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Research Papers: Techniques and Procedures

Relationship between accuracy and number of velocity particles of the finite-difference lattice Boltzmann method in velocity slip simulations

[+] Author and Article Information
Minoru Watari

LBM Fluid Dynamics Laboratory, 3-2-1 Mitahora-higashi, Gifu 502-0003, Japanwatari-minoru@kvd.biglobe.ne.jp

J. Fluids Eng 132(10), 101401 (Oct 06, 2010) (11 pages) doi:10.1115/1.4002359 History: Received February 27, 2010; Revised August 10, 2010; Published October 06, 2010; Online October 06, 2010

Abstract

Relationship between accuracy and number of velocity particles in velocity slip phenomena was investigated by numerical simulations and theoretical considerations. Two types of 2D models were used: the octagon family and the D2Q9 model. Models have to possess the following four prerequisites to accurately simulate the velocity slip phenomena: (a) equivalency to the Navier–Stokes equations in the N-S flow area, (b) conservation of momentum flow $Pxy$ in the whole area, (c) appropriate relaxation process in the Knudsen layer, and (d) capability to properly express the mass and momentum flows on the wall. Both the octagon family and the D2Q9 model satisfy conditions (a) and (b). However, models with fewer velocity particles do not sufficiently satisfy conditions (c) and (d). The D2Q9 model fails to represent a relaxation process in the Knudsen layer and shows a considerable fluctuation in the velocity slip due to the model’s angle to the wall. To perform an accurate velocity slip simulation, models with sufficient velocity particles, such as the triple octagon model with moving particles of 24 directions, are desirable.

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Figures

Figure 3

Unit vectors of moving particles of the octagon family: from left to right, octagon, double octagon, and triple octagon models

Figure 5

D2Q9 model particles

Figure 6

A flow between parallel plates

Figure 7

Nodes for the velocity slip simulation

Figure 9

Octagon family: velocity profile for three models with τ=0.05

Figure 10

Octagon family: velocity slip divided by the velocity gradient uy.slip/(duy/dx) versus Knudsen number

Figure 11

Octagon family: Knudsen profile for three models with τ=0.05

Figure 13

Octagon family: Effect of angle to the wall for three models with τ=0.05

Figure 14

D2Q9 model: Knudsen profile by simulations for various τ

Figure 1

Typical velocity slip profile

Figure 2

Normalized Knudsen profile Y0

Figure 4

Average particle speed of the octagon family and the theoretical value

Figure 8

Grid dependence study: rms error for the octagon model

Figure 12

Relative angular situation between the model and the wall

Figure 15

D2Q9 model: Velocity slip divided by the velocity gradient uy.slip/(duy/dx) versus the relaxation time constant τ

Figure 16

D2Q9 model: effect of the angle θ on the velocity slip for τ=0.05

Figure 17

Steady Knudsen solutions for the octagon model

Figure 18

Knudsen profile comparison between the simulation in Sec. 5 and the Knudsen simulation for the octagon model

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