H_{\rm p}$, $\tilde{\alpha}$ is an arbitrary parameter and $H_{\rm p}$ is the local pressure scale height. Since the 1960’s, several authors have worked directly with the hydrodynamic equations (see [@spiegel71] and references therein). By far, besides the various MLTs, the most practical stellar convection theories are those based on the moment method ([@xiong77; @xiong86; @xiong89a; @canuto93]). The moment approach was introduced first by [@keller24] and its first application in stellar convection was discussed by Castor (1968, unpublished, see the review by [@baker87]). In this method, the flow variables are split into an average and a fluctuating part. Manipulation of the hydrodynamic equations based on such splitting gives a set of equations for the moments of the turbulent fluctuations. In these equations, higher order moments appear and need to be approximated by the so-called closure models. Most of these models use a gradient type approximation, e.g. Xiong’s theory. Along with the development of digital computer and computational fluid dynamics (CFD), numerical simulations have become an essential tool for the astrophysical fluid dynamic community. The pioneer work in the numerical study of stellar type convection is the two-dimensional simulations of compressible convection done by [@graham75]. Since the late 1980’s, three-dimensional numerical experiments 