A generalized eddy viscosity function σ, is introduced in order that the Reynolds stress in an incompressible fluid be expressible as a linear combination of the Kronecker and rate-of-strain tensors. A transport equation for the eddy viscosity is derived from the general turbulence energy equation, thereby introducing two additional functions, the specific turbulence kinetic energy q, and a scale variable s. To determine the three variables, a transport equation for s is postulated, and a modified Prandtl—Wieghardt relation among the three variables is accepted. The theory is expressed in universal, invariant form, and validity is demonstrated by application to several problems.