  How locally injected turbulence, spreads in space is investigated with direct
numerical simulations. We consider a turbulent flow in a long channel generated
by a forcing that is localised in space. The forcing is such that it does not
inject any mean momentum in the flow. We show that at long times a
statistically stationary state is reached where the turbulent energy density in
space fluctuates around a mean profile that peaks at the forcing location and
decreases fast away from it. We measure this profile as a function of the
distance from the forcing region for different values of the Reynolds number.
It is shown, that as the Reynolds number is increased, it converges to a
Reynolds-independent profile implying that turbulence spreads due to
self-advection and not molecular diffusion. In this limit therefore, turbulence
plays the simultaneous role of cascading the energy to smaller scales and
transporting it to larger distances. The two effects are shown to be of the
same order of magnitude. Thus a new turbulent state is reached where turbulent
transport and turbulent cascade are equally important and control its
properties.
