  Degenerate ignition of helium in low-mass stars at the end of the red giant
branch phase leads to dynamic convection in their helium cores. One-dimensional
(1D) stellar modeling of this intrinsically multi-dimensional dynamic event is
likely to be inadequate. Previous hydrodynamic simulations imply that the
single convection zone in the helium core of metal-rich Pop I stars grows
during the flash on a dynamic timescale. This may lead to hydrogen injection
into the core, and a double convection zone structure as known from
one-dimensional core helium flash simulations of low-mass Pop III stars. We
perform hydrodynamic simulations of the core helium flash in two and three
dimensions to better constrain the nature of these events. To this end we study
the hydrodynamics of convection within the helium cores of a 1.25 \Msun
metal-rich Pop I star (Z=0.02), and a 0.85 \Msun metal-free Pop III star (Z=0)
near the peak of the flash. These models possess single and double convection
zones, respectively. We use 1D stellar models of the core helium flash computed
with state-of-the-art stellar evolution codes as initial models for our
multidimensional hydrodynamic study, and simulate the evolution of these models
with the Riemann solver based hydrodynamics code Herakles which integrates the
Euler equations coupled with source terms corresponding to gravity and nuclear
burning. The hydrodynamic simulation of the Pop I model involving a single
convection zone covers 27 hours of stellar evolution, while the first
hydrodynamic simulations of a double convection zone, in the Pop III model,
span 1.8 hours of stellar life. We find differences between the predictions of
mixing length theory and our hydrodynamic simulations. The simulation of the
single convection zone in the Pop I model shows a strong growth of the size of
the convection zone due to turbulent entrainment. Hence we predict that for the
Pop I model a hydrogen injection phase (i.e. hydrogen injection into the helium
core) will commence after about 23 days, which should eventually lead to a
double convection zone structure known from 1D stellar modeling of low-mass Pop
III stars. Our two and three-dimensional hydrodynamic simulations of the double
(Pop III) convection zone model show that the velocity field in the convection
zones is different from that predicted by stellar evolutionary calculations.
The simulations suggest that the double convection zone decays quickly, the
flow eventually being dominated by internal gravity waves.
