In nearly every area of fluid mechanics, our understanding is limited by the onset or presence of turbulence. Although recent years have seen a great increase in our physical understanding, a predictive theory of turbulent flow has not yet been established. Aside from certain results that can be derived through dimensional reasoning, it is still not possible to solve from first principles the simplest turbulent flow with the simplest conceivable boundary conditions. Our continuing inability to make accurate, reliable predictions seriously limits the technological advancement of aircraft design, design of turbomachinery, combustors, mixers, and a wide variety of other devices that depend on fluid motion for their operation.

Anyone who is introduced to the subject of turbulence for the first time quickly encounters the decomposition of the unsteady flow first proposed by Osborne Reynolds in 1895. Various flow variables are divided into a mean and fluctuating part, and upon substitution into the Navier-Stokes equations the result is a system of equations identical in form to the original system except for convective stress terms, which arise from averaging products of velocity fluctuations. In order to close the system of equations, a second relation is needed between the convective stresses and the mean velocity field. Until recently, much theoretical and experimental effort was focused on finding relationships that could be applied to larger and larger classes of mean flows with the ultimate hope of finding a universal constitutive relation for" turbulent fluid." There was never any guarantee that such a relation actually exists and the goals of this effort remain largely unrealized. Hope for a universal turbulence model has been slowly replaced by the growing