Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface

In this paper, a century-old problem is solved; namely, to find a unified analytic description of the non-uniform distribution of mean velocity across the entire domain of turbulent flow for all Reynolds numbers within the framework of the Prandtl mixing length theory. This study obtains a closed form solution of the mean velocity profile of plane turbulent flow for the Prandtl theory, and as well an approximate analytical solution for the van Driest mixing length theory. The profiles of several useful quantities are given based the closed form solution, such as turbulent viscosity, Reynolds turbulent stress, Kolmogorov's scaling law, and energy dissipation density. The investigation shows that the energy dissipation density at the surface is finite, whereas Landau's energy dissipation density is infinite. Strictly speaking, the closed form solution reveals that the universality of the turbulent velocity logarithmic profile no longer holds, but the von K\'arm\'an constant is still universal. Furthermore, a new formulation of the resistance coefficient of turbulent flow in pipes is formulated in implicit form.