The Boundary Layer Equations for Turbulent Flow

Compared to the x-wise Navier-Stokes equation for two-dimensional laminar flow, the equivalent equation for turbulent flow has, when the time averaged values of the 

If the turbulent momentum equation is expressed in nondimensional form in the same way as was done in deriving the laminar boundary layer equations then the additional term becomes  

the rest of the terms retained in the boundary layer equations have the order of magnitude of unity and, therefore, for the boundary layer equations to apply, the dimensionless turbulence terms (u 2lu ) and (u v /u ), which are assumed to have the same order of magnitude, will have the order of magnitude of (8/L) at most. The first term in Eq. (2.154) is, therefore, negligible compared to the rest of the terms in the boundary layer equations. Therefore, the x-wise momentum equation for turbulent boundary layer flow is  

Consider next the y-momentum equation. Expressing this in dimensionless form in the same way as was done with the laminar flow equation and neglecting the appropriate mean flow terms, this equation then gives  

2/u2 must have the same order of magnitude as the turbulence terms in the x-momentum equation so its order of magnitude is (8/L). Hence, it follows that in turbulent flow, as in laminar flow  

---