Kelvin-Helmholtz不稳定性的格子Boltzmann研究：速度和密度梯度的作用
Lattice Boltzmann study on Kelvin-Helmholtz instability: the roles of velocity and density gradients
摘要：
提出了一个求解可压缩Euler方程的具有19个离散速度的二维格子Boltzmann模型（D2 V19-LBM）。采用五阶加权基本无振荡差分格式（5th-WENO）计算格子Boltzmann方程的对流项。通过对Sod激波管的模拟结果与相应的解析解的比较，验证了模型的正确性。利用该模型研究了速度和密度梯度对Kelvin-Helmholtz不稳定性的影响。在我们的模拟中获得了清晰的密度轮廓。结果表明，KHI的线生长速率$γ$随速度过渡层${D_{v}}$宽度的增大而减小，随密度过渡层${D_ρ}$宽度的增大而增大。在初始过渡期之后和旋涡形成之前，线性增长率$γ_v$和$γ_ρ$随${D_{v}}$和${D_ρ}$的变化大致如下：$\lnγ_{v}=a-bD_{v}$和$γ_ρ=c+e\ln D_ρ ({D_ρ}<{D_ρ^{E}})$，其中$a$、$b$、$c$和$e$是拟合参数，${D_ρ^{E}}$是密度过渡层的有效相互作用宽度。当使用${D_ρ}>{D_ρ^{E}}$时，线性生长速率$γ_ρ$不再显著变化。人们可以利用速度和密度过渡层的混合效应来稳定KHI。数值模拟结果与解析结果基本一致。F. Wang，\emph{et al.}，Phys. Plasma \textbf{17}，042103（2010）]。
英文摘要：
A two-dimensional lattice Boltzmann model with 19 discrete velocities for compressible Euler equations is proposed (D2V19-LBM). The fifth-order Weighted Essentially Non-Oscillatory (5th-WENO) finite difference scheme is employed to calculate the convection term of the lattice Boltzmann equation. The validity of the model is verified by comparing simulation results of the Sod shock tube with its corresponding analytical solutions. The velocity and density gradient effects on the Kelvin-Helmholtz instability (KHI) are investigated using the proposed model. Sharp density contours are obtained in our simulations. It is found that, the linear growth rate $γ$ for the KHI decreases with increasing the width of velocity transition layer ${D_{v}}$ but increases with increasing the width of density transition layer ${D_ρ}$. After the initial transient period and before the vortex has been well formed, the linear growth rates, $γ_v$ and $γ_ρ$, vary with ${D_{v}}$ and ${D_ρ}$ approximately in the following way, $\lnγ_{v}=a-bD_{v}$ and $γ_ρ=c+e\ln D_ρ ({D_ρ}<{D_ρ^{E}})$, where $a$, $b$, $c$ and $e$ are fitting parameters and ${D_ρ^{E}}$ is the effective interaction width of density transition layer. When ${D_ρ}>{D_ρ^{E}}$ the linear growth rate $γ_ρ$ does not vary significantly any more. One can use the hybrid effects of velocity and density transition layers to stabilize the KHI. Our numerical simulation results are in general agreement with the analytical results [L. F. Wang, \emph{et al.}, Phys. Plasma \textbf{17}, 042103 (2010)].
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