Theory of the lattice Boltzmann method: discrete effects due to advection

Abstract

Lattice Boltzmann models are briefly introduced together with references to methods used to predict their ability for simulations of systems described by partial differential equations that are first order in time and low order in space derivatives. Several previous works have been devoted to analyzing the accuracy of these models with special emphasis on deviations from pure Newtonian viscous behaviour, related to higher order space derivatives of even order. The presentcontribution concentrates on possible inaccuracies of the advection behaviour linked to space derivatives of odd order. Detailed properties of advection-diffusion and athermal fluids are presented for two-dimensional situations allowing to propose situations that are accurate to third order in space derivatives. Simulations of the advection of a gaussian dot or vortex are presented. Similar results are discussed in appendices for three-dimensional advection-diffusion.

Figures (5)

• Figure 1: Advection factor for main velocity along $\, Ox \,$ axis and wave vector vs angle $\theta$. Dotted line in the absence of anomalous advection. Solid line contribution $h$ for D2Q13, dashed line for D2Q9.
• Figure 2: Advection of an initial Gaussian disturbance simulated with diffusive D2Q9 under conditions described in the text. Top and lower features are isotropic (respectively for $q=-1$ or $12\sigma_1\sigma_4=1$). The middle feature uses conditions that are not tuned for isotropy.
• Figure 3: Simulation with D2Q9. Vorticity of the velocity field from an initial gaussian stream function after 9000 time steps for an advection velocity $\{0.03,0.00\}$. Left with isotropy condition $\, 12 \, \sigma_4 \, \sigma_6=1$. Right: arbitrary conditions.
• Figure 4: Simulation with D2Q13. Vorticity of the velocity field from an initial Gaussian stream function after 2770 time steps. Left with an advection velocity $\{0.10,0.00\}$. Right with no advection.
• Figure 5: Vorticity of the vortex with main velocity at $14^\circ$ from Ox and $r_0=4$ in a domain of size $80\times 80$. Initial state at bottom, final state at top. The advection used is $\, g(k)=1+0.01 \, ( \cos (4 \, \theta) - \cos (2 \, \theta) )) \, k^2$.