Turbulent Convection in Stellar Interiors. I. Hydrodynamic Simulation

(Abridged) We describe the results of three-dimensional (3D) numerical simulations designed to study turbulent convection in the stellar interiors, and compare them to stellar mixing-length theory (MLT). Simulations in 2D are significantly different from 3D, both in terms of flow morphology and velocity amplitude. Convective mixing regions are better predicted using a [dynamic boundary condition] based on the bulk Richardson number than by purely local, static criteria like Schwarzschild or Ledoux. MLT gives a good description of the velocity scale and temperature gradient for a mixing length of $\sim 1.1 H_p$ for shell convection, however there are other important effects that it does not capture near boundaries. Convective "overshooting" is best described as an elastic response by the convective boundary, rather than ballistic penetration of the stable layers by turbulent eddies. We find that the rate at which material entrainment proceeds at the boundaries is consistent with analogous laboratory experiments as well as simulation and observation of terrestrial atmospheric mixing. In particular, the normalized entrainment rate E=$u_E/\sigma_H$, is well described by a power law dependence on the bulk Richardson number $Ri_B = \Delta b L/\sigma_H^2$ for the conditions studied, $20\lesssim Ri_B \lesssim 420$. We find $E = A Ri_B^{-n}$, with best fit values, $\log A = 0.027 \pm 0.38$, and $n = 1.05 \pm 0.21$. We discuss the applicability of these results to stellar evolution calculations.