Weakly-stochastic Navier-Stokes equation and shocktube experiments: Revealing the Reynolds' mystery in pipe flows

Although Reynolds showed the transition to turbulence in pipe flow 100 years ago, instability theories based on deterministic continuum mechanics and numerical simulations based on the deterministic Navier-Stokes equation cannot indicate the transition point in closed pipe flow. Our previous computations (Naitoh, 2007, 2008, 2009, 2010) in a straight and closed pipe using the random number generator have showed the transition in space without applying any stability theories and turbulence models, which suggests the possibility of a stochastic Navier-Stokes equation. The most important point of our approach is a theoretical and philosophical method proposed for determining the stochasticity level, which is deeply related to boundary condition. In this paper, computational analyses performed for the stochastic Navier-Stokes equation with grid systems having high and low resolutions quantitatively reveal the mysterious relation between inlet disturbance and the transition point to turbulence for pipe flows, while consideringsurface roughness of solid walls. Independence of the transition point on grid size implies that stochasticity is dominant rather than numerical discretization. Moreover, a laminarization phenomenon in a straight pipe, including puffs and slugs, are also captured. Finally, we will also show shocktube experiments, which qualitatively clarify the influence of inlet-outlet disturbances on the transition points.