A Continuum Mechanical Gradient Theory with an Application to Fully Developed Turbulent Flows

Abstract

In this study, a gradient theory of grade two, including dissipative boundary conditions based on an axiomatic conception of a nonlocal continuum theory for materials of grade n is presented. 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 so-called porosity tensors. This theory is applied to a velocity profile of turbulent plane Couette flow of water. The velocity distribution parameters are identified by a numerical optimization algorithm, using experimental data of velocity profile of plane Couette flow with tow moving vertical walls from the literature. In an experiment, Hagiwara et al. refined flow visualization technique and carried out a two-dimensional particle-tracking-velocimetry (PTV) measurement for the Couette flow. On the basis of this experimental data the material and porosity coefficients are identified by employing the Levenberg–Marquardt numerical algorithm for nonlinear optimization.

Keywords:

• Couette flow
• grade two
• gradient theory
• slip
• turbulent flows