A survey of the mean turbulent field closure models.

C solutions to the differential boundary-layer equations have for some years now been applied to turbulent boundary layers where relief from the difficulty of solution permits more concern for the physical elements of models which purport to simulate some statistical features of turbulent flowfields. A first step has been accomplished; that is, accurate, quite versatile, and practically/useful computer programs have been combined with rather simple and successful empirical statements" which allow one to estimate the Reynolds shear stress in the equations for the mean velocity field. We call this Mean Velocity Field (MVF) closure since it predicts only the mean velocity field in addition to the mean shear stress. For boundarylayer flows a set of empirical constants must be selected. However, it is then possible to accurately predict flows with wall transpiration, heat transfer, and a variety of other boundary conditions, and, remarkably, with no adjustment in the constants. However, the constants must be adjusted for, say, pipe or channel flow or free shear flows. By moving on to the more complicated Mean Turbulent Field (MTF) closure there is some hope of discovering increasingly universal models and a greater range of predictability. There are two other incentives: First, it is rather comforting actually to compute the turbulent kinetic energy; it is, after all, the premier property that distinguishes turbulent from laminar flow. Second, it appears possible to include body forcelike effects such as curvature, buoyancy, and Coriolis effects with no further empiricism. The latter is a line of thought that is not new,' and it has occupied the present authors' interest for some time. However, in this paper we avoid these topics in order to simplify discussion of an already complicated field. We also assume that the fluid is incompressible. Extension to high Mach number flows does not, however, seem to be a major problem. A basis for MTF calculations began appropriately enough with the semiheuristic models of Kolmogoroff and Prandtl in the early 1940's; they include the turbulent kinetic energy transport equation, a turbulent-energy-related eddy viscosity, and either a prescribed length scale function or a differential equation for a length scale. We wish to call this Mean Turbulent Energy (MTE) closure which together with Mean Reynolds Stress (MRS) closure forms two subsets of MTF closure.* MRS closure implies a closed set of equations which include equations for all nonzero components of the Reynolds stress tensor. Chou' seems to be the first to initiate a study of the full set of equations with an eye towards closure. However, it was Rotta in 1951 who laid the foundation for almost all of the current models. In the Reynolds stress tensor equation (the tensor equation for the single-point, double-velocity correlations, the trace of which is the kinetic energy equation) there appear pressure-velocity gradient correlations, (pdujdxj), which Rotta called the energy redistribution terms and which he argued should be proportional to the deviation from isotropy — dtj(uky/3. On the whole, the assumption seems physically correct, but of further importance is the fact that it provides a unity that was lacking, say, in the 1940's.f Thus, the Reynolds shear stress is now determined as a part of the whole; MTE closure can be obtained as an analytic simplification of MRS closure, and, furthermore, MVF closure (that is, eddy viscosity or mixing length concepts) can be viewed as a further simplification. We shall follow this process of simplification in this paper. Despite the unity of thought provided by Rotta's basic assumption, it is, of course, an approximation to nature and is subject to modification in the hands of investigators eager to achieve agreement with data. Furthermore, there are other terms in the Reynolds stress equations, such as the dissipation and diffusion terms, which are modeled differently by different investigators and represent some impass to a consensus theory such as is the near state of MVF closure. In the present development we have attempted to present the basic ideas and a core model for MRS and MTE closure and