Renormalization group model of turbulent bioconvection of gyrotactic microorganisms

Abstract

The problems connected with processes of turbulent bioconvection are considered. These processes are typical for technologies of microbiological and other industries. The concept of using renormalization groups in the theory of turbulence is shown. A step-by-step algorithm for applying the renormalization methodology to deriving the equation for mass transfer in bioconvective processes, which includes renormalization of the turbulent diffusion coefficient, is presented. The dependence for the turbulent Schmidt number was obtained. It has been shown that turbulent Schmidt number is function of six parameters: molecular Schmidt number, turbulent viscosity, molecular viscosity, parameter of geometric shape of microorganisms, average velocity of motion of microorganisms relative to the fluid, orientation parameter. The turbulent Schmidt number enables closing the diffusion equation. The effect of the geometric shape of microorganisms on the development of turbulence in a bioconvective flow in the presence of gyrotactic microorganisms is analyzed. Analysis shows that microorganisms with spherical symmetry make a small contribution to the formation of turbulent mass transfer processes. The results of the analysis are in a good agreement with data from studies of the nonlinear hydrodynamic stability of bioconvective processes based on the Lorentz model.