Comparison of the lattice Boltzmann equation and discrete unified gas-kinetic scheme methods for direct numerical simulation of decaying turbulent flows

The main objective of this work is to perform a detailed comparison of the lattice Boltzmann equation (LBE) and the recently developed discrete unified gas-kinetic scheme (DUGKS) methods for direct numerical simulation (DNS) of the decaying homogeneous isotropic turbulence and the Kida vortex flow in a periodic box. The flow fields and key statistical quantities computed by both methods are compared with those from the pseudospectral method at both low and moderate Reynolds numbers. The results show that the LBE is more accurate and efficient than the DUGKS, but the latter has a superior numerical stability, particularly for high Reynolds number flows. In addition, we conclude that the DUGKS can adequately resolve the flow when the minimum spatial resolution parameter $k_{\text{max}}\eta >3$, where $k_{\text{max}}$ is the maximum resolved wave number and $\eta$ is the flow Kolmogorov length. This resolution requirement can be contrasted with the requirements of $k_{\text{max}}\eta >1$ for the pseudospectral method and $k_{\text{max}}\eta >2$ for the LBE. It should be emphasized that although more validations should be conducted before the DUGKS can be called a viable tool for DNS of turbulent flows, the present work contributes to the overall assessment of the DUGKS, and it provides a basis for further applications of DUGKS in studying the physics of turbulent flows.

Figure 1

Contours of normalized velocity magnitude $\parallel \mathit{u}\parallel /u_0^{\prime }$ (left column) and normalized vorticity magnitude $\parallel \mathbf{\omega }\parallel L/u_0^{\prime }$ (right column) on the $xy$ plane at $z=L/2$ at time $t^{\prime }=0$, 1.21, 6.08, and 12.16 (from top to bottom) with $N^3={128}^3$. The solid red, green, and blue lines denote results of the PS, LBE, and DUGKS, respectively.

Figure 2

Contours of normalized velocity magnitude $\parallel \mathit{u}\parallel /u_0^{\prime }$ (left column) and normalized vorticity magnitude $\parallel \mathbf{\omega }\parallel L/u_0^{\prime }$ (right column) on the $xy$ plane at $z=L/2$ at time $t^{\prime }=0$, 1.21, 6.08, and 12.16 (from top to bottom) with $N^3={256}^3$. The solid red, green, and blue lines denote results of the PS, LBE, and DUGKS, respectively.

Figure 3

The energy spectra $E(k,t)$ with different mesh resolutions of (a) $N^3={128}^3$ and (b) $N^3{256}^3$ at ${\text{Re}}_{\lambda }=26.06$.

Figure 4

The dissipation rate spectra $D(k,t)$ with different mesh resolutions of (a) $N^3={128}^3$ and (b) $N^3={256}^3$ at ${\text{Re}}_{\lambda }=26.06$.

Figure 5

The energy spectra difference $\mathrm{\Delta }E(k,t^{\prime })$ with different mesh resolutions of (a) $N^3={128}^3$ and (b) $N^3={256}^3$ at ${\text{Re}}_{\lambda }=26.06$.

Figure 6

The dissipation rate spectra difference $\mathrm{\Delta }D(k,t)$ with different mesh resolutions of (a) $N^3={128}^3$ and (b) $N^3={256}^3$ at ${\text{Re}}_{\lambda }=26.06$.

Figure 7

Evolutions of the normalized total kinetic energy $K\left(t\right)/K_0$ and the normalized dissipation rate $\epsilon \left(t\right)/{\epsilon }_0$ with different mesh resolutions of (a) $N^3={128}^3$ and (b) $N^3={256}^3$ at ${\text{Re}}_{\lambda }=26.06$.

Figure 8

Evolutions of the enstrophy $\mathrm{\Omega }$ with different mesh resolutions of (a) $N^3={128}^3$ and (b) $N^3={256}^3$ at ${\text{Re}}_{\lambda }=26.06$.

Figure 9

Evolutions of the Kolmogorov length $\eta$ with different mesh resolutions of (a) $N^3={128}^3$ and (b) $N^3={256}^3$ at ${\text{Re}}_{\lambda }=26.06$.

Figure 10

Evolutions of the Taylor microscale length $\lambda$ with different mesh resolutions of (a) $N^3={128}^3$ and (b) $N^3={256}^3$ at ${\text{Re}}_{\lambda }=26.06$.

Figure 11

Evolutions of velocity-derivative skewness $S$ with different mesh resolutions of (a) $N^3={128}^3$ and (b) $N^3={256}^3$ at ${\text{Re}}_{\lambda }=26.06$.

Figure 12

Evolutions of velocity-derivative flatness $F$ with different mesh resolutions of (a) $N^3={128}^3$ and (b) $N^3={256}^3$ at ${\text{Re}}_{\lambda }=26.06$.

Figure 13

Evolutions of smoothed velocity-derivative skewness $S$ with different mesh resolutions of (a) $N^3={128}^3$ and (b) $N^3={256}^3$ at ${\text{Re}}_{\lambda }=26.06$.

Figure 14

Evolutions of smoothed velocity-derivative flatness $F$ with different mesh resolutions of (a) $N^3={128}^3$ and (b) $N^3={256}^3$ at ${\text{Re}}_{\lambda }=26.06$.

Figure 15

The energy spectra $E(k,t)$ at (a) ${\text{Re}}_{\lambda }=52.12$ and (b) ${\text{Re}}_{\lambda }=104.24$.

Figure 16

The dissipation rate spectra $D(k,t)$ at (a) ${\text{Re}}_{\lambda }=52.12$ and (b) ${\text{Re}}_{\lambda }=104.24$.

Figure 17

The energy spectra difference $\mathrm{\Delta }E(k,t^{\prime })$ at (a) ${\text{Re}}_{\lambda }=52.12$ and (b) ${\text{Re}}_{\lambda }=104.24$.

Figure 18

The dissipation rate spectra difference $\mathrm{\Delta }D(k,t)$ at (a) ${\text{Re}}_{\lambda }=52.12$ and (b) ${\text{Re}}_{\lambda }=104.24$.

Figure 19

Evolutions of the normalized total kinetic energy $K\left(t\right)/K_0$ and the normalized dissipation rate $\epsilon \left(t\right)/{\epsilon }_0$ at (a) ${\text{Re}}_{\lambda }=52.12$ and (b) ${\text{Re}}_{\lambda }=104.24$.

Figure 20

Evolutions of the enstrophy $\mathrm{\Omega }$ at (a) ${\text{Re}}_{\lambda }=52.12$ and (b) ${\text{Re}}_{\lambda }=104.24$.

Figure 21

Evolutions of the Kolmogorov length $\eta$ at (a) ${\text{Re}}_{\lambda }=52.12$ and (b) ${\text{Re}}_{\lambda }=104.24$.

Figure 22

Evolutions of the Taylor microscale length $\lambda$ at (a) ${\text{Re}}_{\lambda }=52.12$ and (b) ${\text{Re}}_{\lambda }=104.24$.

Figure 23

Contours of normalized vorticity magnitude $\parallel \mathbf{\omega }\parallel L/u_0^{\prime }$ on the $xy$ plane at $z=L/2$ at time (a) $t^{\prime }=0$, (b) $t^{\prime }=0.12$, (c) $t^{\prime }=0.6$, and (d) $t^{\prime }=3$ for ${\text{Re}}_{\lambda }=52.12$. The solid red and blue lines denote results of the LBE and DUGKS, respectively.

Figure 24

Contours of normalized vorticity magnitude $\parallel \mathbf{\omega }\parallel L/u_0^{\prime }$ on the $xy$ plane at $z=L/2$ at time (a) $t^{\prime }=0$, (b) $t^{\prime }=0.1$, (c) $t^{\prime }=0.5$, and (d) $t^{\prime }=1.5$ for ${\text{Re}}_{\lambda }=104.24$. The solid red and blue lines denote results of the LBE and DUGKS, respectively.

Figure 25

Contours of the normalized vorticity magnitude $\parallel \mathbf{\omega }\parallel L/u_0^{\prime }$ on the $xy$ plane at $z=L/2$ at time (a) $t^{\prime }=0$, (b) $t^{\prime }=0.9765$, (c) $t^{\prime }=1.9531$, and (d) $t^{\prime }=2.9297$ for $\text{Re}=2000$ with the mesh of ${256}^3$. The solid blue and red lines denote results of the LBE and DUGKS, respectively.

Figure 26

Evolutions of (a) the normalized total kinetic energy $K$, dissipation rate $\epsilon$, and (b) enstrophy $\mathrm{\Omega }$.

Figure 27

Evolutions of (a) Kolmogorov length scale $\eta$ and (b) the relative difference $R\left(\eta \right)$ between the results of DUGKS and LBE on the mesh of ${128}^3$.

Figure 28

Comparisons of (a) the two-point longitudinal correlation $\left({\rho }_{11}\right)$, transversal correlations $\left({\rho }_{22},{\rho }_{33}\right)$, and (b) the pressure-velocity correlations (PU, PV, and PW) at $t=1.95$.