Figure 1

Top: An airfoil shaped solid object placed in a 2D cartesian grid (fluid nodes marked by hollow circles, boundary fluid nodes marked by solid circles) with a nine velocity lattice boltzmann model. The discrete boundary points (marked by stars) are placed exactly halfway between the fluid-solid links for a simple bounce-back boundary condition.

Bottom: The two algorithms are displayed next to each other for comparison. The surface normal pointing into the fluid is denoted by $n_i$, and $\partial \mathrm{\Omega }$ represents the boundary grid points.

Figure 2

Coefficient of drag vs convection time for flow past a two dimensional circular cylinder of diameter $D$ solved using a D2Q9 model at $Re=100$ using 20 grid points per diameter. The aspect ratio of the computational domain is taken to be 8:1 with $120D\times 15D$ points for the channel length and width, respectively. The cylinder center is symmetrically placed at $30D$ from the inlet. $L_x$ represents the length of the computational channel and $U_x$ represents the inlet velocity. The lattice Boltzmann results are compared with established steady state drag values in literature [35].

Figure 3

Coefficient of drag vs convection time for flow past a 3D NACA0012 airfoil ($12\%$ thickness to chord length ratio) at $\mathrm{AoA}=0^{\circ }$, using a D2Q9 model at Re = 5000, using 320 grid points per chord (ppc) length. The size of the computational domain is taken to be $(80\times 8\times 4)\mathrm{ppc}$ with the airfoil symmetrically placed at $24\mathrm{ppc}$ from the inlet. The lattice Boltzmann results are compared with established steady state drag values in literature [36].

Figure 4

A D1Q3 lattice with grid $\mathrm{points}(X_1,X_2,X_3,X_4)$, where $X_1$ is the boundary grid point of the control volume and $X_2$, $X_3$, $X_4$ are the bulk grid points. In the toy example, we assume that the grid is unbounded on the right end and continues to $X_5$, $X_6$, and so on.

Figure 5

(a) Energy shells of the D2Q9 model. (b) Energy shells of the RD3Q41 model: fcc-1 (blue), sc-1 (red), bcc-1 (light green) shown on a regular lattice along with sc-2 (orange), bcc-1/2 (dark green).

Figure 6

Top: A solid object submerged in a fluid marked by $\mathrm{\Omega }$ on a Cartesian grid. The circular dots represent the fluid nodes that make up the discrete boundary ($\partial {\mathrm{\Omega }}^D$) analogous to the continuous solid boundary ($\partial \mathrm{\Omega }$).

Bottom: system evolving from $t$ to $t+\mathrm{\Delta }t$.

Figure 7

A complex shaped solid object placed inside a control volume. The solid surface is marked by $\partial {\mathrm{\Omega }}_1$ and the outer rectangular control volume surface is marked by $\partial {\mathrm{\Omega }}_2$.

Figure 8

The flow past a circular cylinder in two dimensions is a good bench-marking test case for lattice Boltzmann methods [34]. (a) depicts the time evolution of coefficient of drag at $Re=100$ and $D=20$, for MEA, SI, and DRTT. (b) depicts convergence, comparing the drag coefficient ($C_d$) as a function of number of grid point per diameter. (c) The computational domain size is kept fixed with $D=20$, while the drag coefficient ($C_d$) is measured for several values of Reynolds numbers. The results of MEA and DRTT are then compared with literature.

Figure 9

Coefficient of pressure vs $\theta$ for flow past a two dimensional circular cylinder of diameter $D$ solved using a D2Q9 model at $Re=100$ using 20 grid points per diameter. The aspect ratio of the computational domain is taken to be $8:1$ with $120D\times 15D$ points for the channel length and width, respectively. The cylinder center is symmetrically placed at $30D$ from the inlet. The lattice Boltzmann results are compared with established steady state drag values in literature [35].

Figure 10

Flow past a NACA0012 airfoil in two dimensions for Reynolds=1000 [37]. The coefficient of pressure ($C_p$) is calculated over the boundary fluid nodes for a computational domain size of 90 points per chord length. This is repeated for three distinct angles of attack (${\alpha =7,11,15)}^0$ and the results are matched with literature [49].

Figure 11

Flow past a NACA0012 airfoil in two dimensions, using a D2Q9 lattice model and 90 points per chord length. The coefficient of drag ($C_d$) plotted against angle of attack ($\alpha$) at Re=1000.

Figure 12

Flow past a NACA0012 airfoil in two dimensions for Reynolds=1000. The coefficient of lift ($C_l$) plotted against angle of attack ($\alpha$).

Figure 13

The coefficient of drag from a three dimensional simulation of NACA0012 using RD3Q41 lattice Boltzmann model. Converged results have been plotted.

Figure 14

Total number of computational points required to correctly measure the drag values for NACA0012 using the DRTT approach. The $x$ axis is the Reynolds number and the $y$ axis is the points per chord length of NACA0012.