On the use of the theory of v-fluids for the construction of models of turbulence

The class of $\text{v-}\text{fluids}$, defined by Proudman1] to be non-Newtonian fluids all of whose properties depend on a single dimensional constant whose dimensions are those of viscosity, is examined further as a source of possible models of turbulence at high Reynolds number. It is found that, while the general definition is valuable, certain aspects of Proudman's associated concept of a regular $\text{v-}\text{fluids}$ have a number of features that bring its value as a source of models of inhomogeneous turbulence into question. The nature of these problems suggests a suitable modification to the definition of a regular $\text{v-}\text{fluids}$. A number of constraints on the form and solution of the equations of a model of turbulence are proposed with the intention of improving the generality of models of a given degree of complexity, and their justification and significance discussed. In the light of these proposals the simplest modified regular $\text{v-}\text{fluids}$ that can be used to model homogeneous turbulence at infinite Reynolds number is described, and it is seen to include all the most desirable features of earlier models of this type, and is in addition capable of extension in a natural way to describe inhomogeneous turbulence.