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TECHNICAL PAPERS

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

[+] Author and Article Information
Gerhard Silber

Institute for Materials Science and Center of Biomedical Engineering (CBME), University of Applied Sciences, Frankfurt am Main, Germanysilber@fb2.fh-frankfurt.de

Uwe Janoske

University of Cooperative Education, Mosbach, Germany; and Center of Biomedical Engineering (CBME)

University of Science and Technology, Teheran, Iran

Guenther Benderoth

Institute for Material Science and Center of Biomedical Engineering (CBME), University of Applied Sciences, Frankfurt am Main, Germany

J. Fluids Eng 129(5), 643-651 (Dec 22, 2006) (9 pages) doi:10.1115/1.2720476 History: Received August 25, 2005; Revised December 22, 2006

## Abstract

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.

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## Figures

Figure 1

One-dimensional channel flow with moving upper wall

Figure 2

Discretization of the gap (channel flow)

Figure 3

Comparison of experimental data (dots) and theoretical predicted velocity profile (solid line) of a steady channel flow with velocity slip

Figure 4

Comparison of experimental data by Reichardt (32) (dots) and theoretical predicted velocity gradient (solid line) of steady channel flow with velocity slip

Figure 5

Velocity distributions of steady channel flow with moving upper wall for varying Θ

Figure 6

Comparison of mean velocity profiles by experiments and gradient theory with wall slip condition

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