Vol.06 No.09(2017), Article ID:23097,9 pages
10.12677/AAM.2017.69137

The Computation of Lee-Tarver Detonation Based on Lattice Boltzmann Model

Bo Yan, Jianchao Wang

College of Civil Engineering, Jilin Jianzhu University, Changchun Jilin

Received: Nov. 28th, 2017; accepted: Dec. 14th, 2017; published: Dec. 21st, 2017

ABSTRACT

In this paper, we present a high speed compressible lattice Boltzmann model coupled with Lee-Tarver reaction rate function for detonation. Two distribution functions are used to describe the density, momentum and energy of reactant and product in the lattice Boltzmann scheme, which gives consistent results with the Navier-Stokes equation in the continuum limit. Due to the separation of time scales in the chemical and thermodynamic process, the operator-splitting scheme is employed to solve Lee-Tarver reaction rate function. To indicate the validity of the model, we studied the collision between detonation and shock waves, the Richtmyer-Meshkov instability by detonation. The numerical examples show that the scheme can be used to compute the detonation phenomena.

Keywords:Lattice Boltzmann Model, Detonation, Richtmyer-Meshkov Instability, Lee-Tarver Reaction Rate Function

1. 引言

2. 格子Boltzmann模型

2.1. 描述宏观流动的格子Boltzmann模型

$\left\{\begin{array}{l}{v}_{0}=0,\\ {v}_{ki}={v}_{k}\left[cos\left(i\text{π}/4\right),sin\left(i\text{π}/4\right)\right].\end{array}$ (1)

$\frac{\partial {f}_{ki}^{\sigma }}{\partial t}+{v}_{ki}\cdot \frac{\partial {f}_{ki}^{\sigma }}{\partial r}=-\frac{1}{{\tau }^{\sigma }}\left[{f}_{ki}^{\sigma }-{f}_{ki}^{\sigma ,eq}\right]$ . (2)

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

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

$T$ 表示局域平均温度，有

$T=\frac{{\rho }^{r}{T}^{r}+{\rho }^{p}{T}^{p}}{{\rho }^{r}+{\rho }^{p}}$(5)

$\sum _{ki}{f}_{ki}^{\sigma ,eq}={\rho }^{\sigma }$ , (6)

$\sum _{ki}{v}_{ki}{f}_{ki}^{\sigma ,eq}={\rho }^{\sigma }{u}^{\sigma }$ , (7)

$\sum _{ki}\frac{1}{2}{v}_{vi}^{2}{f}_{ki}^{\sigma ,eq}={e}_{therm}^{\sigma }+\frac{1}{2}{\rho }^{\sigma }{\left({u}^{\sigma }\right)}^{2}={\rho }^{\sigma }{T}^{\sigma }+\frac{1}{2}{\rho }^{\sigma }{\left({u}^{\sigma }\right)}^{2}={P}^{\sigma }+\frac{1}{2}{\rho }^{\sigma }{\left({u}^{\sigma }\right)}^{2}$ , (8)

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

$\sum _{ki}{v}_{ki\alpha }{v}_{ki\beta }{v}_{ki\gamma }{f}_{ki}^{\sigma ,eq}={e}_{therm}^{\sigma }\left({u}_{\gamma }^{\sigma }{\delta }_{\alpha \beta }+{u}_{\alpha }^{\sigma }{\delta }_{\beta \gamma }+{u}_{\beta }^{\sigma }{\delta }_{\gamma \alpha }\right)+{\rho }^{\sigma }{u}_{\alpha }^{\sigma }{u}_{\beta }^{\sigma }{u}_{\gamma }^{\sigma }$ ,(10)

$\sum _{ki}\frac{1}{2}{v}_{k}^{2}{v}_{ki\alpha }{f}_{ki}^{\sigma ,eq}=2{e}_{therm}^{\sigma }{u}_{\alpha }^{\sigma }+\frac{1}{2}{\rho }^{\sigma }{\left({u}^{\sigma }\right)}^{2}{u}_{\alpha }^{\sigma }$ , (11)

$\sum _{ki}\frac{1}{2}{v}_{k}^{2}{v}_{ki\alpha }{v}_{ki\beta }{f}_{ki}^{\sigma ,eq}=\left[2{T}^{\sigma }+\frac{1}{2}{\left({u}^{\sigma }\right)}^{2}\right]{e}_{therm}^{\sigma }{\delta }_{\alpha \beta }+\left[3{e}_{therm}^{\sigma }+\frac{1}{2}{\rho }^{\sigma }{\left({u}^{\sigma }\right)}^{2}\right]{u}_{\alpha }^{\sigma }{u}_{\beta }^{\sigma }$ . (12)

$\begin{array}{c}{f}_{kiI}^{\sigma ,new}={f}_{kiI}^{\sigma }-\frac{{c}_{ki\alpha }}{2}\left({f}_{kiI+1}^{\sigma }-{f}_{kiI-1}^{\sigma }\right)-\frac{\Delta t}{\tau }\left({f}_{kiI}^{\sigma }-{f}_{kiI}^{\sigma ,eq}\right)+\frac{{c}_{ki\alpha }^{2}}{2}\left({f}_{kiI+1}^{\sigma }-2{f}_{kiI}^{\sigma }+{f}_{kiI-1}^{\sigma }\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{c}_{ki\alpha }\left(1-{c}_{ki\alpha }^{2}\right)}{12}\left({f}_{kiI+2}^{\sigma }-2{f}_{kiI+1}^{\sigma }+2{f}_{kiI-1}^{\sigma }-{f}_{kiI-2}^{\sigma }\right)+\frac{{\theta }_{\alpha I}^{\sigma }|{k}_{\alpha }^{\sigma }|\left(1-|{k}_{\alpha }^{\sigma }|\right)}{2}\left({f}_{kiI+1}^{\sigma }-2{f}_{kiI}^{\sigma }+{f}_{kiI-1}^{\sigma }\right).\end{array}$ (13)

2.2. 描述化学反应的Lee-Tarver反应率函数

$\frac{\text{d}\lambda }{\text{d}t}=\left\{\begin{array}{l}a\left(1-\lambda \right)+b\left(1-\lambda \right)\lambda ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}T\ge {T}_{th},且0\le \lambda \le 1\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其它\end{array}$ (14)

$\frac{\partial \lambda }{\partial t}+u\nabla \lambda =0$ (15)

$\frac{{\lambda }_{I}^{n+1}-{\lambda }_{I}^{n}}{\Delta t}=-\left\{\begin{array}{l}\frac{u\left({\lambda }_{I}^{n}-{\lambda }_{I-1}^{n}\right)}{\Delta x},u\ge 0\\ \frac{u\left({\lambda }_{I+1}^{n}-{\lambda }_{I}^{n}\right)}{\Delta x},u<0\end{array}$ (16)

$\frac{\partial \lambda }{\partial t}=a\left(1-\lambda \right)+b\left(1-\lambda \right)\lambda$ , (17)

${\lambda }_{I}^{n+1}=\frac{{\text{e}}^{\left(a+b\right)\Delta t}+a\left({\lambda }_{I}^{n}-1\right)/\left(a+b{\lambda }_{I}^{n}\right)}{{\text{e}}^{\left(a+b\right)\Delta t}+b\left(1-{\lambda }_{I}^{n}\right)/\left(a+b{\lambda }_{I}^{n}\right)}$ . (18)

2.3. 化学反应与流动的耦合

$\stackrel{˙}{e}={\stackrel{˙}{e}}_{therm}+{\stackrel{˙}{e}}_{chem}$ , (19)

${\stackrel{˙}{e}}_{chem}=\stackrel{˙}{\lambda }\rho Q$ . (20)

${\rho }^{r,new}={\rho }^{r}-\stackrel{˙}{\lambda }\rho$ ,(21)

${\rho }^{p,new}={\rho }^{p}+\stackrel{˙}{\lambda }\rho$ .(22)

3. 数值例子

${\rho }_{0}\left(D-{u}_{0}\right)={\rho }_{1}\left(D-{u}_{1}\right)$ , (23)

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

${e}_{1}-{e}_{0}=0.5\left({P}_{1}+{P}_{0}\right)\left(1/{\rho }_{0}-1/{\rho }_{1}\right)+\lambda Q$ . (25)

${\left(\rho ,u,v,T,\lambda \right)}_{L}={\left(1.26,0.35,0,1.26,0\right)}_{L}$ ,

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

${\left(\rho ,u,v,T,\lambda \right)}_{R}={\left(5.04,0,0,0.19,0\right)}_{R}$ .

(a) (b) (c) (d)

Figure 1. Physical quantity profiles for before and after collision of detonation and shock. (a) Density $\rho$ , (b) pressure $P$ , (c) x-component of velocity $u$ , (d) mean temperature $T$

Table 1. The Hugoniot relations before and after the transmitted detonation wave

Table 2. The Hugoniot relations before and after the transmitted shock wave

(a) (b) (c) (d)

Figure 2. Snapshots of density field for the Richtmyer-Meshkov instability in the case of detonation wave travels from heavy to light media. (a) t = 0, (b) t = 0.05, (c) t = 0.2, (d) t = 1.0

4. 结论

The Computation of Lee-Tarver Detonation Based on Lattice Boltzmann Model[J]. 应用数学进展, 2017, 06(09): 1126-1134. http://dx.doi.org/10.12677/AAM.2017.69137

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