New scenario of turbulence theory and wall-bounded turbulence: Theoretical significance

Tsutomu Kambe

Published 2016-10-19Version 1

New general scenario of turbulence theory is proposed and applied to wall-bounded turbulence. Significance of the theory rests on a mathematical theorem closely related to the fundamental conservation law of current flux of fluid flow, expressed in a form of 4d physical space-time representation, which predicts a system of Maxwell-type equation and supports transverse waves traveling with a phase speed c_t. In streaky wall flows it is remarkable that there exist both dynamical mechanism exciting transverse waves and an energy channel of exchange between flow field and transverse wave field. In developed state of the wave field, energy is supplied from the flow field to the transverse wave field if wavelengths are sufficiently large. The waves are accompanied with a new mechanism of energy dissipation, i.e. an internal friction analogous to the Joule effect. Energy is supplied from the main flow to the wave field, and some part of the energy is dissipated into heat. Thus, there exists a sustaining mechanism, which implies that the streaky structure of wall-bounded turbulence is a dissipative structure. The predictions are consistent with characteristic features of wall turbulence found in experimental studies, with respect to three points in particular. 1. Existence of traveling waves and their relation to streamwise streaks; 2. Existence of two large scales (LSM and VLSM) observed in turbulent state. Traveling waves predicted by the theory are characterized by two scales of wave-length and damping-length d, whose significance is discussed in relation to the two large scales; 3. Energy dissipation is enhanced in turbulence. Its bulk rate of energy dissipation takes a form resembling the models of eddy-viscosity, and its coefficient \nu_J is estimated to be of the order of the product c_t * d in the present study. No self-contradiction is incurred by the new field.