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Modified lattice Boltzmann solution for non-isothermal rarefied gas flow through microchannel utilizing BSR and second-order implicit schemes

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Abstract

Thermal microscale gas flow was simulated into a coplanar microchannel was simulated at a broad range of Knudsen numbers. Attempts were made to improve the accuracy of slip velocity on walls using a modified model with two relaxation times based upon the mesoscopic method. The temperature jump of fluid flow at the wall was captured by a model with a single relaxation time using a second-order implicit method. The Zou–He boundary conditions were employed at both inlet and outlet boundaries, and bounce-back/specular reflection distribution functions were applied to the impermeable walls. The non-equilibrium distribution functions were also used as the inlet temperature boundary condition. A fully developed temperature profile was considered at the microchannel outlet. A pressure ratio of 2 was considered in the simulations, and various parameters such as dimensionless pressure, pressure deviation from the linear pressure, dimensionless velocity at various Knudsen numbers, centerline velocity and slip velocity of the fluid, centerline temperature and fluid temperature on the wall, Nusselt number with changing Knudsen and Prandtl numbers, parameter k along the microchannel length and Cf·Re values were evaluated in the slip and transition flow regimes. The results of the direct simulation Monte Carlo were used to evaluate the correctness of the numerical model. The consistency of the two methods indicated the accuracy of the proposed method.

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Abbreviations

B:

Molecular slip coefficient

BBC:

Bounce-back

BSR:

Bounce-back/specular reflection

\(c\) :

Lattice speed (m s−1)

\(C^{*}\) :

Temperature jump coefficient

\(C_{\text{f}}\) :

Skin friction coefficient

\(\varvec{c}_{\text{i}}\) :

Discrete velocity vectors (m s−1)

\(c_{\text{s}}\) :

Sound speed (m s−1)

\(\bar{c}\) :

Mean molecular velocity (m s−1)

\(d\) :

Molecular diameter (m)

DSMC:

Direct simulation Monte Carlo

Ec:

Eckert number

f :

Local distribution function (for fluid flow)

g :

Local distribution function (for thermal fluid flow)

H :

Height of the microchannel (m)

Kn:

Knudsen number, Kn = λ/H

lu:

Length unit

\(L\) :

Length of the microchannel (m)

LBM:

Lattice Boltzmann method

Ma:

Mach number

\(m\) :

Molecular mass (kg)

mu:

Mass unit

\({\text{Nu}}\) :

Nusselt number

\(P\) :

Pressure (\({\text{N}}\, {\text{m}}^{ - 2}\))

Pr:

Prandtl number

R:

Gas constant (\({\text{J}}\, {\text{K}}^{ - 1} \, {\text{mol}}^{ - 1}\))

SRT:

Single relaxation time

\(r\) :

Bounce-back fraction parameter

Re:

Reynolds number

T:

Temperature (K)

\(T^{*}\) :

Non-dimensional temperature

\(T_{\text{B}}\) :

Bulk temperature (K)

\(T^{\text{jump}}\) :

Temperature jump (K)

\(T_{\text{mean}}\) :

Mean temperature (K)

t :

Time (s)

TRT:

Two relaxation times

tu:

Time unit

Tu:

Temperature unit

\(U^{*}\) :

Non-dimensional velocity

\(U_{\text{mean}}\) :

Mean velocity (m s−1)

\(U_{\text{s}}\) :

Slip velocity (m s−1)

V :

Velocity vector (m s−1)

X :

Particle location at x-direction (m)

X*:

Non-dimensional parameter at x-direction

Y :

Particle location at y-direction (m)

Y*:

Non-dimensional parameter at y-direction

α:

Thermal diffusivity (m2 s−1)

\(\delta t\) :

Time step (s)

\(\delta x\) :

Step by step (m)

\(\lambda\) :

Molecular mean free path (m)

µ :

Dynamic viscosity (\({\text{N}}\,{\text{s}} \,{\text{m}}^{ - 2}\))

\(v\) :

Kinematic viscosity (m2 s−1)

\(\varPi\) :

Pressure ratio

\(\rho\) :

Density (kg m−3)

\(\sigma\) :

TMAC coefficient

\(\tau_{\text{a}}\) :

Antisymmetric relaxation time (based on slip boundary)

\(\tau_{\text{f}}\) :

Symmetric relaxation time for fluid flow

\(\tau_{\text{g}}\) :

Symmetric relaxation time for thermal fluid flow

\(\tau_{\text{s}}\) :

Symmetric relaxation time (based on viscosity)

\(\varOmega\) :

Collision operator

\(\omega_{\text{i}}\) :

Mass factors

a:

Antisymmetric

aeq:

Antisymmetric equilibrium

e:

Effective

eq:

Equilibrium

i:

Discrete lattice directions

in:

Inlet

l:

Linear

out:

Outlet

s:

Symmetric

seq:

Symmetric equilibrium

w:

Wall

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Authors and Affiliations

  1. Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, 91775-1111, Iran

    Ehsan Kamali Ahangar

  2. Faculty of Mechanical Engineering, Semnan University, Semnan, 35131-19111, Iran

    Morteza Izanlu

  3. Department of Biomedical Engineering, Amirkabir University of Technology, Tehran, 15916-34311, Iran

    Shabnam Dolati Khakhian

  4. Department of Mechanical and Manufacturing Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB, T2N 1N4, Canada

    Abdulmajeed A. Mohamad

  5. Institute of Research and Development, Duy Tan University, Danang, 550000, Vietnam

    Quang-Vu Bach

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  1. Ehsan Kamali Ahangar
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  2. Morteza Izanlu
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  3. Shabnam Dolati Khakhian
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  4. Abdulmajeed A. Mohamad
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  5. Quang-Vu Bach
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Kamali Ahangar, E., Izanlu, M., Dolati Khakhian, S. et al. Modified lattice Boltzmann solution for non-isothermal rarefied gas flow through microchannel utilizing BSR and second-order implicit schemes. J Therm Anal Calorim 144, 2525–2541 (2021). https://doi.org/10.1007/s10973-020-10129-8

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