.2 iter = 20,000 9.99e+06 9.995e+06 1e+07 1.0005e+07 1.001e+07 1.0015e+07 1.002e+07 0 0.05 0.1 0.15 0.2 1.0055 1.0056 1.0057 1.0058 1.0059 1.006 1.0061 0 0.05 0.1 0.15 0.2 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0 0.05 0.1 0.15 0.2 1.0052e+07 1.0054e+07 1.0056e+07 1.0058e+07 1.006e+07 1.0062e+07 1.0064e+07 1.0066e+07 1.0068e+07 1.007e+07 1.0072e+07 0 0.05 0.1 0.15 0.2 iter = 30,000 9.99e+06 9.995e+06 1e+07 1.0005e+07 1.001e+07 1.0015e+07 1.002e+07 1.0025e+07 0 0.05 0.1 0.15 0.2 1.009 1.0092 1.0094 1.0096 1.0098 1.01 1.0102 0 0.05 0.1 0.15 0.2 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0 0.05 0.1 0.15 0.2 1.009e+07 1.0092e+07 1.0094e+07 1.0096e+07 1.0098e+07 1.01e+07 1.0102e+07 1.0104e+07 1.0106e+07 1.0108e+07 1.011e+07 1.0112e+07 0 0.05 0.1 0.15 0.2 iter = 40,000 9.985e+06 9.99e+06 9.995e+06 1e+07 1.0005e+07 1.001e+07 1.0015e+07 1.002e+07 1.0025e+07 1.003e+07 0 0.05 0.1 0.15 0.2 1.0124 1.0126 1.0128 1.013 1.0132 1.0134 1.0136 1.0138 1.014 1.0142 1.0144 1.0146 0 0.05 0.1 0.15 0.2 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0 0.05 0.1 0.15 0.2 1.0132e+07 1.0134e+07 1.0136e+07 1.0138e+07 1.014e+07 1.0142e+07 1.0144e+07 1.0146e+07 1.0148e+07 1.015e+07 1.0152e+07 1.0154e+07 0 0.05 0.1 0.15 0.2 iter = 50,000

Figure 7: Evolution of density, temperature, velocity and pressure profiles in force-driven microchannel flow: ¯_{n = 1.0 × 10}7

