  The injection of hydrogen into the convection shell powered by helium burning
during the core helium flash is commonly encountered during the evolution of
metal-free and extremely metal-poor low-mass stars. With specifically designed
multidimensional hydrodynamic simulations, we aim to prove that an entropy
barrier is no obstacle for the growth of the helium-burning shell convection
zone in the helium core of a metal-rich Pop I star, i.e. convection can
penetrate into the hydrogen-rich layers for these stars, too. We further study
whether this is also possible in one-dimensional stellar evolutionary
calculations. Our hydrodynamical simulations show that the helium-burning shell
convection zone in the helium core moves across the entropy barrier and reaches
the hydrogen-rich layers. This leads to mixing of protons into the hotter
layers of the core and to a rapid increase of the nuclear energy production at
the upper edge of the helium-burning convection shell - the hydrogen injection
flash. As a result a second convection zone appears in the hydrogen-rich
layers. Contrary to 1D models, the entropy barrier separating the two
convective shells from each other is largely permeable to chemical transport
when allowing for multidimensional flow, and consequently, hydrogen is
continuously mixed deep into the helium core. We find it difficult to achieve
such a behavior in one-dimensional stellar evolutionary calculations.
