The venerable 1/7th power law turbulent velocity profile: a classical nonlinear boundary value problem solution and its relationship to stochastic processes

The 1/7th power law turbulent velocity profile, originally estimated from pipe flow data, provided and continues to provide a simple but effective relationship for turbulent mean velocity profiles in moderate favorable pressure gradient regime flows. Here we show that the power law profile is not only a viable empirical relationship but the analytical solution of a nonlinear boundary value problem based on a large Reynolds number asymptotic closures. By extending the length scale closure to include related elementary models we obtain other mean velocity profiles solutions that are in good agreement with the more well-known power-law solution. The interest in these other closure models is that they exhibit a direct connection to classical, normally distributed stochastic process behavior. This relationship is further explored by considering a discrete single step difference equation governing the “random walk behavior” of fluid particle in a wall bounded shear flow. Thus, from these simple analytical solutions, we can relate the success of the empirical 1/7th power law model to a more fundamental understanding of turbulent flow, turbulence modeling closures and their connection to stochastic processes.