  Singularity of Navier-Stokes equations is uncovered for the first time which
explains the mechanism of transition of a smooth laminar flow to turbulence. It
is found that when an inflection point is formed on the velocity profile in
pressure driven flows, velocity discontinuity occurs at this point. Meanwhile,
pressure pulse is produced at the discontinuity due to conservation of the
total mechanical energy. This discontinuity makes the Navier-Stokes equations
be singular and causes the flow to become indefinite. The analytical results
show that the singularity of the Navier-Stokes equations is the cause of
turbulent transition and the inherent mechanism of sustenance of fully
developed turbulence. Since the velocity is not differentiable at the
singularity, there exist no smooth and physically reasonable solutions of
Navier-Stokes equations at high Reynolds number (beyond laminar flow). The
negative spike of velocity and the pulse of pressure due to discontinuity have
obtained agreement with experiments and simulations in literature
qualitatively.
