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ABSTRACT
A model of turbulence is proposed to solve Reynolds equations for fully developed flow in a wall-bounded straight channel. We show that for the channel flow the Reynolds number can be defined as a ratio of flow kinetic energy to the work of friction/dissipation forces. Then, we introduce a turbulent Reynolds number as a balance between energy losses due to the momentum exchange by turbulent vortices traveling from lowto high-velocity areas and wall friction. The main idea of the model is expressed in the following phenomenological law: The minimal energy dissipation rule requires that a local deformation of the axial velocity profile can and, in the presence of finite-size instabilities, should generate turbulence with such intensity that it keeps the local turbulent Reynolds number below the critical value. Thus, the only empirical parameter in the model is the critical Reynolds number.
The model is applied to several basic channel flows such as the fully developed flow in a circular tube, in an infinite plane channel, and in an annulus. The application of the minimal energy dissipation rule requires an additional integral equation, and this can be considered as an integral-equation algebraic model of turbulence.
Nomenclature
| u | = | Flow velocity |
| d | = | Pipe diameter |
| ρ | = | Fluid density |
| μ | = | Dynamic viscosity |
| ν | = | Kinematic viscosity, μ/ρ |
| K | = | Kinetic energy of flow |
| Kf | = | Difference between kinetic energy of flow with velocity profile and kinetic energy of the flow with flat profile, |
| V | = | Volume of integration |
| W | = | Work of friction forces |
| τ, τw | = | Shear stress, Wall shear stress |
| Re | = | Reynolds number Re = ud/ν |
| Ree | = | Energy-balanced Reynolds number, |
| Ret | = | Turbulent Reynolds number, |
| Recr | = | Critical turbulent Reynolds number |
| y | = | Distance to the wall |
| y+ | = | Wall coordinate, y+ = yuτ/ν |
| u+ | = | Dimensionless velocity, u+ = u/uτ, |
| uτ | = | Friction velocity, |
Nomenclature
| u | = | Flow velocity |
| d | = | Pipe diameter |
| ρ | = | Fluid density |
| μ | = | Dynamic viscosity |
| ν | = | Kinematic viscosity, μ/ρ |
| K | = | Kinetic energy of flow |
| Kf | = | Difference between kinetic energy of flow with velocity profile and kinetic energy of the flow with flat profile, |
| V | = | Volume of integration |
| W | = | Work of friction forces |
| τ, τw | = | Shear stress, Wall shear stress |
| Re | = | Reynolds number Re = ud/ν |
| Ree | = | Energy-balanced Reynolds number, |
| Ret | = | Turbulent Reynolds number, |
| Recr | = | Critical turbulent Reynolds number |
| y | = | Distance to the wall |
| y+ | = | Wall coordinate, y+ = yuτ/ν |
| u+ | = | Dimensionless velocity, u+ = u/uτ, |
| uτ | = | Friction velocity, |