Refined similarity hypotheses in shell models of turbulence

TL;DR

This paper investigates the validity of refined similarity hypotheses in shell models of turbulence, confirming Kraichnan's ideas about local energy transfer rate variations causing anomalous scaling in turbulent flows.

Contribution

It provides the first validation of Kraichnan's refined similarity hypothesis and its extension to active scalars within shell models of turbulence.

Findings

Kraichnan's refined similarity hypothesis is valid in shell models.

Extensions to active scalars also hold true.

Supports the role of local energy transfer rate in turbulence scaling.

Abstract

A major challenge in turbulence research is to understand from first principles the origin of anomalous scaling of the velocity fluctuations in high-Reynolds-number turbulent flows. One important idea was proposed by Kolmogorov [J. Fluid Mech. {\bf 13}, 82 (1962)], which attributes the anomaly to the variations of the locally averaged energy dissipation rate. Kraichnan later pointed out [J. Fluid Mech. {\bf 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 the 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, like the temperature in turbulent convection. In this paper, we examine the validity of these refined similarity hypotheses and their extensions to an active scalar in shell models of turbulence. We find that Kraichnan's refined similarity hypothesis and its extension are valid.