Boundary-layer equations for turbulent flows

For the applications considered here, namely two-dimensional boundary layers (more generally, two-dimensional shear layers), only one of the Reynolds stresses is significant, namely the Reynolds shear stress, —pu’v’. Thus for two-dimensional turbulent boundary layers the time-averaged boundary-layer equations (c. f. Eqns 7.7 and 7.14), can be written in the form

dv

+ ^~ = 0
dy

_du _du dp df

U dx^V dy dx dy

The chief difficulty of turbulence is that there is no way of determining the Reynolds stresses from first principles, apart from solving the unsteady three-dimensional Navier-Stokes equations. It is necessary to formulate semi-empirical approaches for modelling the Reynolds shear stress before one can begin the process of solving Eqns (7.108a, b).

The momentum integral form of the boundary-layer equations derived in Section 7.6.1 is equally applicable to laminar or turbulent boundary layers, providing it is recognized that the time-averaged velocity should be used in the definition of momentum and displacement thicknesses. This is the basis of the approximate methods described in Section 7.7 that are based on assuming a l/7th. power velocity profile and using semi-empirical formulae for the local skin-friction coefficient.