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, Kraichnans 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 Kraichnans refined similarity hypothesis and its extension are valid.
Anomalous scaling in the statistics of an active scalar in homogeneous turbulent convection is studied using a dynamical shell model. We extend refined similarity ideas for homogeneous and isotropic turbulence to homogeneous turbulent convection and
attribute the origin of the anomalous scaling to variations of the entropy transfer rate. We verify the consequences and thus the validity of our hypothesis by showing that the conditional statistics of the active scalar and the velocity at fixed values of entropy transfer rate are not anomalous but have simple scaling with exponents given by dimensional considerations, and that the intermittency corrections are given by the scaling exponents of the moments of the entropy transfer rate.
Human heart rate is known to display complex fluctuations. Evidence of multifractality in heart rate fluctuations in healthy state has been reported [Ivanov et al., Nature {bf 399}, 461 (1999)]. This multifractal character could be manifested as a de
pendence on scale or beat number of the probability density functions (PDFs) of the heart rate increments. On the other hand, scale invariance has been recently reported in a detrended analysis of healthy heart rate increments [Kiyono et al., Phys. Rev. Lett. {bf 93}, 178103 (2004)]. In this paper, we resolve this paradox by clarifying that the scale invariance reported is actually exhibited by the PDFs of the sum of detrended healthy heartbeat intervals taken over different number of beats, and demonstrating that the PDFs of detrended healthy heart rate increments are scale dependent. Our work also establishes that this scale invariance is a general feature of human heartbeat dynamics, which is shared by heart rate fluctuations in both healthy and pathological states.
Numerical simulations of turbulent channel flows, with or without additives, are limited in the extent of the Reynolds number Re and Deborah number De. The comparison of such simulations to theories of drag reduction, which are usually derived for as
ymptotically high Re and De, calls for some care. In this paper we present a study of drag reduction by rodlike polymers in a turbulent channel flow using direct numerical simulation and illustrate how these numerical results should be related to the recently developed theory.
