Spatial transition point from laminar flow to turbulence in a pipe with injection revealed by solving a weakly-stochastic navier-stokes equation

In order to simulate the spatial transition point from laminar flow to turbulence in a closed pipe, we proposed the numerical method called “stochastic determinism” that based on the stochastic Navier-Stokes equation. The important point is that the stochasticity level is decided by inlet disturbance intensity. The finite difference method based on the multi-level formulation is also included which can calculate spatial derivatives of physical quantities and integrated quantities accurately. In this report, we compute the transition phenomenon for incompressible flow with injection on the pipe wall. The computations fairly agree with Hof’s experimental results.