text: The logarithmic law of the wall is a self similar solution for the mean velocity parallel to the wall, and is valid for flows at high Reynolds numbers — in an overlap region with approximately constant shear stress and far enough from the wall for (direct) viscous effects to be negligible:
  :  with      and  
  where 
  From experiments, the von Kármán constant is found to be  and  for a smooth wall.
  With dimensions, the logarithmic law of the wall can be written as:
  where y is the distance from the boundary at which the idealized velocity given by the law of the wall goes to zero. This is necessarily nonzero because the turbulent velocity profile defined by the law of the wall does not apply to the laminar sublayer. The distance from the wall at which it reaches zero is determined by comparing the thickness of the laminar sublayer with the roughness of the surface over which it is flowing. For a near-wall laminar sublayer of thickness  and a characteristic roughness length-scale ,
  Intuitively, this means that if the roughness elements are hidden within the laminar sublayer, they have a much different effect on the turbulent law of the wall velocity profile than if they are sticking out into the main part of the flow.
  This is also often more formally formulated in terms of a boundary Reynolds number, , where
  The flow is hydraulically smooth for , hydraulically rough for , and transitional for intermediate values.
  Values for  are given by:
  Intermediate values are generally given by the empirically derived Nikuradse diagram, though analytical methods for solving for this range have also been proposed.
  For channels with a granular boundary, such as natural river systems,
  where  is the average diameter of the 84th largest percentile of the grains of the bed material.