Effects of Truncation Error of Derivative Approximation for Two-Phase Lattice Boltzmann Method

Abstract

We verify the accuracy and the truncation errors of approximation to the first derivatives and to Laplacian operator in the lattice Boltzmann method. The truncation errors are calculated by the Taylor series expansion, and the influences are analytically and numerically examined in the simulation of two-phase flow. We propose the 4th-order accurate approximations to derivatives that utilize the property of the tensor unlike the finite difference. The numerical solutions with various approximations agree well with the theoretical ones within the error of 7% in the test of the Laplace's law. Application of the 4th-order accurate approximation scheme to two-phase flow simulation reduces the spurious current around the interface of a stationary droplet about to one-half of the results with the 2nd-order accuracy. The small spurious velocity in the vicinity of the interface of the 4th-order approximation increases the speed of a moving droplet, and distorts the shape of the droplet little. In the simulation of phase separation, the growth rate of the typical domain size calculated with FFT is strongly affected by the truncation error in surface-tension driven regime for short time. It is seen that the 4th-order approximation saturates at 2nd-order accuracy in simulation of a decaying Taylor vortex flow, because the convergence rate of the LBM is 2nd-order in space. It is shown that the approximation method affects the important physical values, such as velocity, moving speed, or domain size in numerical simulations.

Keywords

• lattice boltzmann method
• computational fluid dynamics
• derivative approximation
• truncation error
• two-phase flow