Refined similarity hypotheses in shell models of homogeneous turbulence and turbulent convection

A major challenge in turbulence research is to understand from first principles the origin of the anomalous scaling of velocity fluctuations in high-Reynolds-number turbulent flows. One important idea was proposed by Kolmogorov [J. Fluid Mech. 13, 82 (1962)], which attributes the anomaly to variations of the locally averaged energy dissipation rate. Kraichnan later pointed out [J. Fluid Mech. 62, 305 (1973)] that the locally averaged energy dissipation rate is not an inertial-range quantity and a proper inertial-range quantity would be the local energy transfer rate. As a result, Kraichnan's idea attributes the anomaly to variations of the local energy transfer rate. These ideas, generally known as refined similarity hypotheses, can also be extended to study the anomalous scaling of fluctuations of an active scalar, such as the temperature in turbulent convection. We examine the validity of these refined similarity hypotheses and their extensions to an active scalar in shell models of homogeneous turbulence and turbulent convection. We find that Kraichnan's refined similarity hypothesis and its extension are valid.