Discrete Boltzmann trans-scale modeling of high-speed compressible flows

We present a general framework for constructing trans-scale discrete Boltzmann models (DBMs) for high-speed compressible flows ranging from continuum to transition regime. This is achieved by designing a higher-order discrete equilibrium distribution function that satisfies additional nonhydrodynamic kinetic moments. To characterize the thermodynamic nonequilibrium (TNE) effects and estimate the condition under which the DBMs at various levels should be used, two measures are presented: (i) the relative TNE strength, describing the relative strength of the ($N+1$)th order TNE effects to the $N\mathrm{th}$ order one; (ii) the TNE discrepancy between DBM simulation and relevant theoretical analysis. Whether or not the higher-order TNE effects should be taken into account in the modeling and which level of DBM should be adopted is best described by the relative TNE intensity and/or the discrepancy rather than by the value of the Knudsen number. As a model example, a two-dimensional DBM with 26 discrete velocities at Burnett level is formulated, verified, and validated.

Figure 1

Flow chart for the discrete Boltzmann trans-scale modeling of compressible flows.

Figure 2

Schematic of the D2V26 discrete-velocity model, where ${\mathbf{v}}_{25}$(=$-{\mathbf{v}}_{26}$) is a flexible vector, adjusted to guarantee the existence of ${\mathbf{C}}^{-1}$.

Figure 3

Comparisons between DBM simulations and the exact solutions for the Sod shock tube, where $t=0.1$ and $\gamma =1.4$. (a) Density, (b) pressure, (c) velocity, and (d) temperature.

Figure 4

Comparisons between DBM simulations and the exact solutions for the modified Lax shock tube, where $t=0.07$ and $\gamma =2$. (a) Density, (b) pressure, (c) velocity, and (d) temperature.

Figure 5

Comparisons between DBM simulations and the exact solutions for the collision of two strong shocks, where $t=0.05$ and $\gamma =1.67$. (a) Density, (b) pressure, (c) velocity, and (d) temperature.

Figure 6

Viscous stress calculated from D2V16 (left column) and D2V26 (right column) DB simulations (scatters) for the weak (I), moderate (II), and strong (III) cases, where dashed and solid lines indicate analytical solutions with first- and second-order accuracies, respectively.

Figure 7

Hydrodynamic quantities calculated from the D2V26 model and the corresponding differences between the D2V26 and D2V16 models at $t=0.025$ for case III. (a) Density, (b) pressure, (c) velocity, and (d) temperature.

Figure 8

Viscous stress for the very strong case (a) and the local Knudsen numbers calculated from pressure, density, and temperature (b).

Figure 9

Effects of shock intensity (a) and interface width (b) on TNE effects.

Figure 10

Heat flux calculated from D2V16 (left column) and D2V26 (right column) DBM simulation(scatters) for the weak (I), moderate (II), and strong (III) cases, where dashed and solid lines indicate analytical solutions with first- and second-order accuracies, respectively.