Leray-alpha model and transition to turbulence in rough-wall boundary   layers
  In the present study, we use a fifth-order ordinary differential equation, a generalization of the Blasius equation derived from the Leray-alpha model, to examine the transition to turbulence in rough-wall boundary-layer flows. This equation, combined with a weaker formulation of the von Karman log law modified to include the effects of surface roughness, provides a family of turbulent velocity profiles with two free parameters: the momentum thickness based Reynolds number and the roughness function. As a result, a skin-friction correlation is obtained and predictions of the transitional Reynolds numbers and maximal skin-friction values are made based on the roughness function. 
Alexey Cheskidov and Diana Ma