Turbulence as a nonequilibrium phase transition

The transition from laminar to turbulent flow is studied on the basis of an exact equation for the averaged velocity and an approximate nonlinear equation for the Reynolds stress $\tau$. The stationary state can be determined from the condition of minimum of a functional that is analogous to the Landau functional in the theory of phase transitions. The Reynolds stress plays the role of a parameter. It is shown that a nontrivial solution for $\tau$ corresponding to a steady turbulent regime exists only for Reynolds numbers $R$ that exceed a certain critical value $R_\mathrm{cr}$. The results of a numerical calculation of the profile of the averaged velocity, the friction coefficient, and the Reynolds stress in a wide range of values of $R$ agree well with experimental data for channel flow.