extends to the deep stellar interior. When the base of the convection zone overlaps with the exhausted nuclear reaction area, the turbulent convection will produce an efficient mixing and dredge up the processed material to the surface of star. This mechanism is used to explain observed surface chemical peculiarities at the surfaces of certain stars. In this paper, we present a preliminary attempt to simulate overshooting below the convective envelopes of some post-main-sequence stars. Besides the EOS and the radiation opacity (modified to accomodate an enhanced flux), the radiative pressure, radiative energy were also taken into account. In Section 2, a brief introduction to the adopted hydrodynamic code and input physics is given. The construction of the initial hydrostatic models, the simplifications and the parameters of the computational models are also specified there. In Section 3 we analyze the numerical results with a focus on the statistical properties of the turbulent fluctuations, energy fluxes, overshooting, and moment closure. A summary will be given in the Section 4 where the limitation and extensibility of the current study are also discussed.

Description of the Computational Model
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For simplicity, the magnetic field, rotation and nuclear reaction are neglected. The motions of fully 