The advected grid explicit (AGE) method for direct numerical simulation of 'incompressible' turbulent shear flows is presented. The Navier-Stokes equations are used for momentum in a velocity -pressure formulation. Mass continuity and an equation of state link pressure with density (which is not assumed identically constant). Time advancement is entirely explicit, and spatial representation is localized (e.g. finite difference) and centred. Magnitudes of non-linear terms are reduced on advected grid(s), and numerical instabilities are efficiently reduced by 'targeted diffusion'. Computation time scales directly on the number of grid points (virtual memory issues aside), and is very short for a DNS method. A spatially developing two-stream mixing layer was simulated as an example, reaching a vorticity thickness Reynolds number \20 000. Comparison with experimental results from self-similar mixing layers is satisfactory in terms of growth rate and Reynolds stress profiles. Turbulent vortical structures are visualized by means of pressure surfaces.