﻿ 一种用于双组份爆轰的格子Boltzmann模型

# 一种用于双组份爆轰的格子Boltzmann模型A Lattice Boltzmann Model for Binary Components Detonation

Abstract: In this paper, we present a lattice Boltzmann model for simulating the binary components detonation phenomena. For modeling the flow behavior in the detonation process, we employ two distribution functions for the density, momentum and energy of reactant and product, respectively. The Lee-Tarver model is selected to describe the chemical reaction kinetics. The reaction heat is naturally coupled with the ﬂow behavior. The numerical examples show that the scheme can be used to simulate the detonation phenomena.

1. 引言

2. 格子Boltzmann模型

2.1. 描述流动的双组份格子Boltzmann模型

$\frac{\partial {f}_{ki}^{\sigma }}{\partial t}+{v}_{ki}\cdot \frac{\partial {f}_{ki}^{\sigma }}{\partial r}=-\frac{1}{{\tau }^{\sigma }}\left[{f}_{ki}^{\sigma }-{f}_{ki}^{\sigma ,eq}\right]$ . (1)

${\rho }^{\sigma }={m}^{\sigma }\underset{ki}{\sum }{f}_{ki}^{\sigma }$ . (2)

${n}^{\sigma }{u}^{\sigma }=\underset{ki}{\sum }{f}_{ki}^{\sigma }{v}_{ki}$ , (3)

$u=\frac{{\rho }^{r}{u}^{r}+{\rho }^{p}{u}^{p}}{{\rho }^{r}+{\rho }^{p}}$ . (4)

${\rho }^{\sigma }{T}^{\sigma }={P}^{\sigma }={e}_{therm}^{\sigma }={m}^{\sigma }\underset{ki}{\sum }\frac{1}{2}{v}_{vi}^{2}{f}_{ki}^{\sigma }-\frac{1}{2}{\rho }^{2}{\left({u}^{\sigma }\right)}^{2}$ , (5)

$T=\frac{{\rho }^{r}{T}^{r}+{\rho }^{p}{T}^{p}}{{\rho }^{r}+{\rho }^{p}}$ . (6)

$\sigma$ 组份的局域平衡态分布函数 ${f}_{ki}^{\sigma ,eq}$ 由下式给出

$\begin{array}{c}{f}_{ki}^{\sigma ,eq}={n}^{\sigma }{F}_{k}\left\{\left[1-\frac{{u}^{2}}{2{\theta }^{\sigma }}+\frac{{u}^{4}}{8{\left({\theta }^{\sigma }\right)}^{2}}\right]+\frac{{v}_{ki\epsilon }{u}_{\epsilon }}{{\theta }^{\sigma }}\left(1-\frac{{u}^{2}}{2{\theta }^{\sigma }}\right)+\frac{{v}_{ki\epsilon }{v}_{ki\pi }{u}_{\epsilon }{u}_{\pi }}{2{\left({\theta }^{\sigma }\right)}^{2}}\left(1-\frac{{u}^{2}}{2{\theta }^{\sigma }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{v}_{ki\epsilon }{v}_{ki\pi }{v}_{ki\upsilon }{u}_{\epsilon }{u}_{\pi }{u}_{\upsilon }}{6{\left({\theta }^{\sigma }\right)}^{3}}+\frac{{v}_{ki\epsilon }{v}_{ki\pi }{v}_{ki\upsilon }{v}_{ki\xi }{u}_{\epsilon }{u}_{\pi }{u}_{\upsilon }{u}_{\xi }}{24{\left({\theta }^{\sigma }\right)}^{4}}\right\}\end{array}$ , (7)

${m}^{\sigma }\underset{ki}{\sum }{v}_{ki\alpha }{v}_{ki\beta }{f}_{ki}^{\sigma ,eq}={e}_{therm}^{\sigma }{\delta }_{\alpha \beta }+{\rho }^{\sigma }{u}_{\alpha }^{\sigma }{u}_{\beta }^{\sigma }$ , (8)

${m}^{\sigma }\underset{ki}{\sum }{v}_{ki\alpha }{v}_{ki\beta }{v}_{ki\gamma }{f}_{ki}^{\sigma ,eq}={e}_{therm}^{\sigma }\left({u}_{\gamma }^{\sigma }{\delta }_{\alpha \beta }+{u}_{\alpha }^{\sigma }{\delta }_{\beta \gamma }+{u}_{\beta }^{\sigma }{\delta }_{\gamma \alpha }\right)+{\rho }^{\sigma }{u}_{\alpha }^{\sigma }{u}_{\beta }^{\sigma }{u}_{\gamma }^{\sigma }$ , (9)

${m}^{\sigma }\underset{ki}{\sum }\frac{1}{2}{v}_{k}^{2}{v}_{ki\alpha }{f}_{ki}^{\sigma ,eq}=2{e}_{therm}^{\sigma }{u}_{\alpha }^{\sigma }+\frac{1}{2}{\rho }^{\sigma }{\left({u}^{\sigma }\right)}^{2}{u}_{\alpha }^{\sigma }$ , (10)

${m}^{\sigma }\underset{ki}{\sum }\frac{1}{2}{v}_{k}^{2}{v}_{ki\alpha }{v}_{ki\beta }{f}_{ki}^{\sigma ,eq}=\left[2\frac{{T}^{\sigma }}{{m}^{\sigma }}+\frac{1}{2}{\left({u}^{\sigma }\right)}^{2}\right]{e}_{therm}^{\sigma }{\delta }_{\alpha \beta }+\left[3{e}_{therm}^{\sigma }+\frac{1}{2}{\rho }^{\sigma }{\left({u}^{\sigma }\right)}^{2}\right]{u}_{\alpha }^{\sigma }{u}_{\beta }^{\sigma }$ , (11)

