A theoretical approach is developed for solving for the Reynolds stress in turbulent flows, and is validated for canonical flow geometries (flow over a flat plate, rectangular channel flow, and free turbulent jet). The theory is based on the turbulence momentum equation cast in a coordinate frame moving with the mean flow. The formulation leads to an ordinary differential equation for the Reynolds stress, which can either be integrated to provide parameterization in terms of turbulence parameters or can be solved numerically for closure in simple geometries. Results thus far indicate that the good agreement between the current theoretical and experimental/DNS (direct numerical simulation) data is not a fortuitous coincidence, and in the least it works quite well in sensible ways in canonical flow geometries. A closed-form solution for the Reynolds stress is found in terms of the root variables, such as the mean velocity, velocity gradient, turbulence kinetic energy and a viscous term. The form of the solution also provides radically new insight on how the Reynolds stress is generated and distributed.
Iwamoto, K., Sasaki, Y. and Nobuhide K. (2002) Reynolds Number Effects on Wall Turbulence: Toward Effective Feedback Control. International Journal of Heat and Fluid Flows, 23, 678-689. http://dx.doi.org/10.1016/S0142-727X(02)00164-9