Two-point closure method for turbulence with reacting and mixing chemical elements of type A+B→ C

Abstract

Turbulence with reacting and mixing chemical elements of type A+B→C was investigated by using the two-point closure method, For implementation of this method, two-point correlation functions and two-point triple correlation functions are defined first. Equations describing the turbulence under study that describe the dynamical behaviour are written in terms of two-point correlation functions and two-point triple correlation functions. These describe the dynamical behaviour of two-point double-reactant fluctuation correlation functions. In each of these equations, two-point triple correlation functions appear. Thus, the characteristic difficulty of indeterminacy in turbulence theory is noticed in these equations too. A simple closure hypothesis for two-point triple correlation functions is proposed with a view to overcoming the indeterminacy. This hypothesis enables one to obtain the closed set of equations for double correlation functions, as desired. The resulting equations for double correlation functions provide theoretical information about the turbulence under investigation. Having obtained the closed set of equations for double correlation functions, the relationships for reactants’ eddy diffusivity functions are derived. Also, the reactants’ energy and transfer functions in fluid space are obtained. Having expressed the Karman–Howarth equations for the present investigation in dimensionless form, these are rewritten in terms of energy functions in fluid space. The system of equations for scales of segregation related to reactants A and B is derived. Various length scales involved in this study can be evaluated on integration of these equations, which in turn generates theoretical information about the turbulence under investigation.