Multiscale semi-Lagrangian lattice Boltzmann method

We present a multi-scale lattice Boltzmann scheme, which adaptively refines particles' velocity space. Different velocity sets of lower and higher order are consistently and efficiently coupled, allowing us to use the higher-order model only when and where needed. This includes regions of high Mach or high Knudsen numbers. The coupling procedure of discrete velocity sets consists of either a projection of the higher-order populations onto the lower-order lattice or lifting of the lower-order populations to the higher-order velocity space. Both lifting and projection are local operations, which enable a flexible adaptive velocity set. The proposed scheme is formulated for both a static and an optimal, co-moving reference frame, in the spirit of the recently introduced Particles on Demand method. The multi-scale scheme is validated with an advection of an athermal vortex and in a jet flow setup. The performance of the proposed scheme is further investigated in the shock structure problem and a high-Knudsen-number Couette flow, typical examples of highly non-equilibrium flows in which the order of the velocity set plays a decisive role. The results demonstrate that the proposed multi-scale scheme can operate accurately, with flexibility in terms of the underlying models and with reduced computational requirements.

Figure 1

Density profile for Sod's shock tube, at time $t^{*}=0.2$, with the PonD $D2Q9$ model.

Figure 2

Deviation of the $xxx$ component of the $D2Q9$ equilibrium third order tensor $Q_{xxx}^L$ (49) from its Maxwell–Boltzmann counterpart $Q_{xxx}^{\text{eq}}$ (47) as a function of Mach number $\mathrm{Ma}=u/c_s$. $\epsilon =(Q_{xxx}^L-Q_{xxx}^{\text{eq}})/Q_{xxx}^{\text{eq}}$. The deviation vanishes for higher-order lattices.

Figure 3

Schematic of the multi-scale semi-Lagrangian propagation. For illustration purposes we use four collocation points for the interpolation and we consider the case of lifting operation. We monitor the propagation step of the high-order population $f_i(\mathit{x},t)$ (point $\mathit{c}$). The departure point ${\mathit{x}}_{\text{dep},i}=\mathit{x}-{\mathit{v}}_i\delta t$ (black circle) is surrounded by the collocation points $\mathit{a}$ to $\mathit{d}$. The lifting operation is applied at the points $\mathit{a}$ and $\mathit{b}$, which use the low-order velocity set. The semi-Lagrangian step concludes with the interpolation of the populations at the departure point.

Figure 4

Velocity set distribution for the Taylor–Green vortex.

Figure 5

(a) Comparison of the $y$-velocity with the analytical solution (64) at three times, $t_1^{*}=0.057$, $t_2^{*}=0.57$ and $t_3^{*}=1.15$. (b) Scaling of $L_2$ error for the $D2Q9$ and the coupled $D2Q9/25$ models. The $L_2$ error is computed based on $x$-velocity.

Figure 6

Vortex advection with the coupled $D2Q9/25$ athermal model. (a), (b): Density; (c), (d): Mach number; (e), (f): Velocity set. Snapshots at $t_1=0$ (left) and $t_2=500$ (right). Black contour in (b), (c) corresponds to the threshold Mach number ${\mathrm{Ma}}_{\mathrm{thr}}=0.3$. The higher-order $D2Q25$ patch follows the advected vortex.

Figure 7

Density contours of an athermal advected vortex simulation with $D2Q9$, $D2Q25$ and the coupled $D2Q9/25$.

Figure 8

Density of an athermal advected vortex simulation with $D2Q9$, $D2Q25$ and coupled $D2Q9/25$. The simulations for the coupled $D2Q9/25$ have been performed with three different Mach number thresholds, ${\mathrm{Ma}}_{\mathrm{thr}}=(0.27,0.3,0.4)$.

Figure 9

Simulation of athermal jet flow with the coupled $D2Q9/25$ model. (Top) Snapshot of the $x$-velocity field; (Bottom) velocity set throughout the domain.

Figure 10

Simulation of athermal jet flow with the coupled $D2Q9/25$ model. (Top) Fraction of time using the higher-order $D2Q25$ velocity set; (Bottom) Time average of the $x$-velocity.

Figure 11

Comparison of athermal jet flow with $D2Q9$, $D2Q25$ and coupled $D2Q9/25$. (a) Mean streamwise velocity profiles along the centerline; (b) Mean streamwise velocity profiles at $x/D_{\text{jet}}=12$; (c) Mean streamwise velocity profiles at $x/D_{\text{jet}}=16$.

Figure 12

Evolution of Knudsen profile (top) and velocity set (bottom) at times $t=100,500,4000$.

Figure 13

Multi-scale $D2Q16/25$, shock-structure for $\mathrm{Ma}=1.6$. Density, temperature, velocity, normal stress and heat flux profiles, compared with results of Ohwada [47].

Figure 14

Multi-scale $D2Q16/25$, shock-structure for $\mathrm{Ma}=1.6$. Velocity set (top), density and temperature profiles (bottom) for two thresholds of Knudsen number, ${\mathrm{Kn}}_{\mathrm{thr}}=(0.02,0.05)$.

Figure 15

Relative error $\epsilon =|u_9-u_{16}|/|u_{16}|$ between the $D2Q9$ and $D2Q16$ models for the Couette flow, where $u_9$ and $u_{16}$ are the analytical velocity profiles [11] for the $D2Q9$ and the $D2Q16$ models, respectively.

Figure 16

Shear driven Couette flow with $D2Q9$, $D2Q16$ and coupled $D2Q9/16$. Profiles of the normalized velocity for Knudsen numbers $\mathrm{Kn}=0.1$ (top); $\mathrm{Kn}=0.2$ (middle); $\mathrm{Kn}=0.4$ (bottom).

Figure 17

Schematic for the computational efficiency study. Computational domain of the size $N\times N$ is decomposed into a high- and low-order regions. Here $r$ corresponds to the ratio of high-order nodes. For the nodes at the coupling interface (inside dashed boxes), the projection and lifting operations are applied.

Figure 18

Speed up ratio $t_{D2Q25}/t_{\mathrm{hyb}}$ as a function of the domain size $N$. Ratio of high-order nodes $D2Q25$ is fixed to $10\%$. The dashed line corresponds to the theoretical maximum speedup, using $10\%D2Q25$. The solid line represents the fitted model.