$\begin{array}{c}{f}_{kiI}^{\sigma ,new}={f}_{kiI}^{\sigma }-\frac{{c}_{ki\alpha }}{2}\left({f}_{kiI+1}^{\sigma }-{f}_{kiI-1}^{\sigma }\right)-\frac{\Delta t}{\tau }\left({f}_{kiI}^{\sigma }-{f}_{kiI}^{\sigma ,eq}\right)+\frac{{c}_{ki\alpha }^{2}}{2}\left({f}_{kiI+1}^{\sigma }-2{f}_{kiI}^{\sigma }+{f}_{kiI-1}^{\sigma }\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{c}_{ki\alpha }\left(1-{c}_{ki\alpha }^{2}\right)}{12}\left({f}_{kiI+2}^{\sigma }-2{f}_{kiI+1}^{\sigma }+2{f}_{kiI-1}^{\sigma }-{f}_{kiI-2}^{\sigma }\right)+\frac{{\theta }_{\alpha I}^{\sigma }|{k}_{\alpha }^{\sigma }|\left(1-|{k}_{\alpha }^{\sigma }|\right)}{2}\left({f}_{kiI+1}^{\sigma }-2{f}_{kiI}^{\sigma }+{f}_{kiI-1}^{\sigma }\right).\end{array}$ (12)

2.2. Lee-Tarver反应率函数

$\frac{\text{d}\lambda }{\text{d}t}=\left\{\begin{array}{l}a\left(1-\lambda \right)+b\left(1-\lambda \right)\lambda ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}T\ge {T}_{th},且0\le \lambda \le 1\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其它\end{array}$ (13)

$\frac{\partial \lambda }{\partial t}+u\nabla \lambda =0$ ,(14)

$\frac{{\lambda }_{I}^{n+1}-{\lambda }_{I}^{n}}{\Delta t}=-\left\{\begin{array}{l}\frac{u\left({\lambda }_{I}^{n}-{\lambda }_{I-1}^{n}\right)}{\Delta x},u\ge 0\\ \frac{u\left({\lambda }_{I+1}^{n}-{\lambda }_{I}^{n}\right)}{\Delta x},u<0\end{array}$ ,(15)

$\frac{\partial \lambda }{\partial t}=a\left(1-\lambda \right)+b\left(1-\lambda \right)\lambda$ , (16)

${\lambda }_{I}^{n+1}=\frac{{\text{e}}^{\left(a+b\right)\Delta t}+a\left({\lambda }_{I}^{n}-1\right)/\left(a+b{\lambda }_{I}^{n}\right)}{{\text{e}}^{\left(a+b\right)\Delta t}+b\left(1-{\lambda }_{I}^{n}\right)/\left(a+b{\lambda }_{I}^{n}\right)}$ .(17)

2.3. 化学反应与流动的耦合

$\stackrel{˙}{e}={\stackrel{˙}{e}}_{therm}+{\stackrel{˙}{e}}_{chem}$ , (18)

${\stackrel{˙}{e}}_{chem}=\stackrel{˙}{\lambda }\rho Q$ .(19)

${\rho }^{r,new}={\rho }^{r}-\stackrel{˙}{\lambda }\rho$ ,(20)

${\rho }^{p,new}={\rho }^{p}+\stackrel{˙}{\lambda }\rho$ . (21)

3. 数值例子

3.1. 带有黏性和热传导的活塞问题

${\left(\rho ,u,v,T,\lambda \right)}_{L}=\left(1.35,0.81,0,2.65,1\right)$ , (22)

${\left(\rho ,u,v,T,\lambda \right)}_{R}=\left(1,0,0,1,0\right)$ . (23)

Figure 1. Physical quantity profiles for the piston problem including effects of viscosity and heat conduction at times 0, 0.05, 0.1, 0.15, 0.2, 0.25 and 0.3: (a) density $\rho$ ; (b) pressure $P$ ; (c) x-component of velocity $u$ ; (d) temperature $T$

${\rho }_{0}\left(D-{u}_{0}\right)={\rho }_{1}\left(D-{u}_{1}\right)$ , (24)

${P}_{1}-{P}_{0}={\rho }_{0}\left(D-{u}_{0}\right)\left({u}_{1}-{u}_{0}\right)$ , (25)

${e}_{1}-{e}_{0}=0.5\left({P}_{1}+{P}_{0}\right)\left(1/{\rho }_{0}-1/{\rho }_{1}\right)+\lambda Q$ , (26)

3.2. 爆轰波的正规反射和Mach反射

$\left(\rho ,u,v,T,\lambda \right)=\left(1.36,0.82,0,2.60,1\right)$ ，在A点和B点处，

$\left(\rho ,u,v,T,\lambda \right)=\left(1,0,0,1,0\right)$ ，其他位置.

Figure 2. Sketch of regular and Mach reflections of detonation wave

Figure 3. The transition process from regular reflection to Mach reflection: (a) Ignition arises and detonation waves generate; (b) The regular reflection of detonation wave; (c) Conversion from the regular reflection to Mach reflection; (d) The Mach reflection of detonation wave

4. 结论

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