Statistical Theory of Turbulence, III Extension of Similarity Theory of Turbulence

抄録

As the second stage of developments of our statistical theory of turbulence, extenston of the initial-period similarity-law is attempted in order to give general explanations to distributions of turbulent quantity across the mean direction of flow, particularly in decaying and nondecaying shear turbulence. In (A), as the statistical hypothesis in our theory, we first give the Gaussian form to P-function. Then, as a simple theoretical result of the above extended similarity law, it can be proved that the distribution of an intermittency factor γ, introduced by A. A.Townsend in his study of turbulent wake, takes the form of Gaussian integral function, which is surveyed further by experimental observations. In (B), by introducing the above result to our fundamental expressions of turbulent intensity, it is interpreted that in all the cases of shear turbulence u⁄\sqrtu²+v², ‾uv⁄uv and w⁄u have respective nearly constant values in the y-direction. These characters are investigated by experimental measurements. In (C), as the first step to study ‾V_*-function in shear turbulent state, the problem of distortion of isotropic turbulence in a contracting stream is taken up. By rather qualitative hydrodynamical discussion on ‾V_*-function in the shearless turbulence with a non-uniform mean velocity, our fundamental formula in this case gives a result that v⁄u=‾U(x)⁄‾U₀, which is also checked in our measurements. In (D), by using all the results in (A) and (C) we derive theoretically the Reynolds stress distribution across a symmetrical shear turbulence, which is the simplest shear turbulence and corresponds nearly to a free mixing flow. Mean velocity distribution is also determined by the Reynolds equations. They are compared with our experimental measurements.