A coupled lattice Boltzmann and fractal geometry method for laminar liquid flows

A coupled lattice Boltzmann and fractal geometry method is introduced to simulate laminar channel flow. The coupled method is divided into two cases. If the height difference between adjacent grids is less than the grid width, the linear interpolation between two points is taken to replace the curve line with the Weierstrass-Mandelbrot (WM) function. If not, the zig-zag line is applied to replace the fractal wall. With a fractal dimension to characterize the shape of the wall, fractal geometry with WM function is adopted to reveal the rough wall. The coupled method overcomes the error of roughness height when adopting fractal function to lattice Boltzmann method. Then, it is concluded that WM function should be used at every grid to ensure the numerical wall is fractal. The coupled method shows its advantage of accuracy with simulations of 2-D laminar channel liquid flows. In addition, the effects of roughness and the boundary slip are taken into consideration. Compared with the experiment data, the proposed method can simulate the real flow accurately. The slip velocity of liquid flow is reduced by the irregular roughness of WM function. Moreover, a comparison between the fractal wall and regular grooves wall indicates that fractal geometry with the WM function is a better way to reveal the real rough wall for laminar flows.