| Turbulence is ubiquitous in astrophysics, ranging from cosmology, interstellar medium to stars, accretion disks, etc. Large scales and small viscosities combine to form turbulence with extreme large Reynolds numbers which can not be gotten in the laboratories. Stars exist rely on turbulence convection to transport heat when radiate process is insufficient. But common usage of phenomenolog-ical turbulence expressions in stellar convection, just like mixing-length theory and other kind of local theories, makes turbulent convection models perennially unpredictive. At the same time, Reynolds stress models (RSM) are deduced by many researchers, but it is hard to apply them in the stellar structure and evolution, mainly because the calculation of the differential turbulence convection equations is the main obstacle on the way to the better theories. The semi-empirical theory with tunable parameters is flexible to some extent and can be used widely in turbulence, so we deduced this kind of theories and resolved corresponding four equations. It has possibility to be the first non-local theory extensively applied to the stellar structure and evolution.Our study is focused on the semi-empirical theory, especially in different turbulence processes, including the turbulent diffusion, the dissipation of turbulent energy, the radiation dissipation, the "return to isotropic" and the effect of the buoyancy. We apply the theory into two kind of stars during their Main Sequence, which are 1M_⊙star and 1.85M_⊙star.My thesis is organized as follows:In Chapter I, the importance of turbulence is recalled with the development process. The historical and characters of turbulence are introduced, including the difficulty in front of the turbulence research. Then we showed our understanding to turbulence in one section since the turbulence is an unknown and controversial question up to now. At the last section, we compared the laboratory phenomenon and the solar convection observation to confirm that the stellar convection is turbulent flow. We reviewed different stellar convection theories, especially introduce typical one of them.In Chapter II, the numerical method of turbulent convection theory is carefully constructed. We firstly deduce the semi-empirical formula from RSM. Then we analyzed the convection stability and compressibility in this theory since they are two features in the turbulence modeling. Based on these preparations, we analyzed the detail during solving the equations in some important sides, which are boundary condition, estimated values, special condition and nonlinear, and finally constructed an effective numerical method to solve the equations.In Chapter III and Chapter IV, we focus on how much our convection theory can describe turbulence in two kinds of stars. We firstly deduced the medial and local theory from the non-local formula, and then mainly compared the non-local with the local theories. In solar-like star, all the description of turbulence in the models, namely diffusion, dissipation, isotropic and buoyancy, were agreed well with the characters of turbulence. In theδScuti type stars, the present conclusion were reconfirmed and we found the turbulent value in the non-local models are greatly lager than those in the local models. Furthermore, we made models of one of theδScuti type stars, named FG Vir. It is found that the non-local theory shows some exciting frequencies in the nonadiabatic oscillation, but the local one shows all damping. Therefore, the frame rotating frequencies are estimated to identify the observation oscillation frequencies of FG Vir. About nine frequencies are fitted and many others can not be corresponding with the calculation. The oscillation of the FG Vir is far away from being clearly understood.In the last Chapter, conclusion and discussion are made. All of this work showed the semi-empirical convection theory is better than the mixing-length theory.
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