Snellman, Jan
(Helsingin yliopisto, 2015)
The mathematical description of turbulence is one of the greatest unresolved problems of modern physics. Many targets of astrophysical research, such as stellar convection zones and accretion discs, are very turbulent. Especially, the understanding of stellar convection zones is important for the theory of stellar evolution. Therefore, it is necessary to use approximate descriptions for turbulence while modelling these objects.
One approximate method for describing turbulence is to divide the quantities under study into mean and fluctuating parts, the latter of which represent small scale changes present in turbulence. This approach is known as the Reynolds decomposition, which makes it possible to derive equations for the mean quantities. The equations acquired depend on correlations of the fluctuating quantities, such as the correlations of the fluctuating velocity components known as the Reynolds stresses, and turbulent heat and passive scalar fluxes. A mathematically precise way of handling these correlations is to derive equations also for them, but the resultant equations will depend on new, higher order correlations. If one derives equations for these new correlations, a new set of even higher order correlations is involved, and the equation system will not be closed. This is called the closure problem.
The closure problem can be circumvented by using approximations known as closure models, which work by replacing the higher order correlations with lower order ones, thereby creating a closed system. Second order closure models, in which the third order correlations have been replaced by relaxation terms of second order, are studied in this Thesis by comparing their results with those of direct numerical simulations (DNS). The two closure models studied are the minimal tau approximation (MTA) and the isotropising variable relaxation time (IVRT) closure. The physical phenomena, to which the closures were applied, included homogeneous isotropically forced turbulence with rotation and shear, compressible as well as homogeneous Boussinesq convection, decaying turbulence, and passive scalar transport.
In the case of homogeneous isotropic turbulence it was found that MTA is capable of reproducing the DNS results with Strouhal numbers of about unity. It was also found that the Reynolds stress components, contributing to angular momentum transport in accretion discs, can change sign depending on rotation rate, which was seen in studies of compressible convection too, meaning that convection can potentially contribute to accretion of matter. Decaying turbulence studies indicated that the relaxation time scales occurring in the relaxation closures tend to constant values at high Reynolds numbers, and this was also observed when studying passive scalar transport. However, in studies concerning Boussinesq convection no asymptotic behaviour was found as a function of the Rayleigh and Taylor numbers.
The correspondence of the closure models to direct numerical simulations is found to be generally achievable, but with varying quality depending on the physical situation. Given the asymptotic behaviour of the optimum closure parameters for forced turbulence, they can be considered universal in this case. For rotating Boussinesq convection the same conclusion cannot be drawn with respect to the Rayleigh and Taylor numbers.