Shear stress in lattice Boltzmann simulations

A thorough study of shear stress within the lattice Boltzmann method is provided. Via standard multiscale Chapman-Enskog expansion we investigate the dependence of the error in shear stress on grid resolution showing that the shear stress obtained by the lattice Boltzmann method is second-order accurate. This convergence, however, is usually spoiled by the boundary conditions. It is also investigated which value of the relaxation parameter minimizes the error. Furthermore, for simulations using velocity boundary conditions, an artificial mass increase is often observed. This is a consequence of the compressibility of the lattice Boltzmann fluid. We investigate this issue and derive an analytic expression for the time dependence of the fluid density in terms of the Reynolds number, Mach number, and a geometric factor for the case of a Poiseuille flow through a rectangular channel in three dimensions. Comparison of the analytic expression with results of lattice Boltzmann simulations shows excellent agreement.

Figure 1

Relative $L2$ errors of the velocity $u_x$ and the shear stress component $S_{xy}$ (both local and FD) as functions of the dimensionless system size $N$: the velocity slope is close to $-2$ for all boundary conditions, whereas the shear stress slope is between $-1.4$ and $-1.9$ [local via Eq. (45)] and around $-1.5$ (FD). The slope values are collected in Table .

Figure 2

Behavior of the relative shear error $e_S\left(\tau \right)$ [left column: from Eq. (45), right column: from FD] for Reynolds numbers 1 (top), 10 (middle), 100 (bottom).

Figure 3

Behavior of the relative velocity error $e_u\left(\tau \right)$ for Reynolds numbers 1 (top), 10 (middle), 100 (bottom).

Figure 4

Mean density $\overline{\rho }$ (top) in lattice units for $\text{Re}=5$ and $\text{Ma}=0.15$ as a function of time step: The simulation data (theoretical value $C_g{\text{Ma}}^3/\text{Re}=1.17\times {10}^{-4}$) is excellently reproduced by a simple exponential with $C_g{\text{Ma}}^3/\text{Re}=1.26\times {10}^{-4}$. Mass ratios $\text{ln}\mu ≔\text{ln}\left(M_{50000}/M_{10000}\right)$ (bottom) for different Re as function of the Mach number: The slopes are close to the theoretical value of 3. Note that $\text{ln}\mu$ itself is plotted logarithmically.