Log law for unsteady velocity profile
 

Hi
I want to simulate the unsteady wind flow over a house. For steady simulation I use Log law velocity profile but I couldn`t find a formula for the Unsteady simulation where the wind speed is changes with time. so I wonder what is the formula for such problem ?

no analytical law exists for unsteady turbulent velocity, the log-law has statistical meaning for a flow in energy equilibrium (production=dissipation) such that the time derivative statistically vanishes.
If you have a flow that increases in time I suggest using LES with a grid resolving the boundary layer, however this is computational expensive.