Transport phenomena in turbulent flow

Abstract

In this chapter we intend to provide a brief introduction to the study of transport phenomena in the presence of turbulent flows. This is a very large and quite specialized topic, and here we will confine our treatment to some basic features, above all in the context of pipe flow, referring the interested readers to more specialized texts (See, for example, Turbulent Flows by S.B. Pope, Cambridge University Press.). Most of the discussion on convective transport phenomena in the preceding chapters assumes that the flow is laminar, while the case of turbulent flows has been treated so far using empirical correlations, that are often subjected to stringent geometric constraints. The reason for concentrating on laminar flows is quite obvious: transport phenomena involving laminar flows can be treated using appropriate simplifying hypothesis, such as the fact that the process is stationary or uni-directional, so that the problem can be treated systematically, often even finding an analytical solution. No such simplifying features can be assumed to apply to turbulent flows, which are always irregular, transient and three-dimensional, and therefore very complex to treat. These difficulties notwithstanding, turbulent flows pop up everywhere in engineering and the natural world, regardless of whether we are optimizing a heat exchanger or watching a cloud moving in the sky. The importance of this subject has stimulated the interest of many investigators, leading to an impressive amount of experimental and theoretical studies. Unfortunately, while the experiments have produced many useful empirical relationships (albeit often with narrow applicability conditions), a complete theory of turbulence is still lacking, although its fundamental characteristics are well understood. After a brief introduction (Sect. 18.1), in Sect. 18.2 we describe the Kolmogorov scaling arguments, thus determining the characteristic time- and length-scales of turbulence. Then, in Sect. 18.3, we present the Reynolds decomposition and the associated governing equations, involving the turbulent fluxes of mass, momentum and energy. Next, Sect. 18.4, a closure model for the diffusive turbulent fluxes is described, based on the Reynolds analogy and Prandtl’s mixing length model. This approach is applied in Sect. 18.5 to determine the logarithmic velocity profile near a wall, and then to describe turbulent pipe flow. Finally, in Sect. 18.6, more complex models are briefly sketched.