ON ANALYTICAL SOLUTIONS OF TURBULENT PLANE COUETTE FLOW WITH GRADIENT DIFFUSION MODELS
Hernan Tinoco
Forsmarks Kraftgrupp AE, 74203 Osthammar, Sweden
Stefan Ahlinder
Forsmarks Kraftgrupp AB, S-742 03 Osthammar, Sweden
Abstract
A number of gradient-diffusion models of turbulence predict a constant value for the kinetic energy of turbulence in the central region of plane Couette flow, in clear contrast with experimental and DNS results. It is shown in this paper that the gradient-diffusion approach is responsible for this result through a readability condition of no counter-gradient diffusive transport.
In the case of the k−ε model, it is shown the proportionality factor, C_{μ} , used in the definition of the eddy viscosity, is not a constant but a function of the strain rate parameter S* (Hallback et al, 1996) and of the production-dissipation ratio λ = P/ε . By assuming λ to depend only on of the strain rate parameter, C_{μ} may be expressed as a function of S*, in a way similar to that of Shih et al (1995).
Henry & Reynolds (1984), through an analytical solution, have concluded that two gradient-diffusion models really predict the constant energy value. It is shown here that the aforementioned solution is incorrect and that, on the other hand, a correct analytical solution may be obtained if the right level for the value of C_{μ} is adopted.