Turbulent shear stresses and prime velocity distribution in compound channels

The turbulent stresses must be defined to calculate the velocity distributions in open channel flows. In the paper, the turbulent stresses are presented as a sum of the normal and shear turbulent stresses. The normal turbulent stresses act like pressure, i.e., they are isotropic and can be absorbed by the pressure-gradient term in the momentum equations. Therefore, only the shear stresses have to be defined to describe the velocity distribution, e.g., the prime velocity distribution in an open channel flow. The shear turbulent stresses are defined by the 3D mixing length hypothesis. This hypothesis is based on the mixing length tensor (MLT). It is shown how to define the components of MLT for compound channels and how to relate it to the turbulent stress tensor. The components of the MLT are defined based on the concept of the generic mixing length (GML). This concept is presented. Having calculated the generic mixing length, the main components of the MLT as well as the turbulent shear stresses can be calculated. The presented concept is applied to calculate the prime velocity distribution in laboratory open channels with two-stage cross-section. Two channels are considered, one with vertical sidewalls and one with inclined sidewalls. The basic hydrodynamics equations (parabolic approximation of Reynolds equations) together with the turbulence model are solved. The well-known Patankar-Spalding algorithm was used to solve these equations. Some numerical simulations were performed for different components of MLT, i.e. for different structure of turbulence. The results of numerical simulations were compared with the primary velocity distribution measured in the laboratory channel. These comparisons show that the model predicts the velocity field reasonably well.