An Application of a Gradient Theory With Dissipative Boundary Conditions to Fully Developed Turbulent Flows

The paper presents a complete gradient theory of grade two, including new dissipative boundary conditions based on an axiomatic conception of a nonlocal continuum theory for materials of grade $n$⁠. The total stress tensor of rank two in the equation of linear momentum contains two higher stress tensors of rank two and three. In the case of isotropic materials, both the tensors of rank two and three are tensor valued functions of the second order strain rate tensor and its first gradient. So the vector valued differential equation of motion is of order four, where the necessary dissipative boundary conditions are generated by using porosity tensors. An application to hydrodynamic turbulence by a linear theory is shown, whereby fully developed steady turbulent channel flows with fixed walls and one moving wall are also examined. The velocity distribution parameters are identified by a numerical optimization algorithm, using experimental data of velocity profiles of channel flow with fixed walls from the literature. These profiles were compared with others given in the literature. With these derived parameters, the predicted velocity gradient of a channel flow agrees well with data from the literature. In addition all simulations were successfully carried out using the finite difference method.