limitation of the numerical results was discussed as well.'
author:
- |
    Chun-Lin Tian$^{1,2}$[^1], Li-Cai Deng$^{1}$ and Kwing-Lam Chan$^{3}$\
    $^{1}$National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China\
    $^{2}$Graduate University of Chinese Academy of Sciences, Beijing 100049, China\
    $^{3}$Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China
date: 'Accepted 2009 May 30. Received 2009 May 29; in original form 2008 July 15'
title: Numerical simulations of downward convective overshooting in the giants
---

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convection – hydrodynamics – turbulence – methods: numerical – stars: interiors.

Introduction
============

Turbulent convection is one of the major uncertainties in our understanding of stellar properties. As a very efficient mixing process and energy transfer mechanism, it affects the stellar structure and evolution significantly. Because of its high non-linearity and complexity, we cannot solve the turbulent flows analytically at present. Since [@emden07] established the first practical model of stellar structure, a lot of effort has been made to study the problems of stellar convection. A traditional treatment is the mixing-length theory (MLT)([@vitense58]). The basic idea of MLT is that convection is viewed as rising and sinking bubbles which blend into the surroundings after traveling a distance of $l$, where $l=\tilde{\alpha} 