, a = 1.00, δs = 4.0 × 10−4

, δt = 1.0 × 10−5

density temperature velocity pressure 9.985e+06 9.99e+06 9.995e+06 1e+07 1.0005e+07 1.001e+07 1.0015e+07 1.002e+07 1.0025e+07 1.003e+07 1.0035e+07 0 0.05 0.1 0.15 0.2 1.0155 1.016 1.0165 1.017 1.0175 1.018 1.0185 1.019 0 0.05 0.1 0.15 0.2 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0 0.05 0.1 0.15 0.2 1.0172e+07 1.0174e+07 1.0176e+07 1.0178e+07 1.018e+07 1.0182e+07 1.0184e+07 1.0186e+07 1.0188e+07 1.019e+07 0 0.05 0.1 0.15 0.2 iter = 60,000 9.98e+06 9.99e+06 1e+07 1.001e+07 1.002e+07 1.003e+07 1.004e+07 1.005e+07 0 0.05 0.1 0.15 0.2 1.024 1.025 1.026 1.027 1.028 1.029 1.03 0 0.05 0.1 0.15 0.2 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0 0.05 0.1 0.15 0.2 1.0279e+07 1.028e+07 1.0281e+07 1.0282e+07 1.0283e+07 1.0284e+07 1.0285e+07 1.0286e+07 1.0287e+07 1.0288e+07 1.0289e+07 1.029e+07 0 0.05 0.1 0.15 0.2 iter = 100,000 9.97e+06 9.98e+06 9.99e+06 1e+07 1.001e+07 1.002e+07 1.003e+07 1.004e+07 1.005e+07 1.006e+07 0 0.05 0.1 0.15 0.2 1.028 1.029 1.03 1.031 1.032 1.033 1.034 1.035 1.036 1.037 0 0.05 0.1 0.15 0.2 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0 0.05 0.1 0.15 0.2 1.0337e+07 1.03375e+07 1.0338e+07 1.03385e+07 1.0339e+07 1.03395e+07 1.034e+07 1.03405e+07 1.0341e+07 1.03415e+07 1.0342e+07 0 0.05 0.1 0.15 0.2 iter = 200,000 9.97e+06 9.98e+06 9.99e+06 1e+07 1.001e+07 1.002e+07 1.003e+07 1.004e+07 1.005e+07 1.006e+07 0 0.05 0.1 0.15 0.2 1.029 1.03 1.031 1.032 1.033 1.034 1.035 1.036 1.037 0 0.05 0.1 0.15 0.2 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0 0.05 0.1 0.15 0.2 1.03395e+07 1.034e+07 1.03405e+07 1.0341e+07 1.03415e+07 1.0342e+07 1.03425e+07 1.0343e+07 1.03435e+07 1.0344e+07 1.03445e+07 0 0.05 0.1 0.15 0.2 iter = 300,000 9.97e+06 9.98e+06 9.99e+06 1e+07 1.001e+07 1.002e+07 1.003e+07 1.004e+07 1.005e+07 1.006e+07 0 0.05 0.1 0.15 0.2 1.029 1.03 1.031 1.032 1.033 1.034 1.035 1.036 1.037 0 0.05 0.1 0.15 0.2 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0 0.05 0.1 0.15 0.2 1.03395e+07 1.034e+07 1.03405e+07 1.0341e+07 1.03415e+07 1.0342e+07 1.03425e+07 1.0343e+07 1.03435e+07 1.0344e+07 1.03445e+07 0 0.05 0.1 0.15 0.2 iter = 400,000 9.97e+06 9.98e+06 9.99e+06 1e+07 1.001e+07 1.002e+07 1.003e+07 1.004e+07 1.005e+07 1.006e+07 0 0.05 0.1 0.15 0.2 1.029 1.03 1.031 1.032 1.033 1.034 1.035 1.036 1.037 0 0.05 0.1 0.15 0.2 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0 0.05 0.1 0.15 0.2 1.03395e+07 1.034e+07 1.03405e+07 1.0341e+07 1.03415e+07 1.0342e+07 1.03425e+07 1.0343e+07 1.03435e+07 1.0344e+07 1.03445e+07 0 0.05 0.1 0.15 0.2 iter = 500,000

Figure 7: (continued) Evolution of density, temperature, velocity and pressure profiles in force-driven microchannel flow: ¯_{n = 1.0 × 10}7

