### Log law at a smooth plate

#### by Yi Zhang

Log-law is THE most important and elegant analytical result in turbulence, considering many others are dauntingly complicated, though justified. What log law says is the fact that for a turbulent flow over flat plate, there is this section of flow away from the wall, within the turbulent boundary layer, in which the relation between the distance and mean flow velocity is simply of log function. This conclusion could be drawn by an asymptotic analysis matching outer flow and inner flow. Any CFD trying tempted to address turbulence models should recover the “law of the wall“, among which log law is a part of, before moving forward to more advanced problems.

The delicacy of the modeling in this case is the boundary condition (shocking!) at the wall, because log law is only valid when y+ is approximately above 30 ( the other cap depends on Reynolds number ), and the boundary conditions there for unknown variables in turbulence models are generally not known *a priori*, where is all kinds of wall function come into play.