Crenulative turbulence in a converging nonhomogeneous material

Crenulative turbulence is a nonlinear extension of the Bell–Plesset instability, usually observed in a converging system in which there is a nonhomogeneous response of stress to strain and/or strain rate. In general, crenulation occurs in any circumstance in which the mean flow streamlines converge the material more strongly than the compressibility can accommodate. Elements of the material slip past each other, resulting in local fluctuations in velocity from that of the mean flow, producing a type of turbulence that is more kinematic than inertial. For a homogeneous material, crenulation occurs at the atomic or molecular scale. With nonhomogeneous stress response at larger scales, the crenulative process can also occur at those larger scales. The results are manifested by a decrease in the rate of dissipation to heat, and by the configurationally irreversible mixing of nonhomogeneities across any mean-flow-transported interface. A mathematical description of the crenulative process is obtained by means of Reynolds decomposition of the appropriate variables, and the derivation of transport equations for the second-order moments that arise in the mean-flow momentum and energy equations. The theory is illustrated by application to the spherical convergence of an incompressible fluid with nonhomogeneous distribution of kinematic viscosity.