针对金属表面功能结构的多尺度数字化设计问题，基于机械加工表面微观形貌中存在的自仿射分形特性，探索了将Weierstrass-Mandelbrot 分形与三维参数曲面相合成的数学方法．该方法能够将表面功能结构的微观粗糙形貌与宏观模型相集成，从而有效支持表面功能结构的多尺度形貌建模． 以犁削成形的V 型毛细微通道的微流体特性为仿真研究对象，建立了具有多尺度形貌特征的三维实体模型，同时分析了雷诺数、表面高度尺度系数和分形维数等对该微通道流动特性的影响，发现: 阻力系数随雷诺数的增大而减小，随表面高度尺度系数的增大而增大，粗糙表面产生的漩涡对流体的影响不能忽视，且随粗糙度的增加越来越突出; 根据Weierstrass-Mandelbrot 函数构造的表面形貌分形维数的增加将导致通道阻力系数的减小．

In order to implement the multiscale digital design of metal functional surface structures,a mathematical method combining the Weierstrass-Mandelbrot fractal geometry and the three-dimension parametric surface is developed based on the self-affine fractal characteristics of the microtopography of machined metal surfaces. With this method,the coarse microtopography of three-dimension structures can be well integrated with macroscopic models,so that a multiscale microtopographic model of functional surface structures can be established. For experimental investigation,a multiscale 3D solid model is created to simulate the flow characteristics of fluid in a ploughed Vshaped microchannel,and the influences of the Reynolds number,the surface height scale coefficient and the fractal dimension on the flow characteristics are analyzed. The results show that ( 1) the frictional resistance coefficient decreases with the increase in the Reynolds number,but increases with the surface height scale coefficient; ( 2) the influence of the fluid vortex due to surface roughness,which becomes significant with the roughness increase,is non-neglectable; and ( 3) the frictional resistance coefficient of microchannels decreases with the increase of the Weierstrass-Mandelbrot fractal dimension.