, a = 1.00, δs = 4.0 × 10−4

, δt = 1.0 × 10−5

2 FORCE - DRIVEN MICROCHANNEL FLOW ₁₆

density temperature velocity pressure

4.995e+07 4.996e+07 4.997e+07 4.998e+07 4.999e+07 5e+07 5.001e+07 5.002e+07 5.003e+07 5.004e+07 5.005e+07 0 0.05 0.1 0.15 0.2 0.99 0.995 1 1.005 1.01 0 0.05 0.1 0.15 0.2 -1 -0.5 0 0.5 1 0 0.05 0.1 0.15 0.2 4.995e+07 4.996e+07 4.997e+07 4.998e+07 4.999e+07 5e+07 5.001e+07 5.002e+07 5.003e+07 5.004e+07 5.005e+07 0 0.05 0.1 0.15 0.2 iter = 0,000 4.9996e+07 4.9998e+07 5e+07 5.0002e+07 5.0004e+07 5.0006e+07 5.0008e+07 5.001e+07 5.0012e+07 5.0014e+07 0 0.05 0.1 0.15 0.2 1 1.0001 1.0002 1.0003 1.0004 1.0005 1.0006 1.0007 1.0008 0 0.05 0.1 0.15 0.2 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.05 0.1 0.15 0.2 5e+07 5.0005e+07 5.001e+07 5.0015e+07 5.002e+07 5.0025e+07 5.003e+07 5.0035e+07 5.004e+07 5.0045e+07 5.005e+07 0 0.05 0.1 0.15 0.2 iter = 10,000 4.998e+07 4.999e+07 5e+07 5.001e+07 5.002e+07 5.003e+07 5.004e+07 5.005e+07 0 0.05 0.1 0.15 0.2 1.001 1.0012 1.0014 1.0016 1.0018 1.002 1.0022 1.0024 1.0026 1.0028 0 0.05 0.1 0.15 0.2 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.05 0.1 0.15 0.2 5.004e+07 5.006e+07 5.008e+07 5.01e+07 5.012e+07 5.014e+07 5.016e+07 5.018e+07 0 0.05 0.1 0.15 0.2 iter = 20,000 4.996e+07 4.998e+07 5e+07 5.002e+07 5.004e+07 5.006e+07 5.008e+07 5.01e+07 5.012e+07 0 0.05 0.1 0.15 0.2 1.0035 1.004 1.0045 1.005 1.0055 1.006 0 0.05 0.1 0.15 0.2 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0 0.05 0.1 0.15 0.2 5.018e+07 5.02e+07 5.022e+07 5.024e+07 5.026e+07 5.028e+07 5.03e+07 5.032e+07 5.034e+07 5.036e+07 5.038e+07 0 0.05 0.1 0.15 0.2 iter = 30,000 4.995e+07 5e+07 5.005e+07 5.01e+07 5.015e+07 5.02e+07 5.025e+07 0 0.05 0.1 0.15 0.2 1.007 1.0075 1.008 1.0085 1.009 1.0095 1.01 1.0105 0 0.05 0.1 0.15 0.2 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0 0.05 0.1 0.15 0.2 5.035e+07 5.04e+07 5.045e+07 5.05e+07 5.055e+07 5.06e+07 0 0.05 0.1 0.15 0.2 iter = 40,000 4.99e+07 4.995e+07 5e+07 5.005e+07 5.01e+07 5.015e+07 5.02e+07 5.025e+07 5.03e+07 5.035e+07 0 0.05 0.1 0.15 0.2 1.0105 1.011 1.0115 1.012 1.0125 1.013 1.0135 1.014 1.0145 1.015 0 0.05 0.1 0.15 0.2 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 5.055e+07 5.06e+07 5.065e+07 5.07e+07 5.075e+07 5.08e+07 5.085e+07 5.09e+07 0 0.05 0.1 0.15 0.2 iter = 50,000

Figure 8: Evolution of density, temperature, velocity and pressure profiles in force-driven microchannel flow: ¯_{n = 5.0 × 10}7

, a = 1.00, δs = 4.0 × 10−4

, δt = 1.0 × 10−5

density temperature velocity pressure 4.99e+07 4.995e+07 5e+07 5.005e+07 5.01e+07 5.015e+07 5.02e+07 5.025e+07 5.03e+07 5.035e+07 5.04e+07 5.045e+07 0 0.05 0.1 0.15 0.2 1.013 1.014 1.015 1.016 1.017 1.018 1.019 1.02 0 0.05 0.1 0.15 0.2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 5.075e+07 5.08e+07 5.085e+07 5.09e+07 5.095e+07 5.1e+07 5.105e+07 5.11e+07 5.115e+07 0 0.05 0.1 0.15 0.2 iter = 60,000 4.98e+07 5e+07 5.02e+07 5.04e+07 5.06e+07 5.08e+07 5.1e+07 0 0.05 0.1 0.15 0.2 1.022 1.024 1.026 1.028 1.03 1.032 1.034 1.036 0 0.05 0.1 0.15 0.2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 5.15e+07 5.155e+07 5.16e+07 5.165e+07 5.17e+07 5.175e+07 5.18e+07 5.185e+07 5.19e+07 5.195e+07 5.2e+07 0 0.05 0.1 0.15 0.2 iter = 100,000 4.96e+07 4.98e+07 5e+07 5.02e+07 5.04e+07 5.06e+07 5.08e+07 5.1e+07 5.12e+07 0 0.05 0.1 0.15 0.2 1.025 1.03 1.035 1.04 1.045 1.05 0 0.05 0.1 0.15 0.2 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 5.21e+07 5.215e+07 5.22e+07 5.225e+07 5.23e+07 5.235e+07 5.24e+07 5.245e+07 5.25e+07 5.255e+07 5.26e+07 5.265e+07 0 0.05 0.1 0.15 0.2 iter = 200,000 4.96e+07 4.98e+07 5e+07 5.02e+07 5.04e+07 5.06e+07 5.08e+07 5.1e+07 5.12e+07 5.14e+07 0 0.05 0.1 0.15 0.2 1.025 1.03 1.035 1.04 1.045 1.05 1.055 0 0.05 0.1 0.15 0.2 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 5.22e+07 5.225e+07 5.23e+07 5.235e+07 5.24e+07 5.245e+07 5.25e+07 5.255e+07 5.26e+07 5.265e+07 5.27e+07 0 0.05 0.1 0.15 0.2 iter = 300,000 4.96e+07 4.98e+07 5e+07 5.02e+07 5.04e+07 5.06e+07 5.08e+07 5.1e+07 5.12e+07 5.14e+07 0 0.05 0.1 0.15 0.2 1.025 1.03 1.035 1.04 1.045 1.05 1.055 0 0.05 0.1 0.15 0.2 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 5.22e+07 5.225e+07 5.23e+07 5.235e+07 5.24e+07 5.245e+07 5.25e+07 5.255e+07 5.26e+07 5.265e+07 5.27e+07 5.275e+07 0 0.05 0.1 0.15 0.2 iter = 400,000 4.96e+07 4.98e+07 5e+07 5.02e+07 5.04e+07 5.06e+07 5.08e+07 5.1e+07 5.12e+07 5.14e+07 0 0.05 0.1 0.15 0.2 1.025 1.03 1.035 1.04 1.045 1.05 1.055 0 0.05 0.1 0.15 0.2 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 5.22e+07 5.225e+07 5.23e+07 5.235e+07 5.24e+07 5.245e+07 5.25e+07 5.255e+07 5.26e+07 5.265e+07 5.27e+07 5.275e+07 0 0.05 0.1 0.15 0.2 iter = 500,000

Figure 8: (continued) Evolution of density, temperature, velocity and pressure profiles in force-driven microchannel flow: ¯_{n = 5.0 × 10}7

, a = 1.00, δs = 4.0 × 10−4

, δt = 1.0 × 10−5

2 FORCE - DRIVEN MICROCHANNEL FLOW ₁₈

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5

3

3.5

flowrate Q

Knudsen number Kn

thermal LB results

CLL results, Ref.[14]

Eq.(9a)

Eq.(9b)

Figure 9: Normalized mass flux Q as function of Knudsen number Kn: comparison of force - driven simulation results with calculated values in the CLL model [42], as well as with analytical solutions, Eqs.(9)

3

Conclusion

We investigated the two-dimensional microchannel flow using a thermal Lattice Boltz-mann model with appropriate boundary conditions. Two separate cases were considered: the pressure - driven case and the external force - driven case. A characteristics of the Lattice Boltzmann model is the recovery of the density, temperature, velocity and pres-sure fields from the local values of the discretized set of distribution functions, whose evolution is governed by the Boltzmann equation, which is easier to manage than the Navier - Stokes or Burnett equations.

Entrance and exit effects are present in the pressure - driven case and are clearly seen especially in the longitudinal temperature and velocity profiles when the fluid flows in a short channel. In long channels, the non-linear pressure profile along the center line, the rarefaction effects, as well as the existence of the so - called Knudsen minimum in the plot of mass flow rate vs. Knudsen number were found to be in good agreement with literature results.

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