Statistical Theory of Turbulence, II Foundation of Similarity Theory of Turbulence

Abstract

As the first stage of developments of our statistical theory of turbulence, fundamental aspects of the similarity laws are investigated through the three cases of isotropic turbulence (A), turbulent wake (B) and turbulent boundary layer (C). In (A), it is first interpreted as the essential character of shearless turbulence that the distribution function F₀(u, v, w) takes the form of Gauss function without correlation. Next, the hypothesis of isotropic turbulence due to G. I. Taylor is proved to hold in the case of shearless turbulence with a uniform mean velocity distribution. Then, the hypothesis of similarity preservation introduced first by Kármán-Howarth is derived as one of our theoretical result under the supposition of ideal state. Under this supposition we get the diffusion law L²\ backsimt. The decay law in the initial period u²\ backsimt⁻¹ is derived by taking the vortex chaos motion of α=1. Further, some interpretations are made on the problem of energy transmission from large vortices to small ones. In (B), at first the existence of turbulent shearing stress is interpreted attributed to the form of P-function. Next, the initial-period similarity-law in (A) is extended into this case of shear turbulence. Such theoretical results as u², v², -‾uv, U₁²\ backsimt⁻¹ and L²\ backsimt are derived and compared with experimental measurements. The fundamental hypotheses in transfer theory in this case are derived from this exfended similarity law. Then, the transition phenomenon of similarity law from the initial period to the final period is surveyed experimentally. In (C), the existence of laminar sublayer is first interpreted physically attributed to the stability theory of laminar viscous flow. By introducing such an interpretation, it can be proved that Reynolds stress at a point are not determined as functions of only this point, and the previously derived similarity law in the initial period is developed into the non-decaying type. Theth eoretical result under a constant pressure distribution is compared with experiments. As for the distribution of Prandtl’s mixture length, the relation l(η)\ backsim(η−η₀)^1⁄2 is derived where η₀ corresponds to the boundary of laminar sub-layer.