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We present the first 3-dimensional, fully compressible gas-dynamicssimulations in $4\pi$ geometry of He-shell flash convection with proton-richfuel entrainment at the upper boundary. This work is motivated by theinsufficiently understood observed consequences of the H-ingestion flash inpost-AGB stars (Sakurai's object) and metal-poor AGB stars. Our investigationis focused on the entrainment process at the top convection boundary and on thesubsequent advection of H-rich material into deeper layers, and we thereforeignore the burning of the proton-rich fuel in this study. We find that, for ourdeep convection zone, coherent convective motions of near global scale appearto dominate the flow. At the top boundary convective shear flows are stableagainst Kelvin-Helmholtz instabilities. However, such shear instabilities areinduced by the boundary-layer separation in large-scale, opposing flows. Thislinks the global nature of thick shell convection with the entrainment process.We establish the quantitative dependence of the entrainment rate on gridresolution. With our numerical technique simulations with $1024^3$ cells ormore are required to reach a numerical fidelity appropriate for this problem.However, only the result from the $1536^3$ simulation provides a clearindication that we approach convergence with regard to the entrainment rate.Our results demonstrate that our method, which is described in detail, canprovide quantitative results related to entrainment and convective boundarymixing in deep stellar interior environments with veryvstiff convectiveboundaries. For the representative case we study in detail, we find anentrainment rate of $4.38 \pm 1.48 \times 10^{-13}M_\odot \mathrm{/s}$.

Hydrodynamic simulations of H entrainment at the top of He-shell flash convection