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

Abstract

General scenario of turbulence theory is proposed and applied to streaky wall-bounded turbulence. This scenario introduces a new field of transverse waves. Significance of the theory rests on a mathematical theorem associated with the conservation law of current flux, 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 regard to the streaky wall flows, there exist both dynamical mechanism and energy channel which excite transverse waves and exchange energy between flow field and wave field. Energy is supplied from the flow field to the wave field if wavelengths are sufficiently large. The waves are accompanied with a new mechanism of energy dissipation, an internal friction analogous to the Ohm's effect. 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 experimental observations of wall turbulence: (i) Existence of traveling waves: The waves are characterized by two scales of wavelength and a damping-length d. (ii) Existence of two large scales (LSM and VLSM) observed in turbulent shear flows: Those are interpreted by the waves amplified with the transient growth mechanism and maintained by interaction with the new transverse wave field. The waves are robust since they have their own energy and momentum. (iii) Enhanced energy dissipation in wavy turbulence. Its bulk rate of energy dissipation takes a form resembling the models of eddy-viscosity, and its coefficient \nu_D is estimated to be of the order of c_t d and much larger than the molecular viscosity. No self-contradiction is incurred by the new field introduced.