Turbulent Stresses

In order to define stresses in turbulent flows, let us write the x- momentum equation with time averaging and fluctuation terms; i.e.
                      (5)
The three correlation terms are called "turbulent stresses" because they have same dimensions as that of laminar shear stress terms i.e. . The turbulent stresses are unknown and must be related to experimental flow conditions and geometry. However, experiments in pipe flows reveal that the stress associated with in y- direction is dominant. Hence, with reasonable accuracy, the momentum equation is reduced to,
                                            (6)

where
                                            (7)
The typical trend of a turbulent-shear layer for a pipe flow is shown in Fig. 3. It is seen that “laminar shear” is dominant near the wall whereas turbulent shear dominates in the "outer layer". There is an intermediate region called "overlap layer" where both laminar and turbulent shear are important.