The prediction and control of the laminar-turbulent transition are always one of the most concerned frontiers and hot topics.Receptivity is the initial stage of the laminar-turbulent transition process in the boundary layer,which decides the physical process of the turbulent formation.To date,the researches of receptivity in the three-dimensional boundary layer are much less than those in the two-dimensional boundary layer;while most of the real laminar-turbulent transition in practical engineering occurs in three-dimensional boundary layers.Therefore,receptivity under the threedimensional wall local roughness in a typical three-dimensional boundary layer,i.e.,a 45° back swept infinite flat plate, is numerically studied.And a numerical method for direct numerical simulation (DNS) is constructed in this paper by using fourth order modified Runge-Kutta scheme for temporal march and high-order compact finite difference schemes based on non-uniform mesh for spatial discretization:the convective term is discretized by fifth-order upwind compact finite difference schemes;the pressure term is discretized by sixth-order compact finite difference schemes;the viscous term is discretized by fifth-order compact finite difference schemes;and the pressure equation is solved by third-order finite difference schemes based on non-uniform mesh.As a result,the excited steady cross-flow vortices are observed in the three-dimensional boundary layer.In addition,the relations of three-dimensional boundary-layer receptivity with the length,the width,and the height of three-dimensional wall localized roughness respectively are also ascertained.Then, the influences of the different distributions,the geometrical shapes,and the location to the flat-plate leading-edge of the three-dimensional wall local roughness,and multiple three-dimensional wall local roughness distributed in streamwise and spanwise directions on three-dimensional boundary-layer receptivity are considered.Finally,the effect of the distance between the midpoint of the three-dimensional wall localized roughness and the back-swept angle on three-dimensional boundary-layer receptivity is studied.The intensive research of receptivity in the three-dimensional boundary-layer receptivity will provide the basic theory for awareness and understanding of the laminar-turbulent transition.