An algorithm has been developed which extends the scope of spectral methods to include solution of non-canonical channel flows arising from more complicated wall geometries. This significantly broadens the direct numerical simulation data base and its range of application, providing an accurate tool for the investigation of flows over three-dimensional surfaces which move in time. Through a time-dependent, curvilinear transformation a general domain is mapped to one which permits spectral representation of the solution and preserves exact boundary conditions. Beginning with the (\mathrm{Na}-) vier-Stokes equation in general tensor form, application of a metric operator effects the transformation. The primitive variables are represented pseudospectrally (Fourier in the stream- and spanwise directions, Chebyshev wall-normal). Covariant differentiation generates variable coefficient terms in the equations for pressure and velocity, necessitating an iterative solution scheme. Standard benchmark tests validate flat-wall flow simulations. Static and dynamic tests of one-dimensional flow over a perturbed wall confirm the accuracy of the time-dependent transformation. Low Reynolds number simulations replicate the appropriate qualitative features of Stokes flow across two- and three-dimensional wall topographies. Results from a higher Reynolds number simulation of separated flow behind a three-dimensional Gaussian protuberance match well with an independent solution from Mason and Morton who have used a finite-difference method. (c) 1995 Academic Press, Inc.