Modeling of Turbulence in Rapid Granular Flows

Abstract

This chapter is devoted to a phenomenological theory of granular materials subjected to slow frictional as well as rapid flows with intense collisional interactions. The microstructure of the material is taken into account by considering the solid volume fraction as a basic field. This variable enters the formulation via the balance law of configurational momentum, including corresponding contributions to the energy balance, as originally proposed by Goodman and Cowin (Arch Rational Mech Anal 44:249–266, 1972, [25]), but modified here by adequately introducing an internal length. The subgrid motion is interpreted as volume fraction variation in relatively moderate laminar variation and rapid fluctuations, which manifest themselves in correspondingly filtered equations in terms of correlation products as in turbulence theories. We apply an ergodic (Reynolds ) filter to these equations as in classical turbulent RANS-modeling and deduce averaged balances of mass, linear and configurational momenta, energy, turbulent and configurational kinetic energy. Moreover, we postulate balance laws for the dissipation rates of the turbulent kinetic energy. All these comprise 10 evolution equations for a larger number of field variables. Closure relations are formulated for the laminar constitutive quantities and the correlation terms, all postulated to obey the material objectivity rules. To apply the entropy principle, three coldness measures are introduced for capturing material, configurational and turbulent dissipative quantities, they simplify the analysis of muller’s entropy principle. The thermodynamic analysis delivers equilibrium properties of the constitutive quantities and linear expressions for the non-equilibrium closure relations.