A ν-fluid model of homogeneous turbulence subjected to uniform mean distortion

In two previous papers (Proudman 1970; Dowden 1972) it has been shown that some of the phenomena of turbulence a t high Reynolds numbers can be modelled by a suitable chosen member of the class of ν-fluids. These are non-Newtonian fluids all of whose properties depend only on a single dimensional constant whose dimensions are those of viscosity. The purpose of this paper is to construct an equation to model homogeneous turbulence in the presence of a spatially constant rate of deformation in the limit of infinite Reynolds number.The equation employed is that of a doubly degenerate third-order v-fluid (in Proudman's classification) in the limit ν → 0. In such a fluid the stress tensor S is governed by an equation of the form \[ A\dot{S}\ddot{S}+B\dot{S}^2+Cu^{\prime}S\dot{S}+D\dot{u}^{\prime}S^2 + Eu^{\prime 2}S^2 = 0, \] where A, B,…, E are isotropic tensor constants of the fluid, u′ is the total rate of deformation tensor and dots denote time derivatives. A list of properties required of the equation and its solution is proposed, and the most general form of A, B,…, E is given consistent with these requirements. Computed solutions of this equation are compared with the results of experiments on homogeneous turbulence, and are found to agree well with them.