The nonlinear integral transport equations for turbulent flow are expanded in a series of homogeneous kernels, the expansion being in the centroid coordinate. This yields equations for the mean velocity, kinetic energy and shear stress which are non-local in wave number and differential in the centroid variables. A recently developed method is used to approximate the integral equations for the kinetic energy and shear stress. It is shown that the spatial average of the kinetic energy spectrum is given by the Kolmogoroff distribution. If we take this result as an approximation to the general kinetic energy function, a simple solution for the mean velocity profile results, which compares fairly well with experimental results as does the distribution of kinetic energy and shear stress.