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Research Papers: Micro/Nanoscale Heat Transfer

On Variance-Reduced Simulations of the Boltzmann Transport Equation for Small-Scale Heat Transfer Applications

[+] Author and Article Information
Nicolas G. Hadjiconstantinou, Gregg A. Radtke, Lowell L. Baker

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

Within the rarefied gas dynamics literature, the relaxation-time approximation is known as the BGK model (20).

Note that a symmetrized algorithm typically provides higher-order accuracy. See Refs. 8-9 and references therein.

J. Heat Transfer 132(11), 112401 (Aug 13, 2010) (8 pages) doi:10.1115/1.4002028 History: Received January 21, 2009; Revised April 12, 2010; Published August 13, 2010; Online August 13, 2010

We present and discuss a variance-reduced stochastic particle simulation method for solving the relaxation-time model of the Boltzmann transport equation. The variance reduction, achieved by simulating only the deviation from equilibrium, results in a significant computational efficiency advantage compared with traditional stochastic particle methods in the limit of small deviation from equilibrium. More specifically, the proposed method can efficiently simulate arbitrarily small deviations from equilibrium at a computational cost that is independent of the deviation from equilibrium, which is in sharp contrast to traditional particle methods. The proposed method is developed and validated in the context of dilute gases; despite this, it is expected to directly extend to all fields (carriers) for which the relaxation-time approximation is applicable.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

The relative statistical uncertainty in temperature, σT/ΔT, as a function of ε for steady-state heat exchange between two parallel plates at different temperatures with k=1, α=1, and T0=273 K. Simulation results (symbols with error bars on solid line) are presented and compared with the theoretical prediction (15) for DSMC (dashed line), which serves as a canonical case of a nondeviational method. Stars show actual DSMC results verifying that equilibrium theory is reliable up to at least ε≈0.3.

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Figure 2

Comparison between the numerical solution of the linearized Boltzmann equation by Bassanini (27) (solid line) and simulation results (circles) for the heat flux between two parallel, infinite, fully accommodating walls. Some numerical solution data for k>1 have been transcribed from Ref. 20.

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Figure 3

Comparison between the numerical solution of the linearized Boltzmann equation by Bassanini (28) (solid line) and simulation results (circles) for the heat flux between two parallel, infinite walls with α=0.826

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Figure 4

Comparison between the proposed method (dots) and the theoretical results of Ref. 29 (solid lines) for ωL/c0=40π, k=1, α=1, and ε=0.02. Here, ρ0=mn0.

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Figure 5

Impulsive heating problem for ε=0.1 and k=10 at t=0.02τ0, 0.1τ0, 0.2τ0, and 0.4τ0. The solid line denotes the present method, and stars denote DSMC results.

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Figure 7

Impulsive heating problem for ε=0.1 and k=0.1 at t=4τ0, 12τ0, and 40τ0. The solid line denotes the present method, and stars denote DSMC results.

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Figure 8

Impulsive heating problem for ε=0.3 and k=1 at t=0.4τ0, 1.6τ0, and 8τ0. The solid line denotes the present method and stars denote DSMC results.

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Figure 6

Impulsive heating problem for ε=0.1 and k=1 at t=0.4τ0, 1.6τ0, and 8τ0. The solid line denotes the present method, and stars denote DSMC results.

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