Lattice Boltzmann study on Kelvin-Helmholtz instability: Roles of velocity and density gradients

A two-dimensional lattice Boltzmann model with 19 discrete velocities for compressible fluids is proposed. The fifth-order weighted essentially nonoscillatory (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 [G. A. Sod, J. Comput. Phys. 27, 1 (1978).]. 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 by increasing the width of velocity transition layer Dᵥ but increases by 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 γᵥ and γ_ρ, vary with Dᵥ and D_ρ approximately in the following way, lnγᵥ=a-bDᵥ and γ_ρ=c+elnD_ρ(D_ρ<D_ρE), where a, b, c, and e are fitting parameters and D_ρE is the effective interaction width of the density transition layer. When D_ρ>D_ρE 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 et al., Phys. Plasma 17, 042103 (2010)].