An Introduction to Turbulent Flow

Three important classes of statistically steady turbulent flows will be examined in some detail in this chapter, namely jets, wakes, and boundary layers. These flows are of considerable practical interest and are illustrated in Figure 5.1, together with a shear layer, which is not specifically considered here, but whose treatment is not essentially different from that of jets and wakes. Much is known about these flows from numerous experimental and theoretical studies, stretching back over many decades, and the resulting theoretical models are semiempirical, being based on the experimentally observed properties of the flows, such as the small angles of divergence of jets and wakes and the existence of self-similar behavior far downstream of the nozzle, in the case of jets, or body, in the case of wakes. We attempt to give a unified presentation of these models, emphasizing their common origins in the boundary-layer approximation and various types of mean-flow self-similarity.

The basis of the theories developed in this chapter is the mean-flow equations, (4.9), to which simplifying assumptions, similar to those of laminar boundary-layer theory, are applied, leading to neglect of some of the terms in the equations. This turbulent boundary-layer approximation, which applies to jets and wakes as well, is justified by slowness of mean-flow development with streamwise distance, as witnessed by the small angles of divergence of turbulent jets and wakes. Since the approximations used are similar to Prandtl's theory of laminar boundary layers, with which we assume the reader is familiar, and since, despite their quite different mechanisms of growth, the behavior of turbulent boundary layers in many ways resembles that of laminar ones, we devote Section 5.1 to reminding the reader of the principal characteristics and theory of laminar boundary layers, before developing the turbulent analog of Prandtl's theory in Section 5.2.

Having applied the boundary-layer approximation to simplify the mean-flow equations, one is still faced with their lack of closure, apparent from the Reynolds stress terms. Here, empirical information is introduced, whose details depend on the class of flows considered. In the case of jets and wakes, we use the observed fact that the mean flow becomes self-similar sufficiently far downstream, together with an eddy-viscosity approximation and an assumed eddy viscosity that does not vary across the flow, but only with streamwise distance.