Numerical validation of a consistent axisymmetric lattice Boltzmann model

A recently derived axisymmetric lattice Boltzmann model is evaluated numerically. The model incorporates a spatially and temporally varying source term into the evolution equation for the momentum distribution function on a two-dimensional Cartesian lattice. The precise form of the source term is derived through a Chapman-Enskog analysis so that the additional axisymmetric contributions in the Navier-Stokes equations are furnished when written in the cylindrical polar coordinate system. The validity of the model is confirmed by simulating Hagen-Poiseuille flow. Numerical predictions for the drag coefficient in Stokes’ flow over a sphere are presented and shown to be in excellent agreement with analytical results. At larger Reynolds numbers the numerical predictions are shown to approach an empirically derived formula for the drag coefficient.

Figure 1

Analytical and numerical solutions for Hagen-Poiseuille flow.

Figure 2

Computational domain for axisymmetric flow over a sphere.

Figure 3

Streamwise velocity contours for axisymmetric flow over a sphere at $\mathrm{Re}=0.01$.

Figure 4

Normal velocity contours for axisymmetric flow over a sphere at $\mathrm{Re}=0.01$.

Figure 5

Streamwise velocity contours for axisymmetric flow over a sphere at $\mathrm{Re}=100$.

Figure 6

Normal velocity contours for axisymmetric flow over a sphere at $\mathrm{Re}=100$.

Figure 7

(Color online) Graph showing $C_D$ as a function of Re for axisymmetric flow over a sphere.