Finite-difference lattice Boltzmann method with a block-structured adaptive-mesh-refinement technique

An adaptive-mesh-refinement (AMR) algorithm for the finite-difference lattice Boltzmann method (FDLBM) is presented in this study. The idea behind the proposed AMR is to remove the need for a tree-type data structure. Instead, pointer attributes are used to determine the neighbors of a certain block via appropriate adjustment of its children identifications. As a result, the memory and time required for tree traversal are completely eliminated, leaving us with an efficient algorithm that is easier to implement and use on parallel machines. To allow different mesh sizes at separate parts of the computational domain, the Eulerian formulation of the streaming process is invoked. As a result, there is no need for rescaling the distribution functions or using a temporal interpolation at the fine-coarse grid boundaries. The accuracy and efficiency of the proposed FDLBM AMR are extensively assessed by investigating a variety of vorticity-dominated flow fields, including Taylor-Green vortex flow, lid-driven cavity flow, thin shear layer flow, and the flow past a square cylinder.

Figure 1

Block-structured AMR: configuration of the blocks (black lines) containing an interface (red-color contour) for (a) $l_{\max }=0$, the root block with $(n_x\times n_y)=5\times 5$ vertex nodes (green lines); (b) $l_{\max }=1$; (c) $l_{\max }=2$; (d) $l_{\max }=3$; (e) $l_{\max }=4$; and (f) $l_{\max }=5$, final arrangement of the blocks.

Figure 2

Neighboring blocks at different refinement levels: (a) before refinement and (b) after refinement.

Figure 3

Interblock communications via ghost nodes: blue circles (with a +) are (a) the ghost nodes for the fine block to the left, to be filled from the data points of the coarse block ($\times$) to the right. (b) Interpolation points.

Figure 4

Pressure contour in decaying vortex flow: comparison between (a) bilinear and (b) biquadratic interpolations.

Figure 5

Convergence test: order of accuracy of the proposed model using bilinear and biquadratic interpolations.

Figure 6

Lid-driven cavity flow at $\text{Re}=3200$ after $200000$ iterations. Comparison of different error indicators: (a) ${\epsilon }_1$, (b) ${\epsilon }_2$, (c) ${\epsilon }_3$, and (d) uniform mesh.

Figure 7

Configuration of the adaptively created blocks in the cavity flow for different error indicators: (a) ${\epsilon }_1$, (b) ${\epsilon }_2$, and (c) ${\epsilon }_3$.

Figure 8

Centerline velocities inside the cavity flow at $\text{Re}=3200$: (a) dimensionless $x$ velocity along the vertical centerline and (b) dimensionless $y$ velocity along the horizontal centerline.

Figure 9

Vorticity contour levels for Kelvin-Helmholtz instability at $\text{Re}=1500$ and $t^{*}=26$. Comparison of different estimators: (a) ${\epsilon }_1$, (b) ${\epsilon }_2$, (c) ${\epsilon }_3$, and (d) uniform mesh. The vorticity increment between contours is $1/6$.

Figure 10

Comparison of (a) current FDLBM AMR findings with (b) AMR benchmark results in Ref. [43] [Fig. 2 therein] for Kelvin-Helmholtz instability at $\text{Re}=1500$ and $t^{*}=26$ with $l_{\max }=5$.

Figure 11

Flow past a square cylinder: computational domain and boundary conditions.

Figure 12

Vortex shedding past a square cylinder at $\text{Re}=100$ for (a) $l_{\max }=5$ $(D=16)$, (b) $l_{\max }=6$ $(D=32)$, and (c) $l_{\max }=7$ $(D=64)$. Shown on the left are streamlines and pressure contours and on the right are vorticity contours and mesh configurations.

Figure 13

Drag and lift coefficients versus dimensionless time for a uniform flow past a square cylinder at $\text{Re}=100$.

Figure 14

Vorticity contours past a square cylinder at $\text{Re}=100$ with $l_{\max }=6$ for (a) $t_0$, (b) $t_0+\frac{1}{4f_s}$, (c) $t_0+\frac{2}{4f_s}$, and (d) $t_0+\frac{3}{4f_s}$.