Comparison of the lattice Boltzmann and pseudo-spectral methods for decaying turbulence: Low-order statistics

Abstract

We conduct a detailed comparison of the lattice Boltzmann equation (LBE) and the pseudo-spectral (PS) methods for direct numerical simulations (DNS) of the decaying homogeneous isotropic turbulence in a three-dimensional periodic cube. We use a mesh size of N 3 = 128 3 and the Taylor micro-scale Reynolds number 24.35 ⩽ Re λ ⩽ 72.37 , and carry out all simulations to t ≈ 30 τ 0 , where τ 0 is the turbulence turnover time. In the PS method, the second-order Adam–Bashforth scheme is used to numerically integrate the nonlinear term while the viscous term is treated exactly. We compare the following quantities computed by the LBE and PS methods: instantaneous velocity u and vorticity ω fields, and statistical quantities such as, the total energy K ( t ) and the energy spectrum E ( k , t ) , the dissipation rate ε ( t ) , the root-mean-squared (rms) pressure fluctuation δ p ( t ) and the pressure spectrum P ( k , t ) , and the skewness and flatness of the velocity derivative. Our results show that the LBE method performs very well when compared to the PS method in terms of accuracy and efficiency: the instantaneous flow fields, u and ω , and all the statistical quantities — except the rms pressure fluctuation δ p ( t ) and the pressure spectrum P ( k , t ) — computed from the LBE and PS methods agree well with each other, provided that the initial flow field is adequately resolved by both methods. We note that δ p ( t ) and P ( k , t ) computed from the two methods agree with each other in a period of time much shorter than that for other quantities, indicating that the pressure field p computed by using the LBE method is less accurate than other quantities. The skewness and flatness computed from the LBE method contain high-frequency oscillations due to acoustic waves in the system, which are absent in PS methods. Our results indicate that the resolution requirement for the LBE method is δ x / η 0 ⩽ 1.0 , approximately twice of the requirement for PS methods, where δ x and η 0 are the grid spacing and the initial Kolmogorov length, respectively. Overall, the LBE method is shown to be a reliable and accurate method for the DNS of decaying turbulence.