Asymptotic and numerical solutions of the unsteady boundary-layer equations are obtained for a main stream velocity given by equation (1.1). Far downstream the flow develops into a double boundary layer. The inside layer is a Stokes shear-wave motion, which oscillates with zero mean flow, while the outer layer is a modified Blasius motion, which convects the mean flow downstream. The numerical results indicate that most flow quantities approach their asymptotic values far downstream through damped oscillations. This behaviour is attributed to exponentially small oscillatory eigenfunctions, which account for different initial conditions upstream.

A numerical method for integrating the unsteady twodimensional boundarylayer equations using a second-order-accurate implicit method, which allows for arbitrary mesh spacing in the space and time variables, is developed. A unique feature of the method is the use of an asymptotic solution valid at the downstream end of the integration mesh which permits backflow to be taken into account. Newton's iterative technique is used to solve the nonlinear finite-difference equations a t each computation step, using a rapid algorithm for solving the resulting linearized equations. The method is applied to a flow which is periodic in time and contains regions of backflow. The numerical computations are compared with known numerical and asymptotic solutions and the agreement is excellent.

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