Calculation of Unsteady and Three-Dimensional Boundary-Layer Flows

NUMERICAL solution technique has been developed and analyzed for the laminar, time-dependent and threedimensional boundary-layer equations. The method has been applied to a rotating flat plate in forward flight, which is of direct interest to helicopter rotors. The numerical method itself is a modification of the implicit schemes used in two- and threedimensional boundary-layer problems and the most significant difficulties were the initial value problem and reversed flow. The combined problem of both time-dependent influences and three-dimensionality on the flat plate exhibited a much different behavior than either of the effects along. The most interesting behavior occurred near the retreating blade portion of the cycle where retreating blade stall has been a recurring problem. Content In the present paper, a three-dimensional and time-dependent boundary layer has been calculated by a numerical method. The flow problem studied consisted of a rotating flat plate in forward flight, and the flow conditions were laminar and incompressible. This particular problem presents many interesting features, both in the areas of numerical analysis and for understanding the complex interactions in three-dimensional and time-dependent flows. The geometry of the present problem is shown in Fig. 1. During one part of the cycle, the forward flight velocity adds to the rotational velocity, while on the other part the two velocities subtract from each other. Also, it should be pointed out that the cross-flow velocity has components from both the forward and rotational motions. A prerequisite for any boundary-layer problem is an inviscid solution to be used as an outer boundary condition for the viscous flow. The inviscid solution for the present problem has been worked out by McCroskey and Yaggy,1 with various examples given by Dwyer and McCroskey.2 The appropriate laminar, incompressible boundary-layer equations for the present problem are: